me3122 handbook of heat transfer equations 2014 final
TRANSCRIPT
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HANDBOOK OF EQUATIONS, TABLES
AND CHARTS FOR
ME3122/ME3122E HEAT TRANSFER
Department of Mechanical Engineering
National University of SingaporeNovember 2014
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CONDUCTION HEAT TRANSFER
1stlaw of thermodynamics: WQdU
Conduction:
Convection:
Radiation: where -4-28 KWm10675 .
Control Volume:
Surface:
Heat Conduction Equation:
Cartesian:
Cylindrical:
Spherical:
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One-Dimensional Walls
Fin Equations:
kA/hPmmdx
d where02
2
2
which has the general solution mxmx eCeC 21 .
Fin Efficiency:
Fin Effectiveness:
Overall Surface Efficiency:
ftf
t
t
max
t
o A
NA
hA
q
q
q
11
0
whereunfinnedft
ANAA .
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Lumped Capacitance Method:
,
, , ,
Other Equations (Thermal Properties):
Solids:
Free electrons:
Gases:
Joule heating: RIEg2
Interfaces:
Heat wave speed:
Two semi-infinite solids touch:
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CONVECTION HEAT TRANSFER
All symbols have their usual meaning.
ConstantsGravitational acceleration: g = 9.81 m/s2
Specific gas constant for air: R= 287 J/kgK
Definitions
Kinematic viscosity, /
Thermal diffusivity, / pck
Volumetric thermal expansion coefficient,TT p
11
for an ideal gas.
General
Dimensionless Groups
uc
h
PrRe
NuSt
PrGrRa
LTTgGr
k/hLNu
/Pr
/VL/VLRe
p
x
x
xx
LL
sL
L
L
Number,Stanton
Number,Rayleigh
Number,Grashof
Number,Nusselt
Number,Prandtl
Number,Reynolds
2
3
Tcm
y
u
VAm
RTpv
TThq
p
c
s
sectionaghflux throuenergyThermal
stress,Shear
rate,flowMass
:lawgasIdeal
Cooling,ofLawsNewton'
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2D Continuity Equation:
2D x-Momentum Equation:
2D Energy Equation:
where viscous dissipation,
2D Boundary Layer Equations:
x-Momentum Equation:
Energy Equation:
Integral Momentum Equation:
Integral Energy Equation:
Forced Convection Over External Surfaces
Generally,nm
PrReCNu
Forced Convection Over a Flat Plate:
For constant , .
Mean heat transfer coefficient,11
0
L
x
A
x dxhL
dAhA
h
0
y
v
x
u
Xy
u
x
u
x
p
y
u
vx
u
u
2
2
2
2
qy
T
x
Tk
y
Tv
x
Tucp
2
2
2
2
222
2y
v
x
u
x
v
y
u
2
2
y
u
y
uv
x
uu
2
2
y
T
y
Tv
x
Tu
00
)(
yy
udyuuu
dx
d
0
0
yy
TdyTTu
dx
d t
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Uniform Surface Temperature (Isothermal):
For laminar flow (5Re 5 10
x ):
;5 3121 PrRex tx ;
For turbulent flow (5
Re 5 10x ):
Pr02960;05920;370 31
51 5451
xxxx,fxturb Re.NuRe.CRex.
For mixed boundary layer conditions ( 5105LRe ):
Uniform Surface Heat Flux (Isoflux):
For laminar flow (5Re 5 10
x ):
For turbulent flow (5Re 5 10x ):
31
Pr03080 54xx Re.Nu
For Unheated Starting Length,xo, with laminar flow for both isothermal and isofluxconditions:
Forced Convection Across Long Cylinders:
where Cand mare given byReD C m
0.4-4 0.989 0.330
4-40 0.911 0.385
40-4000 0.683 0.466
4000-40,000 0.193 0.618
40,000-400,000 0.027 0.805
31213121 6640;3320 PrRe.kLhNuPrRe.Nu LLxx
)8710370(;1742074080151 3
1
.LLLLL,f Re.Prk
LhNuReRe.C
31
21
4530 PrRe.Nu xx
21
21
3281;66402
2
xL,fxx,s
x,f Re.CRe./u
C
31430
1
x/xNuNu oxxx o
31PrReCk
DhNu mDD
3
12
1
6800
111
000 PrRek.
Lqdx
Nuk
xq
Ldx
h
q
LdxTT
LTT
L
sL
x
sL
x
sL
ss
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(a)Aligned tube rows (b) Staggered tube rows
For : where C2for various is given in the table
below:
20220 LL NDND NuCNu
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Forced Convection in Tubes and Ducts
Friction factor,
PerimeterWettedAreasectional-Cross4Diameter,Hydraulic hD
For thermally fully-developed condition:
Laminar Flow (ReD2300):
Fully developed velocity profile:
where mean fluid velocity,
Friction factor, f= 64/ReD
Nuand ffor Fully-Developed Laminar Flow in Tubes of Various Cross-Sections
dx
dpr
r
mum
8
20
2
0
2
0
2
12)(
r
r
u
ru
m
2
or2
2
2
m
m
u
D
Lfp
/u
Ddx/dpf
0)()(
)()(
xTxT
x,rTxT
x ms
s
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Turbulent Flow (ReD> 2300):
For smooth tubes and ducts, the Dittus-Boelter equation:
with n= 0.4 for heating of fluid, and n= 0.3 for cooling of fluid
Friction factor for smooth tubes: 26417900 .Reln.f D
Friction factor for rough tubes of roughness e: 290745733251 .DRe/.D./eln.f
Reynolds-Colburn Analogy
For flow over a flat plate:
For flow in a tube or duct:
FREE CONVECTION
Generally,
flow.entfor turbul31andflow,laminarfor41with mmRaCPrGrCNu mLm
LL
Laminar Free Convection on an Isothermal Vertical Plate:
Boundary layer momentum equation:
Integral Momentum Equation for Free Convection BL:
Boundary layer thickness,
Critical Ra= 109.
Free Convection from an Isothermal Sphere
n
DD PrRe.Nu hh540230
2;2 3232 /CPr.St/CPr.St L,fLx,fx
832 /fPr.St
2
2
y
uTTg
y
uv
x
uu
00
2dyTTg
y
udyu
dx
d
s
414121 9520933 xGrPr.Prx.
541 101for4302 D/
DD GrPrGr.k
DhNu
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Free Convection from Isothermal Planes and Cylinders
Free Convection from a Vertical Plate with Constant Surface Heat Flux
where
mLm
LL RaCPrGrCNu
Geometry GrL Pr C mCharacteristic
Length
Vertical plane and cylinder104109 0.59 1/4
Height10 10 0.10 1/3
Horizontal cylinder
10-1010-2 0.68 0.058
Diameter
10- 10 1.02 0.148
102104 0.85 0.188
104109 0.53 1/4
1091012 0.13 1/3
Hot surface facing up or
cold surface facing down
104107 0.54 1/4Area/Perimeter
1071011 0.15 1/3
Hot surface facing down or
cold surface facing up1051011 0.27 1/4 Area/Perimeter
161341
11551
10102for170:Turbulent
1010for600:Laminar
Pr*GrPr*Gr.Nu
Pr*GrPr*.Gr.k
xhNu
xxx
xxx
x
2
4
k
xqg.NuGr*Gr sxxx
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RADIATION HEAT TRANSFER
Solid angle: 2/ rAn , ddsind
Radiation:where
-4-28
KWm10675
.
surssursr
sursr
"
rad
TTTTh
TThq
22
Spectral directional Intensity:
Diffuse emitter:
Blackbody: 4)( TTEb
Spectral black body emissive power
).)T/Cexp(
C)T,(E b, m(W/m
1
2
2
5
1
m.K104391and/mmW.107423where 42248
1 .C.C
Weins displacement law: m.K2898max T
Emissivity of real surfaces:
4)()()( TTETTE b
Absorptivity of surface:
GGabs
Semitransparent medium: 1
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Black Body Radiation Functions
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View factors:
32
133122132
AA
FAFAF ,
Radiation exchange between black-body surfaces:
Radiation network approach:
resistancespatial1re whe1
resistancesurface1where1
121
121
2112
FA/FA/
JJq
A/A/
JEq b
Radiation Exchange Network for a Two-Surface Enclosure
22
2
21111
1
4
2
4
112
111AFAA
TTq
,
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View factor for aligned parallel rectangles
View factor for coaxial parallel disks
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Correction Factor Charts
lmTUAFq
Correction Factor for Heat Exchanger with One Shell
Passand Two (or Multiples of Two)Tube Passes.
Correction Factor for Heat Exchanger with Two Shell
PassesandFour (or Multiples of Four)Tube Passes.
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-NTU Charts for Heat Exchangers
Effectiveness of parallel flow heat exchangers Effectiveness of counterflow heat exchangers
Effectiveness of Heat Exchangers with One Shell
Passand Two (or Multiples of Two)Tube Passes.
Effectiveness of Heat Exchangers with Two Shell
PassesandFour (or Multiples of Four)Tube Passes.
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Effectiveness of Single-Pass Cross-Flow HeatExchangers withBoth Fluids Unmixed.
Effectiveness of Single-Pass Cross-Flow HeatExchangers with One Fluid Mixed, and the
Other Unmixed.
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Heat Exchanger Effectiveness Relations
Heat Exchanger NTU Relations
Use the above two equations with