a numerical solution of navier-stokes and energy … · which can be used to generate solutions of...

A NUMERICAL SOLUTION OF

NAVIER-STOKES AND ENERGY EQUATIONS

FOR HEAT TRANSFER FROM PARTICLES

A thesis submitted for the degree of

DOCTOR OF PHILOSOPHY

in the

FACULTY OF ENGINEERING

of the

UNIVERSITY OF LONDON

by

TARA RASOOL AL-TAHA, B.Sc.(Eng.), A.C.G.I., M.Sc., D.I.C.

Department of Chemical Engineering

and Chemical Technology,

Imperial College of Science and Technology,

London, S.W.7.

November, 1969

2

ABSTRACT

A theoretical study of steady-state forced convective heat

transfer from solid spheres and oblate spheroids to Newtonian

incompressible fluids has been carried out at intermediate

Reynolds numbers.

The Navier-Stokes and energy equations for axisymmetric

flows were expressed, in terms of vorticity, stream function,

and temperature, in terms of modified spherical and oblate sphe-

roidal coordinates. For purposes of computation, the equations,

which are elliptic second-order partial differential equations,

were replaced by their finite-difference approximations.

The resultant sets of finite-difference equations were

solved using the explicit extrapolated Gauss-Seidel iterative

method.

Two computer programmes: one for the Navier-Stokes equations

and the other for the energy equation, were developed in a form

which can be used to generate solutions of the finite-difference

equations for spheres and for oblate spheroidal bodies.

Numerical solutions of the Navier-Stokes equations were

obtained for spheres and for oblate spheroids with minor to major

axis ratios of 0.8125 , 0.625 and 0.4375 at Reynolds numbers

between 0.0001 and 500.0 . Numerical solutions of the energy

equation were obtained for the same body shapes at Peclet numbers

between 0.01 and 2000.

The influences of mesh size and the proximity of the outer

boundary on the solutions were investigated.

The distributions of vorticity, stream function, and

temperature which were thus obtained were used to calculate

pressure distributions on the surface, drag coefficients, and

3

local and overall Nusselt numbers.

As the Reynolds number increased, a boundary layer type of

flow was observed upstream and a development of wake downstream.

The commencement of flow separation was found to occur at Reynolds

numbers of 20 15 , 12 and 8 for the sphere and for the oblate

spheroids: e = 0.8125 , 0.625 , and 0.4375, respectively.

The total drag coefficients obtained agreed well with the

standard drag curve for Reynolds numbers greater than 10. High

values of the drag coefficient were obtained for Reynolds numbers

less than 10 due to the proximity of the outer boundary.

For Peclet numbers less than 0.3, the overall Nusselt number

approached an asymptotic value which was close to that attributed

to transfer to stagnant medium of the same extent as that confined

between the body and the outer boundary.

For Peclet numbers greater than 10, the overall Nusselt

number was found to vary with Peclet number alone. The variation

was found to lie between Friedlander's8 and Boussinesq's79

solutions. As Reynolds and Prandtl numbers increased, however,

the overall Nusselt number appeared to depend separately on

Prandtl number and Reynolds number.

Comparison of the predicted values of the local Nusselt

number with boundary layer solutions showed good agreement

upstream of boundary layer separation. The results of the overall

Nusselt number also showed good agreement with the experimental

measurements available in the literature.

ACKNOWLEDGMENTS

I would like to express my sincere thanks to

Dr. A.R.H. Cornish for his continued interest, guidance

and encouragement throughout the course of this research

work.

I am also most grateful to Professor R.W.H. Sargent

for his interest in the work and for providing financial

support by making available a Courtaulds Bursary for the

duration of this work.

Thanks are also extended to all my departmental

collegues for their friendship and company , and to

Dr. Keith H. Ruddock and his wife, Joan , for their

friendship and help in many ways.

Lastly, but by no means least, I express my thanks

and gratitudes to my wife, Kawakib for her sympathy

and encouragement, and to my parents for their sacrifices

and generous help in many ways.

Taha R. Al—Taha

4

5

LIST OF CONTENTS

Page

ABSTRACT

ACKNOWLEDGEMENTS 4

LIST OF CONTENTS 5

LIST OF FIGURES 9

LIST OF TABLES 13

LIST OF SYMBOLS 14

CHAPTER 1. INTRODUCTION 22

CHAPTER 2. LITERATURE SURVEY 28

2.1. Introduction 28

2.2. Basic Equations of Steady-State Forced 30 Convective Heat Transfer

2.3. Solutions of the Navier-Stokes Equations 32

2.3.1. Limiting Solutions - Potential 33 Flow Theory

2.3.2. Limiting Solutions - Stokes Flow 35

2.3.3. Perturbation Methods 37

2.3.4. Boundary Layer Theory 41

2.3,5. Galerkin Method 46

2.3.6. Numerical Methods 47

2.3.7. Experimental Studies of Viscous 50 Fluid Flows

a. Flow Separation and the Critical 51 Reynolds Number

b. Drag Coefficients 53

c. Wall Effects 54

2.4. Convective Heat and Mass Transfer 55

2.4,1. Theoretical Studies 55

2.4.2. Experimental Studies of Heat and 64 Mass Transfer from Spheres

6

Page

CHAPTER 3. THEORETICAL ANALYSIS 69

3.1. Equations of Viscous Flow and Heat 69

Transfer

3.2. Navier-Stokes Equations as Vorticity 75 Transport Equations

3.3. Axisymmetrical Flows 76

3.4. Boundary Conditions 81

3.5. Forced Convective Heat Transfer from 82 Spheres and Oblate Spheroids

3.6. Dimensionless Forms 86

CHAPTER 4. NUMERICAL TECHNIQUES 89 4.1. Types of Second-Order Partial 89

Differential Equations

4.2. The Flow Region 91

4.3. Finite-Difference Equations 94

4.4. Boundary Conditions in Finite- 97 Difference Form

i . Fixed Boundary Conditions 98

ii. Specially - Treated Boundary 99 Conditions

4.5. Iterative Methods 104

4.6. Numerical Differentiation and 107 Integration

4.7 General Procedure for the Solution of 109 the Navier-Stokes and Energy Finite-Difference Equations

CHAPTER 5. DISCUSSION OF RESULTS 118

5.1. Introduction 118

5.2. Numerical Solutions 119

5.3, Vorticity and Stream Function 126 Distributions

5.4. Flow Separation 142

5.5. Surface Pressure Distributions 156

7 Page

167

181

5.6. Drag Coefficients

5.7. Temperature Distributions and Local Nusselt Numbers

5.8. Overall Nusselt Numbers

CHAPTER 6. CONCLUSIONS

APPENDIX A. ORTHOGONAL CURVILINEAR COORDINATE SYSTEM

A.1. Curvilinear Coordinates

A.2. Unit Vectors and Scale Factors

A.3. Calculation of Scale Factors for Orthogonal Curvilinear Coordinates

A.4. Area and Volume in Orthogonal. Coordinate Systems

APPENDIX B. VECTOR RELATIONSHIPS

B.1. Vector Algebra

B.2. Vector Operators in Orthogonal Curvilinear Coordinates

B.3. Vector Relationships

APPENDIX C. SPHERICAL AND OBLATE SPHEROIDAL COORDINATES

C.1. The Spherical Polar Coordinates (r, 04)

C.2. The Oblate Spheroidal Coordinates (z, 9,

C.3. Transformation of the Coordinate Systems

APPENDIX D. PRESSURE DISTRIBUTION AND DRAG COEFFICIENTS

D.1. Physical Components of the Stress Tensor in a Newtonian Incompressible Fluid

1. Stress Tensor in Spherical Polar Coordinates (r204)

2. Stress Tensor in Oblate Spheroidal Coordinates (z194)

D.2. The Equations of Viscous Flow and Heat Transfer in Spherical Polar and in Oblate Spheroidal Coordinates

D.3. Surface Pressure Distribution

D.4. Drag Forces and Drag Coefficients

202

216

222

222

223

224

225

226

226

226

227

229

229

230

233 238

238

239

239

240

243

247

8

APPENDIX E. NUSSELT NUMBER DISTRIBUTION AND MOLECULAR CONDUCTION

E.1. Local and Overall Nusselt Numbers

E.2, Molecular Conduction

APPENDIX F. CONVERGENCE AND STABILITY CRITERIA

F.1. Introduction

F.2. Analytical Treatment of Convergence

F.2.1. Convergence of the Solution of the Energy Equation

F.2,2. Thom and Apelt's Method

F.3. Analytical Treatment of Stability

F.3.1. Error Analysis

F.3.2. Residual Analysis

F.4. Summary

APPENDIX G. COMPUTER PROGRAMMES FOR THE SOLUTION OF THE FINITE-DIFFERENCE EQUATIONS

G.1. Introduction

G.2. Scope and Limitations of the Programmes

G.3. Conventions Used in the Programmes

G.4. List of FORTRAN Symbols Used in the Programmes

G.5. Description of the Subroutines

G.5.1. Programme 1

G.5.2, Programme 2

G.6. The User's Quick-Reference Guide

G.7. Listings of the Computer Programmes

APPENDIX H. TABLES 8 to 16

BIBLIOGRAPHY

Page

251

251

254

258

258

260

260

263

266

266

267

268

270

270

270

272

274

278

279

289

294

297

320

330

LIST OF FIGURES

Fig.

3.1 Orthogonal Curvilinear Coordinates

3.2 Streamlines in a Meridian Plane

3,3 Vorticity and Velocity Directions

4.1 The Flow Region and Computational Stars

5.1 Vorticity Distributions Around the Sphere at Reynolds Numbers of 1 and 10

5.2 Vorticity Distributions Around the Sphere at Reynolds Numbers of 100 and 500

5.3 Vorticity Distributions Around the Oblate Spheroid: 130 e = 0,8125 at Reynolds Numbers of 10 and 100

5.4 Vorticity Distributions Around the Oblate Spheroid: 131 e = 0.625 at Reynolds Numbers of 10 and 100

5.5 Vorticity Distributions Around the Oblate Spheroid: 132 e = 0.4375 at Reynolds Numbers of 10 and 100

5.6 Streamlines Around the Oblate Spheroid: e = 0.8125 133 at Reynolds Numbers of 0.1 1 and 10

5.7 Strepmlines Around the Sphere at Reynolds Numbers 134 of 25 2 50 1 and 100

5,8 Variation of Wake Size Behind the Sphere with 136 Reynolds Number

5.9 Variation of Wake Size Behind the Oblate Spheroid: 137 e = 0.8125 with Reynolds Number



5.12 Wake Dimension as a Function of Reynolds Number 140

5.13 Boundary Layer Separation 142

5.14 Surface Vorticity Distributions for the Sphere at 145 Reynolds Numbers Between 0 and 2




9

Page

76

76

81

93

128

129

10

Fig. Page

5.18 Surface Vorticity Distributions for the Oblate 149 Spheroid: e = 0.8125 at Reynolds Numbers Between 17.5 and 100

5.19 Surface Vorticity Distributions for the Oblate 150

Spheroid: e = 0.625 at Reynolds Numbers Between 10 and 100

5.20 Surface Vorticity Distributions for the Oblate 151


5.21 Angles of Flow Separation 153

5.22 Angle of Flow Separation as a Function of Reynolds 155 Number

5.23 Surface Pressure Distributions for the Sphere at 159 Reynolds Numbers Between 1 and 4



5.26 Surface Pressure Distributions for the Oblate 162 Spheroid: e = 0.8125 at Reynolds Numbers Between 5 and 100

5.27 Surface Pressure Distributions for the Oblate 163 Spheroid: e = 0.625 at Reynolds Numbers Between 5 and 100

5.28 Surface Pressure Distributions for the Oblate 164


5.29

5.30

5.31

5.32

5.33

5.34

5.35

166

168

169

170

171

173

175

Surface Pressure Distribution as a Function of Reynolds Number

Variation of the Drag Coefficients with Reynolds Number (Sphere)

Variation of the Drag Coefficients with Reynolds Number (Oblate Spheroid: e = 0.8125)



Relative Contribution of the Form and Skin-Frictional Drag Coefficients to the Total Drag Coefficient

Comparison of Theoretical and Experimental Drag Coefficients for the Sphere

Isotherms Around the of 24 and 120

at Peclet Numbers

Sphere at Peclet Numbers

Isotherms Around the Sphere of 0.024 0.24 and 2.4

Fig.

5.36

5.37

11

Page

183

184

5.38

5.39

5.40

5.41

5.42

185

186

187

189

Isotherms Around the Oblate Spheroid: e = 0.8125 at Peclet Numbers of 0.24, 2.4 and 24

Isotherms Around the Oblate Spheroid: e = 0.625 at Peclet Numbers of 0.24 , 2.4 and 24

Isotherms Around the Oblate Spheroid: e = 0.4375 at Peclet Numbers of 0.24 , 2.4 , and 24

Local Nusselt Numbers for the Sphere at a Prandtl Number of 0.7 and at Reynolds Numbers Between 0.01 and 10

Local Nusselt Numbers for the Sphere at a Prandtl 190 Number of 0.7 and at Reynolds Numbers of 50 100 and 500

5.43 Plots of Nu / NuA=0 versus A for the Sphere 192

at a Prandtl Number of 0.7 and at Reynolds Numbers Between 0.01 and 500

5.44 Plots of NuA / NUA=0 versus A for the Oblate 193

Spheroid: e = 0.8125 at a Prandtl Number of 0.7 and at Reynolds Numbers Between 0.01 and 50

5.45 Plots of NV • / Nu G=0 versus A for the Oblate 194

Spheroid: e = 0.625 at a Prandtl Number of 0.7 and at Reynolds Numbers Between 0,01 and 10

5.46 Plots of Nu • / NuG=0 versus A for the Oblate 195

Spheroid: e = 0,4375 at a Prandtl Number of 0.7 and at Reynolds Numbers Between 0.01 and 10

5.47 Plots of NV • / NUA=0 versus A for All Shapes 198

at a Prandtl Number of 2.4 and at Reynolds Numbers Between 0.01 and 10

5.48

5 049

5.50

5.51

200

203

205

207

Comparison of Theoretical and Experimental Local Nusselt Numbers for the Sphere

Overall Nusselt Number as a Function of Peclet Number

Overall Nusselt Numbers for the Sphere at Peclet Numbers less than 10

Comparison of Theoretical and Experimental Overall Nusselt Numbers for the Sphere

Fig. I

5.52 Plots of Nu / Pr' versus Re I

5.53 Plots of Nu / Re2 versus Pr

5.54 Plots of versus Re

5.55 Plots of Nu versus Re4 Pr117 for the Sphere

A.1 Curvilinear Coordinates (x x x ) 11 2' 3 C.1 Spherical Polar Coordinates (rt e,()

C.2 Oblate Spheroidal Coordinates (z,01 4)

C.3 Elliptic Coordinates (z 9) in a Meridian Plane

C.4 Elliptic Coordinates (2,0)

C.5 Polar Coordinates (rt 9) in a Meridian Plane

C.6 Modified Polar Coordinates (z1(2)

12

Page

211

212

213

214

222

231

231

235

235

235

235

D.1 Flow Past an Oblate Spheroid 244

D.2 Pressure and Viscous Stress on the Surface of 249 a Sphere

D.3 Pressure and Viscous Stress on the Surface of 249 an Oblate Spheroid

F.1 A Computational Star 263

G.1 Flow Diagram of the Computer Programmes 1&2 - MAIN 280

G.2 Flow Diagram of Subroutine FIELD (Programmes 1&2) 283

G.3 Flow Diagram of Subroutine INPUT (Programme 1) 283

G.4 Flow Diagram of Subroutine COCAL (Programme 1) 285

G.5 Flow Diagram of Subroutine CASE (Programme 1) 285

G.6 Flow Diagram of Subroutine BOUNDC (Programme 1) 285

G.7 Flow Diagram of Subroutine SOLVE (Programme 1) 288

G.8 Flow Diagram of Subroutine NSNSEE (Programme 1) 290

G.9 Flow Diagram of Subroutine INPUT (Programme 2) 290

G.10 Flow Diagram of Subroutine COCAL (Programme 2) 293

G.11 Flow Diagram of Subroutine CASE (Programme 2) 293

G.12 Flow Diagram of Subroutine BOUNDC (Programme 2) 293

G.13 Flow Diagram of Subroutine SOLVE (Programme 2) 295

G.14 Flow Diagram of Subroutine NSNSEE (Programme 2) 295

13

LIST OF TABLES

Table Page

1. Dimensionless Quantities 86

2. Definition of Symbols Used in Section 4.7

112

3. The Effect of Using 3rd and 4th Order Approximations 124 to Evaluate the Vorticity at the Surface on the

4.

5.

6.

Drag Coefficients

Values of Ko for the Sphere

Effect of the Proximity of the Outer Boundary on the Drag Coefficients

Effect of Mesh Size on the Sphere Drag Coefficients

157

178

179

7. Asymptotic Values Nu Pe ---4•0 of as 202

8. Angles of Flow Separation and Wake Dimensions 321

9. Drag Coefficients of the Sphere 322

10. Drag Coefficients of the Oblate Spheroid: e = 0.8125 323



13. Overall Nusselt Numbers for the Sphere 325

14. Overall Nusselt Numbers for the Oblate Spheroid: e = 0.8125

327


328


329

14 LIST OF SYMBOLS

Symbol

Explanation or Defining Equation Dimension*

.11-- a Defined as (d2 - b2 )2

a Arbitrary vector quantity

Coefficients in equation (2.63) a1/a21a31a4

a' Coefficient in equation (4.1)

a''2 a'' a' Coefficients in equation (F.1)

a" Coefficient defined by equation'(F.6)

A Cross-sectional area

Ao Constant in equation (2.111)

A1lA2'A3'A4 Coefficients in equation (4.62)

b Length of the semi-minor axis of an oblate spheroid

Arbitrary vector quantity

b1'b2'

b3'

b4 Coefficients in equation (2.63)

b' Coefficient in equation (4.1)

' 2 b' b'1 3 b'1 4 b'15 blCoefficients in equation (F.2) l

b" Coefficient defined by equation (F.7)

Bo Constant in equation (2.111)

B1(i),B2(i) Coefficients in equations (defined in Table 2)

(4.79) to (4.81)

B3(j),B4(j) Coefficients in equations (4.79) to (defined in Table 2)

(4.81)

Bl(i),B2(i),B7Coefficiets in equations (4.91) & (4.92) -1(deflned in Table 2)

c Symbol denoting cosh z in Appendix D

c' Coefficient in equation (4.1)

c" Coefficient defined by equation (F.8)

MIN

cm(x) (m=o11,2,3,4) Coefficients in equation (2.55)

C1(i),C2(j) Coefficients in equations (4.79) & (4.82) (defined in Table 2)

* M= mass, L=length, T=time, 11=heat, 0 =temperature.

•••

L2

dn.

OFR

ONO

Iwo

Cb1'Cb2(j)

C

CDP CDT (CDT)00

Cf(i)

Cp

d

d'

d"

d. .

d1. . ,3

is d. 113

t 1 C1(i),C2(i) Coefficients in equations (4.91) & (4.92)

(defined by equation (4.93)

.15

Coefficients defined in Table 2

Skin-friction (viscous) drag coefficient

Pressure (form) drag coefficient

Total drag coefficient

Total drag coefficient in infinite medium (i.e. with no wall effects)

Coefficients defined in Table 2

Specific heat at constant pressure

Length of the semi-major axis of an oblate spheroid

Coefficient in equation (4.2)

Coefficient defined by equation (F.9)

Additional terms in equation (4.79) (defined by equation (4.82))

Non-linear terms in equation (4.80) (defined by equation (4.83))

Convective terms in equation (4.81) (defined by equation (4.84))

fon

al*

•••

HM 1e L

Differential operator

AC Maximum diameter of particle (2R or 2d)

DroDf21Df3,Df4 Functions defined by eq-7.1,7-tidns (4.85) gill

DroDf21DolDf4 Functions defined by bqUations (4.86) to (4.88)-

DF Skin-friction drag force MLT-2

D. . Non-linear or convective terms in

113 equations (4.60) to (4.63) •••

DP Pressure drag force MLT-2

DT Total drag force MLT-2

Dv Diffusion constant L2T-1

Dw Size of wake

e Ratio of semi-minor to semi-major axes of an oblate spheroid

e' Coefficient in equation (4.2)

ee2l e3 Unit tangent vectors in the directions x1'x2'x5

16

e. Rate of strain tensor for a Newtonian fluid T-1

e. . Error in the numerical solution at the mesh point (i,j) defined by equations (F.11) & (F.51) -

E

E2

E E"

Ec

f(e)

fn(Re)

f(z)

fn(iPe)

F

Fn(Re)

Fn(ze)

g

g

gr

gij G

Gr

Specific internal energy HM 1

Differential operator defined by equation (3.36) L-2

Maximum absolute error defined by equation (F.18)-

Function defined by equation (F.20)

Eccentricity of an oblate spheroid (a/d) •••

Vorticity function defined by equation (4.6)

Function defined by equation (2.32)

Coefficients in equation (2.40)

Function defined as ez (sphere) or as cosh z/cosh z

s (oblate spheroid)

IMO



ONO

External or body force vector per unit volume ML-2T-2




Gravitational acceleration vector LT-2

Coefficient in equation (4.2)

Euclidean metric tensor L2

Vorticity function defined by equation (4.10) ,

Grashof number for heat transfer (iD 3 2)

h Mesh size in the z-direction

h' Coefficient in equation (4.2)

h1,h2,h3 Scale factors in the directions x1,x2,x3

hl, Overall heat transfer coefficient HL-2T-1

hT(0),hT(x2) Local heat transfer coefficients HL-2T-10-

H3(i,j) Functions defined in Table 2

Is Function defined by equation (C.14)

JAI J-factor for heat transfer (Nu/RePi3)

k Mesh size in the . 9-direction

kc Continuous phase mass transfer coefficient

kT Thermal conductivity

K Dimensionless pressure coefficient defined by equation (D.31)

Ko Dimensionless pressure coefficient at the front stagnation point

Local dimensionless pressure coefficient

1 Arc length

L 2/h2 + 2/k2

m Symbol denoting (sinh2z + cos29 in Appendix D -

M Number of mesh steps in 6-direction

M1 N + 1 See

MM Number of mesh steps in z-direction Fig.

MM1 MM + 1 (4.1)

M Local heat transfer number (Nuc /ReYPr-')

Outward pointing normal vector from the body surface

n1 Exponent of Re in equation (2.111)

n2 Exponent of Pr in equation (2.111)

Td Numerical solution of the finite-difference equations

NP Total number of internal regular mesh points

Npp Number of point values unconverged

N Total number of point values (unknown)

Nu Overall Nusselt number (hTDC/kT)

Nu Molecular conduction Nusselt number in a o finite stagnant medium

Nuoo Molecular conduction Nusselt number in an infinite stagnant medium

Nue Local Nusselt number (hT(9)DC/kT)

17

O1.

INN

IWO

gab

•••

ML-1T-2 p Pressure

Pa

Re (Navier-Stokes) or Pe (energy)

Pe

Peclet number ( UDC/ a, or PrRe)

Pr Prandtl number ( Cp /LT = / a, ) •••

18

Heat flux vector HL-2T-1

Components of heat flux in the directions X1'X2'X3 HL

-2T-1

Volumetric flow rate L3T-1

Heat flow rate HT-1

Spherical polar coordinates (L,-,-)

Position vector

Dimensionless radial coordinate (r/R)

q

q1/c12113

Q

QT (r, 0,4))

r

ro

•••

IRV

100

010

Ratio of the semi-major diameter of the outer boundary of the field to the semi-major diameter of the particle

rm(m=1,3,51..)Coefficients in equation (2.51)

r(x) Radii of sections of body of revolution taken at right angles to the axis of revolution

Radius of sphere

Ro Radius of outer spherical surface

Re Reynolds number (UDC / V )

Rec Critical Reynolds number at which separation of the flow first occurs

R. . Residual at mesh point (i,j) defined by equation (4.63)

R0 Residual at mesh point 0 defined by equation (F.59)

Symbol denoting sinh z in Appendix D

S Surface area L2

31(j)1B2 Functions defined in Table 2 00

Sc Schmidt number (1// Dv )

Sh

Sherwood number (kcDC / Dv )

St Stanton number ( Nu / RePr )

EMI

t

Time T

o't21t41' Coefficients in equation (2.106)

tol 1 t'1 2'3 tl'4 t'Coefficients in equation (2.108)

T Temperature

T*

Dimensionless temperature defined as (T - To)/(T

To)

•••

Tn Temperature functions in the expansions (2.82) & (2.88)

Tn Temperature functions in the expansion (2.89)

To Temperature of the undisturbed flow (bulk) 6)

Ts Temperature at the surface of particle

,LT Defined as (Ts - To)

u Velocity component along the surface (in x-direction) LT

Small perturbation velocity vector LT-1

u* Dimensionless velocity used in Appendix D

um(m=1 3,5,..)Coefficients in equation (2.50)

PO.

U Undisturbed stream velocity LT-1

fT Free stream velocity vector LT-1

U(x) Potential flow velocity at the surface of particle LT

-1

✓ Velocity component normal to the surface of particle (i.e. in y-direction) LT-1

V Velocity vector LT-1

Velocity components in the directions x x_ LT-1 vilv2,v3 1/x 2'

vr/ velv Velocity components in the directions r,9, LT-1 Ar vzi

v l

vir Velocity components in the directions z1 9,95 LT-1

vr * v* Dimensionless velocity components in r andO 9 directions

✓ Volume L3

V Free stream velocity vector (uniform flow) LT-1

w Exact solution of the finite-difference equations (F.2) OOP

W Arbitrary function of z and

tilt Complex potential defined by equation (2.16)

We Wall effect correction factor defined by equation (2.72)

xalz Cartesian coordinates

x,y Coordinates used in the boundary layer theory (Chapter 2)

X1'X2'X3 Orthogonal curvilinear coordinates

19

•••

Y1'72a3

Y2

y3 Y

Y1'.2.211-3

z

zo

20

Rectangular Cartesian coordinates

Distance of a point from the axis of symmetry y3 L

Axis of symmetry

Value of y2 along the outer boundary of the field L

Quantities defined in Table 2

Elliptical coordinate or representing In r* for the sphere

Value of z at the outer surface enclosing the flow field

Value of z at the particle surface

Oblate spheroidal coordinates

Thermal diffusivity ( kT /p cp ) L2T-1

Angle defined by equation (D.51)

Temperature coefficient of volumetric expansion,, in the Grashof number for heat transfer dor

8 Hydrodynamic boundary layer thickness L

al' Thermal boundary layer thickness L

Sij Kronecker delta Oft

Forward difference operator

E Absolute relative accuracy defined by equation (4.101) -

et Small disturbance

Vorticity ( (4)3 ) T-1

S Dimensionless vorticity ( De /2U ) -

8 Angular coordinate Mao

es Angle of flow separation •••

/4 Viscosity 111,-1 T-1

V Kinematic viscosity (P. f) ) L2T-1

P Density

T. Stress tensor in a Newtonian fluid ML-1T-2 lj

q6 Arbitrary scalar quantity used in Appendix B

cP Coordinate representing the angle of rotation about the axis of symmetry y3 .41

042103,44 Fractions used in equations (F.13) to (F.16)

21

Velocity potential

Rate of heat generation per unit volume by viscous dissipation

Stream function

Dimensionless stream function (1P/U(2Dc)2 )

Stream function used in boundary layer flow

L2T-1

HL -3T• -1

L 3T -1

L21,7-1

'Pm (m=1,3,5,..)Coefficients in equation (2.52) Stream function used in equation (2.41)

GO Vorticity vector T-1

W11(432"3 Vorticity components in the directions x1'x2'x3 T

-1

f2 I fill c22 Displacement or relaxation factors for equations (4.81),(4.79), and (4.8o) •••

L-1 Vector operator del

Subscripts

i Coordinate directions 1,2,3 (Appendix A)

j Coordinate directions 142,3 (Appendix A)

k Coordinate directions 1,2,3 (Appendix A)

ij (i)th row and (j)th column (Appendices A and D)

i Mesh point index in the z-direction (Fig. 4.1)

j Mesh point index in the -direction (Fig. 4.1)

i,j Indices of a mesh point in the flow region (Fig. 4.1)

o Free stream (or outer boundary)

r-direction

z z-direction

9-direction (local) )e' 4-direction

s Surface of particle

S Flow separation

m Integer subscripts used in Chapter 2

n Integer subscripts used in Chapter 2

T Thermal

Superscripts

(n) (n)th iteration

Dimensionless variable

22

CHAPTER

INTRODUCTION

The transport processes of heat,mass, and momentum are the

basis of many unit operations in chemical engineering, as for

example : evaporation, humidification, drying, distillation,

absorption, extraction, and fluidization. Any fundamental study

of these unit operations becomes ultimately a study of the

transport processes involved.

Recent developments in engineering and science have made

significant contributions to the study of the basic theory of

transport processes. Furthermore, the practical importance of

the basic theory has been greatly enhanced by the use of high-

speed digital computers. These devices have made possible the

application of theory to complex situations for which it was

formerly necessary to be satisfied with empirical methods.

In many chemical engineering problems, the knowledge of

the heat or mass transfer rate between systems of fluid or solid

particles (such as bubbles, drops, and catalyst particles) and a

continuous surrounding fluid is of considerable importance. In

order to understand the mechanism of transfer in such systems,

it is customary to study transfer from a single solid particle,

drop or bubble. The results of such study may occasionally be

applied to multiparticle systems.

Most researchers in this field have used the spherical

shape as their model for the investigation of transfer processes

from particles. However, often the particles are observed to be

non-spherical and to have shapes which approximate spheroids as

pointed out by Skelland and Cornish who surveyed the relevant

literature.

Forced convective heat and mass transfer data have been

23

correlated by a number of workers and the results for isolated

spheres were recently well simmarized213. Countless correlations

have been proposed to establish the relationship between the

Nusselt number (heat transfer) or Sherwood number (mass transfer)

and the Reynolds number and the Prandtl or Schmidt number. However,

the usefulness of these semi-empirical correlations is in many

cases limited because they fail to provide an adequate explanation

of the underlying mechanisms.

The theoretical problem of forced convective heat transfer

from the surface of a body is expressed by the equations of

motion (momentum), continuity, and energy. These equations form

a complex set of interdependent equations which are enormously

difficult to solve. Because of the complexity of these partial

differential equations it has nearly always been necessary to

introduce large numbers of simplifying assumptions and to obtain

solutions of the simplified equations.

When the changes in the physical properties of the fluid,

which arise because of temperature variations, are very small,

the equations of motion (momentum) and continuity can be solved

independently of the energy equation, and the solution obtained

(velocity distribution) used to solve the energy equation for

the temperature distribution.

The equations of motion are non-linear and it is generally

impossible to find exact analytical solutions even when the fluid

is Newtonian. However, in some limiting cases, these equations

can be simplified by the omission of particular terms and

solutions obtained which have a useful range of applicability.

The only exact analytical solutions available for Newtonian flow

around particles are those of creeping flow and potential flow,

i.e. for the limiting cases when the Reynolds number approaches

zero and infinity, respectively4'5'6 . Many other approximate

24

solutions have been obtained among which those based on the

boundary layer assumptions are the most successful6 . These

analytical or approximate solutions have been used with the energy

equation to solve many heat transfer problems.

An asymptotic solution for transfer from a sphere in creeping

flow has been obtained by Acrivos and Taylor? for the limiting

case when the Peclet number is very much less than unity. At high

Prandtl (or Schmidt) number, however, the energy (or diffusion)

equation may be simplified by the boundary layer approximations

which reduce the original equation to more easily solvable one :

this is known as the thermal (or concentration) boundary layer

equation. Analytical solutions of this equation have been obtained

by Friedlander8 and Levich9 who used Stokes' solution for the

velocity distribution, and also by Hamielec et al1011 who used

approximate velocity profiles at intermediate Reynolds numbers.

At high Reynolds and Prandt1 numbers, the simplifying

assumptions of boundary layer theory have been successfully

applied to both the Navier-Stokes equations and the energy

equation6 4, Exact solutions of the resultant hydrodynamic and

transfer boundary layer equations for an axisymmetric body have

been obtained by FrEssling12 who used power series expansions in

order to replace the partial differential equations by infinite

sets of ordinary differential equations. Green13 generalized

FrUssling's approach and obtained exact solutions of the convective

transfer boundary layer equations expressed in orthogonal curvi-

linear coordinate systems. Forced convective transfer data for

oblate spheroids have been obtained for high values of the

Reynolds number

Most workers have been concerned with the study of transfer

rates from the region between the front stagnation point and the

25

separation zone. Boundary layer theory is not applicable beyond

separation and theoretical predictions are, therefore, lacking

for the wake region. Furthermore existing solutions of the boundary

layer equations do not adequately predict transfer rates at all

points upstream separation.

In practice, drop Reynolds numbers are generally less than

one-thousand. Although this intermediate range of the Reynolds

number includes creeping flow, the application of the boundary

layer theory is not strictly justified. Thus, in order to predict

the local and overall heat transfer rates from axisymmetric

particles at intermediate Reynolds number it is necessary to

obtain solutions of the complete Navier-Stokes and energy equations.

Jenson15'16 expressed the time-independent Navier-Stakes

equations in finite-difference form and obtained numerical solutions

using relaxation methods. He obtained stream function and vorticity

distributions for the flow around a solid sphere at Reynolds numbers

less than forty. A similar method was used by Hamielec et a117'18

who obtained solutions in the intermediate Reynolds number range

using a digital computer. These workers have only considered the

case of steady flows around spheres and the range of the Reynolds

number they considered is very limited compared with the range

experienced in practice.

Based on these considerations, the project to be described

was initiated to study theoretically steady-state forced convective

heat transfer from solid axisymmetric particles to Newtonian fluids.

The theoretical prediction of heat transfer rates over the entire

particle surface requires the solution of the complete equations

of Navier-Stokes, continuity, and energy. In this case, the

Navier-Stokes and continuity equations can be solved independently

of the energy equation and the velocity distribution thus obtained

26

used to obtain a solution of the energy equation. In this study,

solid oblate spheroids, including the limiting case of a sphere,

have been chosen to represent an idealized particle shape which

is met in many transfer operations.

Numerical solutions of the Navier-Stokes and continuity

equations have been obtained for a large range of the Reynolds

number (0-500) and for a range of spheroidal shapes. These

solutions in terms of the stream function are then used to obtain

numerical solutions of the energy equation for various values of

the Prandtl number. It is important to note that solutions at

Reynolds numbers greater than five-hundred have not been obtained

as in this range the wake becomes unsteady and vortex shedding

occurs19 .

The method of solution used is a finite-difference one : a

set of finite-difference equations are obtained by the expansion

of the terms in the original partial differential equations in

Taylor series. Each finite-difference equation relates the values

of the dependent variables, such as vorticity, stream function,

and temperature, at neighbouring mesh points, The solutions to the

problem are thus found at a finite number of mesh points distributed

regularly through the enclosed flow field. The large sets of

simultaneous algebraic equations thus obtained are solved

iteratively using the extrapolated Gauss-Seidel method20121

Initial guesses of the variables are supplied, and the values of

the relaxation factors are appropriately fixed to ensure stability

and rapid convergence.

Computer programmes are developed in a general form which can

be used to solve the finite-difference equations for all particle

shapes considered. The distributions of vorticity, stream function,

and temperature which are thus obtained are used to calculate the

27

local and overall heat transfer rates, the drag experienced by the

particle, the pressure distribution on the surface, the angle of

boundary layer separation, and the size of the wake formed

downstream. These aspects serve as a yardstick for the accuracy of

the numerical solutions as compared with experimental and other

theoretical results. Also the effects on the solutions of the

Navier-Stokes equations of variations in the mesh size and of the

proximity of the outer enclosing boundary (representing the fluid

at infinity) are investigated.

***********

28

CHAPTER 2

LITERATURE SURVEY

2.1. Introduction

Because of its considerable importance in many engineering

applications heat transfer from particles in an extensive fluid

has been the subject of many theoretical and experimental studies.

In convective heat transfers the rate of heat transfer is

dependent on the hydrodynamic flow field. If fluid movement is

induced by density variations resulting from temperature variations

within the fluid, heat transfer is said to be by free or natural

convection. However, if the motion of the fluid is independent of

density differences it is called forced convection. Although the

two modes may interact, it frequently occurs that one mode

predominates.

It has often been shown that in systems for which the square

of the Reynolds number is large in comparison with the Grashof

number, free convective effects are negligible and heat transfer

is considered to take place by forced convection alone22 . Further,

when the physical properties of the fluid may be assumed to be

independent of temperature, the hydrodynamic flow field becomes

independent of the temperature field, although the latter still

depends on the former. Hence, for any theoretical study of forced

convective heat transfer from particles, the equations of fluid

motion must first be solved.

The basic equations for the transfer of heat in a moving

fluid are obtained from mass, momentum, and energy balances on a

differential fluid element. A complicated set of partial differen-

tial equations is obtained in which the equations are interdepen-

dent, non-linear, and, in general, contain three space dimensions

as well as time.

29

When the fluid is Newtonian the hydrodynamic flow field is

described by equations which are known as the Wavier-Stokes

equations. Because of their non-linearity, exact analytical

solutions of these equations are difficult, if not impossible, to

obtain for all but a few cases. It was realized by a number of

workers that solutions would be more readily obtained if the

equations could be simplified. At high Reynolds numbers, for

example, the application of boundary layer theory reduces the

order of the equations for the region of the boundary layer.

Similar simplifications can be made to the energy equation,

provided that the thermal boundary layer is thin6 Typically,

solutions of the simplified Navier-Stokes equations are obtained

and then used in the solution of the simplified energy equation.

Similarly, the equations can be simplified at very low Reynolds

numbers.

Although many previous workers in the field of convective

heat transfer from particles have concentrated upon steady-state

transfer to a Newtonian incompressible fluid in the absence of

free convective effects, there is, however, a severe lack of

theoretical studies when the simplifying assumptions of boundary

layer theory or of creeping flow cannot be applied.

In this chapter previous work in this field is reviewed. To

facilitate the presentation, the review has been divided into two

main sections : the first section is concerned with the hydrody-

namics of viscous flow past particles and the available solutions

of the Navier-Stokes equations. The second section reviews studies

of convective heat transfer from particles.

It is convenient and helpful to start the literature survey

with a brief review of the basic equations which describe steady-

state forced convective heat transfer to a Newtonian fluid,

30

2.2. Basic Equations of Steady-State Forced Convective Heat Transfer

The basic equations of steady-state forced convective heat

transfer in the case of a Newtonian, incompressible fluid with

constant physical properties are :

(1) The continuity equation :

= 0 (2.1)

(2) The equations of motion (Navier -Stokes)

Cr • ‘/7 77. " 2 (2,2)

inertia pressure viscous

terms terms terms

where p is the local pressure relative to the undisturbed

hydrostatic pressure which would occur if the fluid was stagnant

at the point considered.

(3) The energy equation, neglecting viscous dissipation :

(14 )T = 2 T

(2.3)

convective conductive

term term

Since the physical properties of the fluid are assumed to

be constant, equations (2.1) and (2.2) provide a complete

description of the motion of an incompressible Newtonian fluid.

The four unknown quantities are the three velocity components and

pressure. However, these are related by four equations; equation

(2.1) and the three components of equation (2.2). When these

equations are solved the velocity field can then be used to solve

equation (2.3) for the unknown temperature distribution.

It is important to note that because of mathematical

difficulties and limitations on computer storage facilities

numerical solutions have not been obtained for three-dimensional

31

flows except in a small number of special cases. However, it is

possible to express axisymmetric flows as two-dimensional flows,

thus reducing the number of space variables by one. The Navier

Stokes equations can, as a. consequence, be expressed in terms of

the single component of vorticity, , in the x1x2-plane as

follows :

v1, r..... ‘ v2 ( ‘ / r ri li E2 ( ...g..- .-- ) .1. =

h32 'c.., h3) (2,4)

h1 a.'1 h3 h2 S:x2 h3

2 , h3 F. ( h2 i..\ I i hi (1_, 0 (2.5) where E = + , —. ) •

h1h2 L...6xI h1 h3 \x1) c5x2 ( h2 h -x 3 - 2

..k 11

and r = ----- ) -6 (h v ) - . (h v )l

s (2.6) .!..-. 1 Ax _ 2 2 3c;2 1 1 j h h I 2 ! ‘-.) 1 ,. -

xi and. b- are orthogonal curvilinear coordinates and scale

factors, respectively, as defined in Appendix A.

By the introduction of the stream function ti)(x1,x2) such

that :

V1 =

v2 = 1 4)

h1h3 6.1 (2.7)

h2h3 ax2

the continuity equation (2.1) is satisfied automatically.

Subbtitution of equation (2.7) into equation (2.6) gives:

E2 4a = h3 (2.8)

By the use of equations (2.7) and (2.8), the Navier-Stokes

equation (2.4) becomes:

L ± L2,111 1 ) Lk (2 '11 -7 = El kp (2.9)

h1h2 ,f5xi 6x2 h32 6x2 6x1 h3-

Equation (2.9) is a non-linear fourth order partial

differential equation*

32

The boundary conditions are:

At the surface of the body;

v1 = == - 0 (no-slip (2.10)

6x2

v2 6 P

ax 1 eondition)

and i.Jf = constant (usually zero) (2.11)

In the undisturbed bulk flow;

= luY‘2, (2.12)

where Yo is the distance of a point on the outer

boundary from the axis of symmetry, and U is the undisturbed

stream velocity.

2,3. Solutions of the Navier-Stokes Equations

Very few exact analytical solutions of the Navier-Stokes

equations have been found, and these are for special cases for

which the non-linear terms in the Navier-Stokes equations vanish

in a natural way. These solutions are applicable to certain types:

of fluid motions such as the well-known Hagen-Poiseuille flow

(parallel flow in pipes) and the Couette flow (flow between two

N parallel walls or between two rotating co-axial cylinders)6

However, flows corresponding to these solutions are only

observed for values of Reynolds number which do not exceed certain

critical values beyond which the flows become turbulent. For

example, the parabolic velocity distribution given by the Hagen

poiseuille solution is observed in practice for Reynolds numbers

less than 2300. The agreement of these solutions and other

available solutions, to be discussed later, with experiment

verifies that the Navier-Stokes equations can be accepted as the

mathematical model of a Newtonian fluid.

In the problem of flow past an object, the full Navier-

Stokes equations are impossible to integrate analytically because

of their non-linearity. As a result, development in this field

33

has usually depended on the use of one or more approximations in

order to simplify the equations. The validity of each of the

resultant approximate solution is limited to a certain range of

Reynolds numbers. A brief review of these solutions is given in

the following subsections.

2.3.1. Limiting Solutions- Potential Flow Theory.

The first stage in the development of theoretical hydrody-

namics involved the study of the flow of an ideal incompressible

fluid (non-viscous). Flows of this type are termed potential flows.

The viscous terms in equation (2.2) vanish and the following

equations of motion (Eulerts equations) are obtained:

= I6 (2.13)

The flow is irrotational and the velocity vector can be

represented as the gradient of a scalar

t (2,14)

where t is termed the velocity potential, which, because

of equation (2.1), satisfies Laplacets equation:

V <19 = 0 (2.15)

For two-dimensional irrotational motions, solutions are

usually expressed by a network of orthogonal lines consisting of

streamlines and of equi-potential lines5123 The latter are curves

of equal velocity potential while the streamlines are curves

of equal stream function IP

Also the solutions of equation (2.13) can be obtained in

terms of the complex potential of the motion which is defined by

the following relationship:

W = G + (I)

(2.16)

following solutions for IP

For a sphere:

= iUr22 n sin (1 - 3

r3)

34 where 41 satisfies equation (2.8) with

i.e. E2 = 0 (2.17)

Equations (2.15) and (2.17) are linear second order partial

differential equations which can be solved easily to give the

curves:

10(x11x2) = constant, and tr(x1,x2) = constant

(2,18)

For inviscid flows around a sphere and an oblate spheroid,

the shapes considered in this thesis, Milne-Thomson23 gives the

(2.19)

4 and, for an oblate spheroid23 /2 :

sinh z cot-1sinh z cosh2z

- ilia2 cosh2z sing0 1

(2.20) sinh zs , -1 ecru sinh zs cosh2zs

Similar expressions can be generated for 4.1)D and the

velocity components are then obtained directly from equation

(2.7) or from equation (2.14).

It is important to note that soliations (2.19) and (2.20)

do not satisfy the conditions of no-slip at the surface. Only the

velocity normal to the surface v1 vanishes while the tangential

velocity may be derived to give for the sphere,

U sin

= 2 e v2 = vv 2

and, for the oblate spheroid,

1-e2 U sin 0

cot-1 - e (e2 sin20 4. cos20)

(1 e2)2 (1 - e

2)` where e is the ratio of the semi-minor to the semi-major axes of

(2.21)

(2.22)

35

the oblate spheroid.

The failure of these solutions to satisfy completely the

prescribed boundary conditions of no-slip at the surface; together

with the famous D'Alembertts paradox, according to which the total

force acting on a particle located in a potential flow is equal to

zero, shows that the theory of ideal fluids is not adequate to

describe fully the motion of real fluids. However, although no

such fluid exists, it is found that for fast flows most fluids

outside the boundary layer region may be treated as ideal and hence

may be well represented by the potential flow theory.

2.3.2. Limiting Solutions- Stokes Flow

Stokes4 succeeded in solving the equations of motion for the

case of very slow motions past a sphere. In this case the inertial

terms in the Navier-Stokes equations are very small and are

neglected completely, thus the Navier-Stokes equations (2.2)

become linear:

vP = v2 4 (2.23)

Flows described by equation (2.23) are termed Stokes

or creeping flows.

Expressing such flows in terms of the stream function alone,

equation (2.9) gives:

E44, = (2.24)

which is of the same order as the complete Navier-Stokes

equation (2.9) so that it is possible to apply the boundary

conditions (2.10) to (2.12).

The solution of equation (2.24) for the flow past a sphere

is as follows:

iUr2 sin26 (1 - a + 13- ) (2.25)

2 r 2 r3

The total drag force, DT, on the sphere is:

36

DT= 6 irp,Ru (2.20

which can be expressed in terms of the total drag coefficient,

CDT' as follows: 24

where Re is the particle Reynolds number.

The relative contribution of the two components of the

drag coefficient is:

CDF : CDP = 2 :

(2.28)

where CDF and CDP are the skin-friction drag coefficient

and the form drag coefficient, respectively.

The case of Stokes flow around oblate spheroids and other

axially symmetric bodies was treated by Payne and Pe1125 Their

solution for flow around an oblate spheroid is:

CDT = (2.27) Re

r 1 -

sinh zs 1-sinh2z s -1 cot sinh zs cosh zs cosh2z

sink z 1 -sinh2z s -1 cot sink z

cosh2 z cosh2zs tP =

2 cosh2z sin2did

The total drag force is:

DT= 8irilati sinh zs (1 . Binh2zs)cot-1 sinh z

(2.29)

(2.30)

which can be expressed in terms of the total drag coefficient,

CDT' as follows:

32 CDT = f(e)

Re (2.31)

2 3/2 ( e )

where f(e) = e (2,32) -1-

e(1 e2)2 (1 -2e2 )cot-1

and e is the ratio of the semi-minor to the semi-major axes of

the oblate spheroid.

37

As e 1 , f(e) and hence equation (2.31) reduces

to equation (2.27) for the sphere.

Stokes flow is valid for the limiting case when the Reynolds

number is very much less than unity and provides a good description

of the flow field in the neighbourhood of the particle. Far away,

the flow approaches the uniform stream velocity where the neglect

of the inertial terms becomes invalid. However, creeping flow

solutions are useful in the theory of lubrication and in the

suspension of fine particles in a fluid when the motion is very slows

2.3.3. Perturbation Methods

Oseen26 improved upon Stokes' solution by linearizing the

Navier-Stokes equations in such a way as to account for the

inertial terms where they are important (in the region approaching

uniform flow), but to neglect them in the region close to the

surface. Oseen expressed the velocity as follows:

= U (2.33)

where U is the free stream velocity vector, and u is a

small perturbation velocity vector whose square and its products

with its derivatives are negligible. Thus equation (2.2) is

reduced to the linear form:

(U. 7 = - v p v2 a (2.34)

For flows past a sphere equation (2.34) may be written in

terms of the Stokes stream function as follows:

63,3 (E2 1.1 ) = VE441 (2.35)

where y3 = r cose

(2.36)

Equation (2.35) is, as in the original Navier-Stokes

38

equation (2.9), a fourth order partial differential equation and

the construction of its exact solution subject to the boundary

conditions (2.10) to (2.12) is a matter of some difficulty.

As a first approximation, Oseen solved equation (2.34)

satisfying only the boundary condition at infinity (2.12). The

improved expression of the drag coefficient for the sphere is:

24

CDT =3 (1 + Re)

Re (2.37)

which is applicable to a good approximation up to a Reynolds

number of two.

The exact solution of Oseen's equation (2.34) was success-

fully obtained by Goldstein27 Oseen's approximation was taken

to hold at great distance from the sphere and also at its surface

where the no-slip conditions (2.10) were applied to complete the

solution. Goldstein then obtained the following expression for

the drag coefficient, CDT, as a power series of the Reynolds

number Re:

24 CDT = — (1 14- Re Re 1290

Reg 71 3 204E-0 'e

70179 306400 Re4 "6° (2'38)

Goldstein compared his results of the drag coefficient

with the experimental correlation of Zahm28 and found good

agreement for Reynolds numbers less than two, but at Reynolds

number of five his solution exceeded experimental measurements

of the drag by about 10 % .

Tomotika and Aoi29 calculated in detail the flow patterns

around a sphere on the basis of Goldstein's exact solution of

Oseen's linearized equation. They expressed the results in terms

of Stokes' stream function, as follows:

= -iUR2 sing L2 (R

2 Re) (r2 -

R

39

2. R2 + Re(-r 2. --.2-)cos0

R r (2.39)

which reduces to Stokes' solution (2.25) in the limit when the

Reynolds number tends to zero. Their calculated pattern of stream-

lines clearly indicated the formation of a stationary vortex ring

behind the sphere at a Reynolds number as low as 0.1. They also

pointed out that, independent of Reynolds number, CDF:CDP= 2:1 as

given in equation (2.28).

Pearcy and McHugh30 have also carried out a detailed compu-

tation of Goldstein's solution using a digital computer. They

plotted the velocity field around a sphere at low Reynolds

numbers (<10). Pearcy and McHugh found no separation of the flow

even at a Reynolds number of ten, and they criticized the work of

. Tomotika and Aol29 on this point.

Proudman and Pearson31 succeeded in obtaining higher approxi-

mations to the flow around a sphere than those represented by

Stokes4 and Oseen26 . Two expansions of the stream function were

developed: one for the region close tc and the other for the region

far from the sphere. The assumed expaneions are of the form:

For the inner region:

= fn(Re) gin(r)

(2.40)

and for the outer region:

L= 2_,Fn(Re) lyn(rRe,0) (2.41)

which are referred to as Stokes and Oseen expansions, respec-

tively. In these expansions (r,9) are polar coordinates and

fn+1/fn and Fn+1/Fn vanish as the Reynolds number tends to zero.

Substitution of these expansions in the Navier-Stokes

equations then yields a set of differential equations for the

4o

coefficients and NYn but only one set of physical boundary

conditions is applicable to each expansion (the no-slip conditions

for the Stokes expansion and the uniform stream condition for the

Oseen expansion) so that unique solutions cannot be derived

immediately.

However, the fact that the two expansions are (in principle) 1

both derived from the same exact solution leads to a matching

procedure which yields further boundary conditions for each

expansion. It is thus possible to determine alternately successive

terms in each expansion. Two expansions are said to match when they

agree to any prescribed order of accuracy. The leading terms of the

expansions are shown to be closely related to the original solutions

of Stokes and ()seen.

The improved expression for the drag coefficient is:

24

CDT = Re 16 Re+ 70 Re2 ln(Re/2) 0(Re2/4)11 (2.42)

which is valid for Reynolds numbers less than five.

Breach32 generalized the results of Proudman and Pearson

to apply to all ellipsoids of revolution both prolate and oblate

at low Reynolds numbers. He obtained exprcosions for the drag

coefficient - his expression for the oblate spheroid may be

rearranged as follows:

( CDT = 32

f(e) 1 f210e ) if(e)Re + Re

2 ln(Re/2) 0(Re2/4)1 (2.43)

Re

which clearly reduces to equation (2.31) as Reynolds number tends

to zero.

41

2.3.4. Boundary Layer Theory

This method of simplification of the Navier-Stokes equations

was introduced by Prandtl (1904) who regarded the flow at high

Reynolds numbers as being split into two regions. A thin boundary

layer adhering to the surface in which the viscous effects are

confined and the main flow outside this layer in which viscosity is

unimportant and potential flow theory applies.

With these assumptions, the equations of motion are reduced

to forms which can be solved with less difficulty. For flows about

an axisymmetric body of revolution, Boltze33 derived the following

boundary layer equations for steady-state incompressible fluid

flows:

612 6u dU(x)

x 6y dx

62u

+ V T-2— uY

(2.44)

(ur(x))

(vr(x))

dx (y = 0 (2.45)

With the boundary conditions:

y =0 : u= v= 0 (2.46)

y u -4' U (X ) (2147)

where x is the distance measured along the body surface from the

front stagnation point to the base point of the normal to the

surface, y is the coordinate at right angles to the surface, r(x)

is the distance from the base point of the normal to the surface to

the axis of rotation, u and v are the velocity components parallel

to x and y respectively, and U(x) is the potential flow velocity

at the surface.

The methods used for the solution of the hydrodynamic boundary

layer equations are of two types, which are known as exact solutions

and approximate solutions. In the first type, the differential equa-

tions are satisfied for each point in the boundary layer and hence

42

for each fluid particle. In methods of the second type the equations

are used in the momentum integral form which expresses the average

behaviour of the fluid in the boundary layer and which satisfies the

boundary conditions at the body surface and at the edge of the

boundary layer.

Exact Solutions

Methods of obtaining exact solutions of the hydrodynamic

boundary layer equations have been developed by means of series

solutions. Blasius Howarth35 , and Fe8ssling12 have made important

contributions to such methods.

For axially symmetrical boundary layers formed around bodies of

revolution, FrBssling12 used Boltze's33 boundary layer equations, as

given by equations (2.44) and (2.45), and introduced a modified form

of the stream function 111J(x,y) as follows:

u_ a r (x ) 6y

[*(x,Y) r(x)] v = -:(3c) 6 x1 6‘ 5P(x1Y) r(x)] (2.48)

The continuity equation (2.45) is satisfied identically by this

form of the stream function.

Substitution of u and v from equation (2.48) into equation

(2.44) gives, in terms of the stream fun,tion only:

614)(xly) 62 4)(x,y)

CY 6x a Y {kp(xly)

6x

dTJ(x) U(x) // dx

dr(x)

r(x) dx

83 Lp (x,y)

aY3

1 qi x )6 / y

(2.49)

IY)

Equation (2.49) is a non-linear third-order partial differential

equation. Thus, the important simplification resulting from the appli-

cation of boundary layer theory is the reduction of the order of the

equations of motion from a fourth-order (equation (2.9)) to a third-

order (equation (2.49)).

43

Following an identical manner to the series solution for two-

dimensional flow of Blasius34 Fr°6ssling12 expressed the potential

flow velocity, the contour of the body, and the stream function in the

following power series forms:

U(x) u1x ▪ u3x3

r(x)

= r1X r3x3

q)(x,y) kpix tp3x3

• u5x5 41041.0 (2.50)

+ r5x5 + 0009 (2,51)

+ Ip5x5 + 04000 (2.52)

where the coefficients um and rm (for m=113,51...) are assumed to be

known. The coefficients are functions only of y. These series

expansions are substituted into the boundary layer equation (2.49)

and the coefficients of the corresponding powers of the current

length x equated. In this way, an infinite set of ordinary differen-

tial equations are obtained which can be integrated by a step-by.,-step

method to calculate the velocity distribution in the boundary layer.

For the sphere, the functions U(x) and r(x) are given by:

lt sin ?I U(x) = i , (2,53)

R and r(x) = R sin N (2.54)

Hence, the coefficients um and r,a are -readily obtained from the

sine series.

However, for many axisymmetric bodies of revolution, including

the oblate spheroid, the functions U(x) and r(x) are more complicated

and cannot be expressed in simple expansion forms. In such cases,

FrUslingls series solution cannot be applied in its present form.

The difficulty of applying FrBsslingls series solution to

spheroids was overcome recently by Green13 who derived the boundary

layer equations in orthogonal curvilinear coordinates and used a new

series solution in which the velocity function was expanded in powers

of the curvilinear coordinate measured along the surface.

= 0 ;

44 Approximate Solutions

The numerical difficulties involved in the exact solutions of

the boundary layer equations even for simple cases, have lead to the

development of approximate solutions which are quicker to apply and

which predict reliably the overall characteristics of the boundary

layer. The momentum integral equation provides the basis of such

methods. Suitable expressions for the velocity distribution in the

boundary layer are assumed which satisfy the flow boundary conditions

in the boundary layer.

The first approximate solution was carried out by Pohlhausen36

for the velocity distribution in two-dimensional flows. He used

quartic polynomial in y to represent the velocity distribution inside

the boundary layer:

u = co(x) + cl(x)y + 02(x)y2 + c3(x)y3 + c4(x)y4

(2.55)

The coefficients co(x) to c4(x) are evaluated from the following

boundary conditions at the surface and at the edge of the boundary

layer:

eaU(x) dU(x) u = 0 2

(2.56) 6Y 1.1 dx

au 2

u = U(x) , = = 0 (2.57) OY (ay

where 0 is the boundary layer thickness. The velocity profile of equation (2.55) was then used in Vol:1-

Karman's momentum integral form of the combined hydrodynamic boundary

layer equations (i.e. equations (2.44) and (2.45)), in order to derive

a single ordinary differential equation for the boundary layer

thickness (1) .

The approximate solutions for axisymmetric flows are very

similar to the above solutions for two-dimensional flows. Millikan37

integrated Boltzets boundary layer equations,(2.44) and (2.45), to

derive the following momentum integral equation:

45

d 1 u2dy

0

- U(x)

U(x)

d r8 1 dr(x)

8 _. u dy + _ _ ( u2 dy dx tw 0 r(x) dx i

„.,8 c-, dU(x) &a

u dy ) = 0 U(x) ---_. - Li( -,.....- )y=0 (2.58)

'6 dx 6Y

dx

Tomotika38 solved Millikanis momentum integral equation (2.58)

for the flow around a sphere. He used Pohlhausents36 velocity profile

of equation (2.55) in equation (2.58) in order to determine the

boundary layer thickness around the sphere. The functions U(x) and

r(x) used by Tomotika are given by equations (2.53) and (2.54),

respectively.

Beg14 applied equation (2.58) to the case of fluid flow around

oblate spheroids. He applied equation (2.22) for the potential flow

velocity U(x), and described dx and r(x) in terms of oblate spheroidal

coordinates as:

sin dx = a cosh zs (1 cosh2zs

I ) 2 d 0 (2?59)

r(x) = a cosh zs sin()

(2060)

Using Tomotikals approximate method, Beg solved the modified

form of equation (2.58) for the boundary layer thickness by the

Runge-Kutta method of numerical integration. Values of the boundary

layer thickness were calculated at intervals of one-tenth of a degree

from the front stagnation point to 900 around oblate spheroids having

the ratio of the minor to major axes of 0.8125, 0.625, 0.4375, and 0.25.

It should be noted that boundary layer theory is applicable at

very high Reynolds numbers (Re ---iPcm0) and in the unseparated flow

region only. At low Reynolds numbers, the simplifying assumptions of

the boundary layer theory are not valid as the boundary layer becomes

thick.

46

2.3.5. Galerkin Method

Other attempts to find approximate solutions of the Navier-

Stokes equations have been made by the use of variation methods - in

particular by the Galerkin method.

A trial function for the stream function is assumed with adjus-

table parameters so that it satisfies the partial differential equa-

tion and the boundary conditions as closely as possible. Optimum

values of the parameters or the coefficients of the assumed function

are obtained by the use of the Galerkin orthogonality method. This

method implies that the error resulting from the replacement of the

dependent variable by the proposed trial function, which is represented

by linear combinations of a set of linearly independent and differen-

tiable functions, should be orthogonal to this set. The method and its

application to the solution of equations of change has been recently

given by Snyder et a139

Kawaguti used the Galerkin method to solve the Navier-Stokes

equations (2.9) for the case of fluid flows around a sphere. As a trial

function, he assumed a special series for the stream function in terms

of Legendre polynomials. His series is of the form:

( )k sin2 1.31* r

)3- sin apcos (2.61) II- 2- = 1

with -2 k <4 and -1 1 4 (2.62)

Two sets of values of k and 1 were chosen: one set for the range'

0.<:*1 <10, and another set for the range 10 <Re4‹:70 For the

latter range, the following form was successful:

IP I = I',

...1 it i )2 + al( r — ) + a2 r ( 2 )2 + a in 3( .1r. )3 + a4( .1,1 ))4 s• 20

UR2 - j

, + `b1 R 1 ( r +

R 2 4- , b ( — 4

r 3 r R )3 + bk( R —r ( sin2(.-1 cos 61

(2.63)

47

The constants a1 to a4 and b1 to b4 were determined by the use

of the boundary conditions, and the Navier-Stokes equations (2.9)

employing the orthogonality principles of the Galerkin method.

The low Reynolds number range solution shows no flow separation

at Reynolds numbers less than 10 but does show separation at Reynolds

number of 20, The high Reynolds number solution, however, shows no

separation at Reynolds number of 20.

The drag coefficients obtained by this method may be expressed

as follows: 32

_ ( 300 57a1 (2.64) 87Re

32 CDP = ( 195 24a1 ) (2.65)

87Re

Hamielec et al11141 using an identical method to that of

Kawaguti4o1 carried out the computation of a1 and b1 for Reynolds

numbers up to 5000 . Their results for the drag coefficients agree

very approximately with the standard (experimental) drag curve in'

the region 10 .,:(Re411000 but show several points of inflexion at

intermediate values of the Reynolds number which are not found in the

standard drag curve.

The success of this method depends upon the choice of the trial

form of the stream function and, therefore, cannot be expected to

apply adequately over a large Reynolds number range.

2.3.6. Numerical Methods

Other approximate solutions of the complete Navier-Stokes

equations are obtained by numerical methods. In these methods the

partial differential equations are replaced by equivalent sets of

finite-difference equations which are solved by successive approxi-

mations. These methods were first used by Thom42 for flow round a

circular cylinder at Reynolds number of 10 , He solved equations

CDP

48

(2.4) and (2.8) simultaneously to obtain the vorticity and the

stream function qi . Other solutions at low Reynolds numbers were

obtained by Thom43'4445 .

Kawaguti46 used Thom's method and obtained the solution for the

flow round a sphere at Reynolds number of 20, but no separation of

the flow was found.

The basic theory of these numerical methods has been developed

extensively by Fex474849 Fox and Southwell50 and Allen and

Dennis51 . These workers reduced the amount of the actual calculations

involved and gave rise to the theory under the title of Relaxation

Methods

Allen and Southwell52 applied relaxation methods, with satis-

factory results, to the problem of fluid flow past a fixed cylinder

at Reynolds numbers of 0, 1, 10, 100, and 1000 . Solutions for the

same problem were also obtained by Apelt53 at Reynolds numbers of

4o and 44 .

Jenson15'16 was successful in obtaining solutions to the problem

of streaming flow past a solid sphere at Reynolds numbers of 5, 10,

20, and 40 using relaxation methods. He expressed the vorticity

transport equation (2.4) and the stream function equation (2.8) in

spherical polar coordinates which were modified in exponential form

for the radial variable. This form of coordinates enabled him to use

uniform mesh sizes which correspond to variable grids in the physical

plane. In order to obtain workable boundary conditions the sphere was

assumed to be in a flow contained in a cylinder having a diameter six

times that of the contained sphere. The flow close to the cylinder

surface was assumed to be uniform and parallel.

The results were presented as stream function and vorticity

distributions; from which the pressure distributions at the sphere

surface and the drag coefficients were calculated. The results were

shown to compare favourably with experimental work. The Reynolds

49

number at which separation first occurs is usually termed the critical

Reynolds number Rec, which Jenson15 estimated, in his case, to be 17.

Russell20 has presented methods of obtaining steady-state

solutions of the Navier-Stokes equations on a digital computer, He

discussed the relative merits of various finite-difference formulae,

which approximate the differential equations. The choice of such

formulae is always a compromise between accuracy on the one hand and

ease of solution on the other. The simplest formula is the five-point

approximation (also known as the Liebmann formula), which replaces

the differential operators on a general function, W at a grid point

(i,j) by the values of the function at the grid point and at four

neighbouring points. Russell carried out an investigation of the

iterative methods used to solve the finite-difference equations, and

pointed out that the method of successive optimum displacement by

points converges rapidly. By this method, the new point value of a

general function, W.. is calculated at every point in the grid in 2.1 3

regular succession, using the most recently calculated values at the

neighbouringpoints.Ifthisnewpointvalueisdenotedby and ilj

the corresponding value at a previous iteration is denoted by W(n-1). • 113

then an improvement on the convergence rate can be achieved by taking:

(n) (n-1) W(n) = witi ( RT1fin` vi(n-1) ) 10 10

(2.66)

where 0. is the displacement or relaxation factor,• which varies

between 0 and 2 , and W(in) denotes the modified new point value. ti

Optimum values of ,Q can be estimated for linear problems as

functions of mesh size and dimensions of the region of integration20

4.

For non-linear problems, these optimum values are difficult to

estimate and, as in the present problem, may vary with the position

of the mesh point, However, a single value of f/ may be used and

its value estimated by trial and error.

50

Hamielec et al1718 used Jenson's15 finite-difference equations

and extended the work to higher Reynolds numbers using IBM 7040 and .

7094 Computers. Solutions of the Navier-Stokes equations for the flow

round a sphere were obtained at Reynolds numbers in the range 0.1 to

200, using the method of successive optimum displacement by points

To obtain rapid convergence and stable solutions, relaxation

factors were used and their values were estimated by trial and error*

The flow round the sphere was assumed to be bounded by a large sphere

having a diameter seven times that of the contained sphere. Along this

outer boundary the flow was assumed to be of zero vorticity and

parallel. At low Reynolds numbers, they investigated the influence of

the proximity of the outer boundary on the numerical solutions of the

flow problem around a sphere. Calculated values of the drag coeffic-

ients were high in comparison with experimental values, and closer

agreement was obtained as the outer boundary was moved further away

from the sphere. The effect of the proximity of the outer boundary on

the drag coefficients is referred to as the wall effect.

Hamielec et al reported that flow separation first occurred at

a Reynolds number of 22 . They also made a comparison between their

results for the stream function distributirns and those obtained by

Galerkin's method and found reasonable agreement upstream separation.

2.3.7. Experimental Studies of Viscous Fluid Flows

Experimental studies of viscous fluid flows have been made,

mainly, for the case of flow round single spheres. The results which

are relevant to the present study are those concerning: (a) the

critical value of the Reynolds number, Rec , at which the formation

of the vortex-ring downstream, and hence separation, first occurs,

(b) the determination of drag coefficients, and (c) the wall effect.

51 (a) Flow separation and the critical Reynolds number

Nisi and porter54 investigated the onset of separation by photo-

graphing the smoke-filled flow of air around a sphere supported in a

square section channel. The sphere was illuminated with a powerful

plane beam of light. Steel spheres of various diameters were used in

channels of different sizes, and the following correlation for the

critical value of the Reynolds number, Reel as a function of wall

effect ratio (ratio of containing wall to sphere diameters, ro) was

obtained:

Rec = 8.15 + 68.2 ro-1.5

(2.67)

This correlation gives a critical value of Reynolds number of

8,15 for no wall effect (i.e. ro =00) and of 12.1 for the wall effect

ratio used in the present work (i.e. for ro = 6.686).

These results show early separation which may be attributed to

the influence of the sphere supports.

Williams55 , on the other hand, found no flow separation even

at a Reynolds number of 720.

A similar technique to that used by Nisi and Porter54 was

employed by Taneda56 . The wakes produced by a steel sphere moving in

a tank of water were photographed at Reynolds numbers of 5 to 300. The

photographs showed that the initial wake formation occurred at a

Reynolds number of 24. This value was obtained by extrapolation of the

plot of the size of the vortex-ring formed downstream against the

logarithm of the Reynolds number to zero vortex-length.

No vortex-ring was observed at Reynolds numbers less than 22

whilst at a Reynolds number of 25.5 it was observed to appear near the

rear stagnation point. Taneda also found that the wake behind the

sphere began to oscillate when a Reynolds number of about 130 was

reached.

It is reported19 that some workers feel that this oscillation

52 occurs at a Reynolds number of 20057 and others at 100058

Garner and Grafton59 studied the streamline patterns of water

flow round a stationary solid sphere by injection of red ink at the

surface of the sphere, The traces thus obtained showed clearly the

configuration of the streamlines containing the wake. The value of the

critical Reynolds number for the beginning of vortex-ring formation is

reported to be between 40 and 50.

In their plot of separation angle versus Reynolds number, the

slope of the curve changes twice; once at a Reynolds number between

40 and 60, and secondly at a Reynolds number between 480 and 520,

after which the angle of separation settles down to a constant value

of 103° from the forward stagnation point as the Reynolds number

approaches 1000. The first change in the slope occurs over the range

of the Reynolds number at which circulation within the wake first

commences (critical Reynolds number range), and the second when the

wake becomes oscillatory. They also observed higher angles of separa-

tion from the sphere when an upstream axial support was used.

From the available observations19'561575859 it seems

probable that oscillation first occurs at a Reynolds number of value

approximately 500 for the case of a smooth sphere moving in a steady

motion through a turbulence-free fluid.

Garner and Skelland60 in their study of liquid droplets in

liquid medium with mass transfer taking place across the spherical

interface, found that the critical Reynolds number was between 19

and 23.

It is reported16 that Keey

61 using phosphor-bronze spheres in

a vertical water tunnel obtained a critical value of 14 for the

Reynolds number,

53 (b) Drag Coefficients

In 1926, Zahm28 drew a general curve of the total drag coeff-

icient versus the Reynolds number for solid spheres. He considered

- ,...- all available experimental results in the range 2x10 5 -..., Re <12x106

and proposed the empirical correlation:

CDT = 28Re-"85 o.48 (2.68)

for the range 0.2 < Re < 2x105. This correlation is valid for

the two ranges 0.2 < Re < 200 and 2x104 <Re < 2x105 ,

but it gives slightly higher values than the standard experimental

drag curve for the range 200 Re <Z.: 2x10 .

Davies62 analysed statistically the experimental results of

total drag coefficient obtained by a number of workers, and deduced

expressions for values of the Reynolds number in the form:

Re = f(CDTRe2)

(2.69)

These expressions are useful for the prediction of the ter-

minal falling velocity of a spherical particle.

Using these expressions, Heywood calculated values of

log CDTRe2 and log CDT/Re and the data were presented in two

tables: the first table for log Re as a function of log CDTRe2

and the second for log Re as a function of log CDT/Re.

The product CDTRe2 is independent of the terminal falling

velocity, and the ratio CDT/Re is independent of the diameter of

the particle. Thus, in order to determine the terminal falling vel-

ocity of a particle, CDTRe2 is evaluated and the corresponding value

of Re, and hence of the terminal falling velocity, can be found from

the first table mentioned above. The diameter of a sphere of known

terminal falling velocity can be calculated by evaluating Cy/Re ,

and then finding the corresponding value of Re, using the second

table, from which the diameter can be calculated.

The tabulated data of Heywood are presented in graphical form

by Coulson and Richardson64 .

54

Lapple and Shepherd65 give complete drag curves for discs,

cylinders, and spheres.

Schiller and Naumann64,66 proposed the following empirical

equation for the total drag coefficient of spheres:

24 = (1 -1- 0.15 Re 0.687 CDT

) (2.70)

Re

which is reliable for values of Reynolds number up to about 800.

Another relation in this range, due to Kliachko67768 , is:

24

CDT = (1 Re'/6 )

Re (2.71)

(c) Wall Effects

The wall effect is particularly important at low Reynolds

numbers. The wall enclosing the flow field exerts an appreciable

retarding effect and hence it increases the drag coefficient of par-

ticles in finite cylindrical boundaries. A number of small correction

factors, , may be applied to account for the effect of the walls

of the containing cylindrical pipe on the drag coefficient of the

contained sphere as follows69

CDT = 6 (CDT)0} (2.72)

where (CDT) is the total drag coefficient with no wall effect

(i.e. sphere in an infinite medium).

The correction factors may be functions of the Reynolds number

and the ratio of the diameter of the cylinder to that of the sphere,

r . Thus69

W: = f(Re,ro-1

) (2.73)

Ladenburg647o introduced a correction factor to the Stokes'

solution which may be written as:

We = 1 2.4 r -1

(2.74)

Faxen69 771 proposed a similar correction factor based on

• Oseen's solution which may be written as:

We = 1 2.1 ro-1

(2.75)

55 From a combination of analysis and experimentation the coeff-

icient of drag for a sphere falling freely along the axis of a cyl-

indrical tube has been evaluated by McNown et al69 for the entire

relative diameter range, and for the Reynolds numbers range

< Re <1 1000. Neglecting the inertial effects, they proposed a

correction factor to Stokes' solution for ro > 4 as:

We = 1 2.25 r -1 5.0625 ro-2 (2.76)

2.4. Convective Heat and Mass Transfer

2.4.1. Theoretical Studies

If the velocity fields have been obtained by the methods out-

lined before, the transport problem can be solved by similar proce-

dures. Numerous theoretical investigations of the problem of heat

and mass transfer from solid spheres have been reported with various

simplifying assumptions and range of validity.

The steady-state molecular diffusion (or thermal conduction)

from a sphere into an infinite stagnant medium was treated by

Langmuir72 who showed that the rate of transfer is described by:

(2.77) Nu = 2 00

where Nu is the Nusselt number based on the sphere diameter.

Cornish24 obtained a more generalized theoretical expression

for predicting the rate of molecular diffusion (conduction) from an

oblate spheroid. His expression may be written in terms of the

Nusselt number based on the major diameter of the spheroid as

follows: .1- . 1Td(d2 - b2 ) 2 Nu 00

(2.78) Tr b2 2 S --- tan-1 (

d2 - b )2 )

where S is the surface area of the spheroid given by equation (C.14),

and d and b represent the lengths of the semi-major axis and the

56

semi-minor axis of the oblate spheroid, respectively as defined by

equation (C.8). This expression reduces to equation (2.77) as a

limiting case of sphere. The derivations of the two expressions

(2.77) and (2.78) are given in Appendix E.

For the case of the velocity field at low Reynolds numbera

Stokes' solution may be assumed and the average Nusselt number Nu,

defined in the usual manner, becomes a function of the Peclet number

alone, where the Peclet number Pe is defined as the product of PrRe.

The theoretical determination of the exact functional relation

between the Nusselt number Nu and the Peclet number Pe is then the

main point of interest in such problems.

Kronig and Bruijsten73 developed a perturbation method for

computing the heat transfer rates from a sphere at uniform constant

temperature to a medium flowing past it at Reynolds numbers very

much less than one. The method is restricted to very small values

of the Peclet number (Pe <<l).

The basic equation for the transport of energy is given by

equation (2.3), which for an axisymmetric spherical system may be

written in dimensionless form as:

T* = (v* + ) (2.79) cyT* vilt;

r* r* 60 where T* = (T To)/(Ts To) , v*r

= vr /U , = vs/U, r*=r/R (2.80) •


T* = 1 at r* =1 T* = 0 for r* --> (2.81)

A perturbation solution of the energy equation (2.79) entails

the expansion of the temperature T* in the form:

1 T* = To + iPeTt1 + (1-Pe)2TI2 +

1 'and the determination of the functions To,T1,T2,.... from the

recursion formula:

(2.82)

2 ' V T = r 6Tn.l

(2.83)

r* de

57

The velocity components v* and v*9 are given by Stokes' solution

for creeping flows as described in section 2.3.2 .

This was the method used by Kronig and Bruijsten73 who employed

the following boundary conditions:

To = 1 at r* = 1 $ - To - 0 for r* =

(2.84) P1 = T2 = = 0 for r* = 1 and r* =

The first function To was obtained from equation (2.79) with

Pe = 0. Thus:

2 77 To = 0

which, with the boundary conditions (2.84), gives:

(2.85)

To = 1 / r* (2.86)

i.e. the temperature field for the sphere in a stationary medium.

The remaining functions Tn could not be made to vanish at infinity,

and for n> 2 they even diverged. Using these solutions for Tn,

Kronig and Bruijsten obtained the following relation for the Nusselt

number:

Nu 8 1-Pe 1920- Pet (2.87)

in which the first term represents the contribution from pure

conduction in the absence of any convective effects.

The difficulty of satisfying the boundary conditions at infinity

was overcome by Acrivos and Taylor? who followed a procedure similar

to that used by Proudman and Pearson31 for the perturbation correction

to Stokes' solution. According to this procedure, an inner and an 1 outer expansion T and T respectively are constructed, in sueh a

way that:

(a) The inner expansion T satisfies the boundary conditions at

the solid surface.

(b) The outer expansion T vanishes at infinity.

(e) The two expansions match identically at some arbitrary distance

from the surface.

58 be

For the spherical system, these expansions may^represented by:

The inner expansion is:

00

T*(r*,O) = fn(iPe) Tn(r*1cose)

(2.88)

n=o

with fo(iPe) =

and the outer expansion:

00 T*(iPe r*,9) ▪ Fn(iPe) Tn(iPe r*Icos0)

n=- o

where the functions f n 2

(Pe) and Fn(iPe) are restricted by the

requirements:

lim fn+1 /fn = 0 and lim F11-1-1/Fn 0 Pe -40 Pe -r0


To(1,cos0) = 1 and Tn(1,cose) = 0 for n >1 ; as Tn( co, cos 9 ) = 0

(2.89)

(2.90)

.(2.91)

From the solutions obtained for Tn and Tn Acrivos and Taylor?

derived the following expression for the Nusselt number:

Nu = 2 1-Pe -Pe2 ln Pe + 0.03404 Pet • 7 -Pe3 in Pe

(2.92)

which is valid for the range 0..< Pe ••••-.

Yuge74 considered the case of forced convective heat transfer

at very low Reynolds numbers (Re 1) subject to the assumption

that the velocity distributions of the flow round a sphere is the

same as that obtained by Stokes He also assumed that the dimen-

sionless temperature T4 can be-expressed as a power series of It with coefficients that are functions of r* alone.

i.e. = fo(r*) f2(r*)02

f4(r*)04

(2.93)

On the substitution of this expansion into the energy equation

(2.79) and equating coefficients of powers in 0 a set of ordinary

differential equations are obtained. These equations, subject to

the boundary conditions of equations (2.81), were integrated by the

59

method of Runge-Kutta at values of the Peclet number of 0.3,1,3,

and 10. He evaluated the average Nusselt number for these Peclet

numbers and found that the value of the Nusselt number may be put

to two at values of the Peclet number less than 0.3 .

In the studies mentioned above, only those systems have been

considered for which the convective effects are relatively minor in

comparison with pure conduction. At the other extreme, however, are

the cases with very large Peclet numbers (Pe >j1) for which the

effects of molecular conduction may be neglected everywhere, except

for a thin boundary layer-type region near the fluid-solid interface,

where the temperature variation occurs.

Friedlander75 analysed theoretically mass and heat transfer

from a sphere at very low Reynolds numbers. The solution of the lin-

earised Navier-Stokes equations due to Tomotika and Aoi29 was used

by Friedlander to solve the diffusion (or energy) equation. On com-

parison with the experimental data available for the low Reynolds

number range, he found that these data were from 10 to 40 % higher

than the theoretical predictions. This is because data were taken

in the range 1 < Re <5, which is higher than the limit of appli-

cability of Tomotika and Aoi's solution, and also because angular

diffusion was neglected in his theoretical analysis.

Friedlander's solution may be summarised as:

Nu = 2 Pe <0.1 (2.944,

Nu Pe In ( 1 "cPe ) 0.1 < Pe <1 1 = 2 + -Pe + -r- Pe2 + (2.95?:

Nu 0.89 Pea Pe>1000 (2.96)

The first two terms of equation (2.95) are the same as those

in equations (2.87) and (2.92) obtained by Kronig and Bruijsten73

and Acrivos and Taylor7, respectively.

In a second paper8 , Friedlander modified his earlier work by

application of thermal-boundary layer approximations and Stokes'

60

velocity distribution. Friedlander's8 solution at high values of

the Peclet number (very thin thermal boundary layer) became as: I

= 0.991 Pe-1 Nu Pe > 100 (2.97)

The upper limit of practical application of equation (2.97) to

liquids is at Peclet number approximately 1000, because Stokes' flow

does not occur above a Reynolds number of 1, and the Prandtl numbers

for liquid-phase diffusion are of the order of 1000.

Levich9 also applied thermal boundary layer approximations and

Stokes' velocity distribution and his solution of the diffusion

equation may be written in the form of equation (2.9$) as follows:

Nu = 1.008 Pe4. Pe >> 1 (2.98)

For Peclet numbers less than unity, Levich gives the following

interapolation formula:

Nu = 2 + 1.008 Pe- (2.99)

which reduces to : Nu = 2 at Pe = 0 , (2.100)

and to equation (2.98) for Pe >,:.>1 .

In the Reynolds number range 10-100, Pe is 104-105 for liquids. This led Baird and Hamielec10 to employ the thin thermal boundary

layer approximation to predict theoretically local and overall

Nusselt (Sherwood) numbers for forced convective transfer from solid

and fluid spheres for Reynolds numbers up to 100. They derived the

following expression for the overall Nusselt number:

'Jr ve R(

4,1

r )r=Rsin

U

39

0

A )2del (2.101)

Nu (Sh) = 0.641 Pe'

where ( )the velocity gradient at the surface, is a Or r=R ,

function of (9 and the Reynolds number, Re , and can be obtained

from the approximate velocity profiles of Kawaguti40 and Hamielec

et al 11'41 . In Stokes' flow regime, equation (2.101) reduces to

equation (2.97) of Friedlander8 . The predicted overall Sherwood

(Nusselt) numbers were shown to be in fair agreement with experimental

results.

The boundary conditions ar2: 6 T

6Y2

Y= C5T ; T = To

y = 0 T = Ts (2.103)

(2.104)

= 0

32T 0

637.2

give:

6Y

A series

1 dr(x) 6T 4.,(x ,y) t, r(x) dx

j 63r expansion for T was assumed in the form:

611/(x,y) 6T 62T

y2

(2.105)

a

61

For still higher ranges of the Reynolds number, the velocity

distribution can be obtained from the hydrodynamic boundary layer

equations as described in section 2.3.11. The boundary layer form

of the energy equation (2.3) for steady-state heat transfer from a

solid body of revolution is12

oT u Ox

6T + V

62T a 6y2 (2.102)

where e)T is the thickness of the thermal boundary layer. As in the case of the hydrodynamic boundary layer equations,

there are two kinds of solution of equation (2.102); exact solutions

and approximate solutions.

Exact solutions of equation (2.102) have been developed by

FrUssling12 for the flow around axisymmetric bodies of revolution.

He replaced the velocity components of equation (2.102) by the mod-

ified form of the stream function as defined by equation (2.48) to

T = to t2x2 4. t 4

4- (2.106)

where the coefficients- tolt2it4,.... are functions of y only.

FrOssling12 substituted this expansion and the expansions for

r(x) and 4/(x,y) given by equations (2.51) and (2.52) respectively,

into equation (2.105), and obtained a set of ordinary differential

equations for the functions tolt2it4o.... The expansion for the

stream function was obtained from his earlier solution of the hydro-

dynamic boundary layer equations. He solved the first few equations

62

of the set numerically to obtain an expression for local transfer

rates in a series form. For the particular case of a sphere and for

a Prandtl number of 2.532, which corresponds to sublimation of naph-

thalene in air, the local Nusselt number is of the following form:

Nu(x) Re-1 = 1.862 - 1.369(x/2R)2 + 0.2075(x/212)4 +.... (2.107)

in which x is the distance measured along the surface from the front

stagnation point and R is the radius of the sphere. The local Nusselt

numbers predicted by equation (2.107) were found by Frassling to be

in good agreement with the experimental measurements of mass transfer

from naphthalene spheres.

Green13

derived the boundary layer equations in orthogonal

curvilinear coordinates. Two sets of ordinary differential equations

were developed by the use of new series expansions for the velocity

and temperature functions in powers of the curvilinear coordinate

along the surface. Solutions of the first four or five equations of

each set were obtained for spheres, discs, and oblate spheroids with

minor to major diameters ratio of 0.8125, 0.625, 0.4375, and 0.25 at

Prandtl numbers between 0.7 and 100 .

Green showed that the exact solution gives a good prediction

of local rates of transfer up to 0 = 70° for spheres and up to = 50° for oblate spheroids, where 6) is the parametric angle

from the front stagnation point.

Approximate solutions of the thermal boundary layer equation

(2.102) have been developed by Aksel'rud76 and Grafton77 for the

sphere, and by Rojey78 for the oblate spheroid. All three used a

similar integral method to that employed for the solution of the

hydrodynamic boundary layer equations. They assumed a quartic poly-

nomial for the temperature or concentration distribution of the

form:

T = to ' + t'y + t'y2 + t'y3 + tl4y 1 2 3 (2.108)

63

The coefficients to to tl4 were determined from the boundary

conditions at the surface y=0, and at the edge of the thermal

boundary layer 8T; i.e. from equations (2.103) and (2.104). From

the evaluation of ( 6T/ 6y) at various values of 0 along Y=0

the surface, the local rates of heat transfer were=' calculated.

Green13 found that his exact solution was in good agreement

with Rojey's approximate solution.

Beg14

used the following approximate relationship between the

respective thicknesses of hydrodynamic and thermal boundary layers:

/ 8T = Pr4 (2.109)

to calculate 8T at various values of 0 along the surface of

the oblate spheroid. The hydrodynamic boundary layer thickness

around the surface was calculated from the approximate solution of

the hydrodynamic boundary layer equations as described in section ' "

2.3.4. The local rates of heat transfer were then 'calculated in

terms of the local values of T Green13 found poor agreements

between his exact transfer solutions and Beg's approximate solutions

particularly for flatter spheroids. This may be attributed to the

poor approximation of equation (2.109) which was used by Beg.

Boussinesq79 showed that for potential flow around a sphere,

where no boundary layer exists and the velocity distribution is

given by the potential flow theory, heat transfer rates are given

by:

Nu = 1.13 Pee

(2.110) which is valid for Pe —4 00 .

It is clear, then, that although the exact functional dependence

of the Nusselt number, Nu, on the Peclet number, Pe, has already been

established for the two limiting cases Pe —4 0 and Pe —4 CO , the

behaviour of the function for intermediate values of the Peclet

number is at present unknown.

64

2.4.2. Experimental Studies of Heat and Mass Transfer from S heres

Experimental studies of forced convective transfer have been

primarily directed towards obtaining correlations which relate the

overall rates of transfer to the physical and dynamic properties of

the system. These correlations usually take the following form:

n1 n2 Nu Ao + Bo Re Pr

The determination of the best values of the constants and the

exponents of equation (2.111) has been the object of most of the

experimental studies.

Experimental studies of heat transfer from spheres are less

numerous than the corresponding studies of mass transfer. In mass

transfer studies, FrZssling80 considered the evaporation of drops

of nitrobenzene,aniline, and water, and the sublimation of naphtha-

lene spheres in a hot air stream. Spherical drops of diameter from

0.1 to 2.0 m.m. were suspended in a vertical wind tunnel, and the

rates of mass transfer were measured by a photographic technique.

He correlated his combined data, which covered the range:

2 < Re < 1300 and 0.6 < Pr < 2.7 , as follows:

Nu = 2.0 + 0.552 Rest Pr4 (2.112)

In this correlation, Fx4bssling assumed n2 to be one-third,

Ao to be two (the value of Nu at zero flow rate obtained from

molecular diffusion (conduction) theory), and from boundary layer

considerations he assumed n1 to be equal to one-half. Thus, the

coefficient B0

is the only value which was determined. experimentally.

Ranz and Marshall81 studied rates of evaporation of drops of

water, aniline, and benzene in air at temperature up to 20000 (the

diameters of the drops being about 1m.m.). They also studied rates

of heat transfer to these drops at varying air temperatures. Their

studies were restricted to a range of the Reynolds number of from

65

two to two-hundred, the range usually encountered in spray drying.

The results were plotted in the form of Nu against Re2- Pr' from

which the following linear relationship was obtained:

Nu = 2.0 + 0.60 Re2 Pr3 (2.113)

This is in close agreement with FrBssling's equation (2.112).

The work of Kramers82 was one of the most comprehensive

attempts to obtain some fundamental data of heat transfer from solid

spheres in both air and water as the flowing media. Steel spheres of

diameters equal to 0.71, 0.79, and 1.26 cm. were suspended by a pair

of fine thermocouple wires, with their junction at the centre of the

sphere, in a vertical tube through which air, water or oil were

passed. A high frequency coil surrounded the tube and sphere, so

that the latter could be heated inductively. A total of 80 results

were tabulated as Nusselt numbers and Reynolds numbers with different

values of the Prandtl number for the different fluids used in this

work. The following correlation was obtained:

-9-0 31 Nu = 2.0 + 1.3 Pr0.15 + 0.66 Re2 Pr (2.114)

which fitted his data within ± 10% in the ranges 0.7 < Pr < 1+00

and 0,4 <7 Re < 2000.

Kramers's correlation differs from others in forced convective

transfer, in that for Re=0, the rate of transfer is dependent on the

Pram-1U number, Pr. This indicates that Kramers' results were

effected by free convection at low Reynolds numbers.

Tang, Duncan, and Schewyer83 carried out identical experiments

to those of Kramers using air as the flowing medium, but their

results are 40% lower than his. They used steel spheres 1u, 1T+ u, and

5 IT inch diameter , and their range of Reynolds numbers was from 50

to 1000. They fitted the following correlation to their 150 data

points:

66 Nu 3.1 _A

Stanton number, St = = + 0.55 Re 2 RePr Re

which may be rearranged to the form:

(2.115)

Nu = 3.1 Pr + 0.55 Pr Re2 (2.116)

For dry air (Pr=0.71) this equation reduces to:

Nu = 2.20 + 0.39 Re2 (2.117)

which compares with:

Nu = 3.23 + 0.59 Re

(2.118)

for air from Kramers' equation (2.114). Equations (2.117) and (2.118)

obtained from Tang et al's and Kramers' correlations predict different

results, although exactly the same system was used in both cases. The

only likely difference is that Kramers' data included values below a

Reynolds number of 50 which may have influenced the final form of

his correlation.

Rowe, Claxton, and Lewis2 carried out an extensive literature

survey of heat and mass transfer from spheres. They found that

almost all heat transfer measurements were made in air, while the

bulk of mass transfer observations were made in water. Although the

literature is extensive, the evidence is inadequate to establish

the relationship between Nusselt number and Reynolds number. This

led them to carry out a series of experiments in air and in water.

In the experiments on heat transfer to air and to water, they

used internally heated copper spheres of -a- inch and 14 inch diameter

and the range: 20 < Re < 2000. The data were first correlated

in the form of equation (2.111) with the power of the Reynolds

number (i.e. n1) varying from 0.40 to 0.60 at intervals of 0.02

whilst the exponent of the Prandtl number (i.e. n2) was assumed to

be constant and equal to one-third. The least squares straight line

67 n1 relating the Nusselt number, Nu to Re was then found, together

with the residual error variance about the line. Eleven different

values of the constants Ao and Bo were thus found depending on the

values of n1. None of the eleven equations obtained was significantly

better than any other in the statistical sense. This point is very

important in convective transfer correlations, because it may

account for the differences in values of the coefficients and

exponents which have been obtained.

Rowe et a12 then decided to select values of Ao and n1 on the

grounds of molecular conduction theory and boundary layer theory,

respectively. i.e. Ao = 2 and n1 = 2 . With these values, they

produced the following correlations for air and water, which they

recommend in the range 30 <7 Re < 2000 :

Nu = 2.0 0.69 Re2 Pr-5J.

for air

(2.119)

1 Nu = 2.0 -1- 0.79 Re2 Pr'

for water

(2.120)

In another paper, Rowe and Claxton3 plotted a growth curve

for n1 as b. function of the Reynolds number, Re , from all the

information that was available to them. They suggested the following

relationship for the exponent n1:

2 - 3n1 0.28 - 4.65 Re- (2.121)

3n1 - 1

with asymptotes n1 = j at Re = 0 and n1 = -3- at Re = Q000 .

Jenson, Horton, and Wearing8 are the most recent to make

heat transfer measurements. They used * inch diameter copper spheres

internally heated in a flowing oil (Pr=121) over the range

35 <Z7Re < 180. In accordance with their results, they pointed

out that the lower values of the overall heat transfer coefficients

in the literature were more likely to be correct. They set n1 = -;12-

68

and n2 = 7 and found that the correlation, determined by least

squares , was:

Nu = 4.5 0.48 Rel (2.122)

Extrapolation of this relation to Re 74 0 gives Ao= 4.5

confirming other reports282 of greater intercepts than that due

to molecular conduction (i.e. Ao = 2).

Introducing a term to account for the natural convection,

Jenson et al84 proposed the following correlation:

Nu = 2.0 0.75 Pr. 0.48 Reg PrT (2.123)

The first, second, and third terms in this equation represent

molecular conduction, natural convection, and forced convection,

respectively.

It is to be noted that the data obtained with present equip-

ment and experimental techniques are not precise enough to allow

anything more than approximate relationships to be found.

Heat transfer studies from oblate spheroids are not available

in literature and those for mass transfer114 cover a high Reynolds

number range (200 <CRe <I:32,000) .

***********

69 CHAPTER 3

THEORETICAL ANALYSIS

3.1. Equations of Viscous Flow and Heat Transfer

The theoretical prediction of heat transfer rates from solid

particles in an extensive fluid requires the sOu+ion of the eq1.75.tions

which describe this transport process. These equations are the contin-

uity equation, the equation of fluid motion, and the energy equation.

They are obtained by the application of the conservation laws of mass,

momentum, and energy, respectively, to a control volume V enclosed by

surface SI through which the fluid is flowing. The derivations of

these equations may be found in various texts6,85,86,87

It is convenient at this stage, however, to list the main

assumptions which are usually made in these derivations:

1. Fluid properties are contiuous functions of space and time.

2. The fluid is Newtonian. i.e. there is a linear relationship

between stress and rate of strain. Also, the viscous stress

vanishes with vanishing rate of strain.

3. The fluid is isotropic. i.e. its properties do not depend on

direction.

With these assumptions, the governing equations of the transpo

process of heat may be written in vector notation as follows:

1. The continuity equation

This equation is based on the physical principle of conservation

of total mass:

'7. ( p Tcr ) = 0 (3.1)

It is useful, now, to define the substantial derivative — Dt

to denote the rate of change of some quantity over a path following

the fluid motion.

i.e. D

Dt

(3.2) 6t

Introducing this into equation (3.1), the contiuity equation

P

70 becomes:

p D --- p v •zr ) = 0 (3.3) Dt

2. The equation of fluid motion

From Newton's second law of motion, this equation is as follows:

Dv

Dt V + 1-L (

ic7 /1( 77• -' ) 2(;71-4..7).i V14/070. ; ) (3.4)

This is the general form of the Navier-Stokes equations. The

terms of equation (3.4) represent forces acting on an element of fluid

per unit volume. The term on the left hand side represents the inertial

forces, and the first term on the right hand side represents the exter-

nal or body forces, the following term expresses the pressure forces,

and the remaining five terms represent the viscous forces in which the

variations of the coefficient of viscosity, /J.., are included.

3. The energy equation

This equation is based on the first law of thermodynamics:

DE -- p( ) (414°D Dt

(3.5)

The terms of equation (3.5) represent rates of energy gained by

an element of fluid per unit volume. In the term on the left hand side

E denotes the specific internal energy so that the term expresses the

rate of gain of internal energy per unit volume. On the right hand

side; the first term represents the rate of input of heat by conduction;

the second term refers to the reversible rate of work done on the fluid

element due to compression, and the last term represents the rate of

irreversible conversion of work to internal energy by viscous

dissipation.

Equation (3.5) can be rearranged as follows:

71

Relating the specific internal energy E to the state variables

V (volume), p (pressure), and T (temperature) , it can be shown from

thermodynamics that:

V dE = Cp dT T() dp p dV

6T P

It follows from the substitution of V by 1/f) and the use of

the contiuity equation (3.3) that:

DE DT T p Dp p p c + —( ) P( 17 at .17 ) Dt p Dt f) Dt

(3.7)

Using this result together with Fourier's law of heat conduction,

which is:

= IcT V T (3.8)

equation (3.5) becomes:

DT pc

P Dt = (Ica, V T )

T( ap ) Dp

•

j P Dt + - D (3.9)

Under general, conditions, the flow of a Newtonian fluid is

partly described by the partial differential equations (3.3), (3.4),

and (3.9). These are five equations, three of which are represented

by the vector equation (3.4).

In general, the properties of the fluid depend on temperature

and pressure. The density is related to p and T by the equation of

state:

f(p,p I T )=0 (3.10)

Similarly, the viscosity and the thermal conductivity may be

related to p and T by:

f(112 p,T) = 0 (3.11)

and f(kp,T) = 0 (3.12)

If the external or body forces r are specified, the solution of equations (3.3), (3.4), and (3.9) to (3.12) (eight equations for

the eight unknown variables v1lv2,v3IpITI /Dl i1, and kT) give a

(3.6)

72

complete description of the motion of a compressible Newtonian fluid.

These equations are so complicated that, in order to rake

progress with the solution of a particular problem, it is necessary

to introduce simplifying assumptions such that the equations nol- only

become simpler and easier to solve but also coLtinue to describe

adequately the particular physical situation.

The case to be analysed here is that of forced convective heat

transfer from spheres and oblate spheroids to a fluid flowing at

intermediate Reynolds numbers. These systems are assumed to possess

the following characteristics:

1. Variations in the fluid density are small. i.e. the fluid is

treated as an incompressible fluid.

2. Also, the other physical properties of the fluid are constant.

3. The heat generated by viscous dissipation is negligible.

4. The external or body forces, 11 , in the Navier-Stokes equations

(3.4) refer only to gravitational forces, pg

On introduction of the above simplifications, equations (3.3);

(3.4), and (3.9) reduce to:

The continuity equation:

= 0 (3.13)

The Navier-Stokes equations:

f) i 2 v p + p. ci, -v -(3.14) Dt

The energy equation:

DT

C = kT ,s7 2 T (301) —

P Dt

In the above equations f) , $ and kT are constant and may be

evaluated, for any particular system at specified p and T from

equations (3.10), (3.11) and (3.12) respectively.

73 The dependent variables v1,v2,v3, and p can be found from

equations (3.13) and (3'.14) and the appropriate boundary conditions

and initial conditions without recourse to the energy equation (3.15),

which can now be used to obtain the temperature distribution.

The present project has studied forced convective heat transfer

from solid particles, spheres and oblate spheroids, under steady-state

conditions. In such systems, the gravitational forces in equations

(3.14) may be eliminated from the equations on the understanding that

the local pressure, p , will be measured relative to the undisturbed

hydrostatic pressure which would occur if the fluid was stagnant at

the point considered.

For steady-state conditions, the derivatives with respect to

time i.e. vanish. = 0 . Then, for such systems, equations (3.13) at to (3.15) reduce to:

\-;7' = 0 (3.16)

1 s7 2 (3.17) ); p ts7

f)

)T = aV2 T (3.18)

where 1.0 and a are the kinematic vIscosity and the thermal

diffusivity, respectively.

Equations (3.16) to (3.18) can be solved for the dependent

variables v1,v2,v3,p, and T subject to the boundary conditions which

are imposed on the system. In the case of viscous fluid flow past a

stationary solid body, the boundary conditions of the system are

obtained from the following considerations:

(a) There is no slip of the fluid at the solid wall. i.e. the layer

of the fluid in contact with a solid body has the same velocity

as that of the body.

(b) The temperature at the surface of the body is maintained at

constant value, Ts

74

(c) The flow at a considerable distance from the surface of the body

is assumed to be undisturbed and parallel. The uniform velocity

and temperature are given by Vo and To, respectively.

From these considerations, the boundary conditions of the system

may be written as:

For n=0 ; ; = 0 T = Ts

For ; ; Vo To

(3.19)

where n is the outward pointing normal vector from the surface.

The vector operators in equations (3.16) to (3.18) must now be

expressed in terms of the standard forms of Appendix B. Transformation

of the vector operators to any orthogonal curvilinear coordinates

(described in Appendix A) is immediate. From the vector relationships

given in Appendix B with the replacement of a and t by the velocity vector ;, and 41 by the temperature T, the following relations are

obtained:

From equation (B.9): (% = (v.7i) VA( VA ;) (3.20)

From equation (B.11): 7 2 ; = ( - VA( V/J) (3.21)

However, 7/7.v = 0 from the continuity equation (3.16), so

that equation (3.21) becomes:

s72 = (VA4) (3.22)

From equation cs-r. (B.12): = Tr. V T (3.23)

These expressions for (;.V); 772 ; 2 and (7.77)T are then

substituted into equations (3.16) to (3.18) to give:

v c-,.v = (3.2k)

(3.26)

1 i- V(17.7)..7.A(VAir) = -

—p— VP - VA( C7A Tr)

Tr. VT =aV2 T

75 Equations (3.24) to (3.26) are now in forms which can be

expressed in terms of any orthogonal curvilinear coordinates by the

application of equations (B.1) to (B.8) of Appendix B. Expressions

for these equations have been obtained in terms of the spherical and

the oblate spheroidal coordinates and are giver 4n Appendix D.

3.2. Navier-Stokes Equations as Vorticity Tranuort Equations

The method used to solve the equation of fluid motion necessi-

tates the expression of Navier-Stokes equations as vorticity transport

equations. The vorticity in a fluid, 6D, is a vector quantity having

the same nature as angular velocity. It is defined by:

GJ = curl Tr = V A -17. (3.27)

The vorticity is regarded5 as a measure of rotation of the

element of fluid as a whole about an instantaneous axis; the component

angular velocities of the rotation being ' GO/ 1 i uj2 2 j- 6)3 . The

vector whose components are W1 ' 4.)3 is called the

vorticity of the fluid at the point defined by the tip of the position

vector r. The physical significance of vorticity is best understood by

imagining a small element of fluid to be suddenly frozen. If the

resulting solid element has rotation then the fluid has vorticity at

the point considered.

The curl of a vector expressed in the orthogonal curvilinear

coordinates (x1,x2ix3) is given by equation (B.7). The replacement

of a in equation (B.7) by "N./. gives the vorticity whose components

CAJ1 I

GO2 , and (403 about x1,x2, and x3, respectively, are

given by: 1 , , CAJ 1 = 6 ( h v ) [ -6x2 3 3

- ----k n v 6x3 2 2 )11 h2h3

1 e e , , (3.28a)

Gi2 = 4( 111'71 ) - ----I. . v (3.28b) ex 3 3 )1 h1h3 3 1

GO . 1 e )k x ( h1v1 )1

] (3.28c) h1h2

[1.t7:( h2 v2 ) ".." 2

Fig. 3.1

Orthogonal Curvilinear Coordinates

coordinates

Floc,

of the body are arranged as shown in Fig. 3.1.

/11/Y3 14%

Y

I 1 }y2

,

Streamlines in a Meridian Plane

Fig. 3.2

76

where hi (i=1,213) are the scale factor, defined by equation (A.19)

The curl of equation (3.25) is known as the vorticity transport

equation. By application of relation (B.13), the curl of the first

term on either side of equation (3.25) vanishes. The curl of tha other

two terms in equation (3.25) gives the following equation for the

vorticity transport:

C A( 77 A (7) ) = V 7A( 7A uv )

(3.29)

Equation (3.29) can be expressed directly in terms of any

orthogonal curvilinear coordinates, giving rise to three equations

which are the components of equation (3.29) in x1,x2, and x3 directions.

3.3. Axisymmetrical Flows The general three-dimensional flow discussed in the previous

sections, in which the three velocity components depend on all three

coordinates, presents enormous mathematical difficulties. However, in

the case of flow past axisymmetrical bodies, such as spheres and

oblate spheroids, the equations of fluid motion are further simplified

and the mathematical difficulties encountered in the original

equations are considerably reduced.

In such flows, the fluid is streaming over the body of revolu-

tion parallel to its axis of symmetry. The orthogonal curvilinear

77

Thus, xi is taken normal to the surface of the body, x2 is taken

parallel to the surface in the flow direction, and x3 is taken in the

direction of rotation of the body about the axis of symmetry y3.

An axisymmetrical flow is then one for which the velocity and

all other variables are independent of x3.

= 0 (3.30)

For the particular systems of streaming flow past a stationary

body of revolution with no swirl, the component of velocity in the

x3-direction is everywhere zero.

i.e. v3 = 0 (3.31)

On the basis of equation (3.31), it follows that the streamlines

lie in meridian planes. In conjunction with equation (3.30) this shows

that the stream surfaces are co-axial surfaces of revolution as

represented in Fig. 3.2 . This also applies to the isothermal surfaces.

It is clear in such conditions that the problem of axisymmetrical

flows is reduced to a two-dimensional problem. Hence, the equations of

fluid motion and energy for such flows are obtained by the use of the

expressions for the vector operators given in Appendix B keeping in

mind the conditions of equations (3.30) and (3.31).

The continuity equation (3.24) becomes:

1 h2h3V1 x

) ----( h1h3v2 ) = 0 (3.32) h1h2h3 6x2

From equations (3.28a,b,c), it is clear that there is only one

component of vorticity, W3 in the x3-direction, since =W2 =0

for axisymmetrical flows. Denoting 6)3 by equation (3.28c)

becomes:

i.e. 0 6T

6x3 63c3

E- ( h2v2 ) - h1h2 dox1

= ---( h1v1 ox2

(3.33)

78

which can now be considered as a scalar point function of xi and x2.

The Navier-Stokes equations expressed as the vorticitr transport

equation (3.29) can be expanded giving the following single non-zero

component:

= ( 11 6 ) ) h1h3 ox1

6 ( hi ( h3

6x2 1 h2h3 6x2 %

Equation (3.34) can be rearranged, using the continuity equation

(3.32), to give:

16 6 il — ) = E2 ( h ) — ( 6x., h2 6 x2 h3 h 2 3 h h1 i 3 3

r 6 where 2 = ) -(_ h2 6 h1 ) + ,

h1h2 t 6.1 h1h3 6.1 (5x2 h2h3 6x2

(3.35)

(3.36)

The energy equation (3.26) becomes:

(3.34)

v1 v2 6T a h2 6x2 h1h2h3

I ( h,h3 6., h1 6., A h,h 6T

cx2 h2 6.2

h1

(3.37)

It is convenient to define a quantity called the stream

function, at the point located at the tip of the position vector r,

by the relation:

= tp('r') = Q / 21r

(3.38)

where Q is the volumetric flow rate through a surface generated

by the rotation of the curve joining the point of position vector r

with any point on the axis of symmetry about this axis.

The stream function IP is a unique scalar point function,

and, according to its definition, = 0 along the axis of revolution.

79 This quantity was first introduced into hydrodynamics by Stokes

and is often referred to as Stokes' stream function. This function

exists in all cases of incompressible flow in two-dimensions, and

in the case of three-dimentional motions only when the latter are

axisymmetric.

• It can be shown89 that the following equation relates the

stream function at a point with the local fluid velocity:

= A V \LJ h3

(3.39)

which gives the following expressions for the velocity components

as a function of the stream function in any system of orthogonal

curvilinear coordinates:

V =

I e v2 =

h1h3 6x1 v3 = o (3.4o)

h2h3 ex2

It is clear, by the substitution of vl and v2 from equations

(3.40) into equation (3.32), that the contiuity equation is automa-

tically satisfied. It is equally clear that the conditions of

equations (3.30) and (3.31) are satisfied as the stream function is

independent of x3

On substitution of the values of v1 and v2 from equations (3.40)

into the equations of fluid motion and energy, the following results

are obtained:.

Equation (3.33) which defines the vorticity becomes:

E2 h3 (3.41)

The vorticity transport equation (3.35) becomes:

6x1 ( ) 64 (

6x2 h3 62c2 6x1 h3

h h 1 2 E2( h3 h

3 ) (3.42)

so


( h2h3 6T

6x1 hi 1 ) 6 ( hih, dT

6x2 h2 6x2

dtP c',1T a 6x1 6x2

6tp 6%, - , , ) (3.43)

Ox2 Oxl

Equation (3.42) can be expressed in terms of a single variable,

mainly the stream function, by the substitution of equation (3.41)

into equation (3.42) to give:

6* ( E2 LP -

h

1 61P E2 h1h2 4

.1,

6K1 ex2 h32 ( ---7f h — ) = 11 E (3.44)

x2 xl 3 3

Equation (3.44) is a non-linear fourth order partial differential

equation, which cannot be solved analytically except for some limiting

cases and for a very few simple boundary shapes. Thus, its solution

must generally be obtained numerically.

However, it is more convenient to solve the Navier-Stokes

equations as two simultaneous equations in two dependent variables,

Ali and r , as given by equations (3.41) and (3.42). These

equations are second order partial differential equations where

equation (3.41) is linear and equation (3.42) is non-linear.

Hence, for the case of forced convective heat transfer, it is

necessary to solve equations (3.41) and (3.42) to obtain the stream

function distribution which is required in the solution of the energy

equation (3.43).

wow

4J

81 3.4. Boundary Conditions

Before an attempt can be made to obtain a solution of equations

(3.41) to (3.43), the appropriate boundary conditions for the system

must be prescribed for each of the dependent variables IP, and T.

By definition, the stream function is zero along the axis of

symmetry, and by continuity, 111 is also zero along the body surface.

The no-slip conditions on the body surface, equation (3.19), become:

i.e. alP 0 ox2

The vorticity ED , as defined in section 3.2, is a vector

quantity. In axisymmetrical flows, this vector has a single non-zero

component, with magnitude in the direction of x3, the normal to

the meridian plane x1-x2 as represented in Fig. 3.3 . Y3

X1 = (X1 )s ; v1 = v2 = 0

Fig. 3.3 Vorticity and Velocity

Directions

If the two points P1 and P2 are chosen on opposite sides of the

axis of symmetry y3 then, clearly, the vorticity 63 will act into the

plane at P1, and it will act out of the plane at P2. Therefore, as P1

and P2 coincides on each other along y3, the net vorticity becomes

zero.

Also, the vorticity is zero at large distances from the body as

the flow is assumed to be uniform and parallel. The boundary condition

for at the body surface is to be evaluated from equation (3.41) in

terms of IP by application of the no-slip conditions at the solid

boundary together with the constancy of IP along the surface. This

boundary condition will be developed in section 3.6.

The above conditions together with the bondary conditions of

82 equation (3.19) may be summarized as follows:

On the body surface,

tif _ 0 6x1 6x2

x1 = (xl )8 :

E2 qi = ) h

3

T=T (3.45)

Along the axis of symmetry:

= x2

= 0

T

= 0 _0

6x2 (3.46)

At large distances from the surface of the body, x1

= iu yoa

= 0 2 T = To (3.47)

where Yo is the value of y2 along the outer boundary, and U is

the undisturbed stream velocity.

3,5. Forced Convective Heat Transfer from Spheres and Oblate Spheroids

The sy)herical polar and the oblate spheroidal coordinate systems

described in Appendix C are particular cases of the orthogonal curvi-

linear coordinate system used in sections 3.3 and 3.4. The coordinates

r l et and for the sphere, and z10, and qb for the oblate spheroid,

correspond to x1,x2, and x3 respectively. The scale factors for the

spherical polar and oblate spheroidal coordinate systems are given in

Appendix C by equations (C.2) and (0.12) respectively.

The sphere and the oblate spheroid are axisymmetrical bodies of

revuluijon; hence the governing equations for axisymmetrical flows

derived in section 3.3 apply.

(a) The Sphere

The governing equations of forced convective heat transfer from

a single sphere are derived in terms of spherical polar coordinates by

the substitution of the scale factors from equation (C.2) into the

general equations given in section 3.3:

6 dr

avr OOP ••••••••••••

r r 69 (3.49)

83 The continuity equation (3.32) becomes:

I ............—( v sin 0) —2- --( vr2 ) + = 0 (3.48)

r sinO 69 r (?)r r

The vorticity, given by equation (3.33), becomes:

Introducing the stream function 1p, equation (3.40) becomes:

...1 ‘p 1 ‘P vr = I v = d (3.50)

r2 sine 69 et r sine 6r and, the continuity equation is automatically satisfied.

Equation (3.41) becomes:

E2 tJJ = r sing

(3.51)


sin() r67) ( ) = Tr tEr ae r sine

r sing) 64q) ( - de dr

-------) r sine

(3.52)


6T I 6 ( 6T ) ) sir,

r sine) 60 ( 611) 6T 64, 6T

a sine Or 60 6,09 6r ) (3.53)

The differential operator E2, defined by equation (3.36),

becomes:

E2 62

6r2

sine 6 1 6 r2 60 ( sine )

(3.54)

Because numerical methods are to be used to solve the differen-

tial equations of the system, (3.51) to (3.53), it is desirable to

express these equations in a rectangular coordinate system for the

reasons given in Appendix C. This is achieved by employing the trans.

84 formation, described in Appendix C, in which the radial coordinate,

from equation (C.28), is given by:

r = R ez

(3.55)

Substitution of this expression for r into equations (3.51) to

(5.54) gives:

22e2z E2tp R3e3z sine

(3.56)

R2e2z E2( ez sing )

ez sine 6tdi) 5

Jl R 6z 69 ( z sine ) 6. C4 (

ez sin

1 ez 6z

) 46T )

( sine 60 7i-Tie ez

64)611 a_ T) (3.58) Q R ez sine 6z 69 60 dz

where R2e2z E2 = ez 6 ( ?) 6 ( 6

69 sine ye-) (3.59) "(S; ez ) sine

Equations (3.56) to (3.58) are then the required equations for

the process of heat transfer from a solid sphere expressed in terms of

the new rectangular coordinate system (z1(9).

(b) The Obalte Spheroid

In a similar manner to that in the case of the sphere, the

governing equations of forced convective heat transfer from an oblate

spheroid are obtained by the use of the scale factors given by

equation (C.12) in the general equations derived in section 3.3. The continuity equation (3,32) becomes:

a( sinh2z + cos2 ) 'cosh z 6z

1 -- v ( sinh2z + sing e

( vz ( sinh2z + cos2eAcosh

cos2eAsin ) = 0 J

.) (3.60)

85 The vorticity, given by equation (3.33), becomes:

a( Binh z + cos20) 2) = 2/— ( 1, ( sinh2z + cos20) 6z e

- i-6 (vz ( sinh2 z + cos29) ) (3.61)

Introducing the stream function IP, equation (3.40) becomes:

4) -1 6

a2( sinh2z + cos20)cosh z sin 0 69 (3.62) _

dtli 1

a2( sinh2z + cos2,1 17)cosh z sine 6z

and, the continuity equation is automatically satisfied.


a2( sinh2z + cos28) E2 sinh2z + cos29)cosh z sine

(3.64)


a2( sinh2z + cos2e) E2( cosh sin e )

cosh z sin9

) -4Wi ( , z„,)

cosh z sin, 0 Q Z cosh z sin vp

(3.65)

ti a


(3.63)

C5 ---( cosh cosh z 6z

1 )

sine 66f(

sin 6T

) ee a LP a a cosh sine( 6z

aT 64) 67, de 's 66 16z

) (3.66)

where a2( sinh2 z + cos2O ) E2 =

6 ( 1 6 sine ( 1 cosh z 6 ) (3.67)

6z cosh z 6z 6e sine 6 0 Equations (3.64) to (3.66) are then the required equations for

the process of heat transfer from a solid oblate spheroid expressed

86

in terms of the elliptic coordinates (z1 19) which form a rectangular

system of coordinates as shown in Fig. C.4 of Appendix C.

3,6. Dimensionless Forms

It is convenient to express the equations of the previous

section in terms of dimensionless variables (superscripted by * ) and

dimensionless groups wich are defined as follows:

Variable Sphere Oblate Spheroid

Characteristic dimension

r*

Re

Nu

Pr

T*

R

r/R

P/UR2

,

2RU/1/

2Rh-r/kT

R/U ,..:1:1./U

d = a cosh zs

MD 11•11 0.00

Ilb/Ud2

261J//,

2dhT/kT

Cp /1/kT = 11/ CL

(T - T 0 )/(T8 - To)

Table 1.

Dimensionless Quantities

By the use of these dimensionless quantities, equations (3.56)

to (3.58) for the sphere, and equations (3.64) to (3.66) for the

oblate spheroid become:

R2e2z- E2 e3

z sine (3.68)

R2e2z 61P* ':5:112 e eZ sine ) = e sin

( ) _Re

* ( z E

W:( ez sin ) (3.69)

1 •••1•••••=aboulmois.

cosh z cosh z

6T*

z )

PrRe cosh zs

2cosh z sine

87 1 :. ,, 1 6 z 6

0

T*

) + sin 9 1?;-g( sine

r6;' ez )

............ ----.‘ = 1 z( e

G.) 0

PrRe [ 64/4' 6T* 64/* 6T* --"'

2ez sin (9 6z TO ,,j6? Tz- j

a2 ( Binh2 z + cos2 e) E2 tp * =

2 z + cos 2 g II )cosh z sin 0 r,*

( sinh

cosh3zs

a2( sinh2 z + cos ) E2( ?;* cosh z sin e)

iRe cosh z cosh z sin 0 [ifcosh zsine

- 64; ( )1 69 6z cosh z sin e

(3.70)

(3.71)

(3.72)

1 6 sine 60 ( s ing-Y66

( 641 6T* 614)* 6T*

z 69 - ae az (3.73)

Equations (3.68) to (3.70) and (3.71) to (3.73) are the

governing equations, in dimensionless forms, of the forced convective

heat transfer processes from a single sphere and from a single oblate

spheroid, respectively. It is clear, from equations (3.71) and (3.72),

that the solutions of the dimensionless Navier-Stokes equations are

function of shape as well as function of the Reynolds number. It is

also clear, from equation (3.73), that the solutions of the dimensior.-

less energy equation are function of Reynolds number, Prandtl number,

and shape.

The solutions of these equations are to be obtained numerically

subject to the boundary conditions of equations (3.45) to (3.47). These

boundary conditions take the following forms for the specific cases of

88 spheres and oblate spheroids:

(i) For z = zs (zs = 0 for the sphere) •

6tP * 6**

= 0 T* = 1 (3.74) 6z

Also, all V derivatives with respect to 8 are zero on the surface of the particle. Hence, the vorticity at the surface of the

sphere becomes, from equation (3.68): 1

.(›is = (

sin 6z2 )z = 0 (3.75)

Similarly, the vorticity at the surface of the oblate spheroid

becomes, from equation (3.71): 2

*t

cosh zs s (3.76)

( sinh2z cos 2 ) 6)sine 6zz=z

Equation (3.75) or equation (3.76) gives the .imrticity at the

surface in terms of the distribution of stream function close to the

surface as will be developed in Chapter 4 .

(ii) Along the axis of symmetry ( 0= 0 and 9 =

IP* - 142- 0, 6T*

= 0

- de = 661 (3.77)

(iii) At large distances from the surface of the particle (z —400):

tp* = iro2 sin29 = 0 , T* = 0 (3.78)

where r is the ratio of the semi-major diameter of the outer

boundary to the semi-major diameter of the particle. Thus for the

z sphere: r0 = R0/R = eo

(3.79)

and for the oblate srheroid:

ro = a cosh zo / d = cosh zo / cosh zs (3.80)

The development of the numerical techniques for the solution of

the Navier-Stokes and energy equations, (3.68) to (3.70) or (3.71) to

(3.73), subject to the boundary conditions of equations (3.74) to

(3.78) are given in Chapter 4 .

***********

89 CHAPTER 4

NUMERICAL TECHNIQUES

4.1. Types of Second-Order Partial Differential Equations

The mathematical formulation of most problems in science

involving rates of change with respect to two or more independent

variables - usually time and position - leads to a partial diffe-

rential equation or to a set of such equations. The two-dimensional

second-order partial differential equations such as those derived

in Chapter 3 constitute an important class of partial differential

equations and are the main concern of the present study.

Partial differential equations for W(z1 9), usually represen-

ting scalar fields or components of vector fields, have the general

form 90 :

2 e e2W A II 6W • 611

at 7.-r+ 2bt -IszsD+ of = f(z29,W, (4.1) •Oz 6z 66 )

If equation (4.1) is linear in W* f has the form:

6W f = + e9

t —6w + &VT + hr (4.2) 6 .

where the coefficients in equations (4.1) and (4.2) are functions

of the independent variables z and eonly.

It is important to note that the methods of solution of the

partial differential equation (4.1) depend essentially on its type,

The type of the second-order partial differential equation is

specified by the nature of the coefficients of the higher order

derivatives (i.e. the second-order derivatives) only. Thus equation

(4.1) is called

(a) elliptic, when

alci>obi2 for all values of (4.3) z and 6)

(b) parabolic, when acct = bt2 everywhere (4.4)

(c) hyperbolic, when atct<b12 everywhere. (4.5)

90

Hence, with this classification, the types of the Navier-

Stokes and energy equations (3.68) to (3.73) can be specified as

follows:

By putting:

g = ez sine , f = c.z sin 9 , (4.6)

equations (3.68) to (3.70) for the sphere, may be written:

* 61P* ,o 6** = ge2z (4.7) 787- coteae

2g

6g cot 2 6z * 60 Re ez sing ( 6'1) 6q)* ) (4.8)

z 60- 60 "S;

12T* 02T* 611 mi. cote ±

6612 + 6z 66 PrReeV 6" 647 6T*

2ez sill 9 (a z de 60 Oz Similarly, by putting:

(4.9)

G = cosh z sine F = / cosh z sine (4.10)

equations (3.71) to (3.73) for the oblate spheroid, may be written:

.2_ i* * * 2 . 2 6*, 6\p = G s

ink z + cos-17 W + W

6z2 2 tanh

-cot

6z a e cosh3zs (4.11)

2 2

G G 6: - cot e 6; =

6z2 tanh 6

sin F 6** 6F iRe cosh z cosh z ) (4.12) 6z de 66 dZ

T* AT* „-1 6T* + -----82 + tanh z + cot& 66

PrRe cosh zs 64; 6T* 60* 6T* 2cosh z sine "e7 69 - az

(4.13)

91

By comparison of equations (4.7) to (4.9) and (4.11) to (4,13)

with the general form (4.1), it follows that in all these equations:

a' = c = 1 and b' = 0 (4.14)

i.e. a'c'

b'2

everywhere . (4.15)

The result of equation (4.15) satisfies the condition of

equation (4.3); hence equations (4.7) to (4.9) and (4.11) to (4.13)

are of elliptic type second-order partial differential equations.

Also, equations (4.8) and (4.12) are non-linear in while

equations (4.7) and (4.11), and equations (4.9) and (4.13) are linear

in tp* and T* respectively.

The domain of integration of any of these two-dimensional

elliptic partial differential equations is always an area bounded

by a closed curve. The boundary conditions usually specify either

the value of the function or the value of its normal derivative or

a mixture of both at every point on the boundary. Such a domain

will be referred to as the flow region.

4.2. The Flow Region

The coordinates z and ()used in equations (4.6) to (4.13) form

a rectangular system of coordinates as described in Appendix C. The

flow regions for the sphere and the oblate spheroid are shown in

Figs. C.6 and C.4 respectively. Each of these flow regions is repre-

sented by a rectangular plane, which is bounded by the straight linee

= 0 =7r Z = Zs , and z --40,01x5D.

As finite-difference methods are being applied to solve the

differential equations of the heat transport process, it is imprac-

tical to extend the mesh to infinity in the z-direction. In order

to obtain workable boundary conditions the particle is assumed to

be in a flow which is enclosed by a finite outer boundary which is

at a large distance from the particle. Along this finite outer

92

boundary the conditions of uniform and parallel flow at infinity,

equations (3.78), may be assumed to be valid. In Jenson's15 solution

for viscous flow round a sphere, the sphere was considered to be

situated on the axis of a cylindrical pipe having a diameter six

times that of the sphere. At the pipe surface the flow was assumed

to be uniform and parallel.

In the (z1 0)-plane the surface of such cylindrical pipe trans-

forms to an inconvenient shape for the application of the uniform

flow conditions (3.78). To avoid this difficulty, the flow region

is simplified further by considering that the outer boundary is:

(a) for the case of flow round a sphere, on a concentric spherical

surface which is at large distance from the solid sphere.

(b) for the case of flow round an oblate spheroid, on a confocal

oblate spheroidal surface which is at large distance from the solid

oblate spheroid.

The ratio of the radius (or semi-major diameter) of the outer

boundary to the radius (or semi-major diameter) of the solid particle

is denoted by ro as defined by equations (3.79) and (3.80). In both

cases, the surface of this outer boundary transforms in the (z1 69)-

plane to the vertical line z =zo so that equations (3.79) and (3.80)

for ro give:

z ro = e (sphere)

cosh z

cosh zs

(oblate spheroid) (4.16)

Bence, Figs. 0.4 and C.6 can be combined to give the complete

flow region as shown in Fig. 4.1 This diagram represents a rect-

angular plane which is bounded by the straight lines 6., 0, 0=17-, z = z0 and z = Z • 0

It is required to fix the value of zo

(or ro) so that the

major diameter of the external boundary is large in comparison with the

diameter of the particle.

k 2

93 Z=0 z=z Z=Z

M1

0

6 • •

j +1

X

2 .

i-1 i

Fig. 4.1

The Flow Engion and Computational -Stars

94

Hamielec et al 17,18 found that at low Reynolds numbers their

drag coefficients approached the experimental values when ro was

relatively large, where as at high Reynolds numbers, the results

were not so sensitive to ro. It is to be expected, therefore, that

the accuracy of the numerical solution of the Navier-Stokes equations

when the Reynolds number is small depends on the diameter of the

outer boundary -• the larger this diameters the better the accuracy.

On the other hand, when this diameter is large the flow region is

also large and hence, a large number of mesh points is required.

This, in turn, increases the computing time. Because of limitations

on the storage capacity of computers and the need for minimizing

computing time, it is necessary to fix ro to a value which provides

both a solution of sufficient accuracy and economical in computer

storage requirements and computing time.

The solution of the differential equations of the transport

process, equations (4.7) to (4.9) for the sphere, and equations

(4.11) to (4.13) for the oblate spheroid, can be obtained by the

solution of an equivalent set of finite-difference equations derived

in the following section.

4.3. Finite - Difference Equations

The first step in numerical methods is ususl3y the replacement

of the differential equations by their finite-difference approxima-

tions. In dealing with two-dimensional partial differential equations

uniform intervals of the independent variables are employed, so that

the required function is computed at the nodes of a rectangular

lattice.

Consider the (z2 0)-plane to be covered with a rectangular

lattice of mesh length h in the z-direction and k in the

and label a typical point as shown in Fig. 4.1 4. Neighbouring

points are numbered 1,2,3, ,12, as in the figure. The value of

h2 62

2, 6z2 h2 2

) • 1,3

95 I

the dependent variable at 0 can be represented or approximated in

terms of values at points symmetrically disposed about I 0I end not

too remote from it. The node point, together with the neighbouring

points, form various computational molecules or stars91 , The 5-point

stars are formed by taking the node 0 and the neighbouring points

112,314 or 5,6,7,8 or 9,10,11,12 . On combination of these points a

9-point star or a 13-point star may be constructed. The simplest form

and the most commonly used of these stars is the ,-point star formed

by the points 0,1,2,3, and 4 . It is convenient to introduce the

indices i and j to locate any point in the flow region, arid these

indices will be used as subscripts to indicate that the value of any

quantity, usually W(z19), is to be taken at the point in question.

Thus, the values of the function W(z10) at the four neighbour-

ing points can be expressed in terms of its value at the node as

follows, using Taylor's series expansion, correct to the second order:

W1. . +10

( R:2 62 4,17)

2, a ,92 k2

'2 (7( 512 1 de

By elimination, approximate expressions for the partial deri-

vatives at point (i,j) can be found in terms of the five point values:

6W1‘t = .0 / - W. V21). &1111 = (W. . V2k

60 1.4.10 1-14 ' 60 1,J+1 - W 1,J-1

62it 2

- (t + W. . - n. )/h (4.18) 6z 1+14 1-1,3 lij

2 et wi,:i = + W. - 2W. .)/k

2 662 i, j+1 1Od 0-1 123

0 + h + 6z

= (1 - h 6 + 6z

= (i + k e + 68

= (i - k 6 +

2 + Tto j

2h 2 -

+

2 - h

4- 2h2 11-14

k coti9, Ti .-1

2 + k Cot 0. +

2k2

1 ••, 2( •-•. + z

2k2

T10+1

113

96

When these expressions, i.e. equations (4.18), are substituted

into a second-order partial differential equation, a finite-differe-

nce equation of the form:

f(W. W. W. W. W. ) = 0 1+10 2 1.44 $ I1 2 14 (4.19)

is obtained for each mesh point (i,j).

When using this method, the Wavier.-Stokes and energy equations

(4.7) to (4.9) for the sphere, and (4.11) to (4.13) for the oblate

spheroid, the following finite-difference equations are obtained:

2 h * 2 + h

2h2 Yi+14 ---f 4i -14 2h,, 2 + k cot40 * di.

7'123-1

2- k cote. a 2k 1 1

- 2(h + )

2z • i 0 = 3.0e

(4.20)

+ • 2k2

2 - h 2 + h - k cot0.

- r gi+1,j + gi-1 0 -2h 2h2 +

2k2 .s1 gi,j+1

2 + k cot 0. 1 I

+ 2k I2J- ; g- • 1 .,2( -r + -2 ) h k

Pe z. e 1 sin e [I(** 1p* ) a 1 • • • •

J i+1 - 22 i.10 1,41 8hk * 3.1j-1 1+14 1-10 i. • - (

• fi l j-1) (4.21)

PrRe

[(NP1,0! - +1 )(TT - T! ) 10-1 1+14 z. 8hk e 1 sing

- ( * )(T! T? )l] = 0 (4.22) i+10 i-1 j 3.0+1 14-1

2h 2 +k cot 0

2k2-

ilj-1

2 + h twill Z.

2h2 1 1

-2( 2 k2 h

2 h cot 9 . xp

i,j+1+*

2k2

sinh2z + cos2 401 t../

G. 4 - 1" - cosh3zEi

= 0 (4.23)

h tank z. * . I dr, N' i+10

97

2 - h tanh z. 2 + h tanh z. 1 Gi+l

1 ,j +

2h2 2h2 Gi_14 +

2k2

2 + k cote 1 1 + i ( ) G.

2k2 Gilj-1 - 2. he + k2 10

4 j+1

2 k cot.

Re ----coshzs coshz.sine. )(F. . - F . ) 8hk 1 J 4j1.1-1,J. - C-14 3.01.1 10-1

dvx i,j-1 1 - ( 1,j+1 // )(F. 1 +1,j - F.-1,3 .)I] = 0

(4.24)

2 + h tanh z. 2 - h tanh z. 2 + k cote. 1 i 3

11! . + TI + T! j 1 2h2 1+1,3 2h2 Ti-11j i 2k2

2 - k cotei . 1 Ti

1 + 0 TI -- 2

2( 73 + k ) ! 2k2 .0 - ,3

PrRe cosh zs - lb * 4. - ) (Tt - TI .) 8hk cosh z. sin 9. ( 1, j+1 C0-1 i+14 ,....1, ,

1 J

(4) i+1,j w 1-11j 3. )(TI,j1

T10I .-1 = 0 (4.25)

Each of these finite-difference equations relates the values

of the dependent variables, such as stream function, vorticity, and

temperature, at neighbouring mesh points. By application of these

finite-difference equations at every internal mesh point a set of

simultaneous algebraic equations results, aid these may be solved

by successive approximations.

The solutions to the problem are thus found at a finite number

of mesh points distributed regularly through the enclosed flow region.

These solutions are obtained when the appropriate boundary conditions

are specified everywhere on the boundary.

4.4. Boundary Conditions in Finite-Difference Form

The solution of any of the differential equations (4.7) to (4.9)

or (4.11) to (4.13) involves the determination of the distribution of

a function 11(z2 e) in the given flow region which satisfies the diff-

erential equation and assumes the prescribed values on the boundary.

98 These equations are elliptic type partial differential equations,

as described in section 4.1, hence; it is well known2190 that a

unique solution is obtained for each equation only when the values of

the function (Dirichlet problem) or the derivative of the function

normal to the boundary (Neumann problem) or both (Nixed problem) are

known everywhere on the boundary.

For the case of fluid flow past particles the boundary conditions

imposed on the boundary of the system are given in section 3.6 by

equations (3.74) to (3.78). On considering Fig. 4.1 with the index i

varying from 1 at the surface of the particle to Mr11 at the outer

boundary, and the index j varying from 1 at the upstram axis of

symmetry (e= 0 ) to M1 at the downstream axis of symmetry (0.1r),

then the boundary conditions in finite-difference form can be written

as follows:

(i) Fixed Boundarj Conditions

(a) For z = zs (surface of particle):

T* . = 1 (4.26)

(b) For z = zo (outer boundary):

MM14 2 sing 6) gmmij = 0 fMM1,j = 0

G = 0 F = 0 = 0 MM1,j 1 MM1,j

where ro is given by equation (4.16)

(c) For 8. 0 (axis of symmetry):

(4.27)

= 0 g1, = 0 := 0 1,1 (4.28)

(d) For 9 = 1r (axis of symmetry):

ilEM = 0 (4.29)

/VI M1 = 0 g GilM1 =

(31)2

67,3 21

Y6,2 La dz' (4.32)

99

(ii) Specially- Treated Boundary Conditions

(e) For z = zs ; the vorticity at the surface of the particle is

to be evaluated using the special relationships given by equation

(3.75) for the sphere and by equation (3.76) for the oblate spheroid.

The new surface vorticity is then computed by considering

Taylor's series expansion for the stream function in the vicinity of

the surface. For example, the values of the stream function at points

(24), (3,j), and (40), removed from a grid point (1,j) on the

surface by one, two, and three grid spacings in the z-direction,

are:

de* h2 a2 p* h3 63** alb = h (4.30) 21i 1:j ez 21 6z2 31 6z3 + —674—

(2h )2 a2* (2h)3 . a +

21 az 31 4132j = +21,,, 64/

101Z 63V (2h)k exp*

4t (4.31)

z3

All the derivatives in equations (4.30) to (4.32) are

considered at the surface grid point (10), But from the boundary 6tit conditions of equation (3.74), both and are zero az

at the surface.

If third and higher order derivatives are neglected, then

from equation (4.30):

(4.33)

If third order terms are included, then after their elimination

from equations (4.30) and (4.31), the following, relationship is

obtained:

eV ) 6z2 11j = (8 qJ* 2,j 31i )/ 2h2 (4.34)

100 Uhen the fourth order terms are also included, then equations

(4.30) to (4.32) give:

)10 20 (108 - 27 30

. 4 4, J . )/18h2

41 (4.35)

The vorticity at the surface given by equations (3.75) and

(3.76) becomes in terms of g, f, G, and F, the vorticity functions

defined by equations (4.6) and (4.10), as follows:

624; gio =

, (4.36) 6z2

cosh3z 2

G1 j 2 2 in s (4.37)

sinhzs + cos 40 6z2 1 1j

Then, for the sphere:

6,j1 = / sin

and for the oblate spheroid:

1,j . = G10 . / cosh z sin 9.

(4.38)

=* . s

. / cosh z sin (4.39) ,J 10

Hence, the vorticity and the vorticity functions at the surface

of the particle can be evaluated at the start of a new cycle from 7..20

equations (4.36) to (4.39)2 using the approximations for (le lr-) . az2 11 3

given by equation (4.33)tor (4.34)1 or (4.35) . The computational

star is shown in the left hand side of Fig. 4.1 •

The first approximation (4.33) is the simplest but the least

accurate one. fore accurate is the approximation (4.34) which has

been used by Jenson15 and Hamielec et al17'18 The last approxi-

mation (4.35) is introduced in order to compare its influence on the

vorticity at the surface with that obtained by the use of approxi-

mation (4.34)

101

(f) Alongthe axis of symmetry 8. 0 ( j = 1 ); special treatment is needed forfi because although = 0 at 19. 0, sine= 0 also. ll

Using L Hospital's rule92 , then:

fill = 1

--- lin

ezi 0 sin 0

v * .* 01

. ezi 6 0 ice

z 1

(4.40)

A similar treatment for the oblate spheroid gives the result:

Si .2 k cosh z. 1

To solve the energy equation (4.9) it is necessary to make use

Fi,1 = (4.41)

of conditions (3.77) from which gi* and all its

derivatives with respect to z vanish at the axis of symmetry 9. 0. * Although 1, 40 and Mr = 0 at 0= 0, sine= 0 also, hence,

60 66 it is necessary to rearrange equation (4.9) by the limiting process as follows:

lim ....--- 97* 0 sin

.....0te 6T* y!.. _

tk 6T4:6 * -67 6 a* 6T*

OF

) = 602

(4.42)

(4.43)

Substitution of these results into equation (4.9) gives:

T* 1?_T* NT* PrRe- 6T* 2.41* *Mb..

Z 2 + 2 02 z 2ez 6z 6e2 (4.44)

which gives the temperature distribution along the upstream axis of

symmetry of the sphere.

Similarly, the corresponding relationship for the oblate

spheroid can be obtained from equation (4.13):

A2T* 462T* 46T* + 2 -s--61- + tanh 6z

z

PrRe cosh Zs 6T*

az 62,1" — (4.45) 602

2cosh z

Because of boundary conditions (3.77), then by application of

Taylor's series approximation, T

and - T can now be 662 2 6(9

102

expressed as follows:

2 1 = 2(Tt ,2 0 - T1 )/k2 (4.46) 69

‘2,b* ° 2 tp 2 /k2

--67 il (4.4 S

7)

The followin equations are the finite-difference forms of

equations (4.44) and (4.43):

2 . h PrRe 1P* 2 + h PrRe i 2 41*. + --.....12--2. ) Tt _..) ) T! -) -' Z. ( 2h` 2 zi 1-1° 4. ( 2h2

2hk2 e z1 1+1,1

2hk e 4 1 2

+ — T112 - 2( 2 + --2, )Ti 1 = 0 k2 h k i

(4.48)

• h tanh z.

1 PrRe i,2 cosh

2h2 2hk2 cosh z.

2 + h T + ( 1-1,1 2h2

.1.* PrRe

2 kivi2 cosh _z 4`4' 1 2

+ T1+1 1 k2 t + ---- Ti' 2 Ti - 2( -2. + ) ! = 0

, 1ll

2hk2 ,cosh z. h k

a. (4.49)

Equations (4.48) and (4.49) can be rearranged in the form:

Tt = f( Ti-1,1,T Ti ) i+1I1, 2 (4.50

which gives the temperature at a point on the boundary = 0 in

terms of the three neighbouring point values, as shown at the bottom

of Fig. 4.1 .

may

(g) Along the axis of symmetry 0= 71-(j = ml); the same treatment be applied as in (f) above with the corresponding results:

• i M fiM1 = z. (sphere) l

k e 1

(oblate spheroid) (4.52) k cosh z.

Also, equations (4.9) and (4.13) become:

T*

2 = 2(TIlM Ti )/k2

6.6

= 2 /k2

(4.55)

(4.56)

103

eT* a2 T* eT* PrRe 6" 62 4)* + 2 - - 662 6z 2ez ez Wf

(4.53)

eT* 62T* &r.* + 2-----2 tanh z 66 ez

PrRe cosh zs

2cosh z

X2e (4.54)

where, now:

Thus, the finite-difference forms of equations (4.53) and

(4..54) for the sphere and the oblate spheroid respectively, become:

2- h PrRe1P1 M 2 h PrRe 14 4137, + -

2 - z73 .

i-11M1 + ( -.-- 7- - ) Tt ( 2h- I 2h . 11-1 1M1

2hk e 2hk2 zI e 4 1 2

4- Ti TT m - 2( 7 + 7 ) Tt,111 = 0 (4.57) k /

(

-1. i,M 2 - h tanh z.

4- PrRe 1/ 4 cosh ze ) T

2 4- h tanh z.

1 3.

2h2 2hk2 cosh z.

T. 10 + ( 2h2 i

PrRe W. cosh z 4 1 2

2I'M s ) Tt + ---- TI - 2( 7 + 7 ) Ti = 0 2 2hk cosh z. 1+1,M1 k 2' ,M1

1 (4.58)

Equations (4.57) and (4.58) can be rearranged in the form:

Ti, _1(T* T Ti ) 11M 1-1,M1 1+1,M1 IM (4.59)

which, as shown at the top of Fig. 4.1, gives the temperature at a

point on the boundary e= iT in terms of the three neighbouring point values.

The boundary conditions are to be computed at the beginning of

each new iteration, so that all the required boundary conditions are

104

,prescribed everywhere on the boundary. Then, the set of the finite-

;difference equations for the internal regular mesh points can be

solved. At the end of each iteration, the boundary conditions are

recalculated and the procedure repeated until the desired accuracy

is obtained.

4.5. Iterative Methods

Each of the finite-difference equations (4.20) to (4.25) may

be rearranged to give the value of the dependent variable at the

centre point as a function of the surrounding four point values.

Thus, the finite-difference equations can be written for each internal

regular mesh point in the following general form:

W. . = f (W. . W. .W. . 120 2.4.10 1 1-10 , 11J+1 W. , D. .) 1/3-1 113

(4.60

where W stands for , g , and T*, and D1. 23 . stands for

the non-linear and convective terms.

If W is known at the boundaries the problem is to find W. at 1,3

every internal regular mesh point of the field. In a field with dimen-

sions as that of Fig. 4.1 the total number of internal regular mesh

points, Np 2 is given by the product (B-1)(MM-1). The product of Pip

with the number of dependent variables of the problem gives the total

number of the unknown point values, Np21. These unknown point values

of the dependent variables are represented by an equal number of

simultaneous algebraic equations of the form of equation (4.6o).

The major difficulty with the use of numerical methods for the

solution of problems of the type discribed arises because of the

large number of simultaneous algebraic equations which are obtained

by the replacement of the partial differential equations by finite-

difference equations. The number of algebraic equations Npro is

seldom less than fifty and it is often several thousand. Fortunately

105

eachecIllationinvolvssonlyasmallnumberofthe NpRi unknown values

so that many of the elements of the coefficient matrix are zero. To

take advantage of this fact, the only efficient methods for solution

of these equations are iterative methods. An iterative method for the

solution of simultaneous algebraic equations 5 -; pne in which a first

approximation is used to calculate a second approximation which in

turn is used to calculate a third and so on. Each step or approximation

is called an iteration. If the iterations produce approximations that

approach the solution more and more closely, then the iterative method

is said to be convergent.

The extrapolated Gauss-Seidel method is chosen for the work

descibed here because it is the best available method with regard to

its rate of convergence and the simplicity with which it may be

applied to an automatic digital computer2O2193 . The terms extra- r I

polated Liebmann successive overrelaxation (S.O.R.) , and succe-

ssive optimum displacements by points (Succ.o.d.p.) are sometimes

used for this method. This method uses the latest iterative values

as soon as they are available and scans the mesh points systematically

from left to right along successive rows (see Fig. 4.1). When the -

(n)th iterative values have been calculatea along the rows j = 2,3,

4,....j-1, and as far as the point (i-10) along the (j)th row, then,

when the (n)th value at the point (i,j) is the next point value to be

calculated, equation (4.60) gives the following iterative formula:

W = f(W W (n) (n-1) (11) W W (n-1) (n) D(n) . . . . . 1 . 1,i 1+1,j ' 1-1,j ' 1,j+1 ' ,j-1 ' 1,j (4.61)

Introducing A1lA2,A3,A4 as the coefficients of the finite-

difference equations, equation (4.61)' becomes:

W(n) = A W + A W(n) w(n....1) A w(n) 1,j 1 1410 2 1.-10 "5-ilj+1 -4"i0-1 D(n) . .

1,3 (4.62)

In this form the iterative procedure is often called the

unextrapolated Liebmann method or the Gauss-Seidel method.

106

With the addition and subtraction of W(n-1) . 1,3

equation (4.62)

can be rearranged to the form:

(n) (n-1) (n-1) (n) 1 W.

,3 . = W.

1,3 . + (A 1Wi+14 + A2W1-11j

= 141 n-.1) + R21). 113 ,3

+ A W. (n-1) + A W. (n) . . 3 1,3+1 4 1,3-1

(n-1) (n) - W. + D. . ) 1,3 113 (4.63)

where the residual R.(n) is the amount by which the value of

( W.changesforoneGauss-Seideliteration.R.n) is zero at complete

convergence. With good initial guesses of W.R becomes very • R. • 11J 123 small, but, because of the desire to achieve a fast rate of convergence

and also to stabilize the computations, a relaxation factor i/ is

often introduced. This leads to the extrapolated methods of 1 Liebmann

I 1

1 I 1 , Gauss-Seidel or the successive overrelaxation (S.O.R.) methods in

) n-1) whichalargerorsmallerchangethanR(.n.3 is given to W.( . . The

11 113

iteration is defined by:

(n)= W (n-1) . 113 1,3

a R 113).

11

,(n-1) = . + 0111-(lp (4.64)

113 1,3 1,3

where W.+(n) . denotes the value of it(n) as calculated by equation (4.62), 193

i.e. by Gauss-Seidel method.

Equation (4.64) may be written as:

/4/ n) = (1 . )/ii n-1 1,3

+(n) W. . 1,3

(4.65)

which shows that the (n-1)th iterative value and the Gauss-Seidel

iterative value are combined linearly to give the (n)th iterative

value. The maximum rate of convergence can be achieved by the use of

an optimum value of Q While optimum values of a can be evaluated

for simple linear problems, the case is more complicated for non-linear

problems. The best value of S1 is to be chosen simply by trial and

error, so as to give a stable solution for the problem in hand. For

overrelaxation f2 is greater than 1, and for underrelaxation 2 is

107

less than 1. In practice f lies between 0 and 2.

The nuterical solutions of the Ravier-Stokes equations and the

energy equation depend largely on the value of the parameters Re and

Pe (=PrRe) respectively. As discussed in Appendix F, the mesh sizes

h and k have to satisfy special conditions in order to obtain conver-

gent and stable solutions. The conditions are given by equations (F.67)

to (F.69) which are: 4

h (or k) <

(4.66)

Pa f(z)

where Pa denotes Re or Pe, and f(z) denotes ez for the sphere and

cosh z / cosh zs for the oblate spheroid. Also, an upper limit for

h and k is given by: 12.648

h (or k)

Pa f(z) (4.67)

The step size and the position of the outer boundary must be

chosen to eliminate, or reduce to an insignificant level, oscillation

of the vorticity or temperature near the outer boundary. This is dis-

cussed in Chapter 5.

4.6. Numerical Differentiation and Integration

The solutions of equations (4.20) to (4.25) give numerical values

of tp *

( , and T* at each mesh point of the flow region(Fig.4.1),

From these distributions, other quantities, such as drag coefficients,

pressure distribution and Nusselt number, need to be evaluated as

described in Appendices D and E. These evaluations require first the

evaluation of the first derivatives of the variables with respect to

z or 0 , and also the evaluation of integrals over some range of z or 6 . The values of the first derivatives and integrals are to be calculated in terms of the point values obtained in the numerical

solutions.

Consider a general function W(x) and let

w(x0 ) = w(xo'h) = w2 (4.68)

108

Define the forward difference operator • and the different-

iation operator D as follows:

AW1 = W1 (4.69)

DW(xo) = W (xo) = ( )x=x (4.70)

6x

By Taylor's series:

' h2 " h3 = w(x +h) = w(x0) + hW (xo) + 20 W (x0) + 3s W

From equations (4.68) to (4.71):

h2D2 h3D3 W2 = (1 + hD + + + )111 ehD W 2: 31 1

(x0) +...(4.71)

(4.72)

Thus, from equations (4.69) and (4.72) the following relation

is obtained:

ehD = 1 + (4.73)

Then, hD = ln(1 + a )

= 2 + 1* 3

6.4 + 5

6 +.0.0 -(4.74) which, gives, on operating on W1 , the following:

hW1 =

where:

Aw1 7 4-1

,A2 w1 + 7 v

1 A LI

v. 1 ..

3 A4 z-1

A = "2

Li • 5 A6 1/

-00

GV (4.75) (4.76a)

& WI = W3

2W2 + Wl (4.76b)

& W1 = W4 - 3W3 + 3W2 W1 (4.76o)

d+ w1 = W5 - 4w4 + 6w3 - 4w2 + w1 (4.76a)

etc....

By substitution of equations (4.76) into equation (4.75) the

following equation is obtained when OS W1 and higher differences

are omitted:

ew 6 x )x=x0

= 1-E(-25"1 + 48w2 - 36W3 + 16w4 - 3w5) (4.77)

Thus, the derivative of the function W(x) can be evaluated at

109

a given value of x in terms of the five successive equally spaced

values of the function 141 to W5 •

The integral of the function W(x) over the limits x=xa and

x=xb can be evaluated by the use of the trapezoidal rule. If the

function is known at equally spaced intervals of x between the limits,

then the integral can be evaluated by the summation of the areas of

the trapezoidal increments. Thus,

x la n-1

ef

W1 + Wn W(x)dx h( +

2.] =

wJ ) (4.78) 2

xa J=2

where h is the size of the interval in x and n is the number of

ordinates.

4.7. General Procedure for the Solution of the Navier-Stokes and

Energy' Finite-Difference Equations

In the previous sections, the finite-difference equations have

been derived from the partial differential equations which describe

the process of heat transfer from spheres and oblate spheroids.

Because of the similarity of the equations for the two cases, it is

convenient to replace the two sets by a single set as given below.

The finite-difference equations (4.20) to (4.25) for the stream

function, vorticity and temperature may be written as follows:

B (i) . ▪ B (j) .

q):,J • B2(i) qii_i,i 3 11J+1

B (j) 41. . I. sO . (4.79)

G. . = B1 (i) Gi+14 B2(i) G. 10 . + B

3 (j)

0 - Giii+1

• B (i)Giti_l + di ,j ,3 . (4.80)

= B (i) T1 . + B (i) T! . + B (j) Tvf . 1 1-1,3 2 1+113 3 131

It

+ B4(j) T! + d. . (4.81)

1,j+1 1,0

D1.3 =

= Fi,j+1 - Filj-1

Df4 = Fi+10

PrY Df1 = 2

H3(i'j)

PrY 2 Df2 =

Fi_10

13-1)

5.+14 i-10) H3(i,j)

where CI(i) = PrY M••••••MMOMMENOMMI and Cs2 qi! C2(i) = -PrY3 (4.93)

cosh z. cosh z

1 1 t t

Ti = (B1 (i) - C1 1(i))Tt + {B2(i) + C1 1 (i))Tt + B TT -1,1 +1,1 3 1,2 (4.91)

Ti ,M1 = (Bt(i) - Ci(i))T + (B(i) + C (i))T! 1

+ B Tt 1,M1 1 2 -1,M1 2 1+1,M1 3 I,M (4.92)

110 11

where d.11i

, I

d. and d.sj represent the non-linear or convective

terms. Thus,

where:

di al = -( C1(i) + 2(j) ) G. .

diaj = ReY H (i j) (D D 1 3 2 f1 f2

s d. =DD -D 1,j f1 f3 f2D f4

D = T3.t . Ti-1,j

+1,3 1-1,j

D = Tt . T* f4 3.13+1 ilj-1

(4.82)

(4.83)

(4.84)

(4.85a)

(4.85b)

(4.85c)

(4.85d)

(4.86)

(4.87)

(4.88a)

(4.88b)

The boundary conditions of section 4.4(ii) are evaluated at the

start of each new iteration from the following equations:

At the particle surface:

G. lo = Cb2(j) (8 LIJ*213 (4.89)

Along the axes 0= 0 and 0= 1r :

F. 1 = Cf (i) and Film = Cf(i) ?;* ilM (4,90)

111

From the solutions of equations (4.79) to (4.81), other important

quantities may be calculated as described in Appendices D and E. These

quantities are:

The dimensionless pressure coefficient at the front stagnation

point, given by equation (D.35) for the oblate spheroid and by equation

(D.36) for the sphere. These may be generalized as:

8 zo*

Ko = 1 Re dz (4.94) 66

zs

Similarly, the dimensionless local pressure coefficient for the

two cases are given by equations (D.41) and (D.42). These may be

written as:

4 K9 = Ko + Re

z=z + e .*] d6)(4.95) s

0

The drag coefficients are given by equations (D.57) to (D.60)

which may be written as:

The skin friction drag coefficient: 7r

CAF = 8q DF Re sin29 dO (4.96)

The pressure (form) drag coefficient: 7r

CDP = Ke sin 20d0

0

CDT = CDF CDP

The local Nusselt number is given by equations (E.7) and (E.8)

for the sphere and oblate spheroid, respectively, and may be written eT*

as: Nue = S1 (j)( —ez )z=z (4.99)

The overall Nusselt number given by equations (E.15) and (E.17)

16" Nu = S2 Li {-(. )z=z sinej Oz

The evaluation of all derivatives and integrals in the above

0

(4.97)

The total drag coefficient is thus given by:

(4.98)

may be expressed as:

(4.100)

Z) cost t/ / cosh3z L

Cbl/( sinh2 z cos2 t1 Ai

4)

-2cosh zs/( sinh2zs

2 1/1 cos ) - ir

2k cosh zs / Iss Is is given by equation (C.14)

-cosh zs / 8hkL

Re cosh zs / 8hkL

Re cosh zs / 2hk2L'

0 Cbl -2

k

112

equations are obtained as described in section 4.6. The various symbols

introduced in this section may have different meanings for t'rie two

cases of heat transfer from a single sphere and from a single oblate

spheroid. The meaning of these symbols is best explained in the

following table:

Symbol Sphere Oblate Spheroid

0

sinh z

sinh zs cosh z

cosh zs tanh z

tanh zs ( I)

zs 0

sinh z ez

sinh zs 1

cosh z ez

cosh zs 1

tanh z 1

e = tanh zs 1

L

L

B1 (i)

B2(i)

B3(j)

B(j)

B1 (i)

B2(i)

B3 C1 (i) cf(i)

Cb1 H3(i0)

G. ,

F. 3J Y1 Y2 Y3

C (j) cb2(j) S1 (j)

S2

2( 1/h2 + 1/k2 )

2( 1/h2 2/k2 )

(2 - h tanh zi)/2h2L

(2 + h tanh zi)/2h2L

(2 - k cot 9 . )/2k2L (2 + k cot 61.)/2k2L

h tanh z.)/2hL

(2 h tanh zi)/2h2Lt

4/k2L1

sinh2z. /cosh3zsL

1 / k z ' cosh3zs / 2h

2

cosh z. sin 8.

(. lj H3 (i j) a.

I. I./H3 (i

' j)

Table 2.

Definition of Symbols Used in Section 4.7

113

This general presentation of the equations enables the procedure

to be generalized for all shapes of particles. The fixing of the single

parameter e specifies the shape for which the solutions are to be

found. This is also helpful in writing the computer programmes as

described in Appendix G.

Before describing the procedure by which the solutions are

obtained, it is necessary to define the criterion for the acceptance

of an approximation or the degree of the desired accuracy. This is

defined for a general function W at a given mesh point as follows:

W - W (n) (n-1) I,j .

4/ 11) i,j

(4.101)

( where W.n) 1 3 (n-1) and W. are the (n)th and (n-1)th iterative values of W ilj

respectively, and 45; the accuracy desired, is a small quantity which

is prescribed in advance( say 10-3). If E is very small this will

increase n, the number of iterations required to achieve convergence.

When the values of W at all mesh points satisfy equation (4.101) then

full convergence has been achieved. However, in some cases, solutions

can be accepted as being good approximations even if some of these

values have not converged— provided the number of these values is

relatively small in comparison with the total number of mesh points

in the field.

(a) Procedure for the Solution of the Navier-Stokes Equations

The iterative procedure for obtaining solutions of the Navier-

Stokes finite-difference equations consists of the following steps:

1. Specify the shape factor e. i.e. the shape of the particle for

which solutions are required.

2. Specify the position of the outer boundary. i.e. the ratio ro as

defined by equations (3.79) and (3.80). This sets the size of the

flow field.

3. Specify the Reynolds number. Re for which the solution is required.

114 4. Specify the mesh sizes h and k keeping in mind the conditions of

equations (4.66) and (4.67). This fixes the number of mesh points

in the flow field.

5. Specify E the degree of the desired accuracy.

6. Set an upper limit to the number of iterations allowed, say n max 1

so that if the method has not converged after nmax iterations the

procedure is terminated.

7. Calculate NPR1 the total number of unknowns in the finite-differe-

nce equations. This is equal to the number of irregular points(along

the boundaries which require special treatment) added to the product

of the number of internal regular mesh points with the number of the

dependent variables.

8. Calculate the field variables which are functions of the independent

variables z and for all values of z and 0 which correspond to Y1

mesh points. e.g. H3(i,j) sin 7, cos e, cot &, sinh z, cosh z, and

tanh z

9. Specify the relaxation factors C2.1 and 0,2 for Iv and G respec-

tively.

10. Assume an array of initial guesses for the dependent variables

and as close to the required solution as possible. i.e. Iv i,J and * (0)

(0) y) 11. Calculate G 0 and F. . from the guessed values of (0 using the 2)i 113 relations given in Table 2.

12. Specify the constant boundary conditions from equations (4.26) to

(4.29). i.e. *

and 4fil1

l'mmili , Gmmi,j , F1,5110 for j=2,31 ....M„

1 G1 . 1 1 G. M1 for i=1,2,....MM1. ,,

13. Calculate the coefficients of the finite-difference equations. i.e.

131(i),B2(i),C1(i),Cf(i) for i=2,3,....MM, and B3(j),B4(j),C2(j),

Cb2(j) for j=2,3,....M.

14. Initiallize the iterative scheme by setting n, the counter of the

number of iterations performed, to zero.

115

15. Add 1 to n so that a new iteration begins.

16. Set NPR, the counter of the unconverged point values, to NPR1

initially (NpRi is calculated in step 7). A test for point conver-

gence is made by application of equation (4.101) as soon as a new

point value has been calculated. If the test is satisfied, NPR is

reduced by 1.

17. Calculate the new boundary conditions for Gili for j=2,31....M from

equation (4.89). Then calculate the values of ils and F1,j from

the new values of G

18. CalculateF

and for i=2,31....MM from eqtations (4.90). 121

At this stage, all boundary conditions will be prescribed along

the boundaries enclosing the flow field. The following steps calculate

new estimates of the dependent variables at all the regular internal

mesh points. i.e. at points (i,j) for i=2,3„....MM and j=2,31....M.

19. Calculate new values of G. . from equation (4.80)—This is followed 11J

by extrapolation using the relaxation facto fl2 20.Calculate andF.from the new values of G. .

1,3 21. Calculate new values of 4). from equation (4.79)—This is follo-

wed by extrapolation using the relaxation factor al •

22. Test for convergence. NpR is zero for full convergence. The solution

may be accepted if NpR is very small in comparison with NpRi and if

the values do not change appreciably with further iterations.

23. If the calculations have not converged, the procedure is repeated

from step 15 provided that n has not exceeded nmax, the upper limit

to the number of iterations specified in step 6. 24. When convergence has been achieved calculate the pressure distrib-

ution and the drag coefficients from equations (4.94) to (4.98).

25. Print out the results.

116 (b) Procedure for the Solution of the Energy Equation.

The iterative procedure for obtaining solutions of the energy

finite-difference equations is similar to that outlined for case (a).

It consists of the following steps:

1. and 2. as in (a)

3. Specify the Reynolds number for which the solution of the Navier-

Stokes equations is known.

4. Specify the Prandtl number for which the solution of the energy

equation is required.

5.-9. As in steps 4.-8. of case (a). 10. Specify the relaxation factor fl

11. Assume an array of initial guesses for T*. i.e. T!(0) .

12. Fix the constant boundary conditions from equations (4.26) and

(4.27). i.e. and TAmio for j=1,2,....M1.

13. Calculate the coefficients of the finite-difference equations. i.e.

131(i),B2(i),34(i),B2(i) for i=2,3,....MM, and B3(j),134(j),S1(j) for

j=1,21....M1 and also B3 .

14. Supply the required values of41* . .for the Reynolds number speci- 1 0

fied in step 3. These values are obtained in case (a).

15. Calculate Dfl(ij) and Df2(i'j) for i=2,3,....MM1 and j=213,....M

from Jquations (4.86) and (4.87).

16. Calculate C1(i) and C2(i) for i=2,31....M from equations (4.93).

17. Initiallize the iterative scheme by setting n, the counter of the

number of iterations performed, to zero.

18. Add 1 to n so that a new iteration begins.

19. Set NPR, the counter of the unconverged point values, to NFyi

initially (NpRi is calculated in step 8). A test for point conver-

gence is made by application of equation (4.101) as soon as a new

point value has been calculated. If the test is satisfied, NPR is

reduced by 1.

20. Calculate the new boundary conditions for M1 1 17 and 11": for i=2, 11

3,....MM from equations (4.91) and (4.92).

117

At this stage, all boundary conditions will be prescribed along

the boundaries enclosing the flow field. The following steps calculate

new estimates of T! . at all regular internal mesh points. i.e. at

points (i,j) for i=213,....MM, and j=2,31....M.

21. Calculate new values of T!. from equation (4.81). This is followed 193

by extrapolation using the relaxation factor S1

22. Test for convergence. Nprt is zero for full convergence. As in case

(a), the solution may be accepted if NpR is very small in compari-

son with NPR1 and if the values do not change appreciably with

further iterations.

23. If convergence has not been achieved, the procedure is repeated

from step 18 provided that n has not exceeded nm ,the upper limit

to the number of iterations specified in step 7.

24. When convergence has been obtained calculate the local and overall

Nusselt numbers from equations (4.99) and (4.100).-

25. Print out the results.

The description of the above steps is given in more detail

together with flow diagrams of the procedure and computer programmes

in Appendix G.

************

118

CHAPTER 5

DISCUSSION OF RESULTS

5.1. Introduction

The procedures for the numerical solution of the finite-diff-

erence equations which correspond to the Navier-Stokes and energy

equations for heat transfer from a particle are given in Chapter 4.

Based on these procedures, the computer programmes of Appendix G are

arranged so that solutions for the vorticity, stream function, and

temperature distributions can be generated for a wide range of

Reynolds and Peclet numbers.

The particles considered in the present study are spheres and

oblate spheroids with ratios of minor to major axes (e) of 0.8125,

0.625 and 0.4375 . The finite-difference equations which correspond

to the Navier.-Stokes equations are solved for the vorticity and

stream function distributions at Reynolds numbers between 0.0001

and 500 for flow around a sphere (e = 1), and at Reynolds numbers

between 0.01 and 100 for the case of flow around the oblate spheroids

considered. The stream function obtained at Reynolds numbers between

0.01 and 500 for the sphere, and between 0.01 and 50 for the other

particle shapes are used to obtain solutions of the energy finite-

difference equations at values of Peclet number in the range

0.01 - 2000.

The accuracy of the numerical solutions depends on the finite-

difference approximations, the numerical errors involved in the

solution of the sets of algebraic equations, and the stability and

rate of convergence of the iterative method of solution. These

factors are examined in the following section. In addition to these

factors, the accuracy with which the numerical solution of the

energy equation predicts the rates of heat transfer depends on the

accuracy achieved in the numerical solution of the Navier-Stokes

equations.

119

From the resultant distributions of vorticity, stream function

and temperature, other quantities which characterize the fluid flow

and convective heat transfer are calculated- thus comparisons with

other methods of solution and experimental data can be made. The

quantities which are related to the viscous flows past particles are

the pressure distribution at the surface, the drag coefficients, and

the phenomenon of flow separation and wake formation.

The vorticity and stream function distributions are discussed

in section 5.3. The phenomenon of flow separation is explained and illustrated in section 5.4. The pressure distributions at the surface

of particles are presented and discussed in section 5.5. The drag coefficients obtained are presented and discussed in section 5.6.

An account of the effects of mesh size and proximity of the outer

boundary is also given.

The results obtained for the convective heat transfer from

spheres and oblate spheroids are treated in sections 5.7 and 5.8. In section 5.7 the temperature distribution around the particles

considered and the local rates of heat transfer are examined. In the

final section, 5.8 the overall rates of heat transfer and their

dependence on the hydrodynamic and physical properties of the flowing

fluid, i.e. on Re, Pr, and Pe, are investigated. These solutions are

compared with other theoretical and empirical relationships.

5.2. Numerical Solutions

The development of numerical methods of solution of partial

differential equations and their application to many practical

problems are increasing in importance. Furthermore, the advent of

modern computers has made it feasible to attack problems that could

not be solved without the aid of a computer.

The accuracy of numerical solutions depends on many factors

which are to be discussed briefly in this section.

120

As described in section 4.3, the first step in numerical

methods is usually the replacement of the differential equations by

their finite-difference approximations. Whenever a continuous operator:

such as eV1 /6z is replaced by a finite-difference approximation

an er'or, called the truncation error, is introduced. For equations,

such as (4.7) to (4.9) with mesh intervals h and k, the truncation

error is 0(h2 + k2). When the boundary conditions are not of Dirichiet

type, they must be approximated by finite-difference forms, thereby

introducing a truncation error in the boundary conditions.

The truncation error, called also the discretization error,

tends to zero as the mesh intervals tend to zero as described in

Appendix F. For this reason, small mesh sizes are usually used in

order to achieve the desired accuracy. This leads to a large number

of algebraic equations which are best solved by iterative methods.

These iterative methods consist of repeated application of a simple

algorithm, such as that described in section 4.7. However, these

methods yield results rather slowly and may yield the exact answer

only after very large number of iterations (n --400,0). The advantage

of these methods is that they tend to be self-correcting and their

structure allows modifications such as over- and under-relaxation.

In any iterative method an arbitrary ini ial approximation to

the solution is assumed, as described in section 4.7, and then

successively modified. Convergence of the method is, of course,

required. However, the method is not considered to be effective

unless the convergence is rapid and the method is stable.

In addition to the truncation error, which results from the

replacement of the continuous problem by a discrete model, there is

an additional error whenever the discrete equations are not solved

exactly. This error, called round-off error, is present in computer

iterative solutions since the calculations involved are only continued

until there is no change greater than a previously specified degree

of accuracy.

121

The interval sizes h and k affect the discretization error and

round-off error in the opposite sense. The first decreases as h

decreases, while the round-off error generally increases. It is for

this reason that one cannot generally assert that decreasing the mesh

size always increases the accuracy. From this discussion we can infer

that the growth of error, outside certain limits, cannot be tolerated

in a numerical technique.

The extrapolated Gauss-Seidel method was chosen to solve the

finite-difference equations which correspond to the Navier-Stokes

and energy equations because of its good rate of convergence and the

simplicity with which it may be computerized as described in section

4.5. In the solution of the Navier-Stokes finite-difference equations,

the stability and rate of convergence of this method depend on the

mesh sizes, Reynolds number, and on the proximity of the outer

boundary (Appendix F). Since vorticity varies more rapidly with z

than with 0 and has high values at the surface of the particle

and decays rapidly as z increases, the solution is, therefore, more

sensitive to the mesh size in the z-direction (h) than in the •••

direction (k). As the dependent variables vary rapidly close to the

surfaces of the spheres and spheroids, it is important to choose h

and k small enough to follow these variations. As shown in Appendix F,

the size of h must satisfy the condition:

h < 4 / Re (5.1)

To ensure stable and reasonably accurate solutions, h and k

were chosen to be 0.1 and 12o for Re < 5, 0.05 and 6o for

5 <Re 100, and 0.025 and 3° for 100 Re S 500. These mesh

sizes were found to be adequate for the range of Reynolds numbers

considered. However, for regions far away from the surface the errors

were found to increase with Reynolds number unless the mesh sizes

were reduced further, but it was found that these errors (instability

near the outer boundary) have little effect on the solution close to

122

the surface; particularly when small values of the relaxation factors

were used. The method converged rapidly at low Reynolds numbe,rs and

convergence was accelerated by the use of relaxation factors in the

range 1 - 2. At high Reynolds numbers the rate of convergence was

slow and it was necessary to use very small rel.ixaUon factors

(between 0 and 1) to ensure stability of the solutions.

Although optimum values of the relaxation factors cannot be

found analytically for such problems, various values were tried and

the best relaxation factors were found by trial-and-error. For all

the solutions obtained, the values of the relaxation factors a l

and 0,2 for the stream function and vorticity fields were taken to

be equal. It is likely that this need not be so, and, as only the

vorticity equations are non-linear, values of cli greater than fl2 may be satisfactory. The values of the relaxation factors used are

given in Tables 9 to 12.

The degree of accuracy, E required of the numerical solutions

for the stream function and vorticity, defined by equation (4.101),

needs to be specified to an acceptable value-- the smaller this value

the better the accuracy of the numerical solutions . However, a small

value of E requires a large number of iterations to achieve conver-

gence. As the desire was to minimize cmputational time, E was

set to 0.001 for Reynolds number less than 100 and to 0.005 for

Reynolds numbers higher than 100. These values are sufficiently

small for the present purpose.

Full convergence was easily achieved for Reynolds numbers less

than 20 when the maximum absolute relative change in the point values

of

and G was less than E . At higher values of the Reynolds

number, full convergence was more difficult to obtain because of

oscillations in the values of vorticity close to the outer boundary.

In these cases the solution was accepted when the number of uncon-

verged point values was not greater than 3% of the total number of

123

the point values in the field. For the largest grid field used

(dimensions 65 x 61), there were 7493 unknown point values and it

was reasonable to assume that the solution was acceptable when less

than 200 (say) point values were still unconverged provided that

these values had no (or negligible) effect on the point values close

to the surface. The computations were continued until the number of

unconverged point values decreased only very slowly in comparison

with the number of iterations performed.

When a solution was obtained for a particular Reynolds number,

the initial approximation to the solution supplied was usually the

solution generated previously for the nearest value of the Reynolds

number. For example, when the solution for Re = 30 was obtained, the

solution for Re = 20 was used as the initial approximation.

At low Reynolds numbers solutions were obtained, applying

overrelaxation (2 > a1 (= SI2) 2> 1), after 20 - 50 iterations which required 1 - 5 seconds of computing time. At high Reynolds

numbers, however, underrelaxation was necessary (04<i-2 (= C22)<:1) in order to obtain stable solutions and hence the rate of convergence

was slower than before. For example the solution for Re = 500 was

obtained after about 1200 iterations and about 20 minutes of comp-

uting time. This solution was achieved in stages, at the end of each

stage the rate of convergence and the changes in the point values

close to the surface were examined. The run was terminated when no

significant changes of the point values could be noticed and the

majority of these values had converged to the desired accuracy.

Another important factor that was examined is the effect on

the solution of the finite-difference approximation to the vorticity

boundary condition at the particle surface. The vorticity at the

surface is given in terms of the value of the second-order derivative

of with respect to z at the surface. This derivative was

approximated by equations (4.34) and (4.35) using Taylor series

124

correct to the third and fourth order respectively (section 4.4).

The truncation errors were very small in the two approximatjons

provided that the mesh size, h was small. To compare the two

approximations, (4.34) and (4.35), solutions were obtained for Re

0.01, 0.1, 1, and 10. The differences between the resulting values

of the drag coefficients were found to be small (less than 0.2% for

Re<1, and less than 1% for Re = 10) as shown in Table 3.

Table 3

The Effect of Using 3rd and 4th Order _Approximations

t,ovaluate the Vorticity at the Surface

on the Drag Coefficients

Re h 17.° r CDF CDP CDT Order

•••••••••••.4 ar.41.••••••

••••••••.....11111.••••••

0.01 0.1 12 6.686 1.0

0.1 0.1 12 6.686 1.0

1.0 0.1 12 6.686 1.0

10.o 0.05 6 6.686 1.0

2174.560 2179.300

217.923 218.057

22.121 22.133

3.030 3.020

1066.790 3241.350 3rd 1067.100 3246.400 4th

106.89$ 324,821 3rd 106.810 324.867 4th

10.888 33.009 3rd

10.878 33.011 4th

1.687 4.717 3rd 1.674 4.694 4th

0.01 0.1 12 6.068 „1961.940 0.81251966.420o

lic71966.420 1231250 3193.190 3rd 1233.610 3200.030 4th

0.1 0.1 12 6.068 0.8125

1.0 0.1 12 6.063 0.8125

10.0 0.05 6 6.068 0.8125

195.006 196.758

19.908 19.849

2.680 2.678

124.125 319.131 3rd 124.:136 320.994 4th

12.568 32.476 3rd 12.535 32.384 4th

1.899 4.579 3rd

1.969 4.647 4th

These results show that the improvement introduced by the

use of equation (4.35) is Small. Thus, the approximation given by

equation (4.34), or generally equation (4.89), is quite satisfactory.

This approximation was also used by Jenson15 and Hamielec et a117t

18

125

From this discussion, it can be said that the numerical

solutions of the Navier-Stokes equations obtained using the mesh

intervals mentioned earlier were satisfactory as the errors, both

truncation and round-off2 were very small. The results obtained

from these solutions are presented and discussed in the following

sections.

The same arguments can be applied to the solution of the

energy equation although it should be noted that the accuracy of

the numerical solutions of the energy equation depends on the

accuracy of the solution of the Eavier-Stokes equations. The values

of the stream function obtained from the Navier-Stokes equations for

a given Reynolds number and mesh size were used to obtain the

temperature distribution to a degree of accuracy, e defined by

equation (4.101), of 0.001. Solutions were obtained at various

values of the Prandtl number using overrelaxation at low Peclet

numbers in order to increase the rate of convergence, and under-

relaxation at high Peclet numbers in order to stabilize the solution.

The values of the relaxation factors used are presented in Tables

13 to 16.

The solution for a given Prandtl number was used as an initial

approximation for the solution at the next higher value of the

Prandtl number. The results of these solutions are presented and

discussed in sections 5.7 and 5.3.

126

5.3. Vorticity and Stream Function Distribution

The vorticity distributions around the sphere and the three

oblate spheroids considered are shown in Figs. 5.1 to 5.5 for Re = 1, 10, 100, 500 for the sphere, and for Re = 10,100 for each of the

oblate spheroids:e = 0.8125, 0.625, and 0.4375. These equi-vorticity

lines are sufficient to give a qualitative picture of the flow

patterns and their variations with Reynolds number.

At a Reynolds number of unity, the vorticity distributions

shown in Fig. 5.1 indicate that there is vorticity in the whole

region of the flow field around the sphere. On the otl-er hand for

a Reynolds number of 500, Fig. 5.2 shows that the vorticity is

confined to a small layer along the surface of the sphere and to a

wake region behind the sphere.

The diffusion of vorticity from the surface affects the flow

immediately to the front of the sphere. As the Reynolds number

increases, vorticity is swept rearward more rapidly than it diffuses

forward. In fact, a boundary layer type of flow was found to be

formed at a Reynolds number as low as 10.

It can be shown that the vorticity around the sphere in Stokes

flow is given by: 3

----E sin (9 (5.2) 2r*

which indicates that the flow is symmetrical about 0= 900.

Fig. 5.1 shows that the vorticity distribution at a Reynolds

number of 1 is less symmetrical than that of Stokes flow but is

similar in form to Oseen's solution in which the vorticity is

contained in a roughly paraboloidal surface. The departure from the

symmetrical Stokes flow becomes more evident as the Reynolds number

increases.

It is shown in Figs. 5.1 to 5.5 that the most intense vorticity

is generated on the upstream surface and is subsequently swept

127

rearward by the stream and persists at considerable distances down-

stream, particularly at high Reynolds numbers. The vorticity becomes

confined to an increasingly thinner layer along the surface with

increasing the Reynolds number. From Fig. 5.2 the thickness of the

boundary layer is about 0.06 of the sphere diameter at a Reynolds

number of 500 --this thickness increases slowly with angle measured

from the front stagnation point. To a great extent the vorticity is

transported downstream in the wake whereas the rest of the flow field

remains free from vorticity.

The figures for the vorticity distributions around the oblate

spheroids:e = 0.8125, 0.625, and 0.4375 at Re = 10 and 100 are shown

in Figs. 5.3 to 5.5, which are similar to those of the sphere and

also give qualitative pictures of the flow patterns.

The stream function distributions around the oblate spheroid:

e = 0.3125 and the sphere are shown in Figs. 5.6 and 5,7 respectively..

The figures which are for Reynolds numbers between 0.1 and 100 were

chosen to represent the range of Reynolds numbers considered in

this work.

Streamlines patterns around the oblate spheroid:e = 0.8125 for

Reynolds numbers of 0.1, 1, and 10 are presented in Fig. 5.6. The

changes in the flow patterns which occur with increasing Reynolds

numbers are not clearly apparent except that the streamlines which

correspond to high values of the stream function (eg. * = 4 )

are slightly more curved at low Reynolds numbers than at higher

Reynolds numbers. Since the outer boundary enclosing the flow field

has a straightening effect on the outer streamlines, this suggests

that the proximity of this boundary is more important at low Reynolds

numbers than at high Reynolds numbers. This is discussed later.

The streamlines drawn in Fig. 5.7 show the flow patterns around

the sphere for Reynolds numbers of 25, 50, and 100. These figures

were chosen to represent the range of Reynolds numbers at which flow

128'

Fig. 5.1

Vorticity Distributions Around the Sphere at Reynolds

Numbers ot 1 - and 10

Fig• 5.2 Vorticity Distributions Around the Sphere" at Reynolds Numbers of 100 and 500

130

Fig. 5.3 Vortioity Distributions Around the .Oblate Spheroid: e = 0.8125

at Reynolds Numbers of 10 and 100

131

Fig. 5.4

Vorticity Distributions. Around the Oblate Spheroid: e = 0.625

at Reynolds Numbers of 10 and 100

= 0.1

= 0.5

132

Fig. 5.5 Vorticity Distributions Around the Oblate Spheroid: e = 0.4375

at Reynolds . Numbers of 10 and 100

133

= 2.0

= 1.0

= 0.05

Re = 0.1

= 4.0

= 2.0

= 1.0

=0.05_ = '10

Fig. 5.6

Streamlines Around the Oblate Spheroid: e = 0.8125

at Reynolds Numbers of 0.1 1 s and 10

134

5.7

Streamlines Around the Sphere at Reynolds Numbers of 25.502 and 100

135

separation occurs. The changes in the flow patterns which occur with

increasing Reynolds numbers are more evident than it is for +he low

Reynolds number range.

It has been shown earlier in this section that vorticity

persists far downstream of the body. This laersibtance of vorticity

causes the stream function to be reduced in the region far downstream

of the body (term d.. in equation (4.82)). The stream function

becomes zero where the streamline adjacent to the body surface

( 4/* = 0 ) separates from the surface of the body. The region

following separation of the main flow from the body is termed the

wake region. The vorticity is largely confined to the fluid which

constitutes the wake. The wake consists of fluid in regular motion

which can be described by streamlines as shown in Fig. 5.7.

As expected the streamline = 0.01 approaches the upstream

surface more closely at a Reyholds number of 100 than at 25. Down-

stream, however, this streamline deflects and tends to leave the

vicinity of the surface of the sphere earlier at higher Reynolds

numbers, thus permitting the streamline = 0 to enclose a larger

wake region. The circulating motion of the fluid inside the wake is

indicated by the negative streamlines enclosed within the wake region.

The length of the wake region increases with increasing Reynolds

numbers. The streamlines IP* = 0 for various Reynolds numbers are plotted in Fig. 5.8 for the sphere and in Figs. 5.9 to 5.11 for the

oblate spheroids:e = 0.8125, 0.625, and 0.1 375. These plots present,

to a good approximation, the dimensions of the wake regions formed

downstream.

The length of the wake region Dw, defined as shown in Fig. 5.3,

was divided by the characteristic dimension, D0 (2R for the sphere

and 2d for the oblate spheroids). The values of Dw / Dc are given

in Table 8 and plotted against log Re in Fig. 5.12.

hr.0.05 kr.6°

r 6-686

Fig. 5.8 Variation of Wake Size Behind the Sphere with Reynolds Number

rn

Fig. 5.9 Variation of Wake Size Behind the Oblate Spheroid: e_0.6125 with

Reynolds Number

-Fig. 5.10

Variation of Wake Size Behind the Oblate Spheroid: e = 0.625 with Reynolds Number

r4 03

Fig. 5.11

Variation of Wake Size Behind the Oblate Spheroid: e =0.4375

with Reynolds Number

1 - 5 '= 6.636

1 40

Shape

0 Sphere TaneuA a56 —•—•—

e = 0.3125

A e = 0.625

V e = 0.4375

Interpolation line

— Extrapolation line

D / DOw

1.0 = 5..0_

0.5

0 VI 7 10

I A 1_1_1_1

20 30 50 70 1 00 I l I

500 700 1000 200 300

Fig. 5.12 Wake Dimension as a Function of Reynolds Number.

141

The following points are observed from Fig. 5.12

1. The dimension of the wake region relative to the particle's major

diameter is nearly proportional to the logarithm of the Reynolds

number.

2. The wake regions behind the more eccentric (thLaner) spheroids

are more extensive (at the same Reynolds number) than those behind

the sphere and the less eccentric spheroids.

3. Extrapolation of the plots to zero length of wake region gives

the critical Reynolds numbers at which flow separation first

occurs. These critical Reynolds numbers are 8, 12, 15, and 20

for the oblate spheroids:e = 0.4375, 0.625, 0.8125, and the

sphere, respectively.

4. The proximity of the outer boundary affects the results - the

wake size becomes smaller as r decreases because of the restric- 0

tive effects of the outer boundary on the wake thus preventing

it from aeveloping to its full size. For the sphere the values

of Dw / D with r0 = 5 become progressively lower as the Reynolds

number increases than those with ro = 6.686.

5. The experimental data of Taneda56 in the range 30 .( Re<:2001

represented by the broken line in the figure, show values about

20 % larger than the present resalts with ro = 6.686. This also

can be attributed to the influence of the outer boundary on the

present results.

For the purpose of a quantitative evaluation of the present

solutions, other results such as angles of flow separation, pressure

distributions along the surface, and drag coefficients were calcu-

lated from the vorticity distributions. These results are presented

and compared with other theoretical and experimental works in the

following sections.

142

5.4. Flow Separation

The phenomenon of flow separation was mentioned briefly in

the previous section. This phenomenon, which at high Reynolds numbers

is called boundary layer separation, is described in various texts6'23

from which the following outline was extracted..

Consider a thin layer of fluid adjacent to the wall (Fig. 5.13)

and wholly inside the boundary layer. This layer is urged forward

by the viscous pull of the superincumbent fluid, and is retarted by

the friction at the wall. Along the front half of the body the

pressure gradient is favourable (pressure decreasing in the direction

of flow), and the thin layer continues to move forward. Near the

wall the forward velocity is smAJ, and, therefore, the momentum of

the fluid will be insufficient for the fluid to force its way for

very long against an adverse pressure gradient over the rear half

of the body (pressure increasing in the direction of flow). This

circumstance brings the fluid, in not very slow motions, to rest

and a slow back-flow sets in. The forward going stream leaves the

surface.

dividing /oro,e--Itrearline

>I

limit of back-flow

Fig. 5.13 Boundary Layer Separation

Fig. 5.13 shows some velocity profiles and the dividing

streamline springing from the point S on the'wall at which the

velocity gradient normal to the wall vanishes. Beyond S a thin

143

layer of fluid in back-flow leaves the wall and enters the interior

of the main flow. Because of the reversal of the flow there is a

considerable thickening of the boundary layer and the boundary layer

approximations cease to apply. The point S on the wall at which the

velocity gradient normal to the wall vanishes iz called the point

of separation.

At S one streamline intersects the wall at a definite angle,

and the point of separation itself is determined by the following

condition. At the point S :Sve /6r is zero for the sphere and

6ve is zero for the oblate spheroid. Since at the surface

vv and vr (or vz) vanish (no-slip conditions) then, from equations

(3.49) and (3.61), the vorticity at the surface becomes:

( )

ll v ve (sphere) 1

) z=z s r r= a(sinh zs cos )- C.J7J

(oblate spheroid) (5.3)

Hence, at the point of separation:

0 (5.4)

Thus, the point of separation can be determined from a plot

of the surface vorticity versus the angle measured from the normal

to the body surface at the front stagnation point. The point at

which the curve vs 9 cuts the 9-axis corresponds to

zero vorticity.

Surface vorticity distributions are shown in Figs. 5.14 to 5.20

for the sphere and for all the spheroids considered in this work.

In these figures, except Figs. 5.14 and 5.17, the distribution of

Re4 is plotted against 6) Figs. 5.14 and 5.17 show the

distribution of against 8 for the sphere at 0 <Re < 2

and at 100" Re 500, respectively. The factor Re4 was

introduced to enable the curves for various Reynolds numbers to be

drawn on common axes for purposes of comparison. The choice of this

144

factor was made on the basis of Jenson's work15 which, in accordance

with boundary layer theory, predicts that the distribution of y*

s Re 4 should be independent of the Reynolds number if the

velocity distributions are approximately the same.

In Stokes flow, the vorticity at the surfaca the sphere,

which is independent of Reynolds number, is, from equation (5.2):

3 — sin (5.5) 2

Equation (5.5) is plotted as bs versus 6) in Fig. 5.14

and is represented by the broken curve. The present rusults of the

surface vorticity for Reynolds numbers of 0.01, 0.1, 0.5, 1, and 2

are also plotted in this figure. The curves for Reynolds numbers

of 0.01, 0.1 and 0.5 are quite close together which indicate that

in this range of the Reynolds number the surface vorticity is nearly

independent of Reynolds number. The curves for Reynolds numbers of

and 2, however, are clearly dependent on Reynolds number and, as

expected, deviate from Stokes' flow.

Another important point to note from Fig. 5.14 is the influence

of the proximity of the outer boundary. The surface vorticity for

a Reynolds number of 0.1 and for ro = 99.484 is plotted in this

figure and shows close agreement to the symmetrical Stokes' solution.

The agreement is closer than when ro = 6.686. This point will be

discussed in more detail in section 5.6.

)"* s On the basis of equation (5.5), the values of t_ z) Re

should decrease with increasing Re. This point is confirmed in

Fig. 5.15 for the sphere where the curve for Stokes' flow (at Re=0.1)

lies above the others for Re = 1,2,3,4,5,7.5, and 10, which are in

correct order below it. The curves for the other values of the

Reynolds number and for the sphere and the oblate spheroids studied

are given in Figs. 5.16 to 5.20 and show similar patterns to those

in Fig. 5.15.

2.2

2.0

1.8

Re = 0.1 (ro =-99 484) 1-6

/..*- Stokes Flow (Re = 0)

1.4

1.2

1.0

0.8

0•G

Fig. 5.14

Surface Vorticity Distributions for the Sphere at Reynolds

Numbers Between 0 and 2

145

30 60 90 120 150 180

(9 (decrees)

Stokor.:Is Solution (Re - 0.1)

120 150 180 • 0 -

0 30 GO 90 .

0 (degrees)

146

Re

1.5

• 1-0

I

0.5

Fig. 5.15

Surface Vorticity Distributions for the Sphere at

Reynolds numbers Between 1 and 10

Stresis Solution (Re = 0.1) 2.0

Re 30

40

50

60

80

/00

147 1.0

0.8

0.6

0.2

0.4

0

-005

Re 17

20

25

Fig. 5.16

Surface Vorticity Distributions for

the Sphere at Reynolds Numbers Between

17 and 100

I • 30 60 9

I0 - 120

0 (degrees) •

148

Fig. 5.17


the Sphere at Reynolds Numbers

Between 100 and 500.

• 1.0

0.8

60 0.6 .80

• 100

0.4

Pig. 5.18

0.2


the Oblate Spheroid: e 0.8125 at

Reynolds Numbers Between 17.5 and 100

11)

30 60 90 -0-05 (degrees)

0

t* s Re 4 50

14,

Re 17.5

20

25

30

40

150

1.5

-• Fig&

Surface Vorticity Distributions for the Oblate Spheroid: e = 0.625

at daubers Between 10 and 100

1.5

100 1.0

0.5

gs Re- 5,0

30 60 90 (degrees) -0.1

0 150

Re

10

20

30

151

.Surface Vorticity Distributions for the Cblate Spheroid: e = 0.4375

at Rnnolds Numbers Between 10 and 100

152

The curves for Re = 17,20,25,30,40,50,60,80, and 100 for the

sphere are shown in Fig. 5.16. These are superimposed in the range

0°<0 <:35° which shows that the prediction that Re-+

should be independent of the Reynolds number (according to boundary

layer theory) is fulfilled in this region.

For the oblate spheroid:e = 0.8125, the curves for Re = 17.5,

20,25,30,40,50,60,80, and 100, shown in Fig. 5.18, are superimposed

in the range 0° 48°. For the oblate spheroids:e = 0.625

and 0.4375 the curves for Re = 10,20,30,50, and 100, shown in

Figs. 5.19 and 5.20 respectively, are almost exactly superimposed

in the regions 0 54° and 0° < < 60°, respectively.

The angles of flow separation were doteratined from the points

of intersection of the surface vorticity distribution curves with

the 671 -axis. The values of these angles measured from the front

stagnation point are given in Table 8 and plotted against (log Re)

for the sphere and the three spheroids considered here in Fig. 5.21.

The lowest values of the Reynolds number where negative surface

vorticities appeased in the present solutions were 25,17.5,20, and 10

for the sphere and the spheroids:e = 0.8125, 0.625, and 0.4375,

respectively. In the solutions for the sphere at Reynolds numbers

of 17 and 20, very small values of the surface vorticity (close to

zero) were obtained around 6= 180° at a Reynolds number of 20 but

were not obtained at a Reynolds number of 17 (see Fig. 5.16). This

suggests that the critical value of the Reynolds number at which

flow separation first occurs is 20 and not 17 as reported by

Jenson15 or 22 as reported by Hamielec et al17 118 . The plots also

show that the angles at which the surface vorticity is zero obtained

in the region 50 < Re ( 500 with mesh sizes h = 0.025 and k = 3°

are slightly higher than those obtained with h = 0.05 and k = 6°.

The differences are between 1° and 30 .

I I I I I 1 1 I

r

•

184

1701-

160

t 150 0 (de tees)

S1401-

1301-

120_

110

100

Shape

Jenson15 Sphere o h =0.05 l<=6° o h0.025 k.3°

.▪ ; er-0•8125

A e 0.625 - e 0.4375 _

• Interpolation curve Extrapolation curve.,

2 1 I I I 1 1 1 1

10 20 30 50 70 100 200 300 500 700 1000 Re

Fig. 5.21 Angles,..Of :•Flow Separation

154

6) By extrapolation of the curves in Fig. 5.21 to = 180°,

the critical Reynolds numbers are shown to be 15,12, and 8 for the

oblate spheroids:e = 0.8125, 0.625, and 0.4375, respectively. These

values are the same as those obtained from Fig. 5.12 in the previous

section. These results show, as expected, that flow separation

occurs at lower Reynolds numbers for flatter bodies.

Experimental results for the sphere discussed in section 2.3.7,

give the critical Reynolds number at which flow separation first

occurs (Rec) to be between 196o and 2456. These values are in good

agreement with the present result.

The variation of the angle of separation with Reynolds number

for the sphere is shown again in Fig. 5.22 together with the avail-

able theoretical and experimental results. In this figure, the

ordinate (0 - 83°) was chosen after Linton and Sutherland94 who

fitted their results by the relations:

es - 83 = 660 Re 1 for 10 <Re < 2000 (5 . 6 )

and es - 83 = 191 Re' for 200 <Re < 104

(5.7 )

These relations are asymptotic to Os = 83° for large Reynolds

numbero; the value of 83° was measured by Fage95 at Re = 1.57 x 105,

These relations are shown in Fig. 5.22 from which it is seen that

in the range 15 <I Re < 1000, relation (5.7) is in good agree-

ment with the observations made by Garner and Grafton59 and by

Taneda56 The results of Hamielec et a111 obtained by the Galerkin

method vary widely around the two relations, casting doubt, for

higher values of the Reynolds number, on the validity of the assumed

form of the stream function, given by equatiOn (2.63).

The present results agree well with Jenson's15

numerical

results and lie between the two curves expressed by relations (5.6)

and (5.7) above. The present results are represented approximately

by the following relation:

es - 83 = 240 Re-' (5 .8 )

155

a 191 Re-'

660 Re-1

SPHERE

-- Linton and Sutherland94

10

240 Re- (Present Relationship)

Y '''

1 --,..

A-, 1

AA A - ---._. yx. I A A -E---Ait...im

Vs- 1 T ------ gab

h = 0.05 k = 6° Present Solution 0 h = 0.025 k = 3°

O Jenson

Hamielec et alli Taneda- 56

A Garner and Graf ton59

20 30 50 70 100 200 300 500 700 10 - t 11111

Reynolds Number

Fig. 5.22

Angle of Flow Separation as a Function of Reynolds Number

A

156

5.5. Surface Pressure Distributions

From the vorticity distributions, the pressure distributions

at the surface of the sphere and oblate spheroid were calculated as

described in Appendix D. The dimensionless pressure coefficient at

the front stagnation point and at other points on the surface were

calculated using equations (4.94) and (4.95), respectively.

Experimental pressure distributions are not available in the

range of Reynolds numbers considered here, and hence the accuracy

of the present results can only be assessed by comparison with

approximate solutions of the Navier-Stokes equations and solutions

obtained for limiting cases. For the sphere, Stokes's flow is rep-

resented by equation (2.25), which, using equations (4.94) and (4.95),

gives: 6

Ko = 1 4. (5.9) Re 6

and K0 = 1 cos 6 (5.10) Re

For the sphere and the oblate spheroid, the potential flow

expressions are given by equations (2.21) and (2.22) respectively.

Subtitution of these equations into Bernoulli's equation gives the

theoretical pressure distribution at the surface as:

For the sphere, 9 K = 1 - sin2O (5.11)

4 and for the oblate spheroid,

e , cot-1 --------T e )• (1 e2)

sin26

2 2/-1 e sin iv cos26

(5.12)

K, = 1 1 e2

1

(1

where Ibis the dimensionless pressure coefficient defined by

equation (D. 1), and K ig-the--10eal-vaIue-of-K. 0

Another point of interest is to compare the pressure coeffi-

cient at the front stagnation point, Ko , with the corresponding

157

results obtained by Homann96 who used boundary layer theory to obtain

the static dimensionless pressure coefficient at the front stagnation

point of a sphere in terms of the Reynolds number. Homann's expression

is:

Ko = 1 + 12

(5.13) Re + 0,643 Reg

where his definition of Re has been altered to agree with the defin-

ition used in this work.

Jenson15 compared the values of Ko predicted by Stokes's and

Homann's formulae with his numerical results and found that his

solutions indicated a trend from Stokes's solution to Homann's as

the Reynolds number increased. This fact, as expected, is confirmed

by the present results as shown in Table 4.

Table 4

Values of Ko for the Sphere

Re K 000110.111111.111•1•401.

Homann Present Work

Jenson Stokes's Solution

0.01 163.162 819.240 1•••••••••••••••11raboollsr.

601.0 0.1 40.603 82.802 61.0 0.5 13.565 17.346 13.0 1.0 8.303 9.267 7.0 2.0 5.125 5.239 4.0

5.0 2.863 2.822 2.742 2.20 10.0 1.997 1.981 1.962 1.60 20.0 1.524 1.488 1.508 1.30 40.0 1.272 1.255 1.264 1.15 50.0 1.219 1.207 1.12

100.0 1.112 1.109 1.06 200.0 1.057 1.057 1.03 300.0 1.038 1.039 1.02 400.0 1.029 1.031 1.015 500.0 1.023 1.026 1.012

158

Table 4 shows that the present results of Ko are about 33 5

higher than Stokes's predictions at Re<0.1, and are in good

agreement with Homann's predictions at Re >100. These results

suggest a steady trend from Stokes's flow towards boundary layer

flow as the Reynolds number increases.

The calculated pressure distributions are plotted in Figs.

5.23 to 5.28 in the form of against , the angle measured from

the normal to the surface at the front stagnation point, for the

sphere and the three oblate spheroids considered in this study. For

the sphere, Figs. 5.23, 5.24, and 5.25 show the pressure distributions

at Reynolds numbers between 1 - 4, 5 - 100, and 100 - 500 respectively.

Although neither creeping flow theory nor potential flow theory

applies to the results presented in this section, predictions of Ke.

can be made from both theories and compared with the present results.

For this reason, the potential flow solution given by equation (5.11)

is included in Figs. 5.23 to 5.25. Predictions from Stokes's solution,

i.e. equation (5.10), evaluated at Re = 1,3, and 6 are also plotted

in Figs. 5.23 and 5.24.

The predictions of Stokes's solution at a Reynolds number of

unity are fairly close to the corresponding results of the present

solution. At higher values of the Reynolds number, the differences

between the present results for Ke and those predicted from Stokes's

solution, as expected, become large. Over the front half of the

sphere, particularly over the parts close to the front stagnation

point, the curves in Figs. 5.23 to 5.25 indicate a transition from

Stokes's flow to potential flow with increasing the Reynolds number.

In these figures, the curve for a given Reynolds number lies

above that for a higher value of the Reynolds number upstream and

below it downstream. The curves intersect at angles that are between

80° and 100°

-4

KG vs 0

Stokes Flow (evaluated at 2e = 1) 2 - Kg

i i 30 - ... .....60 \ 120 , - '

es) \ '

-..... Poteztial Flow

L___ 150 180

10 .Re

Re: 4 3

159

Fig. 5.23

SUrface Pressure Distributions for the Srhere at Reynolds

Numbers Eetween 1 and 4

160

Re •

Pig. 5.24

Surface Pressure Distributions for the

Sphere at Reynolds Numbers Between

KG vs 9

• 5 and 100

,4._. Stokes Flow (evaluated at Re = 3)

\ \

. • t•` \ • Stokes Flow • \

(evaluated at re = 6)/ .\•

N • •• \ •

/

12\0 150 1S0 Re: 100

rotential Flow -->'\\

I-

Fig. 5.25

Surface Pressure Distributions for the Sphere

at Reynolds Numbers Between 100 and 500 / - /

Potential Flow

161

162

2.4 -Kg vs

2-0

1.6

Fig. 5.26

SUrface Pressure Distributions for the

Oblate Spheroid: e = 0.8125 at Reynolds


1.2

C.8

K0

30

-

-08 10

-1-2 Potential Flow

/

2.4 KO vs 0

Fig. 5.27

Surface Pressure Distributions for the

Oblate Spheroid: e = 0.625 at Reynolds


12

0.8

163

20

1.6

30 90 120 (degrees)

150 180

- 0.4

Potential Flow -4.‘

-16

-1.2 Potential Flow -4'1

0.8

0.44

30

-0-4

-08

-1.6

164

KA irs 0

• Fig. 5.28

Re Surface Pressure Distributions for the

• Oblate Spheroid: e = 0.4375 at Reynolds

Numbers Between 10 and - 100 1.2

2.0

1.6

165

The present results of K()

for the oblate spheroids:e = 0.8125,

0.625, and 0.4375 at Reynolds numbers between 5 - 100 together with

the potential flow solution of equation (5.12) are plotted in Figs.

5.26, 5.27, and 5.28 respectively. The curves in these figures

exihibit similar patterns to those for the sphere , and the trend

of the present results towards the potential flow solution with

increasing the Reynolds number is again clearly indicated over the

front half of the spheroid.

Calculated values of the dimensionless pressure coefficient

at selected points round the sphere and spheroids are given in Fig.

5.29 as a function of Reynolds number. The results of Jenson15 for

spheres at Re = 5,10,20, and 40 are in good agreement with those

presented here. The potential flow pressure distributions for the

sphere are represented by the dotted lines. For Re 2> 100, the

present values of Ke approach the potential flow line more closely

for 61<30° than for 0 >30°. This result is also indicated in the previous plots.

At the front stagnation point (0=0°), Ko = 1 according to

potential flow theory which is also the pressure coefficient outside

the boundary layer in accordance with the assumptions of boundary

layer theory. The difference between the values of Ko and unity for

Re<500 gives a measure of the magnitude of the error involved by

the application of boundary layer theory for flows at Reynolds

numbers less than 500. From Table 4, Ko = 1.109 at Re = 100 and

Ko = 1.026 at Re = 50O indicating the increasing appropriateness of

the application of boundary layer theory in this range of the

Reynolds number - thus boundary layer theory can be applied to flows

at Reynolds numbers as low as 100 with only an error of the order

of 11%.

Integration of the pressure distributions around the surface

gives the pressure drag coefficients which are given in the next

section.

1 0 = 3o° -1

Reynolds Number 101

K, vs Re oamamm.ammers••••••••

e= 3

0 = 60°

= go° 0

-

0 = 150°

-2 -

-3-

---- Sphere e = 0.8125

A e = 0.625 V e = 0.4375

I I _1

----- Potential Flew for the Sphere

1 I J L.1 I t

102 10

3 i 0

166

Fig. 5.29 Surface Pressure Distribution as a Function of Reynolds Number

167

5.6. Drag Coefficients

Another aspect of fluid flow past a solid particle is +he

resistance to the motion of the fluid caused by the presence of the

particle. This resistance is commonly described in terms of the

coefficients of drag (Appendix D). The total drag coefficient Cpm

consists of two components; the skin-friction drag coefficient Opp

and the pressure (form) drag coefficient Cpp .

The skin-friction drag coefficients were obtained by integration

of the distribution of shear forces around the particle surface. The

form drag coefficients were obtained by integration of the pressure

forces at all points on the surface. The two integrals are derived

in Appendix D and given in section 4.7 by equations (4,96) and (4.97).

The total drag coefficients were found by summation of these two

coefficients.

The variations of each of CDT , CDF , CDP with Reynolds number

are shown in Figs. 5.30 to 5.33 for the sphere and the oblate spheroids

of ratios of minor to major axes (e) of 0.8125, 0.625, and 0.4375

The numerical results are given in Tables 9 - 12 .

For Stokes flow, the total drag coefficient is given by equa-

tion (2.27) for the sphere and by equation (2.31) for oblate spheroids.

These equations give, for the shapes considered in this work, the

following relations:

For the sphere (e =1 ): CDT = 24.0 Re-I

for the oblate spheroid e = 0,8125: CDT = 23.1 Re-1

for the oblate spheroid e = 0.625 : T = 22.2 Re-1

and for the oblate spheroid e = 0.4375:C DT = 21.1 Re-1

The above relations are represented in Figs. 5.30 to 5.33 by

the broken straight lines. The differences between the numerical

solutions and Stokes's solutions at low Reynolds numbers arise because

of the proximity of the outer boundary - this will be discussed later.

168

I 1 i I 1 I I I I I 1 1 1 1 I I 1 I 1 1 1{ 111 I I 1 I Ili 1

10- 0 1

1 0 1 0 1 0 102 10

Present Eolution

Reynolda Number

Fig.. 5.30

Variation of the Drag Coefficients with Reynolds Number

I

169 I 111i111--1.-T-Ttlifil 1 T 1 1111( r I 11111:

Oblate Spheroid

e = 0.8125 anira•••mgmeam.......•••

0 cn Stokes Flow

1111111 I I I r 11111 I 11111111 I I 111111 1 I 111111

102

103

Reynolds 'Number

Fig. 5.31


- 101 i i!Ili 102

L..../_LL_LI.4

10'

1 I Jilt

102 100 1 01

I I 111111

101

•

174 a1 11 1 n—, 1111111 -r"--1 I

Oblate Spneroid

e = 0,625

DT

Stokes Flow

Reynolds Number

Fig. 5.32

Variaticn of the Drag Coefficients with Reynolds Lumber

3 10 102

171 I I 1 1 1 1 11 I lit

Oblate Spheroid

e = 0.4375

1 1 1 1 I 1 l I I 1 1 1 1 1 I 1 1

1 10 100 101 Reynolds -Number

Fig. 5.33

CDT Stokes Flow

1 1 1 I 1 1 1 1 1 ! 1,,


172

The relative contributions of the two components of the drag

coefficient, i.e. CDF and CDp , vary with Reynolds number and

particle shape. The values of CDF/CDT and CDp/CDT are plotted against

Re for all shapes considered in Fig. 5.34. For the sphere, the

proportion of the skin-friction drag to the total drag decreases

from 67% in Stokes flow to about 40% at Re = 500. The curves of CDF

and CDP intersect at about Re = 130. For the oblate spheroids, the

proportion of the form drag to the total drag increases from about

39% in Stokes flow to about 54% at Re = 100 for e = 0.8125, and it

increases from 46% in Stokes flow to 62% at Re = 100 -!or e = 0.625.

CDP and CDF are equal at about Re = 56 and 17 for the oblate spheroids:

e = 0.8125 and 0.625, respectively.

For the oblate spheroid:e = 0.4375, the contribution of CDp

to CDT is always greater than that of CDF. The proportion of CDP

increases from 56% in Stokes flow to about 72% at Re = 100. It is

shown in Fig. 5.34 that the relative contributions of CDP and CDF in

Stokes flow persist to about Re = 1 for all shapes. The variations

of the relative contributions of CDP and CDF with shape are related

to the flatness of the particle and the phenomenon of flow separation.

The proportion of the form drag to the total drag increases with

decreasing e (i.e. more eccentric spheroids).

For stream-line bodies separation, if it takes place at all,

does so very near the rear of the body, and the form drag is very

small. For bluff bodies, on the other hand, the skin-friction drag

at high Reynolds numbers is small compared with the form drag. The

contribution of form drag gets larger for shapes closer to those with

sharp edges, such as circular discs, since flow separation takes

place earlier with smaller e.

Experimental results of drag coefficients are available for

solid spheres only65 . These data together with results from the

present work and those predicted by empirical and theoretical

I F. illif • I 1 1 0.72

0.66

Shape e=10

0.60L =0.8125

0.54 r---- =0625

C / C or C /C DF DT DP D

TY 2Y ..,...a •

CDF /CDT —•— C /C DP DT

0.301 I I. I I I !III

1 10 Reynolds Number

Fig. 5.34 Relative Contribution of the Form and Skin-Frictional Drag Coefficients to the Total Drag Coefficient

100 I I I

SOO

174

relations are plotted in Fig. 5.35 in the form of CDT against Re on

logarithmic coordinates. The solid curve represents the standard

total drag curve, taken from Lapple and Shepherd's data65 in the

range 0.1.‹Re< 10000.

In Stokes flow, equation (5.14) gives, on plotting CDT against

Re on logarithmic scales, a straight line with a slope of -1 and its

predictions are in agreement with the standard drag curve in the

region Re (0.3. The improvements on Stokes's solution obtained by

Oseen26 Goldstein27 , and Proudman and Pearson31, equations (2.37),

(2.38), and (2.42) respectively, show agreement with the standard

drag curve for Re<2 but give excessively large values of CDT

for Re> 2.

The empirical relationships of CDT with Re due to Zahm28

Schiller and Naumann6466 , and Kliachko6768 equations (2.68),

(2.70), and (2.71) respectively, agree well with the standard drag

curve over a large range of Reynolds numbers. Zahm's formula is

valid in the range 0,2 < Re <:200 but gives values which are

21% larger than the standard drag curve at Re = 1000. Schiller and

Naumann's and Kliachko's formulae give reliable results of CDT for

Reynolds numbers up to 800, but are 7% and 8% below the standard

drag curve at Re = 1000, respectively.

The approximate solution of the Navier-Stokes equations obtained

by the Galerkin method by Kawagutiko and extended by Hamielec et al11

uses equation (2.63) for the stream function, the drag coefficients,

CDP and CDF , in term-2 of Re and the constant a1 are given by equa-

tions (2.64) and (2.65) respectively. The sum of these coefficients

gives the total drag coefficient. These results are plotted in Fig.

5.35, which agree approximately with the standard drag curve in the

region 10 CiRe ( 500 but show a point of inflexion at about

Re = 100. The values of CDT deviate widely from the standard drag

curve for Re >1.1000. Also the curves of CDF and CDP against Re

CDT

100 -

\ \ 0 • 0

\ \ \ 0

101 -

Q•'1.

1 1 1 1 1 1 1 1

175

i 11111 1 1 i 1 lirrn-- 1 I ii—rrn7--- i I 'lc

SPHERE vs Re -

4 Proudman and Pearson31

/3 Goldstein27 ,

/ -- . '0.. / . ... \O. .,•.----- • ft -, • '---2 Oseen26

Stokes4 •

I 1 1 I 111 t 'lit I ilfn I 102-1

L 1 0 10 10 10 102

Reynolds Fumber

•

•

• - - • 8

\ \ \ • •

•

•/'• ,4 •

• • DT

11 DP `. Hamielec et al b \

\ \ \

o

6.

7.

8.

Present Work

Kliachko67168 Lapple and Shepherd65 - (experimental)

-Schiller and EauMann64166

- Zahm28

9

LI

103 104

Fig. 5.35

Comparison of Theoretical and Experimental Drag. Coefficients for

the Sphere

176

show points of inflexion around Re = 100. C . and CDF are equal at

about Re = 85, which is lower than the present numerical revats

(i.e. Re = 130) as shown in Figs. 5.30 and 5.34 . The wide variations

of the results of CDT obtained by the Galerkin method from the

standard drag curve cast doubt on the validity of the assumed form

for the stream function, i.e. equation (2.63), at high Reynolds

numbers.

The results of the total drag coefficient obtained from the

present numerical solutions of the Navier-Stokes equations for the

sphere, which are plotted in Fig. 5.35, show, in comparison with

other approximate solutions, very good agreement with the experimental

standard drag curve . The values of the total drag coefficient, CDT I

are in close agreement with the standard drag curve in the region

10 <Re <(::500 but are high for Re <10 particularly in the

Stokes region. As shown in Fig. 5.30, the calculated values of CDT

are in close agreement with the numerical solutions of Jenson15

and Hamielec et al17

The high values of the total drag_eoefficient, CDT , at low

Reynolds numbers predicted in the present work are a result of the

proximity of the outer boundary at which the streamlines are restr-

icted by the outer finite spheroidal boundaries enclosing the flow

field. These effects become very small as the ratio of particle

diameter to outer boundary diameter decreases (ro --40000). To show

this, solutions were obtained for flow around a sphere and an oblate

spheroid (e = 0.8125) for Re = 0.11 1 and 10 at various values of

ro between 6 and 100 .Values of CDT calculated from these solutions

were compared with experimental values of CDT (Lapple and Shepherd65)

from which correction factors, We defined by equation (2.72), were

evaluated. These results are presented in Table 5 and compared with I the theoretical predictions obtained by Ladenburg70 and Faxen71

equations (2.74) and (2.75) respectively, and also with experimental

177

measurements by McNown et al69 .

The results in Table 5 indicate that at a Reynolds number of

o.11 the value obtained for CDT for the sphere becomes closer to

that predicted from Stokes's solution as ro increases. The values

of the correction factors show that the present results with ro =

6.686 give higher drag coefficients than Stokes's solution by about

35% but that the difference is less than 4% when ro is greater than

8o . These values are in good agreement with the theoretical predi-

ctions,for spheres falling inside a cylindrical pipe, particularly

with Ladenburg's correction for Stokes flow.

If the same theoretical estimations of the correction factor,

We are applied to the oblate spheroids, then the influence of ro

on We is the same as in the case of the sphere. At Re = 0.1 with

ro about 20, the value of CDT is higher than its value in Stokes

flow by about 10% for all particle shapes considered in this work.

The influence of ro on the numerical solutions of the Navier-

Stokes equations decreases with increasing Reynolds number. The

values of CDT at Re = 1 and 10 with ro = 6.686 exceed the corres-

ponding experimental values by 24.5% and 15% respectively. The

difference is less than 7% for ro greater than 20 . For Reynolds

numbers higher than 100 with r0 = 5, the difference is reduced to

less than 5%. These results are in good agreement with the results

of McNown et al69

From this discussion it appears that in order to obtain satis-

factory solutions of the Navier-Stokes equations at low Reynolds

numbers, the diameter of the surrounding boundary must be very large

in comparison with the particle diameter. It is important to note

that when the ratio ro is large instability is caused, particularly

at parts of the flow field close to the outer boundary. On the other

hand, solutions for flow past a sphere at intermediate Reynolds

numbers, especially in the region of interest close to the sphere

Table 5

Effect of the Proximity of the Outer Boundary on the Drag Coefficients

178

CDT Present Work

e

Re (r0--.mN0 ro CDT We Ladenburg Fa:er McNown et al

Sphere (e = 1) The second column represents data given by Lapple and Shepherd65

0.1 240.0 6.686 324.821 1.353 1.358 1.313 1.448 11.023 284.734 1.186 1.216 1.189 1.244 20.086 263.211 1.096 1.118 1.103 1.123 54.598 250.701 1.044 1.043 1.038 1.042 81.451 249.598 1.039 1.029 1.025 1.028 99.484 249.528 1.039 1.024 1.021 1.023

0.2 120.0 6.686 162.457 1.353 1.358 1.313 1.448 0.5 49.5 6.686 65.253 1.318 1.358 1.313 1.448 1.0 26.5 6.686 33.009 1.245 1.220

11.023 29.765 1.123 1.110 am, 20.086 28.44o 1.073 1.050

54.598 28.118 1.061 1.020

2.0 14.4 Awl 6.686 17.267 1.199 1.125 3.0 10,4 6.686 12.084 1.161 ••• 1.120 5.0 6.9 6.686 7.90o 1.144 eon 1.110 10.0 4.1 6.686 4.717 1.150 1.080

4•14 11.023 4.443 1.083 1.050 20.0 2.55 6.686 2.927 1.147 ••• 1.045

30.0 2.0 6.686 2.270 1.135 1.040 50.0 1.5 Olos 5.0 1.691 1.127 1.040 100.0 1.07 .011. 5.0 1.122 1.048 1.040 200.0 0.77 SYR 5.0 0.795 1.033 1.035 300.0 0.65 5.0 0.637 0.980 100 1.030 500.0 0.55 5.0 0.518 0.941 1.030

Oblate Spheroid e = 0.8125 0.1 231.0 6.068 319.131 1.381 1.394 1.344 1.505

9.990 277.851 1.202 1.240 1.210 1,276 18.192 255.265 1.105 1.130 1.113 1,136 29.0 246.131 1.065 1.082 1.071 1.083 48.0 241.774 1.046 1.048 1.042 1.047 72.0 240.402 1.040 1.031 1.027 1.030

1,0 6.068 32.476 - - - 9.990 28.880 - - - 18.192 27.413 - - - 29.o 27.078 - - - 48.0 27.025 _ - _ 11.0

10.0 6.068 4.579 _ - - OF*

9.990 4.298 _ _ 18.192 4.237 - - -

Oblate Spheroid e = 0.625 0.1 222.0 20.0 246.017 1,108 1.120 1.105 1.125

Oblate Spheroid e = 0.4375 0.1 211.0 24.o 235.457 1.115 1.10o 1.088 1.103

179

surface, were found to be quite satisfactory with ro = 6.686 for

Re <100 and with ro = 5 for Re>100 . Values of ro of the same

order as above were used in the case of flow around spheroids. The

choice of ro was made so that:

1. It was large enough to minimize the effects of the proximity

of the outer boundary.

2. Its size was not high enough to cause instability in the

numerical solution.

3. The number of mesh points was within the computer storage

limitations.

Based on these considerations, the values of ro were chosen

as given in Tables 9 - 12 (Appendix H).

Another point to consider in this section is the effect of

the mesh sizes on the solution of the Navier-Stokes equations.

Table 6 shows the variations of the sphere drag coefficients with

mesh sizes for Reynolds numbers from 0.1 to 50.

Table 6

Effect of Mesh Size on the Sphere Drag Coefficients

Re h loo ro CDp CDp CDT Change in CDT

0.1 0.2 12 6.050 224.371 109.359 333.730 0.1 12 6.050 227.108 111.494 338.602 + 1.46 %

0.5 0.2 12 6.050 45.073 21.980 67.053 - 0.1 12 6.050 45.316 22.308 67.624 + 0.85 %

1.0 0.2 12 6.050 22.781 11.137 33.918 - 0.1 12 6.050 22.896 11.328 34.224 + 0.90 %

5.0 0.2 12 6.050 5.376 2.714 8.090 - 0.1 12 6.050 5.323 2.727 8.050 - 0.50 %

10,0 0.2 12 6.050 3.182 1.603 4.785 - 0.1 12 6.050 3.102 1.678 4.780 - 0.11 % 0.1 12 6.686 3.054 1.644 4.698 ... 0.05 6 6.686 3.030 1.687 4.717 + 0.32 %

50.0 0.05 6 6.686 0.967 0.717 1.684 _ 0.025 3 6.686 0.969 0.722 1.691 ± 0.35

180

Although more accurate solutions may be obtained with small

mesh sizes, the number of iterations required for point convergence

varies inversely with the mesh size. For a square mesh, the number

of mesh points in the field varies inversely with the square of the

mesh size. Therefore, for a given starting solution, the computat-

ional labour varies inversely with the cube of the mesh size. Thus

on halving the mesh size, the labour is increased eightfold. The

results presented in Table 6 show that when the mesh size was halved

only the third significant figures of the drag coefficients were

changed. The percentage change in the actual values did not exceed

1.5 . As this refinement was small, although the computational

labour was greatly increased, it is probably advisablesin general,

to use a mesh size that is within the stability requirement (cond-

ition (501)) but which is not too small.

The results presented in the previous sections show that the

numerical solutions of the Navier-Stokes equations obtained at low

Reynolds numbers are satisfactory provided that the outer boundary

is chosen to be at large distance from the particle surface. For

high Reynolds numbers, development of the wake behind the particle

is restricted by the proximity of the outer boundary. However, in

comparison with other approximate solutions and experimental studies

of viscous flow past particles, the numerical solutions give quite

reliable results provided that the mesh sizes and the position of

the outer boundary are chosen for each value of the Reynolds number

so that the solutions are stable and converge.

Hence, the distributions of the stream function (obtained from

the numerical solutions of the Navier-Stokes equations) for Reynolds

numbers between 0.01 and 500.0 can be used to obtain solutions of

the forced convective heat transfer problem. The results of the

numerical solutions of the energy equation are presented and discussed

in the following sections.

181

5.7. Temperature Distributions and Local Nusselt Numbers

The vorticity transport equation and the convective energy

equation represented by equations (3.35) and (3.37) respectively,

may be written, when h1 = h2 = h3 =1, as follows:

The vorticity transport equation becomes:

v v 4.

(5.18) ox1 101

1 2 \ x2 ex1 2

and the energy equation becomes:

6T* 6T* T* eT* v1

v2 a(

2 (5.19)

oxl x2 x1 6x2

By inspection it can be seen that equation (5.19) is of the

same form as equation (5.18) for the vorticity . In fact they

become identical if the vorticity is replaced by the dimensionless

temperature difference and the kinematic viscosity j1 by the thermal

diffusivity CL . The boundary condition T* = 0 at a large distance

from the body corresponds to the condition = 0 for the undis•

turbed parsilel flow also at a large distance from the body. Hence

it is to be expected that the solutions of the two equations, i.e.

the distributions of vorticity and temperature around the body will

be similar in character.

Numerical solutions of the energy equation were obtained for

the temperature distribution, when heat is transferred from spheres

and oblate spheroids of ratios of minor to major axes of 0.8125,

0.625, and 0.4375, in the range 0.01 <Pe ( 2000 .

Local rates of heat transfer from a single particle to a

flowing fluid are closely related to the temperature distributions

around the particle, which in turn vary, for a given fluid, with

Reynolds numbers. In order to show the variations of temperature

distributions and local heat transfer rates with Reynolds numbers,

solutions of the energy equation obtained at Prandtl numbers of

182

0.7 and 2.4 are presented in this section. These solutions were

chosen to represent the solutions of the energy equation obtained

in this work.

Typical sets of the isotherms obtained are shown in Figs. 5.36

to 5.40 for the case of Pr = 2.4 . The temperature distributions

around the sphere at low Reynolds numbers, of 0.01, 0.1, and 1, are

shown in Fig. 5.36, and those at Reynolds numbers of 10 and 50 are

shown in Fig. 5.37. The temperature distributions around the oblate

spheroids:e = 0.8125, 0.625, and 0.4375 (at Re = 0.1, 1, and 10)

are shown in Figs. 5.38, 5.39, and 5.40 respectively.

In the limiting case of zero velocity (fluid at rest) the

influence of the heated body will extend uniformly in all directions.

At very small velocities the fluid around the body will still be

affected approximately uniformly in all directions as shown in the

plots at Re = 0.01 and 0.1 (i.e. at Pe = 0.024 and 0.24). At a

Reynolds number of one (Pe = 2.4) the isotherms begin to become

closer to the surface of the body upstream and to extend slightly

further out downstream. With increasing the Reynolds number, however,

it is clearly seen that the region upstream of the body affected by

the higher temperature of the body shrinks more and more into a

narrow zone in the immediate vicinity of the body whereas downstream

the heated region extends into a tail of heated fluid. For the

sphere at a Reynolds number of 50 (Pe = 120) the thickness of the

narrow upstream zone, which may be termed a thermal boundary layer,

increases with angle 0 around the surface and becomes very thick

beyond 9= 90° as shown in Fig. 5.37 . The boundary layer thickness

for the 0.8125 spheroid and the other flatter spheroids decreases

with increasing angle measured from the normal to the surface at the

front stagnation point up to an angle of about 800. The decrease in

boundary layer thickness is greater with the flatter bodies.

• Prandtl Eumber = 2.4

Reynolds Number= 0.1

Peclet Number :; 0.24

.1

Prandtl. Number = 2.4

Reynolds 'Number.= 0.01

Peclet Number- = 0.024

183

Prandtl Number = 2.4

Reynolds Number= 1.0

Peclet Number = 2.4'

Fig. 5.36 Isotherms .a-ound the Sphere at Peclet Numbers of

0.024 , 0.24 and 2.4

Pr; 2-4 Re = 10 Pe = 24

0 5

=0-05 - 0.1

Pr 2.4 Re = 50 Pe r. 120

0.8 :0.5

Fig. 5.37 isotherms Around the Sphere at Peciet Numbers of 24 and 120

185

Fig. 5.38 Isotherms Around the Oblate Spheroid: e = 0c8125

at Peciet Numbers of , • 0.24 2 4 and 24 g

186

Fig. 5.39 Isotherms Around the Cblate Spheroid: e = 0.625

at Poclet la/mbe;.-s of 0.24 7 2.4 ard 24

187

Fig. 5.40 Isotherms Around the Oblate Spheroid: e = 0.4375

at Peclet Uumbers of 0.24 2.4 1 and 24

188

From the temperature distributions around the sphere and

oblate spheroids the temperature gradients at the surface were

evaluated from which local Nusselt numbers were calculated using

equation (4.99). Typical sets of local Nusselt numbers for the

sphere and oblate spheroids at a Prandtl number of 0.7 are shown

in Figs. 5,41 to 5.46

The values of local Nusselt numbers applicable to low Rey-

nolds numbers are shown in Fig. 5.41, and those values applicable

to higher Reynolds numbers are shown in Fig. 5.42 The curves of

local values of the Nusselt number against angle for various Rey-

nolds numbers plotted in Fig. 5.41 show an increase in the values

of the Nusselt number at the front stagnation point with increasing

the Reynolds number. This increase in the value of the Nusselt

number at the front stagnation point arises because of the decrease

of boundary layer thickness with increasing the Reynolds number.

The local value of the Nusselt number is almost constant at

2.33 when Re = 0.01 which shows the uniformity in all directions

of the transfer of heat from the sphere to the very slowly moving

fluid. This value of the Nusselt number is compared with the theo-

retical Nusselt number at Re = 0 in the following section (5.8).

The curves at higher Reynolds numbers show a decrease in the

values of the Nusselt number with increasing the angle measured

from the front stagnation point. The trend of the Nusselt number

to increase at the front stagnation point continues with more

marked increase at the higher Reynolds numbers as shown in Fig. 5.42.

The curves for Re = 50 and 100 tend to become asymptotic to a

value of the Nusselt number of 2 at the rear stagnation point. On

the curve for Re = 50 the Nusselt number varies from a value of

8.9 at 0= 00 to about 2 at 0= 1600_ 180°. The curve for Re =100 shows a decrease in the Nusselt number from a value of 12.148 at

1

0. 5 0-01

189

Re

Figs 5,41- .

- Local NusSelt Numbers for the Sphere at a Prandtl Number of'0.7

and at Reynolds Numbers Between 0.01 and 10

1 I i 30 60 90

9(degrees)

120 150 180

190

EnGI vs G

Fig. 5,42

Local Nusselt Numbers for the Sphere at

a Prandtl Number of 0.7 and at Reynolds

Numbers of 50 , 100 and 500

(degrees)

30 60 90 120 150 18

191

6= 0 to a minimum value of 1.953 at about = 162°. Beyond

this angle, the value of the Nusselt number increases slightly to

a value of 1.98 at 9= 180°. This apparent minimum in the local values of the Nusselt number arises because the boundary layer

thickness is a maximum at the separation point and beyond this

point there is an increasing rate of circulation in the wake. At

a Reynolds number of 500 the minimum value of the local Nusselt

number (2.96) occurs at 0= 138°. This minimum value is more clearly defined than that at a Reynolds number of 100.

At a Reynolds number of 100 the minimum value of the Nusselt

number occurs at 0= 162° whereas boundary layer separation occurs at O. 131° (Table 8). Correspondingly at a Reynolds number of 500 the minimum value of the Nusselt number occurs at

0= 138° whereas boundary layer separation occurs at 0= 113.5°. Thus the angles at which the Nusselt number is a minimum are

significantly higher than the flow separation angles Os eval-

uated from zero surface vorticity (Table 8). This shows that flow

separation occurs upstream of the region of minimum transfer rates.

Plots (Figs. 5.43 to 5.46) are made of the local Nusselt

number divided by the Nusselt number at the front stagnation point

against angle 0, for a Prandtl number of 0.7, for each shape, and for each Reynolds number. Similar plots are made for a Prandtl

number of 2.4 (Fig. 5.47). The broken curves in Figs. 5.43 to 5.46

represent the exact solutions of the thermal boundary layer equation

obtained by Green13 •

Fig. 5.43 shows, as in Figs. 5.41 and 5.42, the variations

of local Nusselt numbers with the Reynolds number. The curve for

Re = 0.01 shows the uniformity of Nucl / Nug=0 with the angle 4.

The curves for Re = 0.1, 0.5, 1, 3, 10, and 50 fall smoothly from

a maximum value of 1 at the front stagnation point to a minimum

180 120 150 /.1 90

t, (decrees)

30 60

192

1.0

0.8

0.6

0.2

SPHERE

0. 5

/ Huo.o

Exact Boundary Layer

- Solution by Green13

Fig. 5.43

Plots of NuG iNuG=0 Versus. G for the •

Sphere at a Prandtl Number of 0.7 and

at Reynolds Numbers Between 0.01 and 500

0.01 Re

193

e = 0.8125 •••••••••••••••1•8

1.2

1.0

Nu0 Nu0=0

0.8 Exact Boundary Layer -

Solution by Green

0

5.44

Plots of Nu -G / Nu'0=0 versus G for the Oblate

Spheroid: e =•0.8125 at a Prandtl Number of 0.7

and at Reynolds Numbers Between 0.01 and - 50

0-G

0.4

le (degrees)

0.2 0 90 150 120 180 30 GO

0.0

1.6

194

e = 0.625

1.4

1.2 01

1.0 Exact Boundary Layer

Solution by. Green

0.8

Fig. 5.45 -

Plots of Nu -/ Nu8 • versus 4) for. the Oblate = 0

Spheroid: e;-.B 0.625-at a Fraudtl Number of 0.7

0.6 and Reynolds Numbers Between 0.01 and 10 •

0.5 0 30 60 0

6) (degrees)

120 150 180

1'u0 j Nug=6

195

e = 0.4375 1

11.••••••••••11.10,..0411,

2.0

Exact.Boundary Layer

Solution by Green`)

1.5

0.01

1.0

0.5

0.4 0

1 t

30 60 90 0 (degrees)

Fig,. 5.46

120 150 180

Plots of ruo / Nu0..0 versus G for the Oblate Spheroid: e = 0.4375 at

a Prandtl Number of 0.7 and at Reynolds Numbers Eetween 0,01 and 10

196

value at the rear stagnation point. As mentioned earlier, the

curves for Re = 100 and 500 show a minimum value at 9= 162° and

138°, respectively. The figure shows that for angles up to 60°

from the front stagnation point, the present solutions at Re =

100 and 500 are in good agreement with Green's exact boundary layer

solution.

The curves in Figs 5.44 to 5.46 for the oblate spheroids

considered show an increase in the local values of the Nusselt

number with angle up to an angle beyond which the values decrease

again. This increase in the upstream portion is because of the

decrease of the boundary layer thickness with distance measured

along the surface from the front stagnation point as well as with

Reynolds number as reported earlier in this section. The figures

show that the maximum values of the local Nusselt numbers occur

at various angles which are not more than 90o from the front stag-

nation point. The angles at which the Nusselt number is a maximum

depend on the Reynolds number and the particle shape.

The curve for Re = 0.01 and for each of the three spheroids

considered is almost symmetrical about e= 90° with a maximum point at O. 90 . As the Reynolds number increases, the curves become less symmetrical and the angles at which the Nusselt number

is a maximum shift to values less than 90° from the front stag-

nation point. For the Reynolds considered, the maximum values of

the Nusselt number occur at angles between 40o - 90o

It is also shown in Figs. 5.44 to 5.46 that the curves for

more eccentric (thinner) spheroids are more symmetrical about

9. 90° than those for less eccentric spheroids. Fig. 5.44 shows that for angles up to 60°, the present solution for the oblate

spheroid: e = 0.8125 and at a Reynolds number of 50 is in good

agreement with Green's exact boundary layer solution. Similarly,

197

for the oblate spheroids: e = 0.625 and 0.4375 and for angles up

to 60° from the front stagnation point, the agreement between

Green's solutions and the present solutions at a Reynolds number

of 10 is also good (Figs. 5.45 and 5.46).

In Fig. 5.47, the local values of the Nusselt number divided

by the value of the Nusselt number at the front stagnation point

for a Prandtl number of 2.4 are plotted against 0 for all particle shapes considered and for Re = 0.01, 1, and 10. The figure shows

similar curves as those of Figs 5.43 to 5.46.

Experimental measurements of heat or mass transfer rates from

oblate spheroids at low Reynolds numbers are not available in lit-

erature. However, Beg14 has made experimental measurements of local

mass transfer rates from naphthalene spheroids in air (Schmidt

number = 2.4) for 200 'Re < 32000. Green13 compared the local

rates of mass transfer predicted from his exact solution with the

experimental measurements obtained from Beg's photographic plates

and found good agreement for angles up to 70° from the front stag-

nation point for the sphere and up to 50° for all oblate spheroids.

Since the present solutions agree well with Green's solutions

upstream (Figs. 5.43 to 5.46), it is expected, therefore to find

similar agreement with Beg's results. Values of Nu / Nu (4 (i) =0 were

calculated from Beg's measurements for the sphere and the oblate

spheroids: e = 0.8125, 0.625, and 0.4375 and for Reynolds numbers

between 1195 and 4611 (the lowest values for which local rates

are given), These values are plotted against in Fig. 5.47.

The figure shows that the present solutions at a Reynolds number

of 10 predict values of local Nusselt number which are in agreement

with Beg's measurements for angles up to 50° from the front stag-

nation point and for all particle shapes considered. At higher

angles, the numerical solutions overpredict the values of the local

0

198

Shape

Sphere

e = 0.8125

e 0.625

e = 0.4575

Present Work Beg's F.::peritental Results

Re = 1195

Re = 3280 A Re = 2560

V Re = 4611 2.0

1.5

Nu / Nu8=0

Pr = 2.4

1.0

•

// \

\': .• .. // \ . • -... .s.• . .• ..• /.• - — — - ., •• •. • . ••,.. • • , ••••

.1.•

4.• ". . \ •

• • , •••' • ••• .* / ... ..' : \

• . '' //, / r • \ ••• \ .. .........,.......---- .............1........\ . ...

\ ‘. . ...----".. ' \*. '..---••• . s • \ '.

,„ • --31._- — — — — — , . •-1).,i _ • -----='' ‘ \ ...,.. '.., • : ,:::-,,s. ,--,T. -- ..--'—— ......„ \ • \ •• ------ . `.. •.• Re=0.01 .. • -_ \ ss ..-.912:—.."--" ;"--- --"--- . , --....„. ..•,• • -.,..---...:.=-•-• V *S----.. . . ...L. . .__ \ --........4.4....— • —,.: - - - — - — •,_.- -...- A7. . . I- -----. \ -..., ., \ S. . • \

------ A` , • -..._ \\ m's.:,.... \ \ -----;-,....... \ "., ....',....

N. *. •..,...N 0

Fig. 5.47

Plots of 1:u8•/. Nu8=0

0.25

0.5

Prandtl Number of 2.4 and at Reynolds Numbers Between 0.01 and 10 _1 30 - 60 - 90 120 150 180

\*%

N•.N.-..

...-'-'*----.----. • ...:,-.... ..

. • . R e.-.10. 0 .____. versus 8 for All Shapes at a • ---74-:---,------..,-

--__

* Re=1. 0

11 (degrees)

199

Nusselt number by increasingly larger values.

The present results for the local Nusselt number for the

sphere and for Pr = 0.7 and Re = 500 are plotted as Nu / Nu 0=0

against 6? in Fig. 5.48 . The exact solution of the thermal boun-

dary layer equation for the sphere at Pr = 2.532 obtained by

FrBssling12 is given by equation (2.107). By the subtitution of

x/R by 9, equation (2.107) may be expressed as follows:

Nu / Nu = 0 G=0 0.1837 02 0.00696 04 + • 0 • (5.20)

Aksellrud76 has calculated the variation in mass transfer

rates around the sphere from the front stagnation point to the nut,Am ,

separation point for a Prandtl tends to infinity using the approxi-

mate polynomial method to express the velocity and concentration

distributions in the boundary layer. His solution may be expressed

as follows :

/ Nuo=0 = - 0.1728 02

0.0114 ei+

(5.21)

Linton and Sutherland has shown that the aboVe equations

are independent of Reynolds and Prandtl numbers and that the local

transfer number M I defined as

M0 = Nu0 Re4 Pr

(5.22)

is independent of Reynolds number and varies only slightly with

the Prandtl number due to the slight variation in MA = 0t the

heat (or mass) transfer number at the front stagnation point, with

the Prandtl number From Green's solution, M0,0 only decreases

by about 11 % as the Prandtl number varies from infinity to 0.7.

On this basis the following comparison between the available theo-

retical and experimental results,which have been made at various

values of Prandtl and Reynolds numbers, can be made.

200

1•

- SPHERE

;•1

0.8

x

Nu / Nu0=0

(t40 / MQ=0)

O

0

Present Solution and at Pr = 0.7

14 : -- • Beg Re = 1195

Frossling12

Akseltrud76

Grafton77 0,2 - 94 Linton and Sutherland.

0 30 60 90 120 8 (degrees)'

Fig. 5.48 Comparison of Theoretical and Experimental Local Nusselt

Numbers for the Sphere

0.4

0.0 150 180

201

Equations (5.20) and (5.21) together with the approximate

boundary layer solution due to Grafton77 and the experimental

results for the dissolution of benzoic acid spheres in water made

by Linton and Sutherland94 are plotted in Fig. 5.48 Beg's14

experimental measurements for the sphere are also added in this

figure. For angles which are less than 60° from the front stag-

nation point, Beg's results are slightly higher than the present

solution by about 3 % . At O. 80°, Beg's results are lower than the present solution by 11 % . Fig. 5.48 also shows that for angles

up to 60° from the front stagnation point, the numerical solution

predicts values of the local Nusselt number which are in good

agreement with those obtained by all investigators mentioned above,

particularly with AkseltrudIs solution.

In the wake region, the experimental measurements predict

higher transfer rates than the present solution. This may be attrib-

uted to the Influence of turbulence and vorticity shedding from

the sphere surface which are caused by the sphere supports, used

in most experimental arrangements.

From the above discussion it appears that for systems at

high Reynolds numbers ( Re>500), the thermal boundary layer

theory gives satisfactory results over the front half of the

sphere. Therefore, boundary layer theory is recommended over the

region upstream of separation as it is much easier to solve than

the case with the full Navier-Stokes and energy equations. Beyond

separation (in the wake region) it is necessary to solve the

full equations nuaerically.

202

5.8. Overall Nusselt Numbers

The integration of the local Nusselt numbers over the entire

surface of the body yields the overall Nusselt numbers. These were

evaluated by equation (4.100) using the temperature gradients at

the surface between 0= 0° and 180° 4. The va2ues of the

overall Nusselt numbers obtained for each particle shape and for

Peclet numbers (Pe = PrRe) between 0.01 and 2000 are given in

Tables 13 to 16. Fig. 5.49 shows the overall Nusselt numbers

plotted against Peclet number, for Peclet numbers between 0.1 and

2000. A study of this figure reveals that the overall Nusselt

number at the low and intermediate values of Peclet number con-

sidered in the present work is a function of the Peclet number

alone.

In the low Peclet number region, Pe <10 the overall

Nusselt number for each shape appears to converge towards an

asymptotic value as the Peclet number becomes small. The asymp-

totic value of the Nusselt number for Pe = 0 is attributed to

conduction into a stagnant medium. Fig. 5.49 shows that the over-

all Nusselt number attains this asymptotic value at Peclet numbers

less than 0.3

For each particle shape considered, the asymptotic value

of the Nusselt number and the corresponding values of the Nusselt

number evaluated from equations (E.41) and (E.42) for molecular

conduction to a stagnant medium are given in Table 7. Equations

(E.41) and (E.42) are derived in Appendix E.

Table 7. Asymptotic Values of Nu as Pe 0

Shape Position of Nu outer boundary Present work Equation Equation

e emmelift.dm•On.

ro (E.41) (E.42)(r0=00)

1.0 6.686 2.33 2.3515 2.0 0.8125 6.068 2.45 2.5232 2.1341 0.625 20.0 2.40 2.3930 2.2878 0.4375 24.0 2.55 2.5742 2.4834

-10 Nu

Nu

e = 0.14575

e = 0.625

7 :5

10

Re =

E

1-TT T11 1 1 l 1 11111 I I i 1

203

TT-T-11-M

Fig. 5.49

Overall Nusselt Number ac a Function of Peclet NUmber

10_ 7= 5 3 - 2 -

e-= 0.8125

Nu

7 5 3

1.13 Pe-

(Boussinesq793- - .....- ,- -•

_--

A . 10 i

V 5

- -- 4, --,V . _.-- - --:. --

(Friped::) 2

SPHERE V er'*7!!.--

Nu

-- ..... -- t - _,......, 9 x-6N't

9--0;=-0-*-1.0.---a-3-10,----- - . ----- , - -- 3 ..!

,-----

100 1 i JLL 1 111111

1 01

I I 1 I 11111 1 _1_1_1_1_1.11J I I u lif t I 1 A 1

102 0

3

40 Peclet Number

0.01 0.1 0.2 0.5 1 2 3 5 10 50 100 500

-- 2 Pe Interpolation Curve Present RelaTionshipsi_ ______1.3 Pe)

204

The present asymptotic values are higher than those for

molecular conduction into an infinite stagnant medium (column 5)1

but are in good agreement with those for conduction into a finite

stagnant medium (column 4). The influence of the boundary condi-

tion T* = 0 at a finite distance on the results at low Peclet

numbers must be considered whenever comparisons with other theo-

retical works are made. Other investigators more generally con-

sider the boundary condition T* = 0 to apply at infinity.

For Peclet numbers greater than 10, Fig. 5.49 shows that

the values of the overall Nusselt number for each shape considered

can be represented approximately by the following relation:

Nu = 2 Pe4 (5.23)

The present results for the sphere in the range 50 <Pe<2000

are seen also to vary approximately according to the relationship:

= 1.3 PewNu (5.24)

which lies between the predictions of Boussinesq79 and Friedlander8

as will be discussed later.

All investigators in the range 0<(::Pe<1:2000 have confined

their studies of heat or mass transfer from single particles to

spheres. For this reason, the following discussion is based mainly

on the results for the sphere.

For the sphere, values of the overall Nusselt number are

plotted against (log Pe) in Fig. 5.50 for Peclet numbers less

than 10 and in Fig. 5.51 for Peclet numbers between 0.01 and

10,000 . In Fig. 5.50 the results of Kronig and Bruijsten73 ,

Acrivos and Taylor7 1 and Yuge74 are plotted for comparison. As

mentioned in Chapter 2 (section 2.4.1)1 Kronig and Bruijstenis

solution is only valid for Peclet numbers very much less than 1.

•

1

3.4

3.2

30

Re O 0.01 • 0.1 • 0.2 .'

0.5 Present • 1.0 • • work

2.0 3.0 Interpolation curve..., Kronig and Bruijsten73

- Acrivos and Taylor7 Yuge74

Si.'HERE

,/ /

2-4--

22 .0" , • —.

- -1- —r 't i I I 1 lilt 100 10

102 161 Pe

Fig. 5.50 Overall Nusselt Numbers for the Sphere at Peclet Numbers less than 10 0 t.n

206

This solution agrees with Acrivos and Taylor's solution up to

Pe = 0.1, and beyond this value the two solutions deviate as shown

in Fig. 5.50 Acrivos and Taylor's perturbation solution is valid

for Peclet numbers between 0 and 1 .

The present solutions predict higher values of the overall

Nusselt number than the perturbation method Of Acrivos and Taylor

for Pe < 0.8, and lower values for Pe 2> 0.8 . The differences,

relative to Acrivos and Taylor's results, lie between 14% at

Pe = 0.1 and 5% at Pe = 1.0 The present values of the overall

Nusselt number are also higher than the approximate solution by

Yugo74 for Peclet numbers less than 10, who used Stokes flow for

the velocity distribution. The differences, relative to Yuge's

results, lie between 17% at Pe = 0.3 and 7% at Pe = 10. As pointed

out earlier the high values of the Nusselt number obtained in this

work near the asympotic value of the Nusselt number (when Pe-->0)

are due to the influence of the boundary condition T* = 0 at the

outer finite boundary. For the sphere, Table 7 shows that the

present asymptotic value of the Nusselt number is higher than

that for the case of transfer to an infinite stagnant medium by

164% . Thus, considering this influence, the present asymptotic

value of the Nusselt number is in good agreement with that obtained

by Acrivos and Taylor and Yuge.

The present solutions for the entire

considered are plotted as Nu vs Pe in Fig.

solutions of Boussinesq79 and Friedlander

data of Kramers82 and Rowe et alt are also included in

For high Peclet numbers Friedlander8 and Levich9

boundary layer assumptions to obtain a simplified form

Fig. 5.51 .

applied

of the

Peclet number range

5.51. The theoretical

, and the experimental

energy equation and used Stokes's solution for the velocity dis-

tribution to derive equations (2.97) and (2.98). The two equations

24.

20_

8_

Nus

selt Num

ber

le[.

15-

14_

12_

1 I 111111 1 1- 111111 ( 1 1 1 l lilt i

Fig. 5.51 Re 00

0 0

0.01 Comparison of Theoretical • 0.1 i' and Experimental Overall 0 °

i— i

1

0 x 0.2 Nusselt Numbers for the ; 0 0

gi /

0.5 Sphere i 000 ci) i / -

a 2.0 • / c5)0 e A 0 i

1.. 1.0 0 I

1 00 0.

3.0 0 i -4

-Present, / 00 / 0

A 10.0 0 /

1 , 0 )13. 66 J) / •

/

A 5a 0 work , / 0 oe/

A o /

V 100.0 Se 0 e(6 o

0 i i /- o 0° + 500.0 0 /

1.4 • 0 1 _interpolation curve 1.13 Pei/2 _0./../ 0' v2 /

e Rowe et ail

00/ /

• /'°-- 0.991 Pe1 /3 - a k

C:Boussinesq79 ) i. - / O Kramers 82 /0

* / (Fried lander8 )

SPHERE

UPeclet Number

0 1 1 1 1 1111 1 1 1 1 1111 1 1 1 1 1 1111 1 1 1 1 1 1111 1 1_1_1 1 1 111 I 1 lit _Lilt 1 02 101 - 1 00 101

102 103 104

24 .0 1-1111111 0

0

®

/A A 0 "4,-' ,•

• Tio3,50;57"3-

"-0- .P4-0 0-0r0.0-00-400 .•••

208

are nearly the same and both show that Nu is proportional to Pe.)

Both equations are represented by the lower broken line in Fig. 5.51.

Using potential flow theory for the velocity distribution, on the

other hand, Boussinesq79 obtained equation (2.110) where Nu is

proportional to Pel . Equation (2.110) is represented by the

upper dotted line in the figure. Neither of these solutions can

be expected to provide satisfactory results for 1‹,‹Re< 500

because of the invalidity of assuming either Stokes flow or

potential flow for this range of the Reynolds number. However, they

provide lower and upper limits to the values of the overall Nusselt

number. Fig. 5.51 shows that the present numerical solutions give

results which lie between the two limits but are closer to the

lower limit for low Reynolds numbers (Re 10) and closer to the

upper limit for Re = 500

The experimental data of Kramers82 and Rowe et al2 are shown

scattered between the two limits with Kramers' data closer to the

lower limit and Rowe et al's data closer to the upper limit.

Kramers' results for low Peclet numbers agree well with the present

results at Re .<50 while the present results at Re = 500 lie

on a line which separates the lower values of Rowe et al and the

upper values of Kramers.

It must be noted that most experimental measurements of heat

or mass transfer rates are influenced by free convection. Further-

more, the physical properties of the system; eg, fluid density,

viscosity and thermal diffusivity, calculated with available

equipments and experimental techniques are not precise enough to

give accurate evaluations of the overall Nusselt number. It can

be said, therefore, that for low Reynolds numbers the lower experi-

mental values of the Nusselt number shown in Fig. 5.51 are most

likely to be correct.

209

The dependence of the overall values of the Nusselt number

on the hydrodynamic and physical properties of the flowing fluid

has been the subject of most investigators in this field. This

is best examined in terms of the overall Nusselt number, the

Prandtl number and the Reynolds number. It is shown in Figs.

5.49 to 5.51 that the Nusselt number depends on the Peclet number

alone for the solutions at low Reynolds numbers (Re <110). The

relationship between the Nusselt number and Peclet number for

Pe >10 is best described by equation (5.23) as mentioned

earlier in this section.

As the Reynolds number increases beyond Re = 10 the overall

Nusselt numbers do not correlate with the Peclet number alone

and they appear to depend separately on the Prandtl and Reynolds

numbers. The thermal and hydrodynamic boundary layers get thinner

with increasing Prandtl and Reynolds numbers, respectively.

Hence, the dependence of the Nusselt number on the Reynolds and

Prandtl numbers as they increase approaches that based on boundary

layer considerations, i.e. the variation of the Nusselt number

with Re2 and Pr-1 as in, for example, Ranz and Marshall's81

correlation (equation (2.113)).

In order to determine how closely the Nusselt number is

proportional to Re-2- and Pr-i Figs. 5.52 and 5.53 have been plotted

on logarithmic coordinates versus Re and Pr. In addition plots

have been made of JH vs Re (Fig. 5.54) and of Nu vs Rea Pr"(Fig.5.55).

The broken straight lines in these figures represent:

1 1 Nu = 0.6 Re2 Pr=" (5.25)

Equation (5.25) represents the forced convective term of

Ranz and Marshall's correlation (2.113). It is used here as a

typical correlation for forced convective heat transfer.

210

In Fig. 5.52, values of Nu / Pr3 for various values of the

Prandtl number are plotted against the Reynolds number. In thf,,

upper part of the figure the curves represent the data for all

shapes at Pr = 2.4 and 100 and the lower part of the figure is

for the sphere alone at Prandtl numbers between 0.7 and 300. The

plots reveal that at low Prandtl and Reynolds numbers the curves

are almost parallel horizontal lines, but as both Reynolds and

Prandtl numbers increase the curves approach the broken line of

equation (5.25) because of the increasing influence of forced

convection.

In the lower part of Fig. 5.53 , values of Nu / Rei are

plotted against the Prandtl number Pr for the sphere for Reynolds

numbers between 0.01 and 500 In the upper part of Fig. 5.53 the

data for all shapes at Re = 1.0 are plotted against Pr The figure

shows the same trend as that of Fig. 5.52 ; with the results for

Re = 500 approaching the line of equation (5.25) very closely.

The J-factor for heat transfer is defined as follows:

JH = Nu / Re Pr3.

(5.26)

Thus the J-factor corresponding to equation (5.25) takes

the following form:

JH = 0.6 Re-1 (5.27)

This relation is represented by the broken straight line

in the plot of JH against Re of Fig. 5.54 . In this figure the

present results for all particle shapes considered and for Prandtl

numbers between 0.7 and 200 are plotted. The figure shows again

that the present results approach asymptotically the line of

equation (5.27) as the Reynolds and Prandtl numbers increase.

It appears from the above discussions that the dependence

I I III -Fr I I -1-1-n-T-T- ► I 1 Tflh1 ►

e e e

0

A V

Ill I. I

1 0 1

211

Nu,/P1 vs Re

-7-71 I Jill I I 1-77-1-ti

Shape

Sphere =_0.8125 = 0.625 = 0.4375

Interpolation

0.6 Reg

-10 Nu / :7 -5 3 Pr 2 2.4

1

.7 ion

-.3 -.2

lu

3 2

1

- • 7 •5; -3 - .2 -

I I I I III I-

Fiz. 5.52 Plots of Nu / Pr7 versus Pe

102 101 1 00 1 01 Number Reynold

0.6 Re.

I I I it"

2

Nu /

tI

SPHD°.E

Yu / Re

-4 0.6 Pr".

100 70

O 61 Re

212 r-- . I 1-0 1-1-1-T-r7.- r rrn ---1 t I run'

Shape

7 5

3 2

: .5]

SPHERE 3 - 2,

0.5

Nu / Re vs Pr Sphere e = 0.8125 + e.= 0.625 A e = 0.4375 y

.10 :7

- 5 Re

-3 1.0 -2

Interpolation line

Nu Rel.

►11111 1 1 1 111111 11 1111111 1 1.1111111 1 1 111.. 1111

• 10 100 101 102 3 10 • 10

Prandtl Number

Fig. 5.53 Plots of Ku / Rel versus Pr

JH

1 Co

. 1 1 11411 -r-- 1 . I I r 111111

213

Pr • •

JR vs Re 0

1 0

-J

0.6 Re— 2-

'Shape

o Sphere

+ e = 0.8125

t c = 0.625

e = 0.4375 Interpolation line

I 1 1 1 1( 1 1 I 1 1 t 1 1 t

102 1 100 — 01 101

Reynolds Number

I I

102 ,03 1'02 t t 1 1 111

Fig. 5.54 Plots of J versus Re

30 Frossling8cy

84 Jenson et al 20

• 10

7 6 5 4 3

— Ru

-0 -41. • 0-P0 0 .0)-GR 2

I I 1 1 1 1 1 1 I•

a. i Re Pr-

1 I I I 100 101 01

214

I I I I I III

Re 41almal..1.••••••

o 0.01 • 0.1 X 0.2

0.5

1.0 Present

2.0 work 100— '14 3.0

A 10.0 A 50.0

50 9 100.0 ± 500.0

SPHERE

111••••••••••••••

Nu vs Rol- Prl

Fig. 5.55 Plots of Nu versus Re' PP" for the Sphere

215

of Nu on Re7 Pr7 becomes evident at high Reynolds and Prandtl

numbers. This is well illustrated in Fig. 5.55 where values of

Nu are plotted against Re7 Pr on logarithmic coordinates for

heat transfer from a sphere at Reynolds numbers between 0.01

and 500 . The figure shows that :

1. For Re Pri < 0.3 the Nusselt number Nu is almost constant

and equal to 2.33 the value for radial diffusion alone to

a stagnant medium.

2. For Rel Pry > 0.3 , the curve branches into separate lines:

one for each Reynolds number.

3. For Reynolds numbers between 10 and 500 , the present numerical

solutions predict values of the overall Nusselt number which

lie nearly on a single curve very close to that of FrOssling8o

(equation (2.112)) and to that of Jenson et a181+ (equation

(2.122)) .

The results presented in the last two sections indicate

that the numerical solutions of the energy equation are quite

satisfactory. The solutions converged to the required accuracy

rapidly at low Peclet numbers but at high Peclet numbers the

convergence was slow. The boundary layer solutions, which are

easier to obtain, are recommended over the front half of the

body for Peclet numbers greater than 500 .

***********

216

CHAPTER 6

CONCLUSIONS

The results of the present study may be summarised as follows;

1. A finite-difference method of solution of the Navier-Stokes

and energy equations has been developed for steady state forced

convective heat transfer from spheres and oblate spheroidal bodies

which have their axis of symmetry aligned with the direction of

flow. The bodies were assumed to be heated to a uniform surface

temperature and to be immersed in a Newtonian fluid of uniform

lower temperature and flowing with a uniform velocity at large

distances from the bodies.

2. Finite-difference approximations enabled the Navier-Stokes

equations to be reduced to two sets of simultaneous algebraic

equations for the stream function and vorticity , and the energy

equation to be reduced to a similar set for the temperature.

3. The sets of simultaneous algebraic equations were solved

iteratively using the explicit extrapolated Gauss..Seidel method.

Initial guesses of the dependent variables were supplied to start

the iterative procedure. Suitable relaxation factors were obtained

by trial-and-error in order to ensure stability of the solutions

and rapid convergence to the required accuracy.

1+. Two computer programmes were written to solve the sets of

algebraic equations; one for the two sets for vorticity and stream

function ( i.e. for and q)* ) and the other for the set

for temperature ( i.e. for T* ) . These programmes can be applied

to the sphere or to any oblate spheroidal body.

217

5.. The computer programmes were used to solve the sets of the

algebraic equations for the sphere and for three oblate spheroidal

shapes. Solutions of the Navier-Stokes finite-difference equations

were obtained for the sphere for Reynolds numbers in the range:

0.0001 < Re < 500 For the oblate spheroids: P = 0.8125 "*"1%.

0.625 and 0.4375 solutions were obtained for Reynolds numbers

in the range: 0.01 < Re < 100 . Solutions of the energy

finite-difference equations were obtained for Peclet numbers in

the range: 0.01 < Pe 2000 . To obtain these solutions

the stream function distributions for 0.01 < Re < 500

were used for the sphere and for 0.01 < Re < 50 for the

oblate spheroids.

6. The numerical method used appears to give satisfactory

solutions to the problem of forced convective heat transfer from

spheres and oblate spheroids. The solutions predict sufficiently

accurately the hydrodynamic situation and the temperature

distribution.

The convergence of the method of solution was rapid when

good initial guesses of the dependent variables were supplied.

For low Reynolds and Peclet numbers solutions were obtained,

using overrelaxation, after 20 - 50 iterations and 1 - 5 seconds

of computing time (IBM 7094), The solution at Re = 500 required

1200 iterations and 20 minutes of computing time (IBM 7094).

7. The accuracy of the results depended on the choice of mesh

sizes and the position of the outer boundary. The mesh sizes

were chosen to be small enough for both truncation and round-off

errors to be minimized. The position of the outer boundary was

chosen to be sufficiently distant to minimize the influence of

the outer boundary and to minimize instability.

218

8. The distributions of vorticity and stream function show

symmetrical flow patterns at low Reynolds numbers and unsymmetrical

patterns at higher Reynolds numbers. A boundary layer type of

flow was observed at a Reynolds number as low as 10 .

For angles less than, 60° from the front stagnation point,

curves of C Re-4 (where Ss is the dimensionless surface

vorticity) for Reynolds numbers greater than 10 are closely super-

imposed in agreement with boundary layer theory.

9. The critical Reynolds numbers at which flow separation first

occurs were found to be 20 15 , 12 and 8 for the sphere and

for the cblate spheroids: e = 0.8125 , 0.625 , and 0.4375 ,

respectively.

For the sphere, the angles of flow separation were found

to vary with Reynolds number approximately according to:

83 = 240 Re4

10. The wake dimension, relative to the major diameter of the

sphere or oblate spheroid, was found to be proportional to the

logarithm of the Reynolds number. The wake regions behind the

more eccentric spheroids were found to be more extensive (at the

same Reynolds number) than those behind the sphere and the less

eccentric spheroids.

At high Reynolds numbers the proximity of the outer boundary,

due to limited storage capacity of the computer, restricted the

development of the wake downstream and gave a restricted wake.

11. Pressure distributions at the surfaces of the bodies were

calculated, At angles less than 30° from the front stagnation

point these approached those of Stokes flow at small Reynolds

numbers and of those of potential flow at higher Reynolds numbers.

219

At angles greater than 30° the pressure distributions

deviated from those of potential flow even at Reynolds numbers

greater than 100. The use of potential flow to predict the pressure

distribution as used in boundary layer theory becomes, therefore,

inaccurate at higher angles from the front stagnation point.

12. Drag coefficients obtained for the sphere are in good agree-

ment with other workers' experimental measurements in the range:

Re >10. However, high values of the total drag coefficient

were obtained for Re <10 due to the proximity of the outer

boundary. At low Reynolds numbers, results very close to experi-

mental values were obtained when the outer boundary was moved

to a greater distance from the sphere.

The relative contributions of the components of drag were

found to vary with Reynolds number and shape: viscous drag was

found to be predominant at low Reynolds numbers while form drag

was found to be predominant at high Reynolds numbers. The two

contributions became equal at Re = 130 56 and 17 for the

sphere and the oblate spheroids: e = 0.8125 and 0.625, respec-

tively. The form drag was always greater than the viscous drag

for the oblate spheroid: e = 0.4375, The increase in the cont-

ribution of form drag with increasing oblateness is attributed

to the earlier flow separation that occurs from the flatter

spheroids.

13. At low Peclet numbers the temperature distributions were

nearly symmetrical around the spheres and oblate spheroids. As

the Peclet number increased, the region affected by the higher

temperature of the body shrank more and more into a narrow zone

upstream and into a tail of heated fluid behind the body. A

thermal boundary layer developed at Peclet numbers greater than 100.

220

Values of the Nusselt number at the front stagnation point

were found to increase with Peclet number due to the decrease of

the boundary layer thickness with increasing Peclet number. At

intermediate Peclet numbers local values of the Nusselt number

for spheres decreased with angle measured from the front stagnation

point. For the oblate spheroids, however, the local values of the

Nusselt number increased initially with angle due to the decrease

of the boundary layer thickness with increasing angle. After a

certain angle had been exceeded the Nusselt number decreased.

At high Peclet numbers the Nusselt number at both the front

and rear stagnation points increased because of the increase of

the temperature gradient at the front and the increase in the

rate of circulation in the wake. Minimum values of the local

Nusselt number were found to occur at angles which were higher

than the corresponding angles of flow separation.

14. At high Peclet numbers, the numerical solution predicted

values of the local Nusselt number, relative to the Nusselt number

at the front stagnation point, which are in good agreement with

Green's13 exact boundary layer solution, Aksellrudis76 approximate

boundary layer solution for the sphere, and with Beg's14

experi-

mental measurements over the region upstream separation (good

agreement up to angles less than 60 from the front stagnation

point).

15. The overall Nusselt number at low Peclet numbers (Pe <10)

appeared to depend on Peclet number alone. The present results

at Pe <0.8 are slightly higher than those obtained by Yuge74

and Acrivos and Taylor7 because of the influence of the proximity

of the outer boundary on the present solutions.

At very low Peclet numbers (Pe <0.5) the overall Nusselt

221

number approached an asymptotic value which was close to that

attributed to transfer to a stagnant medium of the same extent

as that confined between the body and the outer boundary. The

effect of the proximity of the outer boundary on the asymptotic

Nusselt number was to produce higher values (about 16 %) than

those for the case of transfer to a stagnant medium of infinite

extent.

16. For Peclet numbers in the range: 10 <Pe <2000 and for

all shapes studied, the overall Nusselt number was found to vary

with Peclet number according to the relationship:

Nu = 2 Pe

The solutions of Boussinesq79 and Friedlander8 were found

to give an upper and a lower limit to the present solutions.

As Reynolds and Prandtl numbers increased, the overall

Nusselt number ceased to be a function of Peclet number alone

but varied with Re and Pr raised to the powers i and 31 , respec- 1 a

tively. The results approached the correlation: Uu = 0.6 Rem Pr'

with increasing Reynolds and Prandtl numbers. The present results

for the sphere at Reynolds numbers between 10 and 500 agree well

with the correlations of Frassling80 and Jenson et al84 .

17. The numerical solutions predicted values of the Nusselt number

for the sphere which agree well with the lower values of the

experimental measurements of Kramers82 and Rowe et a12. Experi-

mental data are usually affected by free convection and thus the

lower values of the Nusselt number available in the literature

are more likely to be correct.

**********

APPENDIX A 222

ORTHOGONAL CURVILINEAR COORDINATE SYSTEMS .

A.1. Curvilinear Coordinates

It can be shown89,97 that a point in three-dimensional

space can be located (with respect to some origin) by specifYing

its three rectangular Cartesian coordinates (xly,z), or by speci-

fying the position vector if. of the point. It is often more con..

venient to describe the position of the point by other sets of

coordinates such as spherical and spheroidal coordinates.

If the transformation from (xly,z) to a new set of coordi-

nates (x12x2'x3)

x = x(x 2x21x3)

- or

is made, then:

y = Y(x1 23(22x3)2

r = 7.(x x x ) 12 22 3

z = z(x11x x3)

Hence, the point P(x,y,z) can be located when the numerical

values of x19 XPs and .5 x _ are specified. These numbers.(x,l x22.x3 ) -

are regarded as the curvilinear coordinates of the Point P end

are illustrated in Fig. Ant

z

coordinate sun. surface

A.1

Curvilinear

coordinates

(x12x22x3

)

P el

coo - surf

x3- = cons ant y

--

-Curvilinear coordinates consist of three curved surfaces (co-

. crdinatesurfaces)Idlichareformedwhenx.is constant for 1=1,2,

ana 3 . The intersection of any two coordinate surfaces results in

a curve which is one of the coordinate curves x,.

223 A.2. Unit Vectors and Scale Factors

If (x1'x21x3) are the curvilinear coordinates of a point P

whose position vector is F, and (x1 + dx11x2 + dx2,x3 dx3) are

the curvilinear coordinates of an adjacent point Q whose position

vector is r + dr, then:

a; . dx/ +Z;11 dx2 °xi ox2 00c3

dx3 (A.3)

where a are the tangent voctoro to the coordinate curves x..

The magnitudes of these tangent vectors are known as the scale

factors, hi, for the three coordinate directions.

i.e. (A.k)

The differential arc length dli along the coordinate curve

x. is: dl. = h.dx. 3. 1 1 (A.5)

It is now possible to define the unit tangent vectors ei,e2,e3

in the x1'x2' and x3 directions as follows:

9 ter

( A.6 )

(A.7)

h. ex . a. a.

It follows from equations (A.3) and (A.6) that:

dr = h dx ; + h dx ; + h dx ;

1 1 1 , 2 2 2 3 3 3

The differential of arc length dl (the distance between the

adjacent points P and 0 is given by the magnitude of the elemental

vector di. dl and 6 are related as follows:

(d1)2 = 1612 = di7.d; (A.8)

d 6Y.k = fax -k i (Pc.

Hence, equations(A.12) and (A.13) give:

mation (A.1) : (A.13)

224 A.3. Calculation of Scale Factors for Orthogonal Curvilinear

Coordinates

When the system of curvilinear coordinates is such that the

three coordinate surfaces are mutually perpendicular at each point,

it termed an orthogonal curvilinear coordinate system. In this case

theunittangentvectorsei.tothecoordinatecurvesx.1.are also

mutually perpendicular at each point.

Hence, the scale product of two of these orthogonal unit

vectors gives: ;.1.;. = 1 for i = j

(A.9) = 0 otherwise

and the vector product of two of these unit vectors gives:

e. e = 0 for i = j 21,1 j and

; =-; ; = = -e3,, ; = IA 2 e3 2A 1' 2A 3 1 21 3A 1 e2 -e - 1/13

(A.10)

In such coordinate systems, it follows from eauations (A.7)2

(A.8), and (A.9) that: 3 (d1)2 =L hi

2 (dxi)2 (A.11)

1=1 In a rectangular Cartesian coordinate system, for which the

coordinates will be denoted by yk to distinguish them from the

generalcurvilinearcoordinatesx.1l the distance between two points

with coordinates yk and yk + dyk is dl, where

3 (di)2 . clYkdYk (A.12)

k=1

The differential terms dyk are from the original transfor-

3 V'( 6Y ax 6—b1 ax.) = L. &-c. z ix

ilbx • 3

k=1 1 3 = g.. dx.1dx.

13 j

i = 1,2,3

j = 1,2,3

(A.14)

225

gij eY, A

141) egxj

,Yk (A.15)

is called the Euclidean metric tensor since it relates gij

distance to the infinitesimal coordinate increments.

For an orthogonal coordinate system gij = 0 for i j 2 and

only the diagonal terms, gii , are non zero.

Hence, equation (A.14) becomes:

3 (d1)2

gii (d'i )2

(A.16)

It is clear,on comparing equation (A.16) with equation (A.11),

that: h.2 = 1 g11 (A.17)

where equation (A.15) expresses gii as:

g• • =

yk )2 (A.18) k=1

xi

It follows from equations (A.17) and (A.18) that:

where

h. 1 k=1 t

rc k )2 . (A.19)

The scale factors for any orthogonal coordinate system can be

calculated by equation (A.19) provided that yk(k=1,213) are the com-

ponents of a rectangular Cartesian coordinate system.

A.4. Area and Volume in Orthuonal Coordinate Systems

In an orthogonal curviliuoar coordinate system, the element

of area on the surface x1 = constant is given by:

dS = h h dx dx 2 3 2 3 and the element of volume is given by:

dV = h1h2h3dx1dx2dx3 (A.21)

The total surface area and volume of a body of revolution

may be obtained by integration of equations (A.20) and (A.21) over

the values upon which x1,x2, and x3 vary.

*****

(A.20)

226 APPENDIX B

VECTOR RELATIONSHIPS

B.1. Vector Algebra

A vector quantity can be represented as follows :

a = al.41 + a2e2 + a3e3 (B.1)

- 1'

- 2' where e e e3 are the unit vectors in the x1'x2' and x3

directions, and a1'a2' and a3 are the scalar components of the

vector a.

For orthogonal coordinate systems, the scalar and vector

products of two vectors a and E become, using equations (A.9)

and (A.10) :

Scalar product ast = a1b1 + a2b2 + a3b3 (B.2)

;1 e2 e3

a1 a2 a3

b1 b2 b3

Vector product

AE (B.3)

B.2. Vector Operators in Orthogonal Curvilinear Coordinates

It can be shown97198 that the vector operator 7

(pronounced 'nabla' or 'del') has the following form in the

orthogonal curvilinear coordinate

.

1 ;2 6 + 2_2 6

h1 (X1 h2 674 2 h3 &c3

system (xi x2 I x3 )

(B. )

where h. (1=1,213) are the scale factors as defined in

Appendix A ,

Equation (B.4) is used to derive the expressions for the

gradient, divergence and curl operators in orthogonal curvilinear

coordinates.

The gradient of a scalar quantity 4) is given by :

227

grad ;12 (1)(1)+ = _ h1 ox1 h2 6c2 h3 C. 1c3

(B.5)

If a is a vector quantity defined by equation (B.1), then

the divergence and curl of a are given by :

div a = 77.a = 1 11

,T(h2h3a1) 1-t-(h/h3a2) h1h2h3 1 () 2

(B.6)

(B.7 )

Equations (B.5) to (B.7) show clearly that div a is a

scalar quantity, but that grad4) and curl ; are both vector

quantities.

The divergence of grad qS (or V 2 16) is an important function and can be expressed using equations (B.5) and (B.6),

as follows :

76( V C76 ) 24) - 1 f ( h2h3 64c-k

h1h2h3 1 x1 h1 6x1

i h9 34 h_ei2 Lcp) )c2 h2 ex2i x3 h3 ex3

B.3. Vector Relationships

The following relationships between vectors and vector

operators are valid. They are used in Chapter 3 to express

operators in standard forms whose transformations to any

orthogonal curvilinear coordinates are immediate.

(B.8)

h1;1 h2;2

6 6 6xi 6x2 h1 a1 h2a2

Ox3 h3;3

(X3 curl a .7A a

h1h2h3

h3a3

}

228

1. 7(a.t) (a. 7 A + (z. 7 )a aA(7AE)

▪ EA(VAa) (B.9)

2. VA(ant) = a( p.t) t( 7.a) - (a. Q )E

+ (E. V );

3. 2 a = V ( 7.a) - V, (VA a)

IF. (a•V ) = a.

5. curl grad 4.)

6. div curl a

(B.10)

(B.11)

(B.12)

(B,13)

(B.14)

229 APPENDIX C

SPHERICAL AND OBLATE SPHEROIDAL COORDINATES

C.1. The Spherical Polar Coordinates (r09,4))

A sphere is formed by the rotation of a circle about its

diameter. In Fig. 0.1, the position of the point C on any circle

will be known when its polar coordinates (r49) are specified. This

system of coordinates describes a family of concentric circles and

radiating straight lines, which intersect orthogonally at all

_points. Thus, the coordinate system of the sphere can be defined

as the polar coordinate system rotated about its diameter.

The spherical polar coordinates (r1 0, ) are arranged as

shown in Fig. C.1, so that their relations to the rectangular

Cartesian coordinates (y1ly2,y3) are given by :

yi = r sin (9 004, y2 = r sine sin y3 = r cos (C.1)

where r 0 , 0 ‹.-,6?‹Tr, and 0<') <2 71

This arrangement is made so that, if a spherical body whose

surface corresponds to r = R is immersed in a fluid stream which

is flowing parallel to the axis of symmetry y3 with a uniform

speed U in the negative y3 direction, then the forward and backward

stagnation points of the sphere are at the points (R., 0 Jr/2 ) and (R Tr, 7r/2 ), respectively.

The scale factors for general orthogonal curvilinear

coordinates are defined in Appendix A. They can be calculated with

the aid of equation (A.19). The partial derivatives in equation

(A.19) are obtained from the relationships given by equation (C.1)

whichon substitution into equation (A,19), lead to the following

scale factors for the spherical polar coordinates :

hx. = 1 he = r , 0 r sin i9 (C.2)

The element of area on the surface r = R is given, using

equations (A.20) and (C.2), by :

230

dS = R2 sing dO 114) (C.3)

The total surface area of the sphere is obtained by

integration of equation (C.3) to give

S = 4 -n-R2 (C.4)

The element of volume, using equations (A.21) and (C.2),

is given by :

dV = r2 sine dr de d4 (C.5)0

Integration of equation (C.5) gives, for the tctal volume

of the sphere :

4 v 3 irR3 (c.6)

C.2. The Oblate Spheroidal Coordinates (5(74)

An oblate spheroid is formed by the rotation of an ellipse

about its minor axis. In Fig. C.2, the position of the point C on

any ellipse will be known when its elliptic coordinates (z,&) are

specified. The coordinate 9 is the angle y30C1 , where C' is a

point vertically above C and on the circle whose centre coincides

with 0 and whose diameter is equal to the major axis (2d) of the

ellipse. Such a circle is called the auxiliary circle of the

ellipse.

The elliptic system of coordinates represents a family of

confocal ellipses and confocal hyperbolas, which intersect ortho-

gonally at all points, Thus, the coordinate system of the oblate

spheroid can be defined as the elliptic coordinate system rotated

about its minor axis. The auxiliary circle is now represented by

the auxiliary sphere.

The oblate spheroidal coordinates (z, 6,0i) ) are arranged as shown in Fig. C.2, so that their relations to the rectangular

Cartesian coordinates (y1ly2,y3) are given by :

C, \

--30 Y2.

/

Y3

231

= a cos:a z sing cos 4 y2 = a cosh z sine sinCk (C.7)

y3 = a sinh z cos()

where 0 < -‹ Tr , 0 CP < 21T , and a is the distance between the focus and the centre of the oblate

spheroid.

Fig. C.1

Spherical Polar

Coordinates (r s O,

Fig. C,2

Oblate

Spheroidal

Coordinates

(z,02c)

\ /

\ /... . N . _ .

\ _ .... . ....- \

232 This arrangement, as in the case of the sphere, is made so

that, if an oblate spheroidal body whose surface corresponds to

z = zs is immersed in a fluid stream which is flowing parallel to

the axis of symmetry y3 with a uniform speed U in the negative y3

direction: then the forward and backward stagnation points of the

oblate spheroid are at the points (zs, 0 ,7r/2) and (zs, 7ror/2),

respectively.

The lengths of the semi-major axis d, the semi-minor axis b,

and the distance between the centre and the focus a, are as follows:

d = a cosh zs b = a sinh zs (c.8)

a = (d2 - b2) (0.9)

It is useful to define a quantity es describing the shape of

an oblate spheroid, as the ratio of the lengths of the minor to the

major axes. b i.e. e = • = tanh z

(0.10) d

It is important to note that this quantity should not be

confused with the eccentricity E0 of an oblate spheroid which is

defined as: a

(d2 2 b (1 e2)

7 (C.11)

a

Therefore, for the sphere e = 1 and E0 = 0

The scale factors for oblate spheroidal coordinates are, using

equations (A,19) and (0.7) :

hz = h = a (sinh2 z + cos 20)i h = a cosh z sine (0.12)

.0*

The element of area on the surface z = zs is given, using

equations (A.20) and (C.12), by:

dS = a2(sinh2zs + cos (-1W cosh zs sin6 d0 dc(

(C.13)

The total surface area of the oblate spheroid is obtained by

233

integration of equation (C.13) to give:

S = 2 -11" a2cosh z I s s

where Is = cosh zs inh2zs In cosh zs 1

cosh zs

(C.14)

The element of volume is given, using equations (A.21) and

(C.12), by: dV = a3(sinh2z + cos26)cosh z sin& dz dedCP (C,15)

Integration of equation (C.15) gives, for the total volume of

the oblate spheroid whose surface is z = zs :

4 4 v = 7r- a3sinh zs cosh

2zs = bd2 (C.16) 3 3

C.3. Transformation of the Coordinate Systems

The spherical polar coordinates and the oblate spheroidal

coordinates have been arranged so to take the advantage of the axi-

symmetrical nature of the flow about the sphere and the oblate

spheroid as shown in Figs. C.1 and C.2 . In these cases, the flow

is independent of the angle of rotation 40 . The coordinates in the meridian plane are then the only necessary coordinates required to

describe these flows. Thus, the spherical polar and the oblate 1.1

spheroidal coordinate systems are reduced to the polar (r1 0011) and

the elliptic (z1 6) coordinate systems respectively.

The elliptic coordinates (z,61) are shown plotted in the rec-

tangular plane Fig. C.4 0 The transformation

y3

iy2 = a sinh(z + 161) ,

a>0 (C.17)

leads, on equating imaginary and real parts, to the relations:

y2 = a cosh z sine

(c.18)

y3 = a sinh z cos C7 (C.19)

which give the physical plane shown in Fig. C.3

234

On eliminating g from equations (C.18) and (C.19), the following relationship is obtained:

2 2 Y2 Y3

a2cosh2z a2 sinh2 z = (C.20)

which describes, for different values of z, a con3:ocal family of

ellipses having their geometric centre at the origin.

When z is eliminated from equations (C.18) and (C.19), the

following relationship is obtained:

2 2 Y3

a2cos

2

Y2 = 1 (C.21) a2 sin2 0

which describes, for different values of 6.), a confocal family of hyperbolas.

On examining the relations between Figs. C.3 and Co4, the

following are observed: Consider the rectangle in Fig. C.4 which is

bounded by the lines 9= 0, 9=ir,z = 0, and z --4C>0. Then, for all possible values of e l sin& is positive; hence y2 from equation

(C.18) is always positive, and y3

from equation (C.19) varies from

a sinh z to -a sinh z, that is the half of the ellipse to the right

of the y3-axis of Fig. C.3 is included. Hence, the confocal ellipses

and the confocal hyperbolas of Fig. C.3 correspond to the vertical

lines z = constant and to the horizontal lines 0= constant (of Fig. C.4), respectively. The shaded areas of the two diagrams

correspond, and the line z = zs corresponds to the solid surface of

the oblate spheroid. Hence, the external region on the right hand

side of the physical plane, Fig. C.37 is mapped onto the rectangle

of Fig. C.4 which is bounded by the lines 0= 0, 0. 1,0 = zs, and z --40070.

The polar coordinates (r,8) are related to the Cartesian co-

ordinates (y2,y3) by the transformation:

Y3 iY2 = reie (C.22)

-*-4D00 •

235

z1 z2 z

Fig. C.3

Fig. C.4

Elliptic Coordinates ( 0)

Elliptic Coordinates :z 6))

in a Meridian Plane

Fig. C.5

Fig. c.6

Polar Coordinates (3,19)

Modified Polar Coordinates (z 6))

in a.Meridian Plane

z i i9 re = Re

z = In --- R

(C.27)

(0.28) which gives

236 which leads, on equating imaginary and real parts, to:

y2 = r sine

(C.23)

and y3

r cos (,;'

(0.24)

which give the physical plane shown in Fig. C.5 .

On eliminating Ofrom equations (0.23) and (0.24)1 the following

relationship results:

2 2 2 Y2 y3 = (C.25)

which describes, for different values of r, a family of concentric

circles.

When r is eliminated from equations (C.23) and (C.24)1 the

following relationship is obtained:

y2 = tan (9 Y3 (0.26)

which describes, for different values of e l a family of radiating

straight lines.

Equations (0.25) and (C.26) are the limiting cases of equations

(0.20) and (C.21) (when sinh z = cosh z), respectively.

It is desirable, for reasons to be given later, to transform

the polar coordinates to a system similar to the elliptic coordi-

nates by employing the transformation:

The new coordinates (z,0) are plotted in Fig. C.6 which

shows that the vertical lines z = constant and the horizontal lines

0 = constant correspond to the concentric circles and the radie.ing

straight lines of Fig. C.5, respectively. The shaded areas in the

two diagrams correspond, and the line z = 0 corresponds to the solid

surface of the sphere. As in the case of elliptic coordinates, the

external region on the right hand side of the physical plane,

237

Fig. C.5, is mapped onto the rectangle of Fig. C.6 which is bounded

by the lines g= 0, G. T17,z = 0, and z —4000. The reasons for this transformation are:

1. The two systems, spheres and oblate spheroids, can be

represented by similar coordinates.

2. The resultant rectangular coordinate systems have the

advantage of being able to use uniform intervals and

also to avoid irregularities at the curved surfaces.

3. The uniform intervals in the (z,0)-plane ensure that,

in the physical plane, they give smaller intervals near

the solid surface (where greater detail is required)

and increasingly larger intervals further outs

238 APPENDIX D

PRESSURE DISTRIBUTION AND DRAG COEFFICIENTS

D.1. Physical Components of the Stress Tensor in a Newtonian

Incompressible Fluid

For an isotropic Newtonian fluid there is a linear relaticnship

between stress and rate of strain as pointed out in Chapter 3 . For such a fluid, the following equation has been derived99100 which

gives the components of the symmetric Newtonian stress tensor as:

T -p + e.. ij 8ij 13 - V.17. ) (D.1)

where p --- Hydrostatic pressure.

Kronecker delta --- 6 = 1 for i=j 1J = 0 for i/j

eij --- Symmetric rate of strain tensor.

The stress tensor T.. has nine components in the xi-system of

coordinates. The diagonal components 11 , 22 , and 33 are called the normal stresses; the other components are called the tangential or

shear stresses.

For an incompressible fluid, 77.77. = 0 , then equation (D.1)

becomes:

ij I e1. . 3

(D.2)

The symmetric rate of strain tensor can be expressed in terms

of any orthogonal curvilinear coordinate system xi as follows:

(a) For i=j : e.. = 2( v h 6. m 1

m h.1 h 6xm (D.3)

where 1=112,3 and m refers to the two possible values of i other

than that in question. i.e. when 1=1, m=2,3 and so on.

h. 6 v. h4 6 (b) For i/j : e.. = — 2-. ( -.2.. ) + - ( Ii. ) JJ.

13 hj 6.j hi h.1 ox. h. 1 3

(D.4)

h.1 are the scale factors for the orthogonal curvilinear coordinate

239

system xi as defined in Appendix A.

From equations (D.3) and (D.4) and the results of Appendix C,

thesymetricrateofstraintensore..ij can be evaluated for the

spherical polar and the oblate spheroidal coordinate systems, which

on substitution of their values into equation (D.2)2 the following stress tensors are obtained:

(1) Stress Tensor in Spherical Polar Coordinates

T

= rr

1 a.,..:11X .1. Vr )

r 6 T 1

0 n

efvfer v cot ty, Jur= -p.+ 2 4 ( + r + Ir...,

r sine ?)4) r r; r

1 eiv r 6 ve

re = Ts, r = pt,( —

r (0 r

1 Nvr ev.0 -r = -r , .. IA. ---- + r.er -Crr r sin 9 6 ck6r

evo 6i7,0- T T 1

6

-p + 2 (D.5a)

(D.5b)

(D.5c)

(D.5d)

(D.5e)

cot (D.5f)

r

r

9.0-= 21) = 4 r sine (4)

(2) Stress Tensor in Oblate Spheroidal Coordinates (zI P

Denote:

s = sinh z c = cosh z , and m = (sinh2 z + cos (D.6)

Then: r 1 34.v t) z sinC7t cos

7. z = -p 4. 2 II( -- - v am 6. 8 am3 z

1 .Nt k.icrivi Sc

+ -ree -

- -p + 2 4 ( — --- vz am3 ---- ) (D.7b)

am .? 9

1 e v.er s cot 1" = -p + 2 11- ( 0 elm

)(:47c) .0` + vz _ + v —

ac sin e eCk amc

(D.7a)

240 sin Q cos e

...." + Vz am3 am 6z T, = ze ez

X uvz am

11( e

Sc

v am3 (D.7d)

lr = T = zkr z

eie7-= rere =

1 via z ov

ac sine 640 am 6z

ILL ( 1 _

ac sin() 64 am 66

• — amc

cote • r am

(D.7e)

) (D.7f)

D.2. The Equations of Viscous Flow and Heat Transfer in Spherical

Polar and in Oblate Spheroidal Coordinates

The equations of viscous flow and heat transfer have been

derived in general vector forms in Chapter 3. For steady-state

incomressible flows, these equations (the contiuity equation, the

Navier-Stokes equations, and the energy equation) are given by

equations (3.24) to (3.26) as:

77."4 = 0 (D.8)

V( T:17) C7A( 17) P - I/7A( VA (D.9)

rt'r.vT = a V 2 T (D.10)

The above equations are in standard forms whose transformation

to any orthogonal curviliear coordinates can be obtained with the use

of equations (B.1) to (B.8) of Appendix B, provided that the scale

factors for the coordinate system are known.

The spherical polar and the oblate spheroidal coordinates are

defined in Appendix C, and their scale factors are given by equations

(C.2) and (C.12) respectively. Hence, equations (D.8) to (D.10)

become, in terms of these coordinate systems as follows:

(1) Spherical Polar Coordinates (r, 014)) 241

The continuity equation (D.8) becomes:

, vrr2, --- - - ----tv sin ----- 14, ------- = 0 ✓ (or r sin 0 60 9 ) + r sin 671 6 cp (D.11)

The components of the Navier-Stokes equations (D.9) in r; 9, and 0, directions become:

2 2 ✓ 2") k..) )vvr, .1. ....e ).17. v.i2r eivr v + v ' 6P

I" S r I. G Pr =

Or r 69 r sine 600 r TY ar

2v 2 Ava. _ 2vecot 0 2 e vly . + /, 072 vr - 2 2 - )

r r 60 r2 r2 sin ('' el 4), (D,12)

2 &re ,s0 >,,,,, v .?A) v v v cote ----- - r 0 ,er 1 6P vr or r ill /I

r 4- . o r sine 60, r r Pr 6 9 e 2 2 X

1/4.1v r v 2cos & 6v.0,

+ V(v

v + --2. - -

r u? 2 . 2'1g 2 2 r sin 6) 64, i

r sin (D.13)

✓ Iv.e.'1. .av

1.0 lo vim, _.....2- v v v

+ i-:SL + ve JR- v e =_cot 1 6p

r 6r r 60 r sine 64) r r Pr sine 64)

v43( 2 6v 2cos 0 v‘,- ÷ v (72 v

a -

--lr - r2 sin 0,7

. 2/1 + r

r2 sine 60 r2 sin2e 0 (D.14)

where 7 2 7 .._ttr2 =

r2 sine

6

6 ( s in 777 , + LD e , r i 2 s .n20 ms 2

1 2 1 X

(D.15)

The energy equation (D.10) becomes:

v eT vim 612 vo 6T

r 6r r ee r sine 6) (D.16)

(2) Oblate Spheroidal Coordinates (z, O,Ck)

In these equations 's', fct, and 'm' are as given in equation

(D.6). The continuity equation (D.8) becomes:

1 6 , m s + (')a. N --r .3:-(vz mc) -----kv sin 0 4, , = 0

ac sin am c am2 sine 69 e (D.17)

2sin9cos U evz 2vzs sin Ocos 8 +

a2m4z a2m4c 6

2cose elfxr

a2mc sin29 6cp j (D.19)

2 2 - a. m C X

Z v a2m2 sine 68

1 2--(sine

242

The components of the Navier-Stokes equations (D.9) in zi LD C7 2 and 4, directions become:

2 17. +

vz \vz v Xvz v evz vzvesinecose v sc 4ir s ' ---- 4. . _

am &4 am ze ac sin® e c am3 am3 amc

1 P1 2 (c4 + s4 -. cos20) 2sc vc.) -.. =

_ Pam 6z I/

7 vz - vz

a2m4c2 - -a-2-7 1

2v sc cot 2sine cose wG 2s a vb. 2 \ 2 2 L.] ,k a m a c sin(7

(D.18)

21f am

vz _Z vim +

Nre v4., ce &ifs vzve se

..., +

+ am z am 69 ac sin() a0 am3

+ vz2 sine cose

_ MP •••......

1 ....... 6P +

am3 Pam 69 c2 - 2sin cos2e )

v + (4 a2m3 sin

vz e.v.e.r., vo. '..Jer.,. VP 6v.er+ v JRrs

A- Nrisv. c ot 6

am ez am & ac sine 60 amc am

- f)

t

1

ac 61210 6 4C P

+ 2

1, 7

7'0' -

v -1-

2s 6vz a2c2 sin-

a60

e ,-, a2mo2 sine 6

2cos 9 ve 4. (D.20) 'a-, mc sin 9 64)

v cote

am

+ 2sc \ vz

ID a m

where now

62

a2c2 sin26; 6c 2

The energy equation (D.10) becomes:

(D.21)

vz 6T vo 6T via. 6T arommor •••••••

am 6z am 66 ac sin e (4) V 2 T (D.22)

7: 2ve sc cote

a2m4

243

If the flow is symmetrical about the axis F = 0 and there is

no swirl, then v and all derivatives with respect t -er

equations (D.11) to (D.22) reduce to simpler forms which are applica-

ble to all axisymmetrical flow problems.

D.3. Surface Pressure Distribution

The pressure distribution at the surface of the particle can be

determined from the Navier-Stokes equations when the velocity distrib-

ution round the particle is known. For axisymmetric flows without swirl),

there are only two components of the Navier-Stokes equations since 777,r

and all derivatives with respect to 4) vanish, as pointed out previously.

The two components of velocity are related to the single component of

vorticity of the fluid, in the ct -direction. Therefore, a rela-

tionship can be obtained for the pressure distribution in terms of the

vorticity distribution. This relationship is derived below for the case

of fluid flow round an oblate spheroid, as a general case.

The components of the Navier-Stokes equations in z and three--

tions, equations (D.18) and (D.19), are:

vanish. Hence,

V XVz Z Q Ve cvz e Vzvesinecose v sc 2

" -_---- am 6z am .r) 6 am3 am3

(c4 + 84 - cos29 ) 2sc Xv + 1, i7 2 V .- V Z z

e a2 M4 2

%..)

C - -..7. ae 2sin@cos& .Xv e

a2m4 ez

/1 vz &re ye 6v.. vz ve sc vz2 sin cos 1

am 6z + am ee am3 am3 T Pam

f72 v a

2sc

m

,;; v e (c2 2sin29 cos20 )

c) a2m3 sin28

2sin9cose z 2vz s sinecose

a2m a2m4c

(D.23)

(D.24)

v.z =

z + tang

1

613

where: 2 1 ?)

2-2- 7-(c a m c oz 4- 2 2 m Sint-Jo

244

)(D.25)

The continuity equation (D.17) is:

e

MC) + ( V m sine) 0 (D .26 ) am2 c o \ z z 2 LD f) e ain s int - 0 (-7

The vorticity as given by equation (3.61) is:

r= 1 2

ez (vim)

am (D.27)

The pressure at the front stagnation point can be obtained in

the following way:

Flow-

,.../

-N..\ D \

r .9

I Fig. D.1 \L,

1 j Flow Past an Oblate k

\ Spheroid

Along the axis of symmetry 9= 0; IJ/ and all their deri-

vatives with respect to z are zero. Hence, equation (D.23) reduces to:

1 + Vz

6vz Li2v

Tz- oz ••••••••

ac T:r z Oz 62 2vz s2

2s 6vn 2ve s cot '°- ) (D.28)

where s and c are as given in equation (D.6).

On combination of equation (D.28) with equations (D.26) and

(D.27), the following relation is obtained:

Integration of this equation along the axis from A to B gives:

lim C 9 0 tan( 7 e8

to give:

245

1 6r pE - pA ) + -2-(vz

2 )B i(vz

2 )A = V ( 731 + --- /-1

JA 0 (7 tan (7,1 )dz (D.30)

Define the dimensionless pressure coefficient K as follows:

K 2 total - Po

ip

and hence from Chapter 3 (section 3.1) since ptot al = P Po then

K = p j-PU2 (D.31b)

Also, by the introduction of the dimensionless velocity u''

where = vz/U then equation (D.30) becomes, in dimensionless

terms:

KA + u*2 - u*A

2 = -4

•••••••••••••••••••

Re + )dz (D.32)

tan

If the point A is well upstream of the oblate spheroid and the

point B is on the surface at the front stagnation point, then

u-B = 02 A u* = 1 KA = 0 and KB = Ko

So that equation (D.32) becomes:

4 K = 1 o Re 6

dz (D.33) tan&

c>ca Along the axis: e= 0 =0 , and tan& = 0 also

therefore, it is necessary to use the limiting rule of L Rospita192

(D.34)

It follows by the use of this result in equation (D.33) together

with a change in the limits of integration that: or)

8 4g* Ko = 1 + dz (D.35)

Re jzs

66

246

The corresponding equation for the sphere can be derived

similarly, to give:

Ko = 1 ÷ 8

Re(('4 0 •

dz (D.36)

The integrands in equations (D.35) and (D.36) are evaluated

at . The pressure distribution at the surface of the oblate spheroid

can be obtained from equation (D.24) in the following way:

On the surface IP vz v and all their derivatives with respect

vz to eare zero. Also, from the continuity equation, O. Hence, ez

equation (D.24) becomes:

‘2v 6ve )

TS7 c aZ P ani

By the use of the vorticity equation (D.27), equation (D.37)

becomes:

1 Y ( tanh zs ) (D.38) P 66 - 6z

The shape of the oblate spheroid is defined by e which is the

ratio of the minor to major axes of the oblate spheroid (i.e. e =

tanh zs) so that integration of equation (D.38) round the surface

from C to D gives:

D ..,

( - D - C -;-( ) P z iiii,

C In terms of dimensionless functions, equation (D.39) becomes:

4 D N., y* '4D = KC 4. f( --‘°'

I- e t* ) dEll (D.'!.0) Re Oz elfc

If the point C is considered to be at the front stagnation point

and the point D to be on the surface at a reference angle 6 from the axis of symmetry, then K0 = Ko and KD = Ke . Thus, equation (D.40) can

DT_ CD p u2 A 2

(D.43)

be rewritten as:

247

K = K0 + 4

Re ) de (D.41)

where Ko is given by equation (D.35).

The corresponding relationship for the sphere can be obtained

* 4

Re airs & el 0 K = K + ( * (D.42)

0 where K is given by equation (D.36).

Equation (D.42) is clearly the limiting case of equation (D.41)

as for the sphere e = 1. Equations (D.41) and (D.42) give the

variation of pressure round the surface from the front stagnation

point. The relationships for the sphere, equations (D.36) and (D.42),

are the same as those derived originally by Jenson15

D.4. Drag Forces and Drag Coefficients

The drag force on an immersed body is the resultant of the

pressure and viscous forces exerted by the fluid on the surface of

the body. It is convenient to express the drag force in terms of a

dimensionless coefficient CD defined 8699'101. as follows:

similarly, to give:

where: Dm .... Drag force.

Kinetic head.

A Characteristic cross-sectional area of the body

(facing the flow).

CD = CD(Re) Drag coefficient.

The total drag force, DT , can be expressed as the sum of the

skin-frictional drag force, DF , and the pressure(form) drag force, D pQ

i.e. DT = DF + DP (D.44)

248

Define CDF as the skin-friction drag coefficient, CDP as the

pressure (form) drag coefficient, and CDT as the total drag coeffi-

cient, then,from equations (D.43) and (D.44), it follows that:

CDT = CDF CDP (D.45)

In the evaluation of the frictional drag force, it is necessary

to sum up the tangential or shear forces at all points on the surface

of the body. Similarly, the pressure (form ) drag force is obtained

from the summation of the pressure forces at all points on the surface

of the body.

On the particle surface, vz (or for the sphere), ve and all

their derivatives with respect to 9 are zero. Then on the surface of the sphere, from equations (D.5), T is the only non-zero re,

component of the viscous stress tensor, given by:

re vG

45. (from equation (3.49)) (D.46)

Similarly, on the surface of the oblate spheroid, the only

non-zero component of the viscous stress tensor, from equations (D.7),

is given by:

1• ze a( sinh2zs cos29) 6z

6v149

= s (from equation (3.61)) (D.47)

The average normal pressure at a point on the surface is p in

both cases as shown in Figs. D.2 and D.3 .

The horizontal components of the forces alone contribute to

the drag force while the vertical components contribute to the lift

force (zero in the present case).

Thus, for the sphere:

DF = 7;1) cos,e dS f

(1) .48 )

S

249

and similarly, for the oblate spheroid:

DF Tz0 cos/3 dS (D.49) ff Also, for the two cases:

sib

DP = p sin dS (D.50)

S

where ID is the angle between the tangential plane and the direction

of flow far from the body as shown in Figs. D.2 and D.3 and dS is

the element of surface area given by equations (C.3) and (C,13) for

the sphere and for the oblate spheroid, respectively.

P, T P\

re

/IT e

k

\ k

Fig. D.2

Fig. D.3

Pressure and Viscous Stress on the Pressure and Viscous Stress

Surface of a Sphere on the Surface of an Oblate

For the oblate spheroid: Spheroid

cot e = tan -----

and for the sphere (e = 1):

7T

2

(D.51)

(D.52)

The appropriate combination of equations (D.46) to (D.52) gives,

on integration over

the drag forces:

For the sphere:

from 0 to 27r, the following expressions for

( sin2G D.53)

and for the oblate spheroid:

8e CDF Re

sing 6;) de

Ir

0

(D.59)

7r 250

DP 7112 f p sin 49

(D.54)

0

and for the oblate spheroid: Ir

DF = 2 ra2 sinh zs cosh zs sin 0 de (D.55)

7T

DP 17" a2 cosh2zs p sin 29 dO

(D.56)

0

The drag coefficient is defined by equation (D.43) so that when

all functions are made dimensionless the following equations result:

(Note that A =1TR2 for the sphere and 7r a2 cosh2 zs for the spheroid)

For the sphere:

8 CDF = -- Re *I. *

L sin26 Ds

0

(D.57)

7r

CDP = fK sin 2& d (D.58)

0

Tr

CDP K

8 o sin 26 de

(D.60)

0

When e is unity, the equations for the oblate spheroid reduce

to those of the sphere as should be expected.

hT(x2)

T )x1=(x1)s

Ts o h 6x.1 T T

1 1T 1•••••••••••••••• E .3 )

251 APPENDIX E

NUSSELT NUMBER DISTRIBUTION AND MOLECULAR CONDUCTION

E.1. Local and Overall Nusselt Numbers

In convective heat transfer, the quantity of most practical

importance is the rate at which heat transfer takes place from the

surface of the body to the surrounding fluid.

In accordance with Pourier's law of heat conduction, the heat

flux q1 normal to the surface has the value:

1 q1 = -kT ( )x

(E.1) h1 1 1- -1 ) s

1 6T where kT is the thermal conductivity of the fluid and h1 f?)x1 x1=(x1)s

is the local temperature gradient at the surface in the direction of

the normal to the surface.

The local heat flux may also be expressed in terms of the local

heat transfer coefficient hT(x2) by the equation:

q1 = hT(x2) ( Ts - To )

Equating equations (E.1) and (E.2) gives:

(E . )

Thus for the sphere, equation (E.3) becomes:

hT( 0) 1 6T = - ( ....a .4 )

1T s o T - T 6r r=R --

(E.4)

and for the oblate spheroid, equation (E.3) becomes:

hT() 1 T -1(.---) (L.5) 2 yy az a z...-z kT (Ts - To)a( sink zs + cos

Define the local Nusselt number Nue. in terms of the major

diameter of the particle, D as: C , hT(0) DC

kT where Dc = 2R for the sphere, and Dc = 2d = 2a cosh zs for the oblate

spheroid.

Nuo (E.6)

252

Then, in dimensionless terms, equations (E.4) and (E.5) give:

For the sphere:

where r* = ez.

e T* Nu = -2( )

Or*

6T* =-2( ----

z )z=0 (E.7)

For the oblate spheroid: Nu -2cosh zs 6T*

e ----) (E.8)

( sinh2zs + cos26)2 ( z z=zs

When the temperature gradients in equations (E.7) and (E.8)

are evaluated at various values of 9 from the numerical solution of

the energy equation local values of the Nusselt number round the

surfaces of the sphere and the oblate spheroid can be obtained direct13;

From a knowledge of the Nusselt number as a function of 0 at various

values of shape, Reynolds number, and Prandtl number the local rates

of heat transfer can be predicted.

It is also important to determine the overall rates of heat

transfer from the particle surface to the fluid. These can be obtai-

ned in terms of the overall heat transfer coefficient hT or in terms

of the overall Nusselt number, Nu , as follows:

The total heat flow rate from the surface, QT , is evaluated by

summation of the products of heat fluxes with area at all points on

the surface of the body. If dS represents an element of surface area

on the surface x1 = (x1)s, and S represents the total surface area of

the body, then:

QT q1 dS (E.9)

Also, QT may be expressed in terms of hl by:

QT = hTS ( Ts - To ) (E.10)

Equating equations (E.9), with q1 substituted by equation

(E.1), and (E.10) gives:

hT -1 1 6T --‘ )x _tx dS(E.11)

kT S( Ts o hi 16x1 1-' 1's S

Expressions for dS and S are given in Appendix C in which

equations (C.3) and (C.4) apply to the sphere and equations (C.13)

and (0.14) to the oblate spheroid. Hence, for the sphere:

253

hT

kT

-1 7r

T Rsin9 er

dO (E.12) 2(Ts - To)

and for the oblate spheroid: 7r

hT -1 eT ( -;:--- ) sing dO (E.13)

kT

_ aI s (Ts - To)

bz z=z s 0

To make all functions dimensionless, introduce the overall

Nusselt number Nu: 11_ T1) C

Nu - (E.14) kT

Then for the sphere, when r*=ez, equation (E.12)becomes: 7T T.

Nu = - f ( e )z.0 sine de .

0

where Nu = 2Rh / T Its,

and for the oblate spheroid, equation (E.13) becomes: 7T

(E.15)

(E.16)

Nu - 2cosh zs

0 ( ()T*

ez z=z sine dO (E.17) S Is

where Nu = 2a cosh zs hT / kT (E.18)

Combine equations (E.7) and (E.15) to obtain the relationship

between the overall and the local Nusselt numbers for the sphere as: 7r

Nu = I 2 fNub sin 0 d 6 0

(E.19)

Similarly, for the oblate spheroid, equations (E.8) and (E.17)

give the following relationship: IT

2 Nue ( sinh zs co.2eAsino de (E.20)

1 Nu

Is

254 E.2. Molecular Conduction

It is well known that the rate of heat transfer from a single

sphere to an infinite stagnant medium corresponds to a Nusselt number

of two. This can be shown as follows:

Consider a single sphere contained concentrically within a

spherical shell of a stationary isotropic medium of constant thermal

conductivity. It will be supposed that the surface of the sphere is

maintained at a uniform temperature, Ts , and the spherical shell is

also maintained at a further uniform temperature, To that is lower

than that of the surface of the sphere. Under steady-state conditions

no accumulation of heat occurs within the spherical shell so that the

the rates of transfer from the sphere and through the shell are

identical.

In accordance with Fourier's law of heat conduction, the heat

flux passing through the shell is given by:

= -kT V T (E.21 )

From equation (E.21) the total rate of heat transfer through

the shell may be written as:

4111 Timn dS if kT VT.E dS (E.22)

where E. is the -positive unit vector normal to the surface S of

the shell. The unit vector E. is tangent to the r-curve and is equal

in magnitude and direction at every point on the surface to the unit On

r' me

vector er. The scalar product of the vector and the vectors e . .••

foi

and egare, therefore:

er.n = 1 and ee.n = = 0 (E.23)

The scalar product of the gradient of the temperatre and the

unit normal vector becomes: T 77T.R = (E.24)

6r and dS, from equation (C.3), is:

dS = r2 sine ded40 (E.25)

255

gives:

Substitution of equations (E.24) and (E.25) into equation (E.22)

2 7r '7r

qT - kT r2

eT sin& ded4 (E.26)

ar

cf) 0 =0

Equation (E.26) can be integrated and rearranged to give:

QT 1

(E.27)

6r r2

Integration of equation (E.27) with respect to r, setting

T=Ts at r=R and T=To at r=roR, the result becomes:

T 1

Ts - To (1 ) (E.28) Tf kTR ro

where ro is the ratio of the shell to sphere radii.

A heat transfer coefficient hT may be defined in terms of the

temperature difference between the surface of the sphere and that of

the shell as follows:

QT = hT( Ts - To ) S (E.29)

where S is the surface area of the sphere given by equation (C.4).

On substitution of QT from equation (E.28) into equation (E.29)

followed by rearrangement into a form which involves the Nusselt

number as defined by equation (E.16) the following result is obtained:

2 Nuo = 1 1 - ---

ro

(E.30)

Equation (E.30) clearly shows that as the radius of the shell

is increased in comparison with that of the sphere the Nusselt number

decreases progressively and eventually approaches its minimum limiting

value of two when r0 approaches infinity.

The corresponding equation for the oblate spheroid can be

derived in a similar way by consideration of an outer confocal oblate

spheroidal shell (z:›Pzs). In this case the unit vector xi is tangent

to the z-curve and is equal in magnitude and direction at every point

on the surface to the unit vector ;z .

256

and

Hence, ez.ii = 1 and ;0 .E = e .n = 0 (E.31)

1 bT VT.13 =

(E.32) a(sinh2 z + cos26)2 6z

Also dS, from equation (0.13), becomes:

dS = a2( sinh2z + cos20 cosh z sine de dcf) (E.33)

Substitution of equations (E.32) and (E.33) into equation

IT 2 71"

9T T cosh z sine d d4

(J, 9=0

On integration and rearrangement, equation (E.34) gives:

(E.22) gives:

(E.34)

6T QT 1

(E.35) (z 'IrakT cosh z

Integration of equation (E.35) with respect to z, setting

T=Ts at z=z and T=To at z=zo (or cosh zo=rocosh zs), the result

becomes:

QT T - T 8 0 fir akT

( tan-lsinh zo tan-isinh zs ) (E.36)

Again define the heat transfer coefficient, hT as:

QT = hT (Ts - To ) S

(E.37)

where S is the surface area of the oblate spheroid given by

equation (0.14).

On substitution of QT from equation (E.36) into equation (E.37)

and rearrangement in terms of the Nusselt number the following result

is obtained:

Nuo = 2a cosh zs hT / kT 8

z -1 -1

2 cosh +1 (tan sinh zo - tan sinh zs)(2cosh zs

+ sink zslncosh z-1 s

s

(E..38)

257

In terms of the semi-major and semi-minor axes d and b, which

are defined by equations (C.8) and (C.9), respectively, equ'tion

(E.38) becomes:

r2d2 b2

tan-1 d2o- b2 ( • -1 y-0- tan-1( a2 b2

{ b2 d + (d2 b2)2

2.d + (d2 - b2)y

In d (d2 - b2)

(E.39)

where ro is the ratio of the shell to the oblate spheroid

major diameters.

Equation (E.39) shows that as ro increases the Nusselt number

decreases progressively and eventually approaches its minimum value

when ro approaches infinity. In this limiting case equation (E.39)

becomes:

8(d2 2 4. - b 2

bum=

r-

- tan ( b2

2d + in d+(d2- )2*-1

2- b2

)2 (d

2 b2

(E.40)

In terms of the shape factor of the oblate spheroid. e

defined by equation (C.10), equations (E.39) and (E.40) become:

8(1 - e2)2. Nuo =

r e2 e2 1 (1 -e2)-2-

Gall'(°2

an - 2 -1)2-t-'()2[21----f in -27J

2 ' - 1-e 1-e2

(1-e )"7 1 -• (1-e )2

(E.41)

2 7-1- • 8(1 e )2

-1 Nu = 8(d2 - b2)2

Nu 00

iThr tan 1( Lr-- 2

As e tends to unity equation (E.42) reduces

1 (1 -

1 - (1 - e2)l

(E.42)

to Nuay = 2 for

r 12

, In (1 - e2)Y 1 e

the sphere.

258 APPENDIX X'

CONVERGENCE AND STABILITY CRITERIA

F.1. Introduction

This Appendix is concerned with the conditions that should be

satisfied if the solution of the finite-difference equations is to be

a reasonably accurate approximation to the solution of the correspon-

ding partial differential equation. These conditions are associated

with the interrelated concepts of convergence and stability of the

approximate finite-difference equations.

Let W represent the exact solution of the following elliptic

second-order partial differential equation with independent variables

ew (?,w 614 6w 6z2

z 2

a' — + al + a3 0 (F.1) 60

Let w represent the exact solution of the finite-difference

equations used to approximate the partial differential equation in

which the finite-difference equation of equation (F.1) may be written

in the following form:

. = blw. b + blw. + . + (F.2) 113 1 1+10 Awi-1,j 3 1,j+1 -t 1 3 -1

where the coefficients b' to 7315 are functions of zi e l and the mesh

sizes h and k.

By the Gauss-Seidel iterative method (successive displacements

by points) the improved values are used immediately to compute the

improvements for the next mesh point value. To systematize such a

computation a method of 'ordering' the points must be established so

that the point values are calculated according to the ordering of

these points. If the points are ordered so that the iterative method

scans the mesh points from left to right along successive rows as

shown in Fig. 4.1, then the Gauss-Seidel s iterative form of equation

(F.2) on the (n)th iteration becomes20121

z and

259 (n) (n-1) (n) w. = blw. b'w. b'w(n-1) b'w(n) b' lti 1 1-1-10 j-1 5 (F.3)

If w represents the exact solution of the finite-difference

equations (F.2)„ then the finite-difference equations are said to be

convergent when w tends to W as the mesh size (h and k) tends to zero.

The difference W-w is called the discretization error (sometimes called

truncation error). The magnitude of this error at each mesh point

depends on the finite-sizes of the mesh lengths h and k, and on the

number of finite differences in the truncated series used to approxi-

mate the derivatives.

The equations that are actually solved are, of course, the finite

difference equations and if it were possible to carry out all calcula-

tions to a sufficiently large or infinite number of decimal places we

would obtain their exact solution w. In practice, however, each cal-

culation is carried out to a finite number of decimal places, a proce-

dure that introduces a round-off error every time it is used, and

the solution actually computed is not w but N . N will be called

the numerical solution.

Generally, a solution of the finite-difference equations is

stable when the cumulative effect of all the rounding-off errors is

negligible. More specifically, if errors e,, e112 1....eili , are 1

introduced at the mesh points (1,1) , (1,2) ,....(i,j), respectively,

and le 1,11 2 le1,21 lei,i t , are each less than t the maximum

absolute error, then21 the solution of the finite-difference equations

is stable when the maximum value of (w-N) tends to zero as E tends

to zero and does not increase exponentially with the number of columns

or rows of calculation, i.e. with i or j.

The total error by which the numerical solution of the finite-

difference equations differs from the exact solution of the partial

differential equation is given by:

Total error = (W - N) = (W w) (w N)

discretization round-off

error error

260

F.2. .1.122..„3.1 tical Treatmen of .22ixaE22292.

F.2.1. Convergence of the Solution of the Energy Equation

The energy equation for the case of heat transfer from a single

sphere is given by equation (4.9) which can be rearranged into the

form of equation (F.1) as follows:

eW PrRe etif) 11 + + + (1 + ez2 602 2ez sin 6 6. ( cote

PrRe 6 6"

2ez sine 6z ) 66 = 0

(F.4)

where W denotes T*.

The finite-difference form of equation (F.4) can be written in

the form of equation (F.2) as:

w. . = a" + b" + c"w. . + d"w. . (F.5) 10 wi+1,j wi-1,j 1,3+1 10-1

61kr 2 + h PrRe all ,...,., + (F.6) ---7ff

2Lh 4Lh ez sinO 69

* -2- .. z 2 - h PrRe a t P

b" = (r.7) e 2Lh 4Lh sin 9 6

2 + k cot9 PrRe 6%P* ect _ _ (F.8) 2Lk2 4Lk ez sin& 6z

:' d" = 2 - k cote PrRe ;P (F.9)

2Lic2 - + 4Lk ez sine 6z

L = 2/h2 + 2/k2 (p.10)

If W and w denote the exact solutions of equations (F.4) and

(F.5) respectively, then

w. . = W. . e. . (F.11) 113 10

where e. . is the discretization error at the mesh point (i,j). 10

Substitution of equation (F.11) into equation (F.5) leads to:

e. . = a"e . + b" 19J i+1,3 + cue.+ d" + W. . 041 eilj-1 19 3

where:

(F.12) a"W1. . b"Wi-1 0+1, 3 c"W.d"W. . ,i+1 10-1

By Taylor's theorem 92:

elg.. k) , 3k) \62

62 (z , ( (1)4k ) 02 (F.17)

1 2 b" ;3z2

k2 c" c

- 41c2du

261 . )1W . h2 ell(z.4,h,a)

li 1 = W(z.+h = j) W. . + h ...:.--113.1. a. j (F.13) +1,j 1 / 10 ez 20 z2

N114. . h2 2W(oi-Ch, = W(z.-h = 0j) W- - - h ,::- ,37-'3.4- (F.14) Wi-11j 1 7

1 Jez2

?,)I14 „ 2.,.. e2W(z sOi+CIS.zk) = W(z. , 6+k) = W. . + k d (F.15) Ili,

3+1 1 10 --;.1.-...t 1 J + n.

6,9 21 662

),..r k2 eg(z.a. ,aj 414k) ( ....._.1 + . Wz - 1, 0 j -10 = W. k . - C)b'. F01 6, Wi,j-1 i / ilj

) 2t 6 2

--- 1 - -- 1 41 1 where 0‹ , 0 --cp - -3 ° < 1 Substitution of equations (F.13) to (F.16) into equation (F012)

gives:

e. 10

= aue. i+11j + b" c e. . ei-11j 1 ,0+1 + d"e. h(a"-b") 1, j-i

.511- • k(c"-d") 1-13 ih2a" 62„(zi+4h, (9;)

which is the finite-difference equation for e, -0 .

Let E denote the modulus of the maximum error in the field.

d"

i.e. maximum e. 1 = E 1,j 1

When the coefficients aul b",c", and

(F.18)

are positive or zero, then

lei, j I a" e. + b" 1+11j e. -1 j I + c" lei,i+ + d jei2J-1 1

+ E (F.19)

61:I* ew. h(au_b u ) ----11-j + k(c"-d") ---1-'3

0 e 0 \, W(z.+CID ht 6.) - ed(z.-(P211' 6,1j)

+ 1h2a" ------ 1 3 + ih b" --s?-- Oz

2 N.2W(z. 1 + ---fk c- ----- 1 2 " 6 tgz i t .> k + -n-k2 d r1 -- I

7 1 0

X zi 0

2 `-- \. (1 02 k..)

where E

1

It follows from equations (10.18) and (F.19) that:

sine 60 z

67 )

eV 6 w

+ (1 + PrRe 6 stp )

2e z

PrRe

2ez (F.23)

662 + (cote

4ez sin()

PrRe tfr

64)* 6tp * and

z

k

262

I eilj (a" + b" + c" + d" )E r + E tt

(F.21)

Since a" + b" + c" + d" = 1 , then

,j ei . I < E! ti

+ E (F.22)

As h and k tend to zero, the limiting value of E becomes:

which is zero since W is an exact solution of equation (F.4).

Hence, from equation (F.22), the following results:

< W. 13 .

1 w. 3 .1

1 E

1 (F.24)

Also E tends to zero as h and k tend to zero since it depends

on the mesh lengths h and k. Hence, it follows, from equation (F.24),

that w converges to W as h and k tend to zero when a"l bflic", and d"

are positive or zero. These conditions imply that, for small h and k,

equations (F.6) to (F.10) give:

4ez sine

PrRe r* ---- Z6

assume the fluid flow

to be undisturbed and parallel.

To obtain the order of

(F.25)

i.e. 1. 2z = 2e sin2e (F.26)

Then the magnitudes of h and k become:

4 h (or k)

(F.27) PrRe ez

••

Thus if values of h and k are chosen within this limiting value,

convergence of the solution of the finite-difference equations will

be obtained.

263 F.2.2. Thom and Apelt's Method

It is difficult to apply the analysis of the previous section

to non-linear problems such as the Navier-Stokes equations. However,

a convergence criterion can be derived following basically the same

pattern as that used by Thom and Apelt102 . The Navier-Stokes equat-

ions for the case of flow past a sphere are given by equations (4.7)

and (4.8) for the two dependent variables and ( g =

ez sin and f

C* ez sinO)respe:tively.

In finite-difference form, these equations are:

2 - h

2 + h

2 - k cot°

* 2 + k cot()

0 2Lh2 A

2Lh2 C 2Lk2 2Lt2

- 60 e2z

/ L (Ffl28)

2 - h 2 + h 2 - k cot9 2 + k cotO

+ ----- + + 60 = 2Lh2 gA 2Lh2 gC 2Lk

2 gB 2Lk2 gD

Re ez sin9 lit* kif

-,.•

... 1., B ... )(f '-f ).

D A C .j 8hkL (- IkIA - 4jC)(fB - fD) - — (F.29)

in Fig. F.1 . In this where the points A,B,C,D, and 0 are as shown

diagram which represents part of

the field of a general two-dimen-

sional viscous flow, the values

of IP* and at all points are

assumed to be initially, the settled .

values. If at 0 a finite disturbance

is applied to the flow; the value of

at 0 is changed from 0 to

4;0 the value of 4/0 * being

unaltered. The values of tp* and

Fig. F.1

A Computational Star

at the points 1,2,314,5,617,8

are assumed to remain unchanged from the settled values. At points

A,B,C,D equations (F.28) and (F.29) are used to obtain new (i.e.

disturbed) values of first g and then . If these new values are

tioNI denoted by g' and w then

2 + h (ezsin

2Lh

(e2z)A

=A gA)

(ersin 6)A

(ersin 9)0

264

(F.30)

(F.31)

Re

8hkl, iP2* - 418)

* ir A

g' = g A

ez sine e'

4 4096ez sin&

at* P2 "r4 )

2 g6 = g0 +

Re2

gL - gB

2 + k cota z (e sin 61) 0 2Lk2

Re

(')2 - 1/4) 8hkI,

z (0 sin )

(ersin e) (02z)

11)*1 = %f; _ _____B

13 ( g13 _ gB)

1 2 - h Re (ersine)c

g6 = + - go I ----27 (ersin0)0 8hkL ( tr4. - tr6 ) 2Lh (ersin(9)0

, (e2z)0

qjC = 'PC - ---I.,-'1( g6 - gc)

(F.32)

(F.33)

E T

(F.34)

(F.35)

2 k cot (ersin

Re (ersin

gL = gp 0 kiJ*8 kb . 6) z . 2Lk2 8hkL (e sin

(e2z)

**, = 4,* D gi

D gD ) D

(F.36)

(F.37)

From these disturbed values of g and 111 a new value of g at

0 can be obtained from equation (F.29). If it is assumed that the

points A,B,C,D, and 0 are not far apart so that values of z ande at

these points may be taken to be the same and the mesh is small so that

terms of 0(h) and 0(k) and higher orders may be neglected, then the

following new disturbed value of g0 may be obyained:

4. ( 4), _ 4*6 )2 + ( 4/4 - tp*, )2 + ( tP; - J*

)2

+2( 2 - ti); )( qi*„. - qq6 ) +2( qi; - kr6 )4; ti. J*

4- ( Y2 _ 41; + 4/4. - 466 )( 43 - 417 ) i, *

1 **4 + tP8 - /PG )( kPi - ‘P*5 ) + ( 2 xp* - E' (F.38)

This equation may be further simplified in terms of a VI*t

the points A,B,C, and D to eve:

265

4 OM go - ez 2 Re E sinn

512ez sine 4)*

) 2 - kg; )1 (F.59)

This is the resultant disturbance in the value of go and

hence for convergence:

Re2 E

kA - IP; )2 ( 1PB - IP D )2] < Ei (F.4o)

O•olOooOnam

4 512e2z ein2p

The second term of the left hand side of this inequality is

itself always positive and the condition for convergence can be

expressed as:

- kii ezsine p

( ) A - 7- c V' dr* )2 + ki)* B , )2 <640( ----- )- (F.41) Re

Define the components of velocity at 0, v* and le; as: ..*

r * tot/ i* -1 6? -1 IN -

1~D (F.42a) v* = -

r

e2z sinO be e2z sin° 2h

1 : - ; 6ki tii IP v* _

1 ii = (F.4213)

e2z sin e2z sin e 0 6 z 2h

Substitution of equations (F.42) into equation (F.41) gives:

160 h2( v*2 I. v*2 ) (F.43) r pe2;2z

But since Re = 2e0U/ y = 2U/ v , then

h2e2z( v2 v2 )/ v2 < 40 (F.44)

or the local mesh Reynolds number

Remesh 40 (F.45)

This is the form derived by Thom and Apeltto2 and also reported

by Russell20. On the other hand, if

is expressed by equation

(F.26) then the result of equation (F.43) becomes:

12.648

< z (F.46) Re e

By application of this analysis to the energy equation, the

following result is obtained: 12.648

< z PrRe e

266

(P.47)

F.3. Analytical Treatment of Stability

F.3.1. Error Analysis

Consider again the energy equation (F.4) whose finite-difference

approximations are given by equation (F.5). i.e.

w. . = a" + b" + cuw. . 4 + du (P.48) wi+1,j wi-11j 1,34- 1 wi,j-1

This equation takes, on the (n)th iteration, the form of

equation (F.3). i.e.

w(n) = a" (n-1) + b (n) (n-1) (n)

1,3 w. . " . . 1 . i+1,j w

+ cuw + dhw 1-11j i,j+1 ,3-1

(F.49)

By subtraction of equation (F.48) from (F.49) the error equation

is obtained:

(n) (n-1) 1,j e. = auei+1,j

( + b"e(n) + c"e1 1) n- + due n) 1 i-1,j ,j+1 11J-

wheree.C11).istheerrorinvtdefined by: 1,3 (n) (n) ei,j = - w.

j 1 j

(F.50)

(P.31)

It is easy to obtain a sufficient condition of stability by

direct examination of the error equation (F.50). That is:

<tail e(11-1-1J

which, by the use of equation

- I -

all I ctrl E'(n)

I

E1(n-1) (F.53)

1 b" d"

arc + I cul E

I (1 1 - lb"! - Id."1 i.e. ' (n) < )n E'(0) CF.54)

A sufficient condition that E should be bound as n tends to

infinity is that: a"1 + c" (F.55) 1

1 - b" 1 d"1 •

(n) + ctrl e(n-1) 1 b" I ei...1 ti 1,j+1

idllen) I

1,j-1 1 (F.52)

(F.18), gives the following:

82 B

8Rc

267

i.e. I al + 1 b"1 + 1 c"1 + I dul < 1 (F.56) ------.

Provided that a' l b",c", and d" are positive their sum will be

unity and the condition (F.56) will be satisfied. Hence, the procedure

is stable if (taking h and k to be small):

1 PrRe eV 1 ,_ PrRe 6 ti* ad --- ...---- > (F.57) n .--- h kez sine 60 k - kez sine 6z

By use of equation (F.26) to obtain the order of the derivatives

of qv the conditions (F.57) become:

h (or k ) 4

(F.58) PrRe eZ

which is the same result as that obtained in equation (F.27)

for convergence.

F.3.2. Residual Analysis

The vorticity equation (F.29) may be rearranged as follows:

2 - h 2 + h 2 - k cote 2 + k cote gB + gD RO = 2h2 gA -I- 2h2 gC + 2k2 2k2

.1.! 2 2 fB-fD A-fC - (---f + -7)go - Re eZ sin6( 1) --- ) (F.59) h k Oz k 60 h

where Ro is the amount by which the finite-difference equation differs

from zero. If %,* = 1, then the influence equations are given by: 1 8 R0 = -2(-7 + -17 ) (ez sin 6o (F.60)

z 6 ( kii ez sinO)A S 2 + h Re

RA = —2—(e sine, .... ....., `.-'— )

2 A (ez sine9)0 (P.61)

h 4h

2 + k cot(9 B Re ewe (ez sin 6)B ez sing )0 + ( i'="-- )B 2k2 4k 00 z (ez sin (9) 0

2 - h z Re * (eZ sine) ---2-(e sin 0)0 + 0.11.1.• ( .~....1/0/0 )

C

2h 4h 6 0 c (ez sine)0

2 - k cot (9 Re 6 tii* (ez sine) 8RD - D (ez sin e)0 - - -( .7--- ),

2k2

4k oz ' (ez

D

sin 0)0

(F.62)

(r.63)

(F.64)

Pa cosh z cosh zs

4 12.648 and h (or k) < cosh z Pa-------cosh zs

h (or k ) (F.69)

268

Jenson15 used these equations and obtained a_condition:of

stability as follows:

Equation (F.26) was used to obtain the order of magnitude of

w /be , and h was assumed to be small so that Jenson obtained

from equation (F.61):

SRA 1 z 4R e 1 e cos (F.65)

ez sin et

If &A is negative, the vorticity at A will be of opposite

sign to that at 0, and the result will be oscillating values of

vorticity. The value of h must, therefore, be chosen to ensure that

8RA is always positive.

i.e. 4

h < Re ez

(F.66)

F.4. Summary

It is important that the limiting values for h and k obtained

in the previous sections are not exceeded if satisfactory numerical

solutions are to be obtained. The smaller value of the upper limit

ensures both stability and convergence of the solutions and this

should be used as a guide for choosing h and k for various values

of the parameters Re and Pr.

i.e. h (or k )

where Pa is a parameter denoting Re or Pe (= PrRe).

The larger value of the upper limit ensures convergence but

not stability. 12.648

h (or k ) z (F.68) Pa e

Similarly, for the oblate spheroids the corresponding conditions

are given by:

(F.67) Pa ez

269

The larger values of the upper limit may cause the solutions

to be unstable if very small relaxation factors are not used. This

requires equation (F.49) to be used in the following form:

(n) w. = (1 - (1)14,12-1) + a 051.41,7 n-1) + + c"w 11-1) 11..1 1,j 1+1,j 1-12j 12j+1

(n) -1 + d"w ) (F.70) ilj

where 0 <5? <1 . These small relaxation factors, Q. 2 lead to excessively long

computation times.

Also the limitation on spacing in z-direction (i.e. h) is

more important than the case for spacing in e-direction (i.e. k),

because the variation of the dependent variables in z-direction is

of larger magnitudes to that in the 0-direction.

270

APPENDIX G

COMPUTER PROGRAMMES FOR THE SOLUTION OF THE FINITE-DIFFERENCE EQUATIONS

G.1. Introduction

The development of an accurate, stable and economical method

of solving the Navier-Stokes and energy equations for heat transfer

from solid particles, spheres and oblate spheroids has been the

central theme of the present thesis. A computer programme is a

necessary link between the formal description of the method in terms

of symbols and the practically useful predictions in terms of numbers.

Two computer programmes have been developed to solve the finite-

difference equations; one for solving the Navier-Stokes equations and

the other for solving the energy equation. These programmes are

written in a general form which can be used for all particle shapes

considered. In order that the general programmes presented in this

thesis will be understandable to and usuable by other workers, the

programmes are presented in detail. The purpose of the present

Appendix is to make available such detail information.

G.2. Scope and Limitations of the Programmes

The limitations of the calculation method described in Chapter 4

apply also to the computer programmes. The programmes are useful

only when a good approximate solution for the dependent variables

can be generated at all mesh points as an initial estimate of the

solution, so that the final refined solution can be obtained after

a relatively short computational time. For example, the solution of

the Navier-Stokes equations for a given Reynolds number has been

obtained by the use of the available solution for the nearest lower

value of the Reynolds number as an initial approximation. A similar

procedure has been used to solve the energy equation for a given

Peclet number. The boundary conditions for the dependent variables

must be known everywhere on the boundaries.

271

The programmes are written in FORTRAN IV language and can be

run on IBM 7090,7094 Computers under the IBSYS system. Programme 1

is so written that the two elliptic second-order partial differential

equations of the type (3.42) and (3.41) for vorticity, , and

stream function, IP , are solved simultaneously. The old values

of tJ are used to obtain new values of , which in turn are

used to obtain new values of Programme 2 is written so that

a single elliptic secons-order partial differential equation of the

type (3.43) for temperature, T is solved.

The existing DIMENSION and COMMON statements limit the field

of programme 1 to a maximum size of 65 columns and 61 rows and the

field of programme 2 to 50 columns and 50 rows. These are sufficient

for the cases considered in this thesis.

All quantities which are functions of the independent varia-

bles are generated by the programmes only once for a given field.

The programnes handle all the dependent variables and other auxil-

iary quantities in a destructive way; e.g. in the use of the array

U(I,J), the values of the elements stored at any time are replaced

as the calculation proceeds by the corresponding values for the

next iteration. This reduces the necessary storage space and much

of the computation.

The scope of the programmes depends largely upon the ingenuity

of the user. The programmes can be used, with some modification, to

solve other elliptic second-order partial differential equations,

linear and non-linear, which are not discussed in this thesis. For

the problem of interaction between free and forced convective heat

or mass transfer, the three equations of the process, (3.41) to

(3.43), are dependent on each other and hence must be solved simul-

taneously. It is probable that in such cases the two programmes may

be combined into a new one which will solve for the three dependent

variables , , and T simultaneously.

272

The subroutines of the programmes can be divided into two

groups; subroutines of the first group have general validity for

all cases of the type considered, and those of the second group may

vary from one run to another. The main routine, MAIN , and the

subroutines FIELD, COCAL, CASE, BOUNDC, SOLVE, and NSNSEE belong to

the first group, and the subroutines SETUP, INPUT, and RESULT belong

to the second group. When a FORTRAN subroutine is compiled under the

IBSYS system, it is translated into machine language and, if required,

a machine language equivalent of the FORTRAN routine is punched on to

cards. This set of cards, called a binary deck may be used in

place of the FORTRAN deck when the programme is used next. This saves

compilation and printing time and it is, therefore, to the programmers

advantage to use these cards. For this reason, binary decks are

produced for all subroutines of the first group, and used together

with those of the second group whenever the programmes are run.

G.3. Conventions used in the Programmes

Before the details of the various subroutines are presented,

a general familiarity with the programmes can be developed by the

knowledge of the different conventions used therein.

Subscripts for the Mesh Points

The numbering of the lines of constant z and constant

corresponds to that used in Chapter 4 (Fig. 4.1). The subscript I

refers to the line number at fixed z, and the subscript J refers

to the line number at fixed .In this convention, any mesh point

is specified by the values of the two subscripts I and J. I varies

from 1 at the particle surface to MM1 at the outer boundary, while

J varies from 1 at 0=0 to M1 at (9= 1$

Dependent Variables

The arrays U(I,J), V(I,J), and T(I,J) are used for the dimen-

273

sionless variables of stream function, vorticity, and temperature,

respectively at mesh points. Other related functions may be used

similarly.

pagnetic Tapes

The use of magnetic tapes is optional. When information is to

be used several times, and data cannot be provided easily, it is

advisable to use magnetic tapes for reading and storing data. In

each of the programmes, two such tapes have been used; one for reading

information and initial data and the other for storing intermediate

information and the final results achieved. The use of two tapes in

this way ensures that at least one tape holds the information safely

and also avoids time wasted in the rewinding of the tapes. Note that

one or more tapes can be used and the necessary alterations for the

read and write instructions are left to the user.

New Iterative Values

TN is the name of the location at which the new mesh point

( value of the dependent variables is stored. TN is equivalent to W.n)

of equation (4.65), which denotes the value of the dependent varia-

ble, W, at the mesh point (i,j) on the (n)th iteration. All new mesh

point values of all the dependent variables calculated in succession

are stored in this location.

Convergence and Computation Termination

NPR1 is the number of the total mesh point values (number of

algebraic equations) that require solution. NPR is the current number

of unconverged point values which is set to NPR1 at the beginning of

each new iteration. L1 and EX2 are, respectively, the numbers of the

unconverged point values permissible and the maximum number of iter-

ations which may be performed. Whenever a point convergence has been

achieved (i,e. satisfying condition (4.101)), NPR is reduced by 1

until NPR becomes less than or equal to L1, at which full convergence

274

is considered to be achieved. However, if this condition has not been

satisfied during a given run, the computation is terminated after

performing MX2 number of iterations.

G.4. List of FORTRAN Symbols Used in the Programmes

Given below is the list of all the important FpRTRAN symbols

used in the programmes and their algebraic equivalents whenever

possible; where the symbol has a direct connection with a particular

subroutine, a reference to that subroutine is also given. Numeral 0

refers to the MAIN routine and the numerals 1 to 9 refer to the nine

subroutines according to their calling sequences. Programme 1 and

programme 2 are referred to by (i) and (ii), respectively.

FORTRAN Symbol

Meaning Related Subroutine

A

m

L 4

Al zi el*

dz 9(1)

A2 sin2O d 6 9(i)

AK(J) Ye

9(1)

AKNOT Ko 9(i)

AKSN2 (J) KG sin 26 9(1)

ALENDA zo I

AM(J) me 9(ii)

AMRAT(J) Nue / Nue .0 9(ii)

ANU(J) Nu

0- 9(ii)

.. AVHJ JH 9(ii)

AVM Nu / Re' Pr3 9(ii)

I 9(ii) AVNU Nu

AVNUPR Nu / Pr7 9(ii) I

AVNURE Nu / Re2 9(ii)

B1 (I) B1(i) 4,7,9

B2(I) B2(3_ ) 4,7,9

275

FORTRAN Symbol


B3(J) 133(j) 4,719

B4(J) B4(j) 4,7,9

BB1(I) Bi(i) 4,6,9 (ii)

BB2(I) B2(i) 4,6,9 (ii)

BB3 B3 4,6 (ii)

C1(1) C1(i) 4,7,9 (i)

C2(J) c2(j) 4,7,9 (i)

ci(i) C1 (i) 5,7 (ii)

C2(I) C2(i) 5,7 (ii)

CB1 Cbl 4 (i)

CB2(J) Cb2(j) 4,6,9 (i)

CDF CDF 9 (i)

CDP CDP 9 (i)

CDT CDT 9 (i)

CS(J) cos 0 2,4,9

CSH(I) cosh z or ez 2,3,419

CT(J) cote 2,4,9 I I?

D dili or ditj or dio 7

DFI(I,J) 111:1j4-1 - 411,j-1) / H

3(i,j) 3,5 (ii)

DFJ(I,J)

DF1(I,J)

OF2(I,J)

DN1

DN2

E

EPS

F(I,J) rl. io . 3,6,7,9 (i)

rcop(i) Cf(i) 4,6,9 (i)

G(I,J) G. . 10 3,6,7,9 (i)

H3(I,J) H3(i,j) 2,516(i)17(i)19

(qi+1,i - Ilji-1,j) / H3(i,j) 3,5 (ii)

D f1 5,7 (ii)

Df2 5,7 (ii)

Rei Pr' 9 (ii)

Re Pr-11' 9 (ii)

e 1,2,4

€ 1,6,7

276

FORTRAN symbol Meaning Related Subroutine

IIJ(J) Nue / Re Pr' 9 (ii)

2 to 9

INTAPE When INTAPE is set to 1, a 1,217j magnetic tape will be used for reading data.

ITHETA(J) (9 in degrees

2,9

2 to 9

KS k in degrees

1,2

L1 Maximum number of unconverged 0,1 mesh point values permissible

L2 When L2 is set to 1, pressure and drag 1,9 coefficients or Fusselt number will not be computed.

L3 Maximum value of i (dimension) 1,2,31 4,9

L4 When L4 is set to 1, the known boundary 1,3 conditions will not be computed.

L5 When L5 is set to 1, the field variables 1,2,3,4 will not be computed.

TI M 1 to 9

NI M+ 1 1 to 9

MM mm 1 to 9

MM1 MM + 1 1 to 9

MMX1 Initially set to MX1 and when becomes 0 zero, results will be printed out.

MX1 Number of iterations after which the 011,5 current solution is required to be printed out.

1X2 Maximum number of iterations allowed 0,1 per run.

N n 0,5,8

N1 —When 0, MMX1 will be set to MX1 0,5 —Chen 1, MMX1 will have its current value

N2 Total number of irregular mesh points 3

Nj —Equals 0 initially and whenever convergence 0,1 of the current case has been achieved.

—Equals 1 otherwise.

Not used

277

FORTRAN Symbol Meaning Related Subroutine

N5 Total number of cases which require solution 0,1

NDTAPE When NDTAPE is set to 1, a magnetic 1,9 tape will be used to store data.

NPR NPR 0,617,8

NPR1 NPR1 0,3

OMEGA (--1 apt, 5,6,7,8 (ii)

PR Pr 5,8,9 (ii)

PkUIK Pr. Y2 5 (ii)

PRICA PrY3 5 (ii)

RATIOI ro 3 (i)

Ire; Re 5(1) or 3:(ii)

REG 4,5 (i) IT 1 -

REG Y2 4,3 (ii)

REHK Re Y1 5,7 (i)

a PF1 l 5,7,8 (i)

4 RF2 2 5,718 (i)

S1 1 / 12h 9 (i)

S2 1 / 12k 9 (1)

SH h 1 to 9

SIS1 Is 4 (ii)

SIS2 S2 4,9 (ii)

SK k 1 to 9

SN(J) sin © 2 to 9

SNIT (I) sinh z or ez 2 to 9

SRT(J) S1(j) 4,9 (ii)

SSE -1 / 12h 9 (ii)

T(I,J) Tt . 3,6,7,8,9 (ii) 11J

THETA Gin radians 2 n) TN W.(2j 6,7

TNH(I) 2,4,9 tanh z or 1.0

278

FORTRAN Symbol


T* TS ( )s 9 (ii)

TSN(J) sin 9 9 (ii)

U(I,J) ti

9,315(ii),6(i)17(i)18(i)

UNREG Y3

4,5 (ii)

V(I,J) 3,6,7,8,9 (i)

VSN2(J) sin29 9 (i)

VTH(J) -13e (( )s+ e ) 9 (i)

VZ(I) /

)'.0 9 (i)

X1 Equals MX1 (with floating point) 5

Y ( 6c/ z)s 9 (i)

Y 1:: TSN(J) 9 (ii)

Z z 2

ZS zs 1,2

11•Ys.....111•11.1•0 1•••••••••••••••11111MMOMM

G.5. Description of the Subroutines

The two programmes presented in this thesis are generally

identical. Each consists of a main routine (MAIN) in which nine

subroutines can be called to perform various operations. These sub-

routines have been introduced so that operations of same kind can

be confined to specific parts of the programme. Some of these subr-

outines are identical in the two programmes, others differ slightly.

The values of many constants and variables may be used in

various parts of the programme. These quantities can be transferred

to their respective subroutines via COMMON statements. The descriptions

of individual subroutines of Programme 1 will now follow and any

differences in those of Programme 2 will be pointed out later.

Listings of all the subroutines are given in section G.7. The FORTRAN

symbols used therein have been defined in the previous section.

279

G.5.1. Programme 1

1.0 MA IW

Strictly speaking, the main programme cannot be called a 1

subroutine according to FORTRAN terminology; however, this point is

of little significance for the present purpose. Fig. G.1 shows the

flow diagram of the computer programme- MAIN . It indicates the seq-

uence of operations and the connections with various parts of the

programme and the functions of these parts in brief.

The description of MAIN can be best followed with the help of

its flow diagram. MAIN starts the computation by setting the register

N (the current number of iterations performed) to zero. Subroutines

SETUP, FIELD, INPUT, and COCAL are then called upon in turn to perform

the operations indicated briefly in the flow diagram. At this point,

the field has been fully described, initial guesses for the dependent

variables have been generated at all mesh points, boundary conditions

have been fixed, all coefficients of the finite-difference equations

have been calculated, and various control variables have been set up.

The programme is now ready to solve the finite-difference equ-

ations of the Navier-Stokes equations for a specific value of the

Reynolds number. Note that N3 is always set to zero initially, by

subroutine SETUP, so control can be transferred to subroutine CASE at

the beginning of the run in order to specify the value of the Reynolds

number for which the solution is required. N3 is then set to unity in

subroutine CASE, and it stays so throughout the computations for this

value of the Reynolds number, until convergence is achieved, at which

time it is reset to zero by MAIN. A test is then made on the value of

N1 (see subroutine CASE). If it is zero, MMX1 is set to MXI so that a

new group of MX1 iterations are started. Then, or when NI is unity,

NPR is set to NPR1 and a new single iteration starts.

Special treatment is required for the unknown boundary condi-

tions G(1,J) for which new estimates are generated in subroutine

C

280

CALL SETUP ----- -I SETUP describes the size of the field and sets up various control variables.

CALL FIELD

CALL INPUT

TIELD generates field variables which are functions of z &

INPUT sets up an array of initial guesses for the dependent variables and fixes the known boundary conditiorq

Yes CCCAL calculates the coefficients of the finite-difference equations.

CASE introduces a new value of Re (foil

-. -.

the NaVier-Stokes equations) or Pr (for the energy equation).

- V

SOLVE solves algebraic equations by

N+1 Y successive approximation and tests = for point - convergence.

= 0

CALL RESULT {RESULT prints out the results.

N5 = N5-11,

(-CALL NSNSEE ESNSEE calculates the pressure distribution and the drag coeffici-ents (Navier-Stokes equation -3) or the Nusselt number distribution (energy equation).

STOP ) %.•

Fig. G.1 Flow Diagram of the Computer Programn les 1E2 - MJL :N

CALL BOUNDC )7_7

( CALL SOLVE

I

BOUNDC estimates new values of the unknown boundary conditions by solv-ing special algebraic equations.

281

BOUNDC. Then, subroutine SOLVE is called upon to estimate new values

of the dependent variables G(I,J) and U(I,J) at all mesh points. At

the end of this iteration, N is increased by 1, while MMX1 is decre-

ased by 1. A test for convergence follows, by comparison of the cur-

rent value of PPR with L1- if it is less than or equal to L1, conver-

gence has been achieved and the following route of operations is

taken: N3 is set to zero, the results in the form of U(I,J) and V(I,J)

are printed out by the subroutine RESULT, the number of cases to be

considered, N5, is reduced by 1, and subroutine I'ISNSEE is called upon

for the calculation of other results such as drag coefficients.

The run is terminated when there are no more cases to be con-

sidered, i.e. when N5 = 0. If 115 is not equal to zero, the solution

for a new value of the Reynolds number is started and the above

operations repeated. When NPR is greater than. L1, a test is made on

the value of MMX1 - if it is positive, a new iteration is started,

otherwise subroutine RESULT is called upon and a new group of 1X1

iterations started, provided that N has not exceeded MX2. Finally

the run is terminated when N exceeds or becomes equal to MX2.

It is to be noted that MAIN calls subroutines SETUP, FIELD,

INPUT, and COCAL only once before commencing any iteration. It may

call subroutines CASE and NSNSEE N5 times, and subroutine RESULT

possibly more than N5 times. However, subroutines BOUNDC and SOLVE

are called N times. Descriptions of these subroutines will now follow

in more detail.

1.1. SETUP

This is the first subroutine in the programme in which instr-

uctions may vary from one run to another. It sets the particle shape

factor E, the mesh size SH, SKI KS, the field size nm, M, the rela-

tive accuracy criteria EPS, and the acceptable convergence limit L1.

N3 is always set to zero so that a new case can be started by

282

MAIN. The first subroutine (SETUP) also provides the necessary inst-

ructions for the number of cases to be considered, N5, for the use of

magnetic tapes, and other control variables (see listings).

1.2. FIELD

Subroutine FIELD describes fully the flow field The dimen-

sions of the field MM1, Ni and all variables related to the position

of the mesh point in the field are computed by this subroutine. These

variables are required in various parts of the programme and since

the method of solution is one with many iterations, it is, therefore,

necessary to generate these variables only once so that computing time

can be minimized. For this reason, subroutine FIELD generates and

stores all field variables, so they can be used when required.

Fig. G.2 shows the flow diagram of subroutine FIELD. It shows

that field variables are read from the input tape whenever the latter

is used (i.e. when INTAPE = 1). These field variables are corrected

if the tape holds data of another shape (i.e. L5 / 1). In such cases

or when an input tape is not used, the variables are computed as

follows: The variables in I are generated by the use of exponential

functions when E = 1.0 (i.e. sphere) and by hyperbolic functions when

E,<1.0 (i.e. oblate spheroid). The variables in J and the scale

factor H3(I,J) for all I and J are then computed.

1.3. INPUT

The initial values of the stream function, vorticity and other

auxiliary quantities are specified in subroutine INPUT. Fig. G.3 shows

the flow diagram of this subroutine. The number of irregular mesh

points, N2, and th6 total number of the unknown point values, NPR'',

are calculated here. If INTAPE = 1, an input tape is used in which

the data of a previous run is stored. In this case, an array of initial

guesses of U(I,J) and V(I,J) for all I and J are read from the tape.

Yes

283

Reading of SEB,CSH,TNH,ITHETA,/// -SN,CS,CT & H3 for / all I & J from a. magnetic tape

OBLATE SPHEROID

Calculation of

SNH,CSH &. TNH for all I

Calculation of IIHETA,SN,CS & CT for all J

Calculation of H3 at all mesh points

Return to MAIN

C START :1

Calculation of. Ill & MM1

Yes

Fig. 0.2 Flow Diagram of Subroutine FIELD (Programmes 1&2)

Calculation of N2 & NPR1

r \Reading. of

initial guesses of the dependent variables U & V at all mesh points from data cards.

& V at all mesh point: from a magnetic tape

Reading of initial guesses of the

Yes;> dependent variables.0 START

Reading of

\G(I,J) & F(I,J)

for all I & J from \.a. magnetic tape

Fig, 0.3 Flow Diagram of Subroutine INPUT' (Programme 1)

Calculation of G(I,J) & F(I,J) for all I & J

E

Fixing of the known boundary conditions

•

4 Return to MAIN

284

If the data are for the same shape (i.e. L5 = 1), then G(I,J)

and F(I,J) will also be read from the tape; otherwise these must be

calculated. If an input tape is not used, initial guesses of U(I,J)

and V(I,J) are read from data cards, and values calculated for G(I,J)

and F(I,J) for all I and J.

The known boundary conditions are either read or computed in

this subroutine. They will be computed if they are not included with

the initial guesses (i.e. when L4 is set to a value other than 1 in

subroutine SETUP).

In the cases considered in this thesis, all the necessary values

are read in from either data cards or from a magnetic tape. Alternat-

ively, one can use approximate algebraic expressions, if available, to

generate the initial guesses. The necessary alterations are left to

the user.

1.4. COCAL

Subroutine COCAL can be described as the heart of this programme.

Its purpose is to obtain the coefficients B/1B2,B31B4 mentioned in

equations (4.79) and (4.80), and ClIC21Y1,C132,Cf mentioned in equations

(4.82),(4.83),(4.89),(4.90) for all the mesh points. Fig. G.4 shows

the flow diagram of the subroutine COCAL in which all the important

operations are clearly indicated.

'then an input tape is used and L5 = 1, the required coefficients

are read from the tape; otherwise they will be calculated as shown in

the flow diagram and the listings.

1.5. CASE

Subroutine CASE introduces a new value of the Reynolds number,

RE, for which a solution is required. In addition RF1 and RF2, the

two relaxation factors to be used in solving for U(I,J) and G(I,J)

respectively, are specified. The three values are read in from a

i

CalculationF of A & REG START.

285

Calculation of B1, B2.C1,8c FCOF for all _I

1 Calculation of B3 &

B41 for all J

Calculation of CB1

OBLATE SPHEROID/ S

Calculation of

P H

C2 & CB2 E for all J R

B Yes

(..S TART

Reading of RE,RF1 & RF2 from data cards

Calculation cf I 7-1 REHK, Y1 & Y2 Ni = 1 L---T--

(: Return to MAIN N3=1

LIFR = NPR-1

J = J+11

No

\

Reading of B1 ,B2,FCOF & C1 for all I, and B3,B4, C2 & CB2 for all J-

from a \ magnetic tape

(Return to,:) • MAIN

Fig. G.4 Flow Diagram of Subroutine COCAL (Programme 1)

Fig. G. Flow Diagram of Subroutine CASE (Programme 1)

(: START Calculation of TN (the new es-timate of G(1,0)

Convergence condition atisfied

o Yes

Replacing the of

G(1-,(7) by the

old value • value of TN

Calculation. of V(1,J) & F(1,J)

Calculation of F(I,1) & F(I,M1)

for all I

(Return to MAIN

fcs

Fig. G.6 Flow Diagram of Subroutine max: (Fcogramme 1)

286

single data card (see Fig. G.5 and the listings). The value of REHIC

is computed and N3 is set to 1 (i.e. new unconverged case).

The value of N1 is set to 1, then it is reset to zero initially

and whenever MX1 iterations have been performed by the time of calling

subroutine CASE. This allows MAIN to start a new group of MXI itera-

tions. Note that for small values of RE it is possible to obtain sol-

utions for various values of RE in one run. In such a case, the number

of the various values of RE to be considered, N5, is fixed in subrou-

tine SETUP, and CASE is called N5 times to supply one value of RE at

each call.

1.6. BOUNDC

Subroutine BOUNDC generates new estimates of all the unknown

boundary conditions which require special treatment. For the present

problem, the unknown values of G(1,J) on the particle surface are

evaluated using for example equation (4.89). Fig. G.6 shows the flow

diagram of the subroutine BOUNDC. It shows that operations are per-

formed in a loop controlled by a single DO statement, which runs

through J values from 2 to M. At each point on the surface the new

value of G(1,J) is calculated and stored momentarilly under TN. To

determine convergence, as each new value is computed it is compared

with the value for that point in the preceding iteration. If the

convergence condition is satisfied, NPR is reduced by 1. Then or

otherwise, the value of G(1,J) is replaced by the value of TN and

new values of V(11J) and F(1,J) are computed. At the end of the loop,

new values of F(I,1) and F(I,M1) are computed for all values of I.

By this stage all boundary conditions are fixed,and new estimates

for the regular interior mesh point values can then be made in sub-

routine SOLVE.

287 1.7. SOLVE

Subroutine SOLVE performs the mathematical operations necessary

to solve simultaneous algebraic equations of the type given in equ-

ations (4.79) and (4.80). Fig. G.7 shows the flow diagram of the sub-

routine SOLVE. As in subroutine BOUNDC, the operations are performed

in a loop controlled by DO statements. There are two DO statements

one of which runs through J. values from 2 to M, the other through

values from 2 to MM. Each (I,J) pair identifies a mesh point; at each

point equations (4.80) and (4.79) are solved for G(I,J) and U(I,J),

respectively. The important steps are given in the listings and shown

clearly in the flow diagram.

On leaving this subroutine, a whole iteration is completed, and

new values of all the dependent variables are generated. The current

value of NPR is used by MAIN to determine convergence. If the process

has not converged, subroutines BOUNDC and SOLVE are called again and

the whole iteration scheme is repeated.

1.8. RESULT

The instructions for printing out the results are to be contained

in the subroutine RESULT. In the example given in the listings, the

values of RE,RF1,RF2,NI NPR, and the solution obtained for the stream

function and vorticity are arranged to be printed out in this subro-

utine. The solution is printed in the form of a table with M1 rows

and MM1 columns. The user is free to make arrangements here to print

out any available information he chooses. The calling of this subro-

utine is so arranged that the current solution is printed out when-

ever the process converges or after every MX1 iterations.

1.9. NSNSEE

When convergence is achieved for a given value of the Reynolds

number, RE, it is usual to calculate various functions using the

F. J = J4-1

Calculation of TN

(the new estimate of G(I,J) )

NPR = NPR -11 Convergence condition

satisfied 9

No Replacing the old value

of G(I,J) by the value of TN

---/ Calculation of V(I,J) & F(I,J)

Calculation of TN

(the new estimate of U(I,J) )

J= 1

NPR = NPR-1

ZZ

4,Yes

Return to MAIN

288 START

Yes Convergence

condition satisfied .>

No

Replacing the old value -1 of

U(I,J) by the value of TN

Fig. G,7 Flow Diagram of Subroutine SOLVE Programme 1)

289 solution obtained. All these calculations and the instuctions for the

storage of information on a magnetic tape are contained in subroutine

NSNS EE.

In the example provided in section G.7 and shown in Fig. G.8 ,

the drag coefficients and the pressure distribution along the surface

are calculated using equations (4.94) to (4.98). The values obtained

are then arranged to be printed out in this subroutine. These operat-

ions can be omitted when not required by setting L2 to 1 in subroutine

SETUP. When NDTAPE is set to 1 in subroutine SETUP, information of the

type shown in the flow diagram and the listings will be stored on a

magnetic tape.

G.5.2. Profframme 2

Figs. G.9 to G.14 show the flow diagrams of the new subroutines

of Programme 2 0 The individual subroutines and their functions are

closely similar to those of Programme 1. Hence, only the differences

of each subroutine in the two programmes need to be pointed out.

2.0. MAIN

The sequence of operations and the connections of various parts

of the programme are the same as shown in Fig. G.1. The main listing

used in the case of Programme 1 is used again except for the DIMENSION

and COMMON statements. These are replaced by the new statements shown

in the new listings of MAIN.

The description of MAIN made in section G.5.1 holds here again,

if RE and all the dependent variables mentioned there, are replaced

by PR and temperature, T(I,J), respectively. Also, subroutine BOUNDC

is called to generate new estimates of T(I,1) and T(I,M1) along 9= 0

and 0= IT, respectively.

2.1. SETUP and 2.2. FIELD

These are the same as before, but with the new DIMENSION and

COMMON statements (see listings).

290

I Calculation of VTH&VSN2 for all J

[Calculation of AK&AKSN2 for all J

NDTAPE 4-

Calculation of A2,CDF CDP & CDT

Storing of SNH,CSH,TNH,ITHETA,SN,CS,CT, H3,U,V,G,F,B1,B2,FCCF,C1,B3, B41C2 & CB2 for all I & J o a magnetic tape

Printing out the

above results

Fig. G.3 Flow Diagram of Subroutine NSIZEE (Programme 1)

Ye Calculation of N2,NPR1 & RE

Reading of initial guesses .of the

dependent variabl

\

T at all mesh points from a magnetic tape

Reading of initial guesses of the dependent variable T at all mesh point from data cards

Reading of U(I,J) at all 1 & J from

data cards 4,

Calculation of DFI(I,MDFJ(I,J) at all I & J

Reading of U(I,J) DFI(I,J)&DFJ(I,J) at all I & J from a magnetic tape

Fixing of the known boundary conditions

1.es

No

_f Return to MAIN

• Fig. G.9 Flow Diagram of Subroutine INPUT (Programme 2)

No

Calculation of SI & 52

• 4, Calculation of VZ for all I

Calculation of Al & AKNOT

291

2.3. INPUT

In addition to the calculations of N2 and NPR1, the value of

RE for which velocity profiles are known is to be fixed in subroutine

INPUT. Fig. G.9 shows the flow diagram of this subroutine. An array

of initial guesses of T(I,J) for all I and J is read from either data

cards or a magnetic tape, if an input tape is used. Values of the

stream function U(I,J) corresponding to the considered value of RE

are necessary to solve the energy equation.

When both INTAPE and L5 are set to 1 in subroutine SETUP, the

values of U(I,J),DFI(I,J), and DFJ(I,J) will be read from the input

tape. Otherwise, the values of U(I,J) are read from data cards and

the functions DFI(I,J) and DFJ(I,J) computed. The known boundary

conditions, T(1,J) and T(MM1,J) are computed in this subroutine.

The functions DFI(I,J) and DFJ(I,J) are very closely related

to the velocity components vz and ve, respectively. They are called

many times during a run and hence it is advisable to generate them

only once. These functions are used to calculate DF1(I,J) and DF2(I,J)

in subroutine CASE. However, in the case of solution for high RE,

larger dimensions are needed and because the storage space of the

computer is limited, DFI(I,J) and DFJ(I,J) can be omitted from the

programme and new arrangements must be made to calculate DF1(I,J) and

DF2(I,J) in subroutine CASE. In such case, all FORTRAN statements

associated to these functions must be removed and the new DIMENSION

statements excluding these functions must be used in all the

subroutines.

2.4. COCAL

As in Programme 1, all coefficients of the algebraic equations

(4.81),(4.84),(4.86) to (4.88),(4.91) to (4.93) are generated in

subroutine COCAL for all mesh points The flow diagram of Fig. G.10

and the listings in section G.7 indicate clearly all the operations

involved in this subroutine.

292 2.5 CASE

Subroutine CASE introduces the value of the Prandtl number, PR,

for which the solution is required (see Fig. G.11). The values of PR

and the relaxation factor, OMEGA, to be used to solve for T(I,J) are

read in from a single data card. The coefficients C1(I) and C2(I)

mentioned in equations (4.91) and (4.92) are computed using eollation

(4.93).

The parts of the convective term in equations (4.81) and (4.84),

DF1(I,J) and DF2(I,J), do not vary with iterations and their values

are specified only once for a given PR. They are evaluated from the

values of DFI(I,J) and DFJ(I,J), which have been generated in subro-

utine INPUT, by multiplication by the value of PRHIC. Note that for

the case of high RE, DFI(I,J) and DFJ(I,J) are omitted from the

programme as pointed out in subroutine INPUT. In this case DF1(I,J)

and DF2(I,J) are generated from the values of stream function U(I1J)

using equations (4.86) and (4.87) respectively.

Finally the values of N1 and N3 are set as described in the

case of Programme 1.

2.6. BOUNDC

In the present problem, the boundary conditions for the temp-

erature along the axes of symmetry are not fixed but can be generated

using equations (4.91) and (4.92). The values of T(I,1) and T(I,M1)

are evaluated in a loop controlled by a single DO statement, which

runs through I values from 2 to MN (see Fig. G.12).

By this stage all boundary conditions are fixed. New estimates

for T(I,J) at all regular interior mesh points are made in subroutine

SOLVE.

2.7. SOLVE

Fig. G.13 shows the flow diagram of subroutine SOLVE, which

performs the mathematical operations necessary to solve simultaneous

Calculation of A,AA,BB3,

REG & UNREG

293 Calculation of B1,B2, BB1 & BB2 for all I

B3 & B4 for all J Calculation of

= I

Yes V

Calculation of

-1

DF1(I,J)&DF2(I,J) at all I & J

START Calculation of

Cl & C2 for all I

Reading of PR & OMEGA

from data cards

START. = I+1

=1\2+1

NPR = YPR-1

Replacing the old value of T(I,M1) by the value of TN

(

Return to MAIN

Convergence condition satisfied

4. Yes

Convergence condition atisfied - /

Yes c

INPR = NPR-1

No

Calculation of TN (the new estimate

T(I,1) )

Reading of B1,B2,BB1 & BB2 for all I, and B3,134 & SRT for all- J, an SIS2; from a mag-netic tape

OBLATE SPHEROID/ S

Calculation of

P H

SRT for all E J and SIS2 R

E Yes

( Return to MAIN

Fig..G.10 Flow Diagram of Subroutine COCAL (Programme 2)

Calculation o YI & Y2

N3 = 4,

( Return to

Flow Diagram of Subroutine CASE (Prograrme 2 . MAIN

,,) Fig. G.11

Fig. G.12 Flow Diagram of Subroutine BOUNDC (Programme 2)

%,)

294

algebraic equations of the type given in equation (4.81). These ope-

rations are analogueous to those described in Programme 1, except

that there is only one dependent variable, T(I,J), to solve for at

the mesh points.

On leaving this subroutine, a whole iteration is completed. If

the process has not converged, MAIN recalls subroutines BOUNDC and

SOLVE and the whole iteration scheme is repeated.

2.8. RESULT

As before, the instructions to print out the solution obtained

are given in subroutine RESULT. In the example given in section G.7,

the values of T(I,J) are printed out in the form of a table with M1

rows and MM1 columns.

NSNSEE

When convergence is achieved for a given value of PR, various

functions can be evaluated by this subroutine such as the local and

the overall Nusselt numbers and other related functions. The values

of these functions are arranged to be printed out in this subroutine.

As in the case of Programme 1, information can be stored on a

magnetic tape if required, Fig. G.14 shows the flow diagram of the

subroutine NSNSEE.

G.6. The User's Quick Reference Guide

The details of the computer programmes given so far and the

listings given in section G.7 complete the information necessary for

the user to proceed on his own. However, after acquiring a genera3

understanding of the programmes, the user should make sure that he

has taken the following necessary steps every time he wishes to run

the programmes. These steps together with the general flow diagram

of Fig. G.1 serve as a quick-reference guide and they express the

295 START ,)

J 11

Yes

J= J+1

Calculation of TN

(the new estimate of T(I,J) )

NPR = NPR-1 Ye

Replacing the old value. of

T(I,J) by the value of . TN

Yag: G.13

N

= MM

(:-Return to MAIN - Flow Diagram of Subroutine SOLVE (Programme 2)

C

Calculation of SSH,RERT,PR3,

DN1 & DN2

Nn

Yes

C START ) Calculation of ANU,TSN,HJ I AM & AMRAT for all J

Calculation of AVNU,AVHJ,AVM, AL NURE & AVNUPR

Printing out the

above results

Yes

\.-

Storing of N SNH,CSH,TNH,ITHETA,S,CS,CT,H3,

-- TI U,DFI,DFJ,B1,B2,13131,BB2,B3,B4

\L & SRT for all I & J, and SIS2; on a magnetic tape

(: Re;tilitrir, to

Fig. G.14 Plow Diagram of Subroutine NSNSEE (Programme 2)

296

same information in a concise form.

1. Keep MAIN and the subroutines FIELD and SOLVE unaltered.

2. Make sure that in SETUP the mesh sizes and the necessary control

variables together with the conditions for terminating the

computations are suitably specified.

3. Prepare the INPUT subroutine to generate initial guesses for all

the dependent variables as close to their solution as possible.

Also calculate NPR1 and fix all the known boundary conditions

of the dependent variables in the problem.

When solving the energy equation, fix RE for which the distrib-

ution of stream function is known.

4. Arrange the subroutine COCAL to generate all the coefficients

of the algebraic equations.

5. Supply the value of the parameter on which the solution depends

(i.e. RE or PR) and other related functions in subroutine CASE.

Make sure that the relaxation factors together with N1 and N3

are appropriately specified.

6. Prepare the BOUNDC subroutine to solve for the unknown specially-

treated boundary conditions.

7. Arrange for the desired print out in subroutine RESULT.

8. Prepare the NSNSEE subroutine to calculate various functions

related to the converged solution and arrange for their print

out.

9. Prepare the data cards according to the sequence of the READ

statements in the whole of the programme.

10. Obtain binary decks for all the unchangable subroutines to

save compilation and printing time,

297

G.7. Listings of the Computer Programmes

In this section, the complete listings of the two computer

programmes are presented. Here the subroutines appear in the foll-

owing order :

Page

Programme 1

MAIN

SETUP

FIELD

INPUT

COCAL

CASE

BOUNDC

SOLVE

RESULT

NSNSEE

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307

Programme 2

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310

311

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315

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316

317

318

MAIN

SETUP

FIELD

INPUT

COCAL

CASE

BOUNDC

SOLVE

RESULT

NSNSEE

298

PROGRAMME 1 MAIN

DIMENSION u(65,61),v(65,61),G(65,61),r(65,61),H3(65,61),sm(65), icsn(65)19m(65),ITHETA(61),sN(61),cs(61),cT(61),PcoF(65),c132(61), 2131 (65 )032 (65 ) ,B3 (61) ,Bk(61 ),ci (65 ) ,c2 (61 )0,7(65),m-1(6i )x112(61), 3AK(61),AaN2(61) COMMON U, V, G I F , H3 SITH , CSH , TNH , ITHETA , SN CS , CT , FCOF , CB2 , B1 , B2 , B3 ,

1134, C1, C2 , VZ VTH,VSN2 ,AIC,AICSN2 , E,RE,RF1 PF2 , EPS ,SH,SK,M,KM,M1 ,MM1 , 2NPR1,NPR,INTAPEI NDTAPE,MX1,MX2,N,N1,N2,N3,114,N5,L1,12,L3,L41 L5,TN, 3REGI UNREG,RELIC,KB,ZS,ALENDA

C C INITIAL SETTINGS

N = 0 CALL SETUP CALL FIELD CALL INPUT CALL COCAL

10 IF(N.GE.MX2) GO TO 3000 IF(N3.EQ.1) GO TO 30 CALL CASE

C HERE BEGINS THE ITERATION SECTION OF THE PROGRAMME FOR THE GIVEN CASE IF(N1.Eg.1) GO TO 40

C A GROUP OF MX1 ITERATIONS WILL BE STARTED NITH MIDC1=MX1 30 MMX1 = MX1

C A NEW ITERATION BEGINS WITH NPR=NPR1,WHERE NPR1 IS THE TOTAL NUMBER C OF POINT RECORDS TO SOLVE FOR (REGULAR + UNREGULAR) AND NPR IS THE C TOTAL NO. OF POINT RECORDS LEFT UNCONVERGED.

40 NPR = NPR1 • CALL BOUNDC CALL SOLVE

C END OF THE CURRENT ITERATION N = MHX1 = NLEX1-1

C TEST FOR THE OVERALL CONVERGENCE OF THE PROCEES. IT IS CONSIDERED C CONVERGED IF NPR IS LESS THAN OR EITAL L1 .

IF(NPR-L1)1000,1000,900 C IF I.:MX1=0,THE GROUP OF MX1 ITERATIONS HAVE BEEN PERFORMED. OTHERWISE C GO TO 40 AND START A NEW ITERATION 900 IF(MMX1)2000,2000,40

C SET N3 TO ZERO IF THE OVERALL CONVERGENCE CONDITION IS SATISFIED 1000 N3 = 0 2000 CALL RESULT C NOTE THAT N3=1 IF CONVERGENCE CRIT:ERIA HAS NOT BEEN SATISFIED. HENCE C CONTROL WILL BE TRANSFERRED TO STATEMENT NO. 10 TO TERMINATE C COMPUTATION OR START A NEW GROUP OF MX1 ITERATIONS DEPENDING UPON C THE CURRENT VALUE OF N

IF(N3.M.1) GO TO 10 N5 = N5-1 CALL NSNSEE

C N5 IS THE NO: OF CASES LEFT TO BE CONSIDERED IN THIS RUN. IF POSITIVE C CONTROL WILL BE TRANSFERRED TO STATEMENT NO. 10 TO TERMINATE

299

C COM'UTATION OR S TART NEW CASE (PG. FIX NEW RE) . WHEN ZEO, ALL • C CASE Z HAVE BEEN CONSIDEIED AND COMPUTATION WILIJ BE TERMINATED.

IF(N5.NE.0) GO TO 10 3000 STOP

END

SETUP

SUBROUTINE SETUP DIMENSION U(65,61),V(65,61),G(65,61),F(65,61),H3(65,61),SNH(65),

1CSH(65),TNH(65),ITHETA(61),SN(61),CS(61),CT(61),FC0F(65),C132(61), 2B1(65),B2(65),B3(61),B4(61),C1(65),C2(61)JZ(65),VTH(61),VSN2(61), 3AK(61),AKSN2(61) COMMON U,V,G,F,H3,SNH,CSH,TNHIITHETA,SN,CS,CT,FC0FI CB2,B1,B2,B3,

1B4,C1,C2JZI VTH,VSN2,AK,AKSN2,E,REIRF1IRF2,EPS,SH,SKIM,MM,M1,MN1, 2NPR1,NPRIINTAPEI NDTAPE,MX1,1a2,N,E1,N2,N3,N4,N5,L1,L2,L3,L4,L5,TN, 3REG,UNREG,REHK,KS,ZS,ALENDA

C C SETUP GENERATES A DESCRIPTION OF THE FIELD, SPECIFIES THE MESH SIZES C AND PROVIEDS MISCELLANEOUS CONTROL INFORMATION .

E = 1.0 SH =0.1. SK =0.2094 KS = 12 M= 15 MM = 19 EPS = 0.001 MX1 = 50 MX2 = 200 N3 = 0 P5 = 1 L1 = 1 L2 = 0 L3 = 65 L4 = 0 L5 = 0 INTAPE = 0 NDTAPE = 0 ZS - 0. ALENDA = 1.9 RETURN END

300 FIELD

SUBROUTINE FIELD DIMENSION U(65,61),V(65,61),G(65,61),F(65,61),H3 (65,61),SNH(65),

1CSH(65),TNH(65),ITHETA(61),SN(61),CS(61),CT(61),FC0F(65),CB2(61), 2B1(65)032(65),133(61),B4(61),C1(65),C2(61),VZ(65),VTH(61),VS112(61), 3AK (61 ) I AICSN2 (61 ) COMMON U,V,G,F, 113 S NH CSH, TNI-1, 'THETA SN, CS , CT, IPCOF CB2 ,B1,B2,B3,

1B4,C1,C2 tITZ VTR VSN2 AK , AKSN2 RE,171 ,RF2 EPS ,SH,SK ,M,124,141 ,MivI1 2NPR1 , NPR , INTAPE ND TAPE , MX1 , MX2 IN, N1 , N2 I N3 , , N5 L 1 , L2 , L3 , , L5 , TN, 3REGI UNREG,REHK,ICS IZS,ALENDA

C C COMPUTING FIELD VARIABLES C CALCULATE THE FIELD DIMENSIONS NMI AND M1 .

M1 = MA-1 MM1 = r2,1+1

C IF INTAPE=1, VALUES OF THE FIELD VARIABLES WILL BE READ FROM TAPE 7 C OTHERWISE, GO TO 1 TO CALCULATE THEM .

IF(INTAPE.NE.1) GO TO 1 REWIND 7 READ(7)(SNH(I),CSH(I),TNEI),I=1,10) READ(7)(ITHETAW,SN(J),CS(J),CT(J),J=1,M1) READ(7)((H3(I,J),I=1,1,3),j=1,M1)

C IF L5=1, RETURN BECAUSE THE INFORMATION JUST READ IN IS FOR THE SAME C SHAPE FACTOR E AS THE CURRENT ONE. OTHERWISE CALCULATE THE FIELD C VARIABLES FOR THE CURRENT SHAPE .

IF(L5.EQ.1) GO TO 50 C IF E=1.0, THE PARTICLE IS A SPHERE . C IF E LESS THAN 1.0, THE PARTICLE IS AN OBLATE SPHEROID .

1 IF(E.EQ.1.0) GO TO 10 DO 5 I.14Mm1 Z = ZS,-SH*FLOAT(I-1) SNH(I) = SINH(Z) Cal(I) = COSTI(Z)

5 TNH(I) = SNE(I)/CSH(I) GO TO 20

10 DO 11 3=1,MM1 Z = ZS+Sfl*FLOAT(I-1) SNH(I) = EXP(Z) CSH(I) = SNII(I)

11 TNH(I) =1.0 20 SN(1) = O.

SN(M1)= O. CS(1) = 1.0 CS(M1)=-1.0 ITHETA(1) = 0 ITHETA(M1)= 180 DO 22 J=2,M ITHETA(J) = KS*(J-1) THETA = SVFLOAT(J-1) SN(J) = SIN(THETA) CS (J) = COS (THETA)

22 CT(J) = CS(J)/SN(J)

301

C GENEhATE THE'SCALE FACTOR H3(I,J) AT ALL GRID POINTS DO 40 J=1,M1 DO 30 I=1 1 MM1 113(I,J) = CSH(I)*SN(J)

30 CONTINUE 40 CONTINUE 50 RETURN

END

INPUT

SUBROUTINE INPUT DIMENSION U(65,61 ),V(65,61),G(65,61 ) I r(65,61 ),H3 (65 1 61 )1skill(65),

ical(65),TRE(65),ITnETA(61),sN(61) I cs(61),cT(61),Fc0F(65) 1 cB2(61), 2B1(65),B2(65),B3(61),B4(61),01(65),c2(61),vz(65),WH(61),vsr2(61), 3AK(61),AKSN2(61)

COI,IMON U,V,G,F1 H3,SNHI CSH,TNHI ITHETA I SNI CS I CT,FC0F I CB21B1,B2,B3, 1B4,C1 1 C2,VZVTHI VSN2,AK,AMB2,E,RE,RF1 I RF2,EPS,SH,SK,N,MM,M1 I MM1, 2NPR1,NPR,INIAPE,NDTAPEIMX1,MX2,N,N1,N2,N3,N4IN5,L1,L2,L3,L4,L5,TN, 3REG,UNREG,REHK,KS,ZS,ALENDA

C C

SUPPLYING INITIAL GUESSES C N2----THE TOTAL NO. OF IRREGULAR POINT RECORDS WHICH REQUIRE SPECIAL C TREATMENT (VORTICITY AT THE PARTICLE SURFACE) C NPR1--THE TOTAL NO. OF POINT RECORDS TO SOLVE FOR (REGULAR-I-IRREGULAR)

N2 = M-1 NPR') = 2c(M-1)*(MM-1)-02

C SUPPLY INITIAL GUESSES FOR THE DEPENDENT VARIABLES. IF(INTAPE.EQ.1) GO TO 1

C READ FROM DATA CARDS OR USE ALGEBRAIC EXPRESSIONS TO GENERATE THE C

VALUES OF U(I ,J) AND V(I,J) READ(51 100) ((U(I,J),I=1,10),J=1,16) READ(5,100) ((U(I,J),I=11,20),J=1,16) READ(5,100) ((V(I,J),I=1,10),J=1,16) READ(5,100) ((V(I,J),I=11,20),J=1,16)

100 FORMAT(10F8 .5) CO TO 10

C IF INTAPE=1 1 INPUT WILL READ U (I,J) AND V(I,J) FROM TAPE7 C IF L5=1 1 INPUT WILL READ G(I,J ) AND F(I,J) FROM TAPE 7 .

1 READ(7)((U(I,J),I=1,L3),J=1,M1) READ(7)((V(I,J),I=1,L5),J=1,M1) IF(L5.NE.1) GO TO 10 READ(7)((G(I,J),I=1,L3),J=1,M1) READ(7)((F(I,J),I=1,1,3),J=1,111) GO TO 33

302

C INITIAL VALUES OF G AND F ARE CALCULATED FROM-THE GUESSED VALUES OF C VORTICITY AT ALL REGULAR INTERIOR POINTS.

10 DO 22 J=2,14 DO 11 I=2, M11 G(I,J) = v(I,J)*H3(I,J) F(I,J).= V(I,J)/H3(I,J)

11 CONTINUE 22 CONTINUE

C FIXED KNOWN BOUNDARY CONDITIONS . 33 IF(L4.EQ.1) GO TO 99

RATIO1 = CSH(1411)/CSH(1) RATIO2 = 0.5*RATIOI*RATIOI

C ALONG THE AXES OF SYMMETRY THETA = 0 180 . DO 44 i=1,mmi u(I0) = 0. V(I,1) = O. G(I,1) = O. U(I,M1)= O. V(I,M1)= O. G(I,M1)= O.

44 CONTINUE DO 55 J=2,M

C ON THE PARTICLE SURFACE . U(1,J) = O.

C ON THE OUTER BOUNDARY V(MM1,J) = O. G(1011,J) = O. F(MM1,J) = O. U(MM1,J) = RATIO2*SN(J)*SN(J)

55 CONTINUE 99 RETURN

END

COCAL

SUBROUTINE COCAL DIMENSION U(65,61),V(65,61),G(65,61),F(65,61),H3(65,61),SNH(65)t 1CSH(65),TNH(65),ITHETA(61),SN(61),CS(61),CT(61),FC0F(65),CB2(61), 2B1 (65 ) , B2 (65),B3(61),M(61),c1(65),c2(61),vz(65),vm(61),vsx2(61), 3AK(61),AKSN2(61) COMMON U,V,G,F,H3,SNHI CSH,TNH,ITHETA,SN,CS,CTIFC0F,CB2,B1,B2,B3,

1134,C1,C2,VZ,VTH,VSN2,AK,AKSN2,E,RE,RF1,RF2,EPS,SH,SK,M,MM,M1 1EM1, 2NP21,NPR,INTAPE,NDTAPE,EX1,MX2,N,N1,N2,N3,N4,N5,L1,L2,L3,L4,L5,TN, 3REG,UNREG,REHK,KS,ZS,ALENDA

C C CALCULATION OF ALL COEFFICIENTS

303

C REG IS A FACTOR IN THE CONVECTIVE TERM OF THE VORTICITY EQN. ( SEE C SUBROUTINE CASE) . C B1(I),B2(I),B3(J),B4(J) ARE THE COEFFICIENTS OF THE FINITE DIFFERENCE C EQUATIONS TO BE USED IN SUBROUTINE SOLVE . C Cl(I),C2(J) ARE TWO FACTORS IN THE CONVECTIVE TERM OF THE STREAM C FUNCTION EQUATION (SEE SUBROUTINE SOLVE) . C CB2(J),FCOF(I) ARE THE COEFFICIENTS TO BE USED IN SUBROUTINE BOUNDC C TO CALCULATE G(1 01),F(I,1) AND F(I,M1) C IF L5=1, READ ALL COEFFICIENTS FROM TAPE '7 .

SH2 = SH*SH SK2 = SK*SK A = 2.*(1./SH2+1./SK2) REG = -CSH(1)/(8.*SH*SK*A) IF(L5.EQ.1) GO TO 60 All = 0.5/(SH2*A) A22 = 0.5/(SK2*A) CH3 = CSH(1)**3 A3 = A*CH3 DO 10 I=2,MM CAT = SH*TNE(I) B1(I) = (2.-CAT)*All B2(I) = (2.+CAT)*All Cl(I) = SNH(I)*SNH(I)/A3 FCOF(I) = 1./(SK*CSH(I))

10 CONTINUE DO 20 J=2,M SKCT = SK*CT(J) B3(J) = (2.-SKCT)*A22 B4(J) = (2.+SKCT)*A22

20 CONTINUE CB1 = 0.5*CH3/SH2 IF(E.EQ.1.0) GO TO 40

C CALCULATE C2(J),CB2(J) FOR THE OBLATE SPHEROID SUBS = SNH(1)*SNH(1) DO 30 J=2,M CSS = CS(J)*CS(J) C2(J) = CSS/A3 CB2(J) = CB1/(SNHS+CSS)

30 CONTINUE GO TO 100

C CALCULATE C2(J),CB2(J) FOR THE SPHERE . 40 DO 50 J=21M

C2(J) = O. CB2(J) = CBI

50 CONTINUE GO TO 100

60 READ(7)(B1(I),B2(I),FC0F(I),C1(I),I=1,L3) READ(7)(B3(J),B4(J),C2(J),CB2(J),J=1 2M1)

100 RETURN END

301+

CASE

SUBROUTINE CASE DIMENSION U(65,61),V(65,61),G;65,61),F(65,61),E3(65,61),SNH65),

1CSH(65),TNH(65),ITHETA(61),SN(61),CS(61),CT(61),FC0F(65),CB2(61), 2B1(65),B2(65),B3(61),B4(61),C1(65),C2(61),VZ(65),VTH(61),VSN2(61), 3AK(61),AKSN2(61) COMMON U,V,G,F,E3,SNH,CSE,TNE,ITEETA,SN,CS,CT,FC0F,CB2,B1,B2,B3,

1B4,C1,02,VZ,VTEIVSN2,AK,AKSN2,E,RE,RF1,RF2,EPS,SE,SK,M,MMIM1,1T1, 2NPR1I NPR,I=PE,17DTPIPE,a1,MX2,N,N1,N23,N4,115,L1,L2,L3,L4,15,TN, 3REG UNREG , WEEK KS , ZS , ALENDA

C C INTRODUCTION OF A IMW CABS C SPECIFY THE VALUES OF RE AND THE RELAXATION FACTORS RF1 AND RF2 6 RFI IS APPLIED IN SOLVING THE STREAM FUNCTION EQUATION U(I,J) AND C RF2 IS APPLIED IN SOLVING THE VORTICITY EQ.UATION G(I,J)(IN SUB. SOLVE) C CALCULATE REEK THE FACTOR TO BE USED BY SUBROUTINE SOLVE TO EVALUATE C THE CONVECTIVE TERM OF THE VORTICITY EQUATION . C Y1=Y2 IF A MULTIPLE OF MXI ITERATIONS HAVE BEEN PERFORMED . C N1=0 ALLOWS MAIN TO START A NEW GROUP OF NX1 ITERATIONS . C N1=1 ALLOWS MAIN TO CONTINUE THE OLD GROUP OF MXI ITERATIONS . C N3=1 SPECIFIES UNCONVERGED NEW CASE .

READ(5111)RE,1:F1,RF2 11 FORMAT(3F10.5)

REEK = RE*REG XN = N X1 = MXI: Y1 =_N/MX1 Y2 = XN/X1 N1 = 1- IF(Y1.EQ.Y2) N1=0 N3 = RETURN END

BOUNDC

SUBROUTINE LOUNDC DIMENSION U(65,61),V(65,61),G(65161),F(65,61),E3(65161),SNE(65),

1CSE(65),TNH(65),ITHETA(61),SN(61),CS(61),CT(61),FC0F(65),CB2(61), 2B1(65),B2(65),B3(61),B4(61),C1(65),C2(61),VZ(65),VTE(61),VSE2(61), 3AK(61),AKSN2(61) COMMON U,V,G,F,E3ISNE,CSE,TNE,ITEETAISN,CS,CT,FC0F,CB2,B1,B2,B3,

1134,C1,C2,VZ,VTEI VSN2,AK,AKSN2,E,RE,RF1,RF2,EPSISE,SK,MI MM,M1,11M1, 2NPR1,NPR,INTAPE,NDTAPE,MX1,MX2IN,N1,N2,N3,N4,N5,L1,L2,L3,L4,L5,TN, 3REG,UNREG,REHK,KS,ZS,ALENDA

C C SOLUTION OF THE IRREGULAR SPECIALLY-TREATED BOUNDARY CONDITIONS .

305

C NEW ESTIMATES OF THE VORTICITY AT THE PARTICLE SURFACE C TN -- THE NEW ESTIMATE OF G(1,J) C A RELATIVE CONVERGENCE TEST AGAINST THE CRITERION EPS . C IF THE TEST IS PASSED , NPR IS REDUCED BY 1 . C REPLACE THE OLD VALUE OF G(1,J) BY TN AND CALCULATE THE NEW VALUES OF C V(1,J) AND F(1,J) .

DO 200 J=2,M TN = CB2(J)*(8.*U(2,J)-U(3,J)) IF(ABS((G(1,J)-TN)/TN)-EPS)90,90,100

90 NPR = NPR-1 100 G(1,J) = TN

V(1,J) = G(1,J)/H3(1,J) F(1,J) = V(1,J)/H3(1,J)

200 CONTINUE C CALCULATION OF F(I,1) AND F(I,M1) ALONG THE AXFS OF SYMMETRY

DO 400 I=1,MM1 F(I,1) = FCOF(I)*V(I,2) F(I,M1)= FCOF(I)*V(I,M)

400 CONTINUE RETURN END

SOLVE

SUBROUTINE SOLVE DIMENSION u(65,61),v(65,61),G(65,61),F(65,61)05(65,61),sm(65), icsii(65),TNE(65),ITHETA(61),sN(61),cs(61),cT(61),FccR(65),cB2(61), 2B1(65),B2(65),B3(61),B4(61),c1(65),c2(61),vz(65),vm(61)I vsN2(61), 3AK(61),AKsN2(61) COMMON U,V,G,F,H3,SNH,CSH,TNH,ITHETA ISN,CS,CT,FC0FI CB2,B1,B2,B3,

1B4,C1,C2I VZ,VTH,VSN2,AK,AKSN2,E,RE,RF1 IRF2,EPS,SH,SK,M,MM,M1 I MM1, 2NPR1,NPR,INTAPE,NDTAPE,MX1,MX2I N,N1 I N2IN3,N4I N5,L1,L2,L3IL4,L5,TN, 3REGI UNREG,REHK,KS I ZS,ALENDA

C C SOLUTION OF THE FINITE DIFFERENCE EQUATIONS AT ALL REGULAR INTERIOR C GRID POINTS . C THE POINTS IN EACH ROW OF THE FIELD ARE COVERED SUCCESSIVELY STARTING C FROM LEFT TO RIGHT. THE ROWS ARE COVERED FROM BOTTOM TO TOP . C AT EACH GRID POINT, THE NEW ESTIMATE OF THE DEPENDENT VARIABLE IS C CALCULATED AS FOLLOWS = C 1. CALCULATE THE CONVECTIVE TERM D . C 2. CALCULATE THE CONDUCTIVE TERM TN . C 3. ADD D TO TN . C 4. APPLY OVERRELAXATION (OR UNDERRELAXATION) . C TN -- THE LOCATION IN WHICH THE NEW ESTIMATED VALUE OF THE C DEPENDENT VARIABLE IS ACCUMULATED . C 5. TEST FOR THE RELATIVE CONVERGENCE AGAINST THE CRITERION EPS .

306

C 6. IF THE TEST IS PASSED, NPR IS REDUCED BY 1 . C 7. REPLACE THE OLD VALUE OF THE DEPENDENT VARIABLE BY TN .

DO 800 J=2,N DO 700 I=2,MM

6 NEW ESTIMATE OF THE VORTICITY DF1 = U(I+1,J)-U(I-1,J) DF2 = F(I,J+1)-F(I,J-1) DF3 = U(I,J+1)-U(I,J-1) DF4 = F(I+1,J)-F(I-1,J) D = REHK*H3(I,J)*(DF1*DF2-DF3*DF4) TN = B1(I)*G(I+1,J)+B2(I)*G(I-1,J)+B3(J)*G(I,J+1)+B4(J)*G(I,J-1) TN = TN+D TY= RF2*(TN-G(I,J))+G(I,J) IF(ABS((G(I,J)-TN)/TN)-EPS)500,500,600

500 NPR = NPR-1 600 G(I,J) = TN

V(I,J) = G(I,J)/H3(I,J) F(I,J) = V(I,J)/H3(I,J)

C NEW ESTIMATE OF THE STREAM FUNCTION . D = -(C1(I)+C2(J))*G(I,J) TN = B1(I)*U(I+1,J)+B2(I)*U(I-1,J)+B3(J)*U(I,J+1)+B4(J)*U(I,J-1) TN = TN+D TN = RF1*(TN-U(I,J))+U(I,J) IF(ABSUU(I,J)-TH)/TN)-EPS)50,50,60

50 NPR = NPR-1 60 U(I,J) = TN


RETURN END

RESULT

SUBROUTINE RESULT DIMENSION U(65,61),V(65,61),G(65,61),F(65,61),H3(65,61),SNII(65),

1CSH(65),TNH(65),ITHETA(61),SN(61),CS(61),CT(61),FC0F(65),CB2(61), 2B1(65),B2(65)1 133(61),B4(61),01(65),C2(61),VZ(65),VTH(61),VSN2(61), 3AK(61),AKSN2(61) COMMON U,V,G,F,H3,SNUI CSH,TNII,ITHETA,SN,CS,CT,FC0F,CB2,B1,B2,B3,

1134, C1, C2, VZ, ITTH VSN2 AK ,21.E.SN2 , E, RE,RF1 ,RF2 , EPS ,SH,SK,M,IZI,M1,1241 2NPR1,NPR,INTAPEI NDTAPE,1000,MX2,N,111,N2,N3,N4,N5,L1,L2,L3,L4,L5,TN, '3REG,UNREGIREHK,KS,ZS,ALENDA

C C'PRINTING OUT THE RESULTS OF STREAM FUNCTION AND VORTICITY

WRITE(6,10)RE,RF1,RF2-N,NPR 10 FORMAT(1H1,1X,411RE =,110.5,5X,5HRF1 =,F10.5,5X,5HRF2 =1 1110.5,5X,

13HN =,15,5X,5HNPR =,I5)

WRITE(6,400) 400 FORMAT(25H U -- 500 FORMAT(25H V 200 FORMAT(5X,10F10.5

WRITE(6,200)((U(I WRITE(6,200)((U(I WRITE(6,500) WRITE(6,200)((V(I WRITE(6,200)((V(I RETURN END

- STREAM FUNCTION) VORTICITY )

0"),I=1,1o),J=1,m1) ,J),I=11,20),J=1 lm1)

01),I=1,10),J=1,111) 01),I=11,2o),J=1,111)

307

NSNSEE

SUBROUTINE NSNSEE DIMENSION U(65,61),V(65,61),G(65,61),F(65,61),H3(65,61),SNH(65), 1CSH(65),TNH(65),ITHETA(61),SN(61),CS(61),CT(61),FC0F(65),CB2(61), 2B1(65),B2(65),B3(61),B4(61),C1(65),C2(61),VZ(65),VTH(61),VSN2(61), 5AK(61),AKSN2(61) COMMON UI VI GIF,H3,SNHI CSH,TNHI ITHETA,SIT,CS,CT,FC0F,CB2,B1,B2,B3, 1B4,C1,C2,VZ,VTH,VSN2,AK,AKSN2,E,RE,RF1,RF2,EPS,SH,SK,M,MM,M1,1T1, 2NPR1 NPR , INTAPE NDTAPE , MX1 , MX.2 N2 03 , N5 , L2 , L3 , , L5 , TN, 3REG UNREG REIM KS , ZS ALENDA

C C

FINAL RESULTS IF(L2.Eq.1) GO TO 2100 S1 = 1.0/(12.0*SH) S2 = 1.0/(12.0*SK)

C CALCULATION OF AKNOT, THE PRESSURE AT THE FRONT STAGNATION POINT . DO 1005 I=1,MM VZ(I) = 52*(48.0*V(112)-36.0*V(1 ,3)+16.0*V(I,4)-3.0*V(I,5))

1005 CONTINUE YVZ = 0.5*VZ(1) DO 1055 I=2, NCI YVZ = YVZ+VZ(I)

1055 CONTINUE Al = SH*YVZ AKNOT = 1.0+8.0*Al/RE

C CALCULATION OF THE LOCAL PRESSURE DISTRIBUTION RE4 = 4.0/RE DO 2005 J=1,M1 Y=S1*(-25.0*V(1,J)+48.0*V(207)-36.0*V(3,J)+16.0*V(4,J)-3.0*V(501)) VTH(J) = RE4*(Y+E*V(1 01)) VSN2(J) = V(1,J)*SN(J)*SN(J)

2005 CONTINUE BK5 = 0.5*SK AK(1) = AKNOT AKSN2(1) = 0.

308 DO 2055 J=2,M1 AK(J) = AK(J-1)+BK5*(VTH(J-1)+VTH(J)) AKSN2(J) = 2.0*AK(J)*SN(J)*CS(J)

2055 CONTINUE C CALCULATION OF THE DRAG COEFFICIENTS CDF, CDP, AND CDT .

CF = O. CP = O. DO 2066 J=21 M CF = CF+VSN2(J) CP = CP+AKSN2(J)

2066 CONTINUE A2 = SK*CF CDF = 8.0*E*A2/RE CDP = SK*CP CDT = CDF+CDP

C PRINTING OUT ABOVE RESULTS . WRITE(6,2010)

2010 FORMAT(1H1,20X15HTHETA1 7X,3HVTH,7X1 4HVSN2,9X,1HK1 8X,4HKSN2) WRITE(6,2020)

2020 FORMAT(20X,7H ,5X,5H ,5X,6H ,7X,3H- ,6X,6H DO 2040 J=1,M1 WRITE(612030)ITHETA(J),VTH(J),VSN2(J),AK(J),AKSN2(5)

2030 FORMAT(20X,I4,2X,4F11.5) 2040 CONTINUE

WRITE(6,2050) 2050 FORMAT(1X1 ///,17X1 2HRE,8X,2HA1,8X1 2HA2,8X13HCDFOX,3HCDPDXINCDT)

WRITE(6,2060) 2060 FORMAT(10X,3(6X,4H----)1 1X,3(5X,5H ) )

WRITE(6,2070)RE,A1,A2,CDFI CDPI CDT 2070 FORMAT(14X,F8.315(F10.5))

WRITE(6,2080) 2080 FORMAT(1X,////,1X,61•HDERIVATIVE OF VORTICITY W.R.T. THETA ALONG

1THE AXIS THETA=O) WRITE(6,2090)(VZ(I),I=1,MM)

2090 FORMAT(5X,10F10.5) 2100 IF(NDTAPE.NE.1) GO TO 2200 C IF NDTAPE=1, STORE INFORMATION ON TAPE 8 .

REWIND 8 WRITE(8)(SNH(I),CSH(I),TNH(I),I=1,15) WRITE(8)(ITHETAW,SN(J),CS(J),CT(J),J=1,m1) URITE(8)((H3a07),I=1,13)07=1,m1) WRITE(8)((U(I,J),I=1,L3),J=1,m1) wRITE(8)((V(I,J),I=1,13),J=1,m1) WRITE(8)((G(I,J),I=1,L3),J=1,m1) WRITE(8)((F(I,J),I=1,L3),J=1,m1) WRITE(8)(B1(I),B2(i),FC0F(I),c1(I),I=1,L3) WRITE(8)(B3(J),B4(J),C2(J),CB2(J),J=1,M1)

2200 RETURN END

309

PROGRAMME 2 MAIN

DIMENSION U(50150),T(50,50),DF1(50,50),DF2(50150),DF1(50150), 1DFJ(50,50),H3(50,50),SNH(50),CSH(50),TNE(50),ITHETA(50),SN(50)1 2CS(50),CT(50),B1(50),B2(50)1 B3(50),B4(50),C1(50),C2(50),BB1(50), 3BB2(50),SRT(50),ANU(50),TSN(50),HJ(50),AM(50),AMRAT(50) COMMON Ul T,DF1,DF2IDEI,DFJ,H3ISNHICSHI TEH,ITHETA,SNICSI CTI B1I B2,

1B3,B4,C1,C2I BB1,BB2,SRTI ANU,TSNIEJI AM,AMRATI E,RE,PR.OMEGA,EPS,SH, 2SK M MM , M1 MM1 /URI , NPR INTAPE t NDTAPE , MX1 , MX2 t N, N1 , N2 N3 N4 , N5 ' 3L2 L3 L4 L5 , TN1 REG , UNBEG S IS 2 , KS , BB3 , ZS , ALENDA

§C INITIAL SETTINGS N = 0 CALL SETUP CALL FIELD CALL INPUT CALL COCAL

10 IF(N.GE.MX2) GO TO 3000 IE(N3.Eg.1) GO TO 30 CALT, CASE

C HERE BEGINS THE ITERATION SECTION OF THE PROGRAMME FOR THE GIVEN CASE IF(N1.Eq.1) GO TO 40

C A GROUP OF MX1 ITERATIONS WILL BE STARTED WITH mai.kai . 30 MMX1 = MX1

C A NEW ITERATION BEGINS WITH NPR=NPR1,WHERE NPR1 IS THE TOTAL NUMBER C OF POINT RECORDS TO SOLVE FOR (REGULAR + UNREGULAR) AND NPR IS THE C TOTAL NO. OF POINT RECORDS LEFT UNCONVERGED.

40 NPR = NPR1 CALL BOUNDC CALL SOLVE

C END OF THE CURRENT ITERATION . N = NA-1 MMX1 = MMX1-1

C TEST FOR THE OVERALL CONVERGENCE OF THE PROCESS. IT IS CONSIDERED C CONVERGED IF NPR IS LESS THAN OR EQUAL TO L1.

IF(NPR-L1)10000000,900 C IF MM X1=0, THE GROUP OF MX1 ITERATIONS HAVE BEEN PERFORMED. OTHERWISE C GO TO 40 AND START A NEW ITERATION . 900 IF(MMX1)2000,2000440

C SET N3 TO ZERO IF THE OVERALL CONVERGENCE CONDITION IS SATISFIED . 1000 N3 = 0 2000 CALL RESULT

C NOTE THAT N3=1 IF CONVERGENCE CRITERIA HAS NOT BEEN SATSFIED. HENCE C CONTROL WILL BE TRANSFERRED TO STATEMENT NO. 10 TO TERMINATE C COMPUTATION OR START A NEW GROUP OF NIX1 ITERATIONS DEPENDING UPON C THE CURRENT VALUE OF N .

IF(N3.EQ.1) GO TO 10 N5 = N5-1 CALL NSNSEE

C N5 IS THE NO. OF CASES LEFT TO BE CONSIDERED IN THIS RUN. IF POSITIVE C CONTROL WILL BE TRANSFERRED TO STATEMENT NO. 10 TO TERMINATE

310

C COMPUTATION OR START NEW CASE (EG. FIX NEW PR). WHEN ZERO, ALL CAST'S C HAVE BEEN CONSIDERED AND COMPUTATION WILL BE TERMINATED .

IF(N5.NE.0) GO TO 10 3000 STOP

END

SETUP

SUBROUTINE SETUP DIMENSION U(50,50),T(50,50),DF1(50,50),DF2(50,50),DFI(50,50),

1DFJ(50,50),H3(50,50),SNH(50),CSH(50),TNH(50),ITHETA(50),SK(50), 20S(50),CT(50),B1(50),B2(50),B3(50),B4(50),C1(50),C2(50),BB1(50), 3BB2(50) 13RT(50),APU(50),TSN(50),HJ(50),AM(50),AMRAT(50)

COMMON U,T,DF1 IDF2IDFI,DFJ,H3,SNHI CSH,TNH,ITHETA,SN,CS,CT,E1,E2, 1B3,B4,C1,02,BB1,11B2,SRTIANU,TSN,HJI AM,AMRAT,E,REI PR,OMEGA,EPS,SE, 2SK,M,UM,M11MM1,NPR1 I NPRI INTAPEINDTAPE,MX1,MX2IN,N1,N2,N3,N4,N5,L1, 3L2,L3,L4,L5,TNIREG,UNREG,SIS2,KS,BB3,ZS,ANDA

C C SETUP GENERATES A DESCRIPTION OF THE FIRLD, SPECIFIES THE MESH SIZES C AND PROVIDES MISCELLANEOUS CONTROL INFORMATION .

E = 1.0 SH = 0.1 SK = 0.2094 KS = 12 M = 15 MM = 19 EPS = 0.001 - MX1 = 50 MX2 = 200 N3 = 0 N5 = 1 L1 = 1 L2 = 0 L3 = 50 1,4 = 0 L5 = 0 INTAPE = 0 NDTAPE = 0 ZS = O. ALENDA = 1.9 RETURN END

311 FIELD

SUBROUTINE FIELD DIMENSION U(50,50),T(50,50),DF1(50,50),DF2(50,50),DFI(50,50), 1DFJ(50,50),H3(50,50),SNH(50),CSH(50),TNH(50),ITHETA(50),SN(50), 2CS(50),CT(50),B1(50),B2(50),B3(50),B4(50),C1(50),C2(50),BB1(50), 3B132 (50) ,SRT (50) ,ANU (50 ) TSN (50 ) ,HJ (50) , AM(50) AMRAT (50) COMMON U T, DF1 DF2 DFI DFJ, H3, SIN, CSH, TNII, ITHET.A SN, CS „ CT, B1, B2,

1133 1 1314, Cl , C2 BB1 BB2 ISRTI ANU, TSNIHJ,AM, AMRATIE,RE, PR , OMEGAIEPS ,SH, 2SK MINIM, M1 MM1 NPR1 „ NPR INTAPE, NDTAPE MC1 MX2 N N1, N2 N3 , NLF , N5 , Ll 3L2 11,3 Lk, L5 , TN, REG , UNREG ,S IS2 , KS , BE3 , ZS , ALENDA

C C COMPUTING FIELD VARIABLES C CALCULATE THE FIELD DIMENSIONS MM1 AND ml

M1 = M+1 mmi = mm-1.1

C IF INTAPE=1, VALUES OF THE FIELD VARIABLES WILL BE READ FROM TAPE 7 C OTHERWISE, GO TO 1 TO CALCULATE THEM .

IF(INTAPE.NE.1) GO TO 1 REWIND 7 READ(7)(SNH(I),CSH(I),TNH(I)I=1,15) READ(7)(ITHETAW,SN(J),CS(J),CT(J),J=1,M1) READ(7)((115(I,J),I=1,1,3),J=1,M1) .

C IF L5=1, RETURN BECAUSE THE INFORMATION JUST READ IN IS FOR THE SAME C SHAPE FACTOR E AS THE CURRENT ONE. OTHERWISE CALCULATE THE FIELD C VARIABLES FOR THE CURRENT SHAPE .

IF(L5.EQ.1) GO TO 50 C IF E=1.0, THE PARTICLE IS A SPHERE . C IF E LESS THAN 1.0 THE PARTICLE IS AN OBLATE SPHEROID .

1 IF(E.EQ.1.0) GO TO 10 DO 5 I=1,MM1 Z = ZS+SH*FLOAT(I-1) SNH(I) = SINII(Z) CSH(I) = COSII(Z)

5 TNII(I) = SNH(I)/CSH(I) GO TO 20

10 DO 11 I=1, M141 Z = ZS+Sil*FLOAT(I-1) SNH(I) = EXP(Z) CSH(I) = SW')

11 TNH(I) = 1.0 20 SN(1) = O.

SN(M1 )= O. CS(1) = 1.0 CS(M1)=-1.0 ITHETA(1) = 0 ITHETA(M1)= 180 DO 22 J=2,M ITHETA(J) = IBS* (J-1) THETA = SK*FLOAT(J-1) SN(J) = SIN(THETA) CS(J) = COS(THETA)

22 CT(J) = CS(J)/SN(J)

312 C GENERATE THE SCALE FACTOR H3(I,J) AT ALL GRID POINT .

DO 40 J=1,M1 DO 30 I=1,MM1 H3(I,J) = CSH(I)*SN(J)

30 CONTINUE 40 CONTINUE 50 RETURN

END

INPUT

SUBROUTINE INPUT DIMENSION U(50,50),T(50,50),DF1(50,50),DF2(50,50),DFI(50,50), 1DFJ(50,50),H3(50,50),SNR(50),CSH(50),TNH(50),ITHETA(50),SN(50), 2CS(50),CT(50),B1(50),B2(50),B3(50),B4(50),C1(50),C2(50),BB1(50), 3BB2 (50 ) ,SRT(50) ,ANIT(50) ,TSN(50) ,HJ (50 ) ,AM(50),A/CAT(50) COMMON U T DF1 DF2 DFI DFJ H3 , SNH CSH TNH ITHETA , CS , CT, B1 , B2 ,

1B3 , B4 Cl , C2 , BB1 IBB2,SRT,ANU, TSN,HJ , AM, APERAT , EIRE, PR , OMEGA, EPS ,SH 2SK M ,141 M1 NM NPR1 , NPR , INTAPE ND TAPE , MX1 MX2 N N1 , N2 , N3 , N4 $ N5 L 1 , 3L2,L3,L4,L5,TN,REG,UNREG,SIS2,KSIBB3,ZS,ALENDA

C C SUPPLYING INITIAL GUESSES C FIXING RE FOR WHICH THE VALUES OF THE STREAM FUNCTION ARE TO BE USED.

RE = 1.0 C N2----THE TOTAL NO. OF IRREGULAR POINT RECORDS WHICH REQUIRE SPECIAL C TREATMENT (TEMPERATURE ALONG THE AXES OF SYMMETRY) C NPR1--THE TOTAL NO. OF POINT RECORDS TO SOLVE FOR (REGULAR-FIRREGULAR)

N2 = 2*(MM-1) NPRI = (M-1)*(MM-1)+N2

C SUPPLY INITIAL GUESSES FOR THE DEPENDENT VARIABLES. IF(INTAPE.EQ.1) GO TO 1

C READ T(I,J) FROM DATA CARDS . READ(5,100)((T(I,J),I=1,10),J=1,16) READ(5,100)((T(I,J),I=11,20)1J=1,16)

100 FORMAT(10F8.5) GO TO 10

C IF INTAPE=1, INPUT WILL READ T(I,J) FROM TAPE 7 . 1 READ(7)((T(I,J),I=1,L3),J=1,M1)

C IF L5=1, INPUT WILL READ U(I,J), DFI(I,J) AND DFJ(I,J) FROM TAPE 7 . IF(L5.NE.1) GO TO 10 READ(7)((U(I,J),I=1,L3),J=1,M1) READ(7)((DFI(I,J),I=1,L3),J=1,M1) READ(7)((DFJ(I,J),I=1,1,3),J=1,M1) GO TO 33

10 CONTINUE C READ U(I,J) FROM DATA CARDS .

READ(5,100)((U(I,J),I=1,10),J=1,16) READ(5,100)((U(I,J),I=11$20),J=1,16)

313C COMPUTE DFI(I,J) AND DFJ(I,J) TO BE USED IN SUBROUTHTE CASE FOR THEC CONVECTIVE TERH OF THE ENERGY EQUATION 0

DO 22 J=2,MDO 11 I=2,1-'I1-1DFI(I,J) = (U(I,J+1)-U(I,J-1»/H3(I,J)DFJ(I,J) = (U(I+1,J)-U(I-1,J»/H3(I,J)

11 CONTINUE22 CONTINUE

C FIXED BONDARY CONDITIOh~

33 IF(L4.EQ.1) GO TO 99DO 55 J:::1,M1

C ON THE PARTICLE SURFACET(1 ,J) :: 1.0

C ON THE OUTER BOUNDARYT(MM1,J) :: o,

55 CONTINUE99 RETURN

END

COCAL

SUBROUTINE COCALDI~mION U(50,50),T(50,50),DF1(50,50),DF2(50,50),DFI(50,50),1DFJ(50,50),H3(50,50),S~m(50),CSH(50),TNH(50),ITHETA(50),SN(50),

2CS (50) ,cT(50) ,B1 (50) ,B2(50), B3(50), B4(50) , C1 (50), C2 (50), BB1 (50),3BB2 (50 ) ,SRT(50 ) , iJro (50 ) , TSN(50 ) ,HJ (50 ) , M1(50) , AMRAT(50)

COM}10N U,T,DF1,DF2,DFI,DFJ,H3,SNH,csH,Tmi,ITHETA,SN,CS,CT,B1,B2,1B3,B4,c1,c2,BB1,BB2,SRT,Al~,TSN,HJ,AM,A}rnAT,E,RE,PR,OMEGA,EPS,SH,

2SK,1'1,MM,M1 ,Ml·i1 ,NPR1 ,NPR, INTAPE,NDTAPE,MX1 ,MX2,N,N1 ,N2,N3,N4,N5, L1 ,3L2,L3,L4,L5,TN,REG,UNREG,SIS2,KS,BB3,ZS,ALENDA

CC CALCULATION OF ALL COEFFICIENTSC REG IS A FACTOR IN THE CONVECTIVE TER}1 OF THE ENERGY EQUATION (REGC ULAR POINT - SEE SUBROUTINE CASE)C UNREG IS A FACTOR IN THE COEFFICIENTS OF THE FINITE DIFFERENCEC EQUATIONS ALONG THE AXES OF SYI1METRY 0

C B1(I),B2(I),B3(J),B4(J) ARE THE COEFFICIENTS OF THE FINITE DIFFERENCEC EQUATIONS TO BE USED IN SUBROUTINE SOLVE 0

C BB1(I),BB2(I),BB3 ARE THE COEFFICIENTS OF filE FINITE DIFFERENCEC EQUATIONS TO BE USED IN SUBROUTINE BOU1IDC 0

C SRT(J) AND SIS2 ARE COEFFICIENTS TO BE USED IN SUBROUTINE NSNSEE TOC CALCULATE LOCAL AND O'~ALL NUSSELT NUMBERS RESPECTIVELY •C IF L5:::1, READ ALL COEFFICIENTS FROH TAPE 7

SH2 ::: SH*SHSK2 ::: SK*SKA = 2.*(1./SH2+1./SK2)AA= 2.* (1./SH2+2./SK2)

C

All = 0.5/(SH2*A) A22 = 0.5/(SK2*A) A33 = 0.5/(SH2*AA) A44 = 0.5/(SK2*AA) BB3 = 8.*A44 AB = RE*CSH(1)/SH REG = AB/(8.*SK*A) UNREG = AB*A44 IF(L5.EQ.1) GO TO 60 DO 10 I=2,MM CAT = SH*TNH(I) Bl(I) = (2.—CAT)*All B2(I) = (2.+CAT)*Al1 BB1(I)= (2.—CAT)*A33 BB2(I)= (2.+CAT)*A33

10 CONTINUE DO 20 J=2sM SXCT = SK*CT(J) B3(J) = (2.—SKCT)*A22 B4(J) = (2.+SKCT)*A22

20 CONTINUE IF(E.EQ.1.0) GO TO 40

C CALCULATE SRT(J) AND SIS2 FOR THE OBLATE SPHEROID SNHS = SNH(1)*SNH:(1) CSH2 = —2.0*CSH(1) DO 30 J=1,M1 SNC = SITES+CS(J)*CS(J) SNCRT = SUT(SNC) SRT(J) = CSH2/SNCRT

30 CONTINUE SIS1 = CSH(1)+0.5*SNHS*ALOG((CSH(1)+1.0)/(CSH(1)-1.0)) SIS2 = —SK*CSH2/SIS1 GO TO 100

C CALCULATE SRT(J) AND SIS2 FOR THE SPHERE 40 DO 50 J=1,111

SRT(J) = —2.0 50 CONTINUE

SIS2 = SK GO TO 100

C IF L5=1, READ ALL COEFFICIENTS FROM TAPE 7 60 READ(7)(B1(I),B2(I),B131(i),BB2(1),I=1,L3)

READ(7)(B3(J)034(J),SRT(J),J=1,M1),SIS2 100 RETURN

END

314

315

' CASE

SUBROUTINE CASE DIMENSION U(50,50),T(50,50),DF1(50,50),DF2(50,50),DF1(50,50,

1DFJ(50,50),E3(50,50),SNH(50),CSH(50),INIT(50),ITHETA(50),SN(50), 2CS(50),CT(50),B1(50),B2(50),B3(50),B4(50),C1(50),C2(50),BB1(50); 3BB2(50),SRT(50),ANU(50),TSN(50),HJ(50),AM(50),AMIZAT(50) COMMON U,T,DF1,DF2,DFI,DFJ,H3ISNHI CSH,TNH,ITHETAISN,CS I CTI B1,B2,

1B3,B41 C1,C2,BBlIBB2ISRTIANU,TSNIHJ,AM,AMRATIE,RE,PR,OMEGAIEPS,Shl 2SKAMM,M1 IMM1,NPR1,NPRIINTAPE,NDTAPE,Ya1,MX2,N,N1,N2IN3,N4,N5,L1 3L22L3,L4,L5I TN,REG,UNREG,SIS2,KS,BB3,ZS,ALENDA

C C INTRODUCTION OF A NEW CASE C SPECIFY THE VALUES OF PR AND THE RELAXATION FACTOR OMEGA .

READ(5,45)PR,OMEGA 45 FORMAT(2F10.5)

C CALCULATE Cl(I) AND C2(I).SOME OF THE COEFFICIENTS OF THE FINITE C DIFFERENCE EQUATIONS TO BE USED IN SUBROUTINE BOUI'IDC C CALCULATE DF1. (I,J) AND DF2(I,J) THE PARTS OF THE CONVECTIVE TERM. TO C BE USED IN SUBROUTINE SOLVE .

PRIM = PR*REG PRKA = PR*UNREG DO 60 I=2,MM PRKAC = PRKA/CSH(I) C1(I) = PRKAC*U(I,2) 02(I) =-PRKAC*U(I,M) DO 50 J=2,14 DF1 (I,J) = PRHK*DFI(I,J) DF2(I,J) = PRHK*DFJ(I,J)


C Y1=Y2 IF A MULTIPLE OF MXI ITERATIONS HAVE BEEN PERFORMED . C N1=0 ALLOWS MAIN TO START A NEW GROUP OF MXI ITERATIONS . C N1=1 ALLOWS MAIN TO CONTINUE THE OLD GROUP OF MXI ITERATIONS . C N3=1 SPECIFIES UNCONVERGED NEW CASE . C

XN = N XI = MXI YI = N/MX1 Y2 = XN/X1 NI = 1 IF(Y1.EQ.Y2) N1=0 N3 = 1 RETURN END

316 BOUNDC

SUBROUTINE BOUNDC DIMENSION U(50,50) I T(50,50) ,DF1 (50,50) I DF2 (50,50) ,DFI (50,50)

1DFJ (50,50) ,H3 (50,50) ISM (50) CSH(50) THH (50) , 'THETA (50 ) ,SN(50 ) , 2CS (50) , CT (50) B1 (50) B2 (50) (50) Bk (50) Cl (50) C2 (50) , BB1 (50) , 3BB2 (50) ,SRT (50 ) , ANU(50 ) TSN(50 ) HJ(50 ) AM(50) AMRAT (50 )

COMMON U, T, DF1 ,DF2 DFI DFJ H3 ISNH CSH, TNH, 'THETA SN, CS „ CT, B1 I B2, 1 B3 Bi+, C1 , C2 , BB1 BB2 ISRT, ANU, TSN, HJ, AM, AMRAT, E, RE , PR OMEGA , EPS ,SH, 2SK , MM, M1 „NMI ,NPR1 NPR INTAPE, NDTAPE, MX1 , MX2 , N, N1 , N2 N3 , NI+, N5 ,L1 3L2 , Lk ,L5 TN, REG , UNREG,S IS2 , KS BB3 , ZS , ALENDA

C CC SOLUTION OF THE IRREGULAR SPECIALLY-TREATED BOUNDARY CONDITIONS . C TN --- THE NEW ESTIMATE OF T(I l l) OR T(I,M1) . C APPLY THE TEST FOR RELATIVE CONVERGENCE AGAINST THE CRITERION EPS . C IF THE TEST IS PASSED, NPR IS REDUCED BY 1 . C REPLACE THE OLD VALUE OF T(I,1) OR T(I,M1) BY TN .

DO 200 I=2,1414 C NEW ES TIMATM OF T ( I , 1 ) ALONG THE AXIS OF SYMMETRY THETA = 0 .

TN = (BEIM-CI (I))*T(I-1,1)+(BB2(I)+C1(I))*T(I+1,1)+BE3*T(I,2) TN = OMEGA* (TN-T(I,1 ) )+T(I,1 ) IF (ABS ( (T ( I , 1 )-TN)/TN)-EPS )80, 80,85

80 NPR = NPR-1 85 T(I,1) = TN

C NEW ESTIMATES OF T (1 , M1 ) ALONG THE AXIS OF SYMMETRY THETA = 180 . TN = (BEI (I)..C2 (I) )*T(I-1 ,M1 )+(BB2(I)+C2 (I) )*T(I+1 ,M1 )+BB3*T(I ,M) TN = OMEGA* (TN-T (I,M1 ) )+T(I,M1) IF (ABS ( (T ( I M1 )-TN)/TN)-EPS )90, 90,100

90 NPR = NPR-1 100 T(I,M1) = TN 200 CONTINUE

RETURN END

SOLVE

SUBROUTINE SOLVE DIMENSION U(50,50) , T (50,50) ,DF1 (50,50) I DF2 (50,50) ,DFI (50,50) ,

1DFJ (50 150 ) ,H3 (50 150) „SNH(50) , CS H (50 ) TNH(50 ) 'THETA (50 ) ,SN(50) 3

2C6(50) 3 C11 (50)3 131(50),B2(50) 1 B3(50) 3 /34(50) 3 C1(50) 3 C2(50) 1 BB1 (50) 3BB2 (50 ) ISRT(50 ) ,ANU(50 ) TSN(50 ) I HJ(50) AM(50 ) ATHRAT (50 )

COMMON U T DF1 DF2 DFI ,DFJ ,H3 ,SNH ICSH, TNH, ITHETA SN, CS CT B1 B2, 1 B3 Bk, C1 C2, EB1 BE2,SRT, ANU, TSN, HtT, AM, AMRAT E, RE, PR OMEGA, EPS ,SH, 2SK 111,101, M1 I MM1 NPR1 NPR INTAPE , NDTAPE , MX1 I MX2 N , N1 N2 , N3 , „ N5 L1 , 32 ,L3,L4,L5 TN,REG UNREGISIS2 KS , EB3 , ZS ALENDA

C C SOLUTION OF THE FINITE DIFFERENCE EQUATIONS AT ALL REGULAR INTERIOR C GRID POINTS

317

C THE POINTS IN EACH ROW OF THE FIELD ARE COVERED SUCCESSIVELY STARTING C FROM LEFT TO RIGHT. THE ROWS ARE COVERED FROM BOTTOM TO TOP . C AT EACH GRID POINT, THE NEW ESTIMATE OF THE DEPENDENT VARIABLE IS C CALCULATED AS FOLLOWS = C 1. CALCULATE THE CONVECTIVE TERM D . C 2. CALCULATE THE CONDUCTIVE TERM TNN . C 3. ADD D TO TN . C 4. APPLY OVERRELAXATION (OR UNDERRELAXATION) CC TN -- THE LOCATION IN WHICH THE NEW ESTIMATED VALUE OF THE C DEPENDENT VARIABLE IS ACCUMULATED . C 5. TEST FOR THE RELATIVE CONVERGENCE AGAINST THE CRITERION EPS . C 6. IF THE TEST IS PASSED, NPR IS REDUCED BY 1. C 7. REPLACE THE OLD VALUE OF THE DEPENDENT VARIABLE BY TN .

DO 800 J=2,M DO 700 I=21 MM

C NEW ESTIMATES OF THE TEMPERATURE T(I1 J) . DF3 = T(I1-1,J)—T(I-1,J) DF4 = T(I,J+1)—T(I1J-1) D = DF1(I1 J)*DF3—DF2(I1J)*DF4 TN = B1(I)*T(I-1 01)+B2(I)*T(I-1-1,J)+B3(J)*T(I,J-1)+B4(J)*T(I1 J+1) TN = TN = OMEGA*(TN—T(I,J))+T(I,J) IF(ABSUT(I1 J)—TN)/TN)—EPS)500,500,600

500 NPR = NPR-1 600 T(I,J) = TN 700 CONTINUE • 800 CONTINUE

RETURN END

RESULT

SUBROUTINE RESULT DIMENSION U(50,50)1 T(50250),DF1(50150),DF2(50,50),DF1(50,50),

1DFJ(50150),H3(50,50),SNH(50),CSH(50),TNH(50),ITHETA(50),SN(50), 2CS(50),CT(50),B1(50),B2(50),B3(50),B4(50),C1(50),C2(50),SB1(50), 3BB2(50),SRT(50),ANU(50),TSN(50),HJ(50),AM(50)1 AMRAT(50) COMMON U,T,DF1,DF2,DFI,DFJ,H3,SNHI CSH,TNHI ITHETAISNI CS,CTIB1,B2,

1B31 B41 C1 1 C2I BB1 1 BB2ISRT,ANUI TSNIHJ,AM,AMRAT,E1REIPR,OMEGA,EPS,SH, 2SKAMMIM1 IMM1,NPR1,NPR,INTAPE,NDTAPEINX1 IMX2,N,N1 IN2,113,N41115,L1, 3L2sL311,41L51 TN,REGIUNREGISIS21KSIBB31 ZSI ALENDA

C C

PRINTING OUT THE TEMPERATURE DISTRIBUTION WRITE(61 10)RE,PR OWEGAININPR

10 FORMAT(1H1,1X1 4HRE =,F10.515X1 4HPE =,F10.515X17HOMEGA =,F10.515X, 13HN =1 15,5X,5HNPR =115)

C C C C C C

318

WRITE(61200)((T(I1J),I=11 20),J=11 M1) 200 FORMAT(1XIF3.1219F6.5)

RETURN END

NSNSEE

C

SUBROUTINE NSNSEE DIMENSION U(50250)1T(50,50)1DF1(50150)1DF2(50150),DF1(50150),

1DFJ(50150),H3(50150),SNH(50) 1 CSH(50)1TNH(50),ITHETA(50),SN(50), 2CS(50 ),CT(50),B1(50) 2132(50),B3(50),E4(50)1C1\50)1C2(50)1 BB1(50), 3BB2 (50) 1SRT (50 ) ANU(50 ) TSN(50 ) ,HJ(50) t AM(50)11114RAT (50)

COMMON U I T 2 DF1 3 DF2 DPI DFLT2H3 I SNH3 CSHI TITH I ITHETA7SNI CSI CT Bl I B2 1B31 134-1C11 C21 BB1 BB2 I BRT ANU TSN, HJ I AM, ArGIAT EI RE, PRI OMEGA t EPS 1SH 2SIC M 'MK, M1 , MFI1 NPR1 1 NPR INTAPE NDTAPEt MX1 t MX2 t NI N1 I N2 t N3 g Nk 115 L11 3L2 11.3 I L1411 L5 TN, REG t UNREG S IS 2 BB3 t ZS I ALENDA

C

FINAL RESULTS IF(L2.EQ.1) GO TO 21 00 SSH = -1.0/(12.0*SH) RERT = SQRT(RE) P113 = PR**(1./3.) DN1 = PR5*RE DN2 = PWRERT Y = O.

CALCULATION OF = ANU(J) ---- THE LOCAL NUSSELT NO. HJ(J) THE LOCAL J-FACTOR (HEAT TRANSFER) . AM(J) ---- THE LOCAL HEAT TRANSFER NO. AMRAT(J)--- THE RATIO OF THE LOCAL NUSSELT NO. TO THE FRONT STAGNAT-

ION NUSSELT NO. • DO 1500 J=11141

TS = SSH* (25.0-48.0*T(21 J)+36.0*T(31 J)-16.0*T(41J)+3.0*T(514) ANU(J) = TS*SRT(J) TSN(J) =-TS*SN(J) HJ(J) = ANU(J)/DN1 AM(J) = ANU(J)/DN2 AMRAT(J) = ANU(J)/ANU:(1) Y = Y+TSN(J)

1500 CONTINUE C OVERALL VALUES OF NUSSELT NO. , J-FACTOR AND HEAT TRANSFER NO.

AVNU• = SIS2*Y AVHJ = AVNU/DN1 AVM = AVNU/DN2

C CALCULATION OF HU/RE**1/2 AND NU/PR**1/3 . AVNURE = AVNU/RERT AVNUPR = AVNU/PR3

319

C PRINTING OUT ABOVE RESULTS . WRITE(6,1510)AVNU

1510 FORMAT(1H1,1X1 35HOVER-ALL NUSSELT NUMBER NU =1 E10.5) WRITE(6,1520)AVHJ

1520 FORMAT(2X135HOVER -ALL J-FACTCR (HEAT TRANSFER) =,F10.5) WRITE(6,1530)AVM,AVBURE,AVNUPR

1530 FORMAT(2X135HOVER-ALL M -FACTOR (HEAT TRANSFER) =1F10.5151,12HNU/RE 14,41/2 =iE10.5,5X,12HNV/PR**1/3 =1110-.5) WRITE(6,1540)

1540 FORMAT(1X,//120X15HTHETA,7X15HNU(J)16X16HTSN(J)17X15HHJ(J)17X1511 M 1(J),7X19HM(J)/M(1)) WRITE(611550)

1550 FORMAT(19X15(7H 15X)111H DO 1700•J=11 N1 WRITE(6,1600):THETA(J),ANU(J),TSN(J),HJ(J),AM(J),AMRAT(j)

1600 FORMAT(20X114,4X,F10.5,2X,F10.5,F13.5,2X1F10.5,6X,F6.3) 1700 CONTINUE 2100 IF(NDTAPE.NE.1) GO TO 2200 C IF NDTAPE=1, STORE INFORMATION ON TAPE 8 .

REWIND 8 WRITE(8)(Spli(I),CsH(I),TNH(I),I=1,1,3) WRITE(8)(ITHETA(J),SN(J),CS(J),CT(J),J=1,m1) WRITE(8)((H3(I,J),I=1,L3),J=1,m1) WRITE(8)((T(I,J),I=1,L3),J=1,M1) WRITE(8)((U(I,J),I=1,L3),J=1,M1) WRITE(8)((DFI(I,J),I=1,L3),J=1,M1) WRITE(8)((DFJ(I,J),I=1,L3),J=1,M1) WRITE(8)(B1(I),B2(I),BB1(I),BB2(I),I=1,L3) WRITE(8)(B3(J),B4(J),SRT(J),J=1,M1),5IS2

2200 RETURN END

APPENDIX H

320

TABLES 8 to 16

Table 8. Angles o1 Flow Separation and Wake Dimensions

Re ro h k 95 Dw / DC (degrees) (degrees)

Sphere 20 6.686 0.05 6 180 0.0 25 6.686 0.05 6 162 0.08 3o 6.686 0.05 6 156 0.14 40 6.686 0.05 6 149 0.28 5o 6.686 0.05 6 144 0.37 5o 6.686 0.025 3 144

bo 80

6.686 6.686

0.05 0.05

6 6

140 135

0.31

°

0.66 100 6.686 0.05 6 131 o.8o 100 5.0 0.025 3 134 150 6.686 0.05 6 126 1.08 150 5.o 0.025 3 1282 0.66

200 6.686 0.05 6 123 1.26 200 5,0 0.025 3 124 0.76 300 6.686 0.05 6 118A 1.59 300 5.0 0.025 3 119E- 0.88 400 6.686 0.05 6 115 1.85 400 5.0 0.025 3 116 0.98 500 6.686 0.05 6 112 2.01 500 5.0 0,025 3 1132 1.05

Oblate Spheroid: e = 0.8125 17,5 6.068 0.05 6 171 0.055 20 6.068 0.05 6 160 0.111 25 6.068 0.05 6 1511 0.175 30 6.068 0.05 6 146

6.068 0.05 40

6 140 0.401

50 60

6.068 6.068

0. 0 0.05

6 6

136 1321

°O.:427697655 0.60o

8o 6.068 0.05 6 128 100 6.o68 0.05 6 125 0.925

Oblate Spheroid: e = 0.625 10 9.0 0.05 6 - 20 9.0 0.05 6 143 0.207 30 9.o 0.05 6 134 0.430 50 7.3 0.05 6 126 0.690 100 6.o 0.05 6 118 1.125

Oblate Spheroid: e = 0.4375 10 8.o 005 6 160 0.076 20 8.o 0.05 6 131

= 3o 6.5 0.05 6 124 50 6.5 0.05 6 116 0.925 100 6.5 0.05 6 110 1.370

321

AMOON•m111.10 M•0110....1111.111

Table 9. Drag Coefficients of the Sphere 322

Re =s-12 h ko z o ro CDP

0.0001 1.3 0.1 12 1.9 6.686 217690.39344 0.001 1.3 0.1 12 1.9 6.686 21765.52002 0.005 1.3 0.1 12 1.9 6.686 4350.83200 0.01 1.3 0.1 12 1.9 6.686 2174.56000 0.05 1.3 0.1 12 1.9 6.686 434.93439 0.1 1.3 0.2 12 1.8 6.050 224.37078 0.1 1.3 0.1 12 1.8 6.050 227.10797 0.1 1.3 0.1 12 1.9 6.686 217.92320 0.1 1.3 0.1 12 2.4 11.023 191.01200 0.1 1.3 0.1 12 3.0 20.086 176.73680

0.1 1.3 0.1 12 4.o 54.598 168.62880 0.1 1.3 0.1 12 4.4 81.451 167.9764-6 0.1 1.3 0.1 12 4.6 99.484 167.96852 0.2 1.3 0.1 12 1.9 6.686 109.02000 0.5 1.3 0.2 12 1.8 6.050 45.07348 0.5 1.3 0.1 12 1.8 6.050 45.31551 0.5 1,3 0.1 12 1.9 6.686 43.76976 1.0 1.3 0.2 12 1.8 6.050 22.78132 1.0 1.3 0.1 12 1.8 6.050 22.89581 1.0 1.3 0.1 12 1.9 6.636 22.12088 1.0 1.3 0.1 12 2.4 11,023 19.96200 1.0 1.3 0.1 12 3.o 20.086 19.09904 1.0 1.3 0,1 12 4.0 54.598 13.92112 2.0 1.3 0.1 12 1.9 6.686 11.52704 3.0 1.3 0.1 12 1.9 6.686 8.03259 4.0 1.3 0.1 12 1.9 6.686 6.25936 5.0 1.3 0.2 12 1.3 6.050 5.37570 5.o 1.3 0.1 12 1.8 6.050 5.32255 5.0 1.3 0.1 12 1.9 6.686 5.22283 7.5 1.3 0.1 12 1.9 6.686 3.79619 10.0 1.0 0.2 12 1.8 6.050 3.18164 10.0 1.0 0.1 12 1.8 6.050 3.10159 10.0 1.0 0.1 12 1.9 6.686 3.05459 10.0 1.0 0.05 6 1.9 6.686 3.02968 10.0 1.0 0.05 6 2.4 11.023 2.87332 12.5 1.0 0.05 6 1.9 6.686 2.54423 15.0 0.9 0.05 6 1.9 6.686 2.23290 16.0 0.85 0.05 6 1.9 6.686 2.12877 17.0 0.85 0.05 6 1.9 6.686 2.03652 20.0 0.8 0.05 6 1.9 6.686 1.32000

25.0 0.6 0.05 6 1.9 6.686 1.55964 30.0 0.5 0.05 6 1.9 6.686 1.36662 40.0 0.35 0.05 6 1.9 6.686 1.12276 50.0 0.2 0.05 6 1.9 6.686 0.96678 50.0 0.18 0.025 3 1.9 6.686 0.96928 60.0 0.18 0.05 6 1.9 6.686 0.85905 80.0 0.16 0.05 6 1,9 6.686 0.71182 100.0 0.15 0.05 6 1.9 6.686 0.61631 100.0 0.15 0.025 3 1.6 5.0 0.57172 150.0 0.14 0.05 6 1.9 6.686 0.47081 150.0 0.12 0.025 3 1.6 5.o 0.42962

CDP

106841.45020 10682.01147 2134.99327 1066.79332 213.37378

109.35882 111.49405 106.89825 93.72233 86.47449 82.07256 81.62218 81.55971

52-3i:IX514 22.30314 21.48336 11.13714 11.32854 10.88799 9.802::

5.70

4 9.34100 9.19689

3 4.05149.17445 2.71381 2.72669 2.67662 1.99643 1.60317 1.67825 1.64366 1.63759 1.57008 1.46426

1:2= 1.21413 1.10700 0.98503 0.90406 0.79147 0.71757 0.72232 0.66332 0.58493 0.52770 0.55028 0.41411 0.46887

CDT

324531.84766 32447.53149 6485.82727 3241.35330 648.30817

333.72960 333.60201 324.82144 284.73433 263.21128

250.70136 249.59864 249.52823 162.45720 67.05302 67.62366 65.25312 33.91846 34.22435 33.00887 29.76487 28.44004 28.11801 17.26723 12.08407 9.43331 8.08950 8.04924 7.89945 5.7926? 4.78481 4.77984 4.69825 4.71727 4.44340 4.00849 3.53398 3.38232 3.25065 2.92700 2.54467 2.27068 1.91423 1.6('475 1.69160

1.52237 1.29676 1.14401 1.12199 0.38492 0.89850

323 Table

Re

9. continued. f/1 h =Q1

ko zo ro CDF C

DP CDT 200.0 0.13 0.05 6 1.9 6.686 0.39332 0.32273 0.71605 200.0 0.11 0.025 3 1.6 5.0 0.36408 0.43147 0.79555 300.0 0.10 0.05 6 1.9 6.686 0.30433 0.17068 0.47501 300.0 0.10 0.025 3 1.6 5.0 0.27269 0.36463 0.63732 400.0 0.07 0.05 6 1.9 6.636 0.26136 0.07149 0.33285 400.0 0.09 0.025 3 1.6 5.0 0.23369 0.34084 0.57453 500.0 0.05 0.05 6 1.9 6.686 0.23164 0.05848 0.29009 500.0 0.08 0.025 3 1.6 5.0 0.20538 0.31262 0.51800

Table 10. Drag Coefficients of the Oblate Spheroid e = 0.8125

Re f/1

=.111 h

z =1.13 ko zs

r 0 CDF C

DP CDT

0.0001 1.3 0.1 12 3.03 6.068 196194.04883 123129.08789 319323.13672 0.001 1.3 0.1 12 3.03 6.068 19619.46997 12313.01587 31932.48584 0.005 1.3 0.1 12 3.03 6.068 3923.91998 2462.62973 6386.54968 0.01 1.3 0.1 12 3.03 6.068 1961.94049 1231.24605 3193.18652 0.05 1.3 0.1 12 3.03 6.068 392.37250 246.11902 638.49152

0.1 1.3 0.1 12 3.03 6.068 195.00585 124.12504 319.13089 0.1 1.3 0.1 12 3.53 9.990 170.48362 107.36736 277.85097 0.1 1.3 0.1 12 4.13 18.192 156.81954 98.44499 255.26452 0.1 1.3 0.1 12 4.63 29.0 151.40097 94.72981 246.13078 0.1 1.3 0.1 12 5.13 48.0 148.87957 92.89449 241.77407 0.1 1.3 0.1 12 5.53 72.0 148.13590 92.26586 240.40176 0.2 1.3 0.1 12 3.03 6.068 98.42267 62.02890 160.45157 0.5 1.3 0.1 12 3.03 6.068 39.40131 24.74688 64.14819 1.0 1.3 0.1 12 3.03 6.068 19.90872 12.56768 32.47640 1.0 1.3 0.1 12 3.53 9.990 17.73866 11.14189 28.88054 1.0 1.3 0.1 12 4.13 18.192 16.'84944 10.56357 27.41301 1.0 1.3 0.1 12 4.63 29.0 16.67099 10.40719 27.07818 1.0 1.3 0.1 12 5.13 48.0 16.65472 10.37073 27.02545 2.0 1.3 0.1 12 3.03 6.o68 10.31745 6.58098 16.89843 3.0 1.3 0.1 12 3.03 6.068 7.16891 4.62029 11.73920 4.0 1.0 0.1 12 3.03 6.068 5.55937 3.61501 9.17438 5.o 1.3 0.1 12 3.03 6.068 4.64529 3.03446 7.67975 5.0 1.3 0.05 6 3.03 6.068 4.59521 3.04797 7.64318 7.5 1.3 0.05 6 3.03 6.068 3.33422 2.36610 5.70032 10.0 1.2 0.05 6 3.03 6.068 2.67984 1.89905 4.57889 10.0 0.9 0.05 6 3.53 9.990 2.53775 1.76002 4.29777 10.0 0.6 0.05 6 4.13 18.192 2.51504 1.72162 4.23666 12.5 1.2 0.05 6 3.03 6.068 2.27091 1.64791 3.91882 15.0 1.2 0.05 6 3.03 6.068 1.98498 1.47510 3.46007 15.5 1.1 0.05 6 3.03 6.068 1.93565 1.44831 3.38396 16.0 1.05 0.05 6 3.03 6.068 1.89372 1.41950 3.31322 17.5 0.8 0.05 6 3.03 6.068 1.77694 1.34970 3.12664 20.0 0.8 0.05 6 3.03 6.068 1.61644 1.25384 2.87028 25.o 0.6 0.05 6 3.03 6.068 1.38616 1.12/1111 2.51057 30.o 0.5 0.05 6 3.03 6.068 1.22455 1.03192 2.25647 40.0 0.4 0.05 6 3.03 6.068 1.00682 0.91076 1.91758

324

Table 10. continued.

Re h k z CDr CDP CDT

50.0 0.25 0.05 6 3.03 6.068 0.86234 0.36128 1.72362 60.0 0.20 0.05 6 3.03 6.068 0.76219 0.77308 1.53527 80.0 0.18 0.05 6 3.03 6.068 0.62939 0.68976 1.31915 100.0 0.17 0.05 6 3.03 6.068 0.54577 0.63368 1.17946

Table 11.

Re

Drag Coefficients of the Oblate Spheroid e = 0.625 zs=0.733

h k° z o ro CDF CDP CDT 0.01 1.3 0.1 12 3.933 20.0 1326.62115 1131.20706 2457.82822 0.1 1.3 0.1 12 3.933 20.0 132.75677 113.26054 246.01710 1.0 1.3 0.1 12 3.933 20.0 14.15613 12.05740 26.21353 5.0 1.1 0.1 12 3.633 15.0 3.63417 3.18575 6.81992 10.0 0.9 0.05 6 3.133 9.0 2.13502 2.02563 4.16065 20.0 0.7 0.05 6 3.133 9.o 1.31757 1.36510 2.68267 50.0 0.6 0.05 6 3.133 9.0 0.99868 1.12171 2,12039 50.0 0.2 0.05 6 2.933 7.3 0.71121 0.92833 1.63954 100.0 0.16 0.05 6 2,733 6.0 0.44910 0.74022 1.18932

Table 12.

S-2,‘ Re , 2.

Drag Coefficients of the Oblate Spheroid e = 0.4375 zs.o.469

h ko z r Cm) CDT 0 o CDF

0.01 1.1 0.1 12 3.969 24.0 1026,54089 1289.20782 2315.74872 0.1 1.1 0.1 12 3.969 24.0 103.30628 132.15099 235.45727 1.0 1.1 0.1 12 3.969 24.o 11.07264 14.0283? 25.10101 5.0 1.1 0.1 12 3.469 14.5 2.34638 3.71864 6.56502 10.0 0.9 0.05 6 2.869 8.0 1.67292 2.41942 4.09234 20.0 0.7 0.05 6 2.869 8.0 1.04035 1.60887 2.64922 30.0 0.6 0.05 6 2.669 6.5 0.79947 1.35616 2.15563 50.0 0.4 0.05 6 2.669 6.5 0.56317 1.10713 1.67030 100.0 0.13 0.05 6 2.669 6.5 0.35003 0.91140 1.26143

325

Table 13. Overall Nusselt Numbers for the Sphere

Values of the Relaxation Factor Used are Enclosed Between Brackets

Nu

Re 0.01 0.1 0.2 0.5 1.0 2.0 3.0

Pr 0.1 2.32750 2.32297 *Mr ••••

(1.3) (1.3) 0.4 2.32766 2.33344

(1.3) (1.3) 0.7 2.32257 2.33628 2.34329 2.3450o 2.40751 2.47586 2.60252

(1.3) (1.3) (1.3) (1.3) (1.3) (1.3) (1.3) 1.0 2.34327 2.34133 2.35901 2.41589 2.58009 2.75431

(1.3) (1.3) (1.3) (1.3) (1.3) (1.3) 2.0 2.35193 2.36599 2.41205 2.57260 2.88390 3.13835

(1.3) (1.3) (1.3) (1.3) (1.3) (1.3) 2.4 2.32963 2.33816 2.35353 2.44008 2.63902 2.98374 3.26393

(1.3) (1.3) (1.3) (1.3) (1.3) (1.3) (1.3) 5 2.33644 2.36155 2.40784 2.65333 2.99250 3.45743 3.84096

(1.3) (1.3) (1.3) (1.3) (1.3) (0.9) (0.8) 10 2.34220 2.41050 2.59560 2.99013 3.45482 4.03133 4.53741

(1.5) (1.3) (1.5) (1.4) (0.9) (0.6) (0.5) 15 2.48524 2.74209 3.24837 3.77990 4.40646 5.02387

(1.5) (1.5) (1.4) (0.8) (0.45) (0.3) 20 2.34657 2.59222 2.86653 3.44917 4.04124 4.69306 5.40.537

(1.5) (1.5) (1.5) (1.2) (0.6) (0.3) (0.25) 3o 2.34986 2.73091 3.09553 3.77232 4.45727 5.13719 5.99065

(1.4) (1.4) (1.4) (0.9) (0.45) (0.25) (0.2) 40 2.35472 2.86432 3.27667 4.03257 5.49581 6.43690

(1.4) (1.4) (1.4) (0.6) (0.2) (0.15) 50 2.36113 2.97400 3.45560 4.25370 5.06662 5.80761 6.80251

(1.4) (1.4) (1.0) (0.5) (0.3) (0.15) (0.1) 75 2.38195 3.26317 3.77799 4.70091 5.61865 6.46481 7.53966

(1.3) (1.3) (0.8) (0.4) (0.2) (0.1) (0.08) 100 2.41236 3.45126 4.03863 5.05539 6.04306 7.01291 8.17476

(1.3) (1.0) (0.6) (0.3) (0.15) (0.08) (0.06) 150 - 3.77647 4.45060 5.60692 6.68195 .00

(0.8) (0.5) (0.2) (0.1) 200 2.56976 4.02920 4.76827 6.03067 7.17121 -

(1.3) (0.6) (0.4) (0.15) (0.08) 250 - 4.23664 5.05977 - - ••••

(0.5) (0.3) 300 2.72849 4.44877 5.29278 6.66877 7.94935 IMP a•

(1.3) (0.5) (0.25) (0.1) (0.06) 400 2.86649 4.77115 5.69667 7.15413 8.62655 ONO

(1.3) (0.4) (0.2) (0.08) (0.05)


Nu

Re 0.01 0.1 0,2 0.5

Pr

2.98740 (1.3)

3.24308 (1.3)

5.05185 (0.3)

5.60090 (0.25)

6.02808 (0.15)

6.66667 (0.1)

7.55857 (0.06) 8.41065 (0.05)

500

750

1000 3.44572 6.01883 7.15206 9.19381 (1.2) (0.15) (0.08) (0.04)

2000 4.02803 - - - (0.6)

3000 4.44215 - . - (0.5)

4000 4.77208 - (0.4)

5000 5.04957 . - - (0.3)

6000 5.29024 - - - (0.25)

7000 5.50301 . - . (0.2)

8000 5.69295 - - - (0.15)

9000 5.86562 - - -• (0.1)

10000 6.02401 - - (0.08)

15000 6.66295 - - - (0.06)

20000 7.14761 (044)

Re = 10 50 100 500

0.1 - - 3.68497 6.69577 (0.8) (0.2)

0.7 3.34284 5.44341 6.92677 12.91109 (1.3) (0.5) (0.2) (0.1)

1.0 3.61995 6.03186 - 14.52680 (1.3) (o.4) (0.07)

2.4 4.55437 7.74119 - - (0.6) (0.2)

5.0 5.46844 9.42861 - (0.8) (0.1)

10.0 6.60848. ;1.44992 (0.2) (0.05)

.20.0 7.85958 14.27654 IMO

(0.1) (0.03) 30.0 8.28215 14.59871 ••• I•••

(0.06) (0.02) 40.0 - 17.92677 (0.01) - . 50.0 9.96852 (0.05) - - 75.0 10.55120 (0.04) - . - 100.0 11.05428 (0.03) - - -

326

327

Table 14. Overall Nusselt Numbers for the Oblate Spheroid: e=0.8125


Nu

Re 0.01 0.1 0.2 0.5 1.0

Pr

0.7 2.52299 2.43320 2.52308 2.52641 2.54757 (1.3) (1.3) (1.3) (1.3) (1.3)

1.0 2.52277 2,43603 2.52211 2.53255 2.57910 (1.3) (1.3) (1,3) (1.3) (1.3)

2.0 2.52201 2.43516 2.52716 2.57795 2.73014 (1.3) (1.3) (1.3) (1.3) (1.3)

2.4 2.52168 2.5236o 2.53135 2.60347 2.79545 (1.3) (1.3) (1.3) (1.3) (1.3)

5.0 2.52069 2.43714 2.57748 2.80894 3.16063 (1.3) (1.3) (1.3) (1.3) (1.3)

10.0 2.52007 2.49228 2.73252 3.15826 3.64674 (1.5) (1.5) (1.5) (1.4) (0.9)

15.0 2.52037 2.56171 2.89138 3.43026 3.98832 (1.5) (1.5) (1.5) (1.4) (0.8)

20.0 2.52113 2.63381 3.03459 3.64070 4.26339 (1.5) (1.5) (1.5) (1.2) (0.6)

30.0 2.52353 (1.4)

2.79227 (1.4)

3.26855 (1.4)

3.98082 (0.8)

4.(zior)

40.0 2.52765 2.93685 3.46058 4.25486 - (1.4) (1.4) (1.2) (0.6)

50.0 2.53321 3.05161 3.63883 4.48786 5.3476o (1.4) (1.4) (1.o) (0.5) (0.3)

75.0 2.55127 3.2826o 3.98010 4.96026 5.93623 (1.3) (1.3) (0.8) (0.4) (0.2)

100.0 2.57840 3.52473 4.25409 5.33621 6.39077 (1.3) (1.0) (0.6) (0.3)

150.0 - 3.84095 4.69150 5.92386 7(.°071758)6 (0.8) (0.5) (0.2) (0.1)

200.0 2.72798 4.10870 5.04095 6.37801 (1.3) (0.6) (0.4) (0.15) 7(r.gg

300.0 2.88774 4.51686 5.50197 7.06475 8.39193 (1.3) (0.5) (0.25) (0.1) (0.06)

400.0 3.03159 4.83353 6.02400 7.58272 9.03587 (1.3) (0.4) (0.2) (0.08) (0.05)

500.0 3.15662(1.3)5.13173(0.3)6.38035(.15)8.00560(.06) - 750.0 3.41551(1.3)5.70295(.25)7.06623(.10)8.84813(.05) - 1000.0 3.63730(0.9)6.09515(.15)7.58372(.08)9.55901(.04) -

2000.0 4.25210(0.6) - - .. -

5000.0 5.33238(0.3) - - - -

9000.0 6.20621(0.15) - - ... -

328


Nu

Re 2.0 3.0 10.0 50.0

Pr

0.7 2.63501(1.3) 2.75775(1.3) 3.36794(1.3) 5.72938(0.5)

1.0 2.73491(1.3) 2.90993(1.3) 3.70104(1.3)

2.o 3.04386(1.3) 3.30981(1.3) 4.45902(0.9) •••

2.4 3.15149(1.3) 3.46495(1.3) 4.69890(0.85) 8.13854(0.2)

5.0 3.65348(0.9) 4.04505(0.8) 5.71989(0.5) Nog

10.0 4.29277(0.6) 4.77785(0.5) 6.55752(0.1)

15.o 4.73628(0.45) 5.29203(0.3) MO

20.0 5.09003(0.3) 5.69727(0.25) 30.0 5.64672(0.25) 6.32243(0.2) 00

40.0 6.08088(0.2) 6.79972(0.15) 50.0 6.43783(0.15) 7.18763(0.1)

75.0 7.12763(0.1) 7.94704(0.08)

100.0 8.56109(0.06)

Table 15. Overall Nusselt Numbers for the Oblate Spheroid: e = 0.625


Nu ..11.•••••••••••••Iu

Re 0.01 0.1 1.0 5.0 10.0

Pr

0.7 2.41827(1.3) 2.42058(1.3) 2,56295(1.3) 3.14044(1.3) 3.64754(1.3) 2.4 2.40667(1.3) 2.40744(1.3) 2.87706(1.3) 4.02523(0.5) 4.89078(0.5) 5.0 2.39899(1.3) 2.45473(1.3) 3.24937(0.8) 4.77683(0.2) *PO

10.0 2.39385(1.5) 2.58999(1.5) 3.67805(0.4) 5.67399(0.1) 30.0 2.40992(1.4) 2.96549(0.9) 4.71187(0.2) 7.52541(0.08)

50.0 2.45757(1.4) 3.21709(0.8) 5.33335(0.1) 8.55160(0.06)

75.o 2.52346(1.3) 3.45718(0.6) 5.92387(0.08) -

100.0 2.58748(1.3) 3.65866(0.5) 6.38676(.06)10.39316(0.04) Ma

200.0 2.78973(1.3) 4.21488(0.3) 7.50735(.03)14.58753(0.02) 300.0 2.96613(1.3) 4.62962(0.2) 8.07584(.02) - 400.0 3.08323(0.9) 4.95822(0.1) 500.0 3.19450(0.8) 5.24957(0.1) 750.0 3.43587(0.6) - 1000.0 3.63934(0.4) - 2000.0 4.20256(0.25) -

329

Table 16. Overall Nusselt Numbers for the Oblate Spheroid: e = 0,4375


Nu abas..m••••••••...andRamd-ra•-•a•

Re 0.01 0.1 1.0 5.0 10.0

Pr --...

0.7 2.56537 2.56801 2.72645 3.30532 3.90263 (1.3) (1.3) (1.3) (1.3) (1.3)

1.0 2.55914 2.55479 2.75860 3.52514 11110

(1.3) (1.3) (1.3) (0.8)

2.4 2.54987 2.55115 3.06089 4.20850 5.08552 (1.3) (1.3) (1.3) (0.5) (0.5)

5.0 2.54300 2.60917 3.41589 4.96945 (1.3) (1.3) (0.8) (0.2)

10.0 2.53828 2.74693 3.79438 5.87968 (1.5) (1.5) (0.4) (0.1)

30.0 2.55811 3.12370 4.90374 7.82683 (1.4) (0.9) (0.2) (0.07)

50.0 2.61164 3.38033 5.55195 8.88954 (1.4) (0.8) (0.1) (0.05)

75.0 2.67906 3.56259 6.15458 - 11•111

(1.3) (0.5) (0.08)

100.0 2.74373 3.75431 6.62776 10.57907 (1.3) (0.4) (0.06) (0.03)

200.0 2.94281 (1.2)

4.36682 (0.3)

7.92630 (0.03)

13.(vm

300.0 3.10480 4.80720 8.80238 .1•1

(1.1) (0.2) (0.02) 400.0 3.23265 5.13243 9.43791 111•0 ea/

(0.9) (0.15) (.015) 500.0 3.36556 5.42678 - IAA IMO

(0.8) (0.1)

750.0 3.55998 6.02461 - (0.5) (0.08)

1000.0 3.73704 - - (0.3)

2000.0 4.34112 - - (0.2)

3000.0 4.76749 - - - - (0.1)

4000.o 5.10049 - - _ _ (0.08)

5000.0 5.39973 - - _ - (0.6)

....11••••••••subwrownwmgmomft..41....••

*RI&

330

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a numerical solution of navier-stokes and energy … · which can be used to generate solutions of...

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