mixed convective and radiative heat transfer in a...

Journal of Engineering Volume 18 May 2012 Number 5

552

MIXED CONVECTIVE AND RADIATIVE HEAT TRANSFER IN A HORIZONTAL CONCENTRIC AND

ECCENTRIC CYLINDRICAL ANNULI

Ass. Prof. Manal Hadi Al-Hafidh Lina S. Safwat Mech. Engr. Dept. Mechanical Engineer

College of Engineering [email protected] University of Baghdad

ABSTRACT

A numerical investigation has been performed to study the effect of eccentricity on unsteady state, laminar aiding mixed convection in a horizontal concentric and eccentric cylindrical annulus. The outer cylinder was kept at a constant temperature while the inner cylinder was heated with constant heat flux. The study involved numerical solution of transient momentum (Navier-Stokes) and energy equation using finite difference method (FDM), where the body fitted coordinate system (BFC) was used to generate the grid mesh for computational plane. The governing equations were transformed to the vorticity-stream function formula as for momentum equations and to the temperature and stream function for energy equation. A computer program (Fortran 90) was built to calculate the bulk Nusselt number (Nub) after reaching steady state condition for fluid Prandtl number fixed at 0.7 (air) with radius ratio ( =2.6), Rayleigh number (Ra=200), Reynolds number (Re=50) for both concentric and eccentric cylindrical annulus with different eccentricity ratios (ε=0, 0.25, 0.50, 0.75) and angular positions (φo=0o, 45o, 90o, 135o, 180o). The results show a reasonable representation to the relation between Nusselt number and (ε, φo). Generally, Nub decreased with the increase in (ε and φo). Also, results show that the best thermal performance for the inner cylinder was at the angular position (φo=0o) for eccentricity ratio (ε=0.25), while the maximum reduction in the rate of heat transfer for the inner cylinder was at the angular position (φo=180o) for eccentricity ratio (ε=0.75). Comparison of the result with the previous work shows a good agreement.

الخالصة

ع بين اسطوانتين أفقيتين لدراسة تأثير الالمركزية لجريان مختلط مساعد غير مستقر يق عددي بحث تم إجراء

متحدة وغير متحدة المركز، تم حفظ االسطوانة الخارجية بدرجة حرارة ثابتة بينما سخنت االسطوانة الداخلية

بينت الدراسة النظرية حل عددي لمعادالت الزخم والطاقة لمسألة البحـث باسـتخدام .بفيض حراري ثابت

في توليد شـبكة العقـد بعـد (BFC)إلحداثيات المطابقة للجسم المحددة وتم استخدام نظام ا تاطريقة الفروق

.الدوامية -تحويل هذه المعادالت إلى صيغة دالة االنسياب

553

MIXED CONVECTIVE AND RADIATIVE HEAT TRANSFER N A HORIZONTAL CONCENTRIC AND ECCENTRIC YLINDRICAL ANNULI

Ass. Prof. Manal Hadi Al-Hafidh Lina S. Safwat

بعد الوصول إلى حالة االستقرار لقيمة رقم برانتل ) Nu(لحساب رقم نسلت ) 90فورتران ( تم بناء برنامج

لـي ورقم را ) Re=50 (لدولخواص جريان ممثلة برقم رينو ) 2.6= ( قطر نصف لنسبةو) هواء (0.7ثابتة

)Ra=200 (ةلحالتي االسطوانتين المتحدة وغير متحدة المركز لنسب المركزي (ε=0, 0.25, 0.50, 0.75)

) φo =0o, 45o, 90o, 135o, 180o(ولموقع زاوي لالسطوانة الداخلية ضمن التجويف

على قيمة (ε, φo) من كل نسلت بواسطة رسم مخططات تمثل تأثيرأعطت النتائج تمثيال معقوال لمعدل رقم

Nu .وبصورة عامة وجد إن رقم نسلت يقل مع زيادة.(ε , φo) أفضل أداء حراري أيضا انبينت النتائج و

بينما كان أعظم انخفـاض (ε=0.25) ولنسبة المركزية(φo=0o)لالسطوانة الداخلية كان في الموقع الزاوي

. (ε=0.75) ولنسبة المركزية (φo=180o) انتقال الحرارة لالسطوانة الداخلية في الموقع الزاوي في معدل

.تم إجراء مقارنة مع البحوث السابقة وكانت جيدة

KEY WORDS: Flow and Heat Transfer, Laminar, Mixed Convection, Concentric and Eccentric, Horizontal Annulus.


554

INTRODUCTION: The process of heating and cooling of the flowing fluids inside channels was considered as one of the important subjects in heat transfer problems. Many researchers studied the heat transfer and fluid flow through the channels with different cross section areas to attain the best performance of the heat exchanger. Mixed convection heat transfer in horizontal ducts of concentric and eccentric cylindrical annular form has received increased attention due to the interesting feature of specific heat transfer phenomenon and fundamental importance in practical applications. An experimental and theoretical study has been conducted by [Akeel Al−Sudani, 2005] on mixed convection heat transfer of the flow through an inclined concentric annulus with uniformly heated inner cylinder and adiabatic outer cylinder with both fixed and rotating inner cylinder, little researches dealt with mixed convection in an eccentric annulus. [William, 1963], presented a solution for the temperature distribution in a fluid flowing in an eccentric annulus formed with circular cylinders under the assumption of slug flow. [Shu and Wu, 2001], presented an efficient numerical approach of using domain-free discretization method to solve partial differential equations on a doubly connected domain concentric and eccentric annulus. The consideration in the present study is given to laminar unsteady state mixed convection with radiation in concentric and eccentric horizontal annuls with the outer cylinder maintained isothermal while the inner cylinder was subjected to a uniform constant heat flux. Fig. (1) shows the annulus geometry and coordinate system of the problem under consideration

Important applications for mixed convection in an annulus may be summarized as follows: Double pipe heat exchangers, heating processes in nuclear reactors, the cooling of electrical equipments, the design of certain types of solar energy collectors and heating of process fluids [Yasin et.al 2006].

GOVERNING EQUATIONS: Unsteady steady state, quasi three-dimensional, incompressible, fully developed laminar aiding air flow was investigated. Accordingly the governing, continuity, momentum and energy conservation equations were as follows:- Continuity equation:

(1) Momentum Equations:

(2)

(3)

(4)

555



Energy Equation:

(5) The dimensionless parameters:

, De = 2(ro - ri ),

, , , ,

, ,

, , By using these dimensionless forms, the above equations can be written as follows.

(6)

(7)

(8)

(9)

(10)

The governing equations in dimensionless form above were written in terms of dependant variables . The pressure term in the momentum equations will be eliminated in the resulting vorticity equation as can be shown below:

(11) For this flow field, the only non-zero component of the vorticity is:

(12) Also by making use of the vorticity definition of equation (12) and the definition of stream function ( Ψ ), which satisfy continuity equation, the horizontal and vertical velocities can be written as follows respectively:-

(13)

(14)

By substituting the velocity components of equations (13) and (14) in the vorticity definition equation (12 ) , stream function equation resulted as :-

(15) Boundary Nodes: The positions of the inner and outer cylinders can be represented by the eccentricity (ε) and the angular position ( o), where (0o ≤ o ≤ 360°).


556

For an eccentric annulus, the origin was put at the center of the inner cylinder. So, ri(φ) was a constant, but ro(φ) was a function of φ. If the radius ratio is defined as , then the non-

dimensional radii of the inner and outer cylinders are Ri=1/( -1) and Ro= /( -1). Therefore, ri(φ) and ro(φ) are given by [Shu and Wu 2001] :-

(16)

(17) Initial Conditions: Initial conditions may be chosen as zero: At = 0 ,

U = V = W = Ω = Ψ = 0 [No slip

condition] The boundary conditions which defined by [Kotake and Hattori 1985] and [Kaviany 1986], making use that the boundary conditions for a motionless rigid surface which required that both horizontal and vertical velocities components (U and V) to be vanished at surface. This expressed in terms of stream function as follows:- • Inner cylinder surface :

U = V = = 0 ,

= - ,

(18)

• Outer cylinder surface :

U = V = W = = 0 ,

= - ,

(19)

TRANSFORMATION OF GOVERNING EQUATIONS:- Governing equations can be transformed from the Cartesian coordinates (X,Y) to generalized coordinates as shown below: 1- Vorticity-Transport Equation:-

(20) 2- Axial Momentum Equation:-

(21)

3- Energy equation :-

(22) 4- Stream Function Equation:-

(23) 5- Vertical Velocity:-

(24) 6- Horizontal Velocity:-

(25)

557



NUMERICAL SOLUTION: Explicit finite difference technique was the numerical method used for solving the transient behavior of the fluid flow and heat transfer until the steady state was reached by marching out in time steps (∆τ).

1- Discretization of Vorticity Equation:-

(26) Where:-

(27-a)

(27-b)

(27-c)

(27-d)

(27-e)

(27-f)

(27-g) 2- Discretization of Axial Momentum

Equation :-

(28)

Where:-

(29-a)

(29-b)

(29-c)

(29-d)

(29-e)

(29-f)

(29-g)


558

3- Discretization of Energy Equation :-

(30) Where:-

(31-a)

(31-b)

(31-c)

(31-d)

(31-e)

(31-f)

(31-g)

(31-h)

(31-i) 4- Stream Function Solving Method :-

(32) 5- Calculation of Average Axial Velocity :-

(33) 6- Calculation of Bulk Temperature :-

(34) 7- Calculation of Nusselt Number :-

(35) RESULTS AND DISCUSION: Numerical investigations was conducted for different eccentricity ratios ε in different angular positions φo of the inner cylinder within the physical domain. The isotherms and streamlines for different ε and φo and the results for variation of eccentricity with Nusselt number were shown in figures [(2) to (8)], in which five angular positions of (φo=0o, 45o, 90o, 135o, 180o) and four eccentricities of (ε = 0, 0.25, 0.50, 0.75) where considered . Isotherms & Streamlines:- Figs. [(2) to (7)] illustrate the isotherms and streamlines for Ra=200, Re=50, Pr=0.7and =2.6 with different values of ε and φo. Pure conduction heat transfer is increased with eccentricity, which is revealed by the increasing in specific conductivity Kε as a function of ε. Interpretation is as follows: because of

559



the narrowed gap in the large eccentricity annulus thermal convection by large recirculating vorticities become more and more difficult in contrast to the growing influence of the thermal conduction. Fig. (2) shows the isotherms and streamlines for concentric annulus. The streamlines are symmetric with respect to the vertical line, there will be a stagnant region in the lower part of the gap, in this region the natural convection effect will be low and it is identical to the case of thermally steady state fluid flow between two horizontal plates, when the upper plate is much higher than the lower one. Detailed isotherms and streamlines for eccentric annulus are presented in Figs.[(2) to (7)] respectively, the vortex strength current was un symmetric and the vortex strength will be increased in the wider part of the gap. Buoyancy plume will be deviated to the narrowed part of the gap and this deviation will be increased as ε increased due to the limitation in the fluid motion in the narrowed part and the buoyancy force will be unequal on each sides of the gap. As ε increased the buoyancy plume will be separated in the largest part of the gap due to the viscosity force. In the narrowest part of the gap, the conduction dominancy is readily recognizable from the isotherm plots. Also, as seen from the streamline contours, more and more fluid is mobilized in the convection currents with decreasing φo to deliver thermal energy from the inner heated cylinder to the outer cold cylinder. It is noted that the positioned influence on the heat transfer is felt more strongly from the isotherm plots than from the streamlines, since the temperature inversion phenomenon becomes very

distinguished as φo is decreased from (180o to 0o), this clearly indicates that the role of convection increases with lower φo. For high eccentricity the conduction dominating flow region at the narrowest gap of the annuli becomes locally stagnant which results in splitting of the core of the vortex in the constricted region into two sub vortices rotating in the same direction. At first, the vortex core only is halvened, but as the gap is further narrowed local stagnant region grows large enough to bisect the whole vortex even much before the two cylinders come into contact. In the wider part of the eccentric annulus the vortex current is slowed down and location of its core is lowered as eccentricity increased. This clearly indicates that the relative role of convection is steadily decreased with higher eccentricity, whereas the overall heat transfer is changed to increasing pattern after a slight decrease near ε=0.50. This is again surely due to the contribution of conduction for increased eccentricities. It is noted that the decreased degree of plume development and temperature inversion with higher eccentricities and the slowed –down stream speed together with the vortex halvening and all consistently related with the magical interaction between the conduction and the convection discussed so far. Also, the radiation effect plays a significant role with the position of the inner cylinder. Once the medium participates in the absorption and emission of radiation, the medium temperature tends to move uniform. Further more, as the participating medium alternates the radiation more, a direct interaction of the inner hotter cylinder with the cold outer cylinder, i.e., surface radiation, is decreased.


560

This is clearly evidenced by observing a downward shift of the isotherm closer to the outer cold cylinder for both cases of concentric and eccentric annulus as N increased. When the inner cylinder is displaced downward as shown in Fig. (3), the location of the convective cell center barely changes. Moreover compared to the pervious case, N has an insignificant effect in the medium temperature variation; this is derived from the fact that the thermo-fluid dynamics characteristics become buoyancy dominant. In other wards, when the inner cylinder is located at the downward positions the internal buoyancy-induced flow becomes stronger, which in turn results in higher heat transfer rate. The angular positions of φo=0o and φo=180o are two special cases in the eccentric annulus, for these two special cases, there is no global circulation. As a consequence, the flow and thermal fields are symmetric with respect to the vertical line connecting the centers of two cylinders. This can be clearly shown if Figs.[(3) and (7)]. When the inner cylinder is moved near the bottom, the outer cylinder has a boundary layer every where, when the inner cylinder is moved near the top, there is no boundary layer on the bottom portion of the outer cylinder. For φo=0o it is evident that the convective flows are both larger and stronger than the concentric annulus for low eccentricity ratio (ε=0.25, ε=0.5) but for large eccentricity for example (ε=0.75), the only effect which can be recognized is that the reduction in the rate of heat transfer and this is again due to the stagnant region in the narrowest gap. Also, φo=180o provides least favored circumstance for the development of the heat transfer, both the size and

strength of the fluid flow are markedly reduced. On the other hand, it was found that the global circulation of the flow does exist around the hot inner cylinder for eccentricity cases of (φo=45o, 90o and φo=135o) as shown in Figs. [(4), (5) and (6)] . For these cases, the computed Ψmax has a relatively large value. The magnitude of the circulation varies form zero for a concentric annulus to a maximum value for an intermediate eccentricity and back to zero for ε≈1. This is because for ε = 0 the flow field is symmetric, and no global circulation exists. When ε tends toward 1, the two cylinder surfaces are very close at some point so that there is no sufficient space for fluid flow. Therefore, the global circulation for this case is very weak. As can be shown from Fig. (6) that for φo=135o and ε = 0.50 and ε = 0.75, there will be a small vortex in the upper part of the gap. This small vortex will cause a deviation to the buoyancy plume to the largest part of the gap. Effect of Eccentricity on Nu: Fig.(8) illustrates the variation of Nu with angular location φo for different ε. For fixed eccentricity ε for example (ε=0.25), and the inner cylinder is moved circumferentially, by increasing φo , Nu will be decreased. This clearly indicates that the role of convection increases with lower φo . For fixed angular position for example (φo=90o) and different eccentricity, Nu will be decreased as ε increased from (0.25 to 0.75). That the relative role of convection is steadily decreased with higher eccentricity, this is again surely due to the contribution of conduction for increased eccentricity.

561



A correlation equation for the plotted curve of Nu for any eccentricity ratio and angular position had been written to show the eccentricity effect on the rate of heat transfer. Curve fitting method (Least square method) with two programs (Statistica and DGA) which were used to form this equation.

(36)

The above equation is valid for Re=50, Pr=0.7, Ra=200, =2.6 and

. Where, a1, b1 and c1 are constants and there values are as follows:-

Parameter Estimate

a1 6.80 for 6.03 for 5.94 for

b1 0.5 c1 -0.64

Comparison of Results: A comparison was made with the isotherms and streamlines resulted from the work of [Ho, Lin and Chen 1989] for natural convection heat transfer in an eccentric horizontal annulus with (Ra=106, ε=0.625 and φo=180o) as shown in Fig.(9), the results show a good agreement . CONCLUSIONS: For fixed eccentricity ratio and radius ratio, the overall heat transfer increased due to the expanded convection as the angular position φo of the inner cylinder decreased. And for a fixed angular position, when the inner cylinder was moved outward from the concentric position along a horizontal line, convection heat transfer decreases contrary to the conduction heat transfer which grows

with a faster rate. It was found that at ε=0.25 and φo=0o, maximum heat transfer will be recognized. REFERENCES: Akeel A. M. Al−Sudani, 2005, "An Investigation Into Laminar Combined Convection Heat Transfer Through Concentric Annuli"، PhD. Thesis, University of Technology, Mechanical Engineering Department . Ho C. J., Lin Y. H. and Chen T. C., 1989, "A Numerical Study of Natural Convection in Concentric and Eccentric Horizontal Cylindrical Annuli with Mixed Boundary Conditions", Int. J. Heat and Fluid Flow, Vol. 10, No. 1, PP. 40-47. Kaviany M., 1986, " Laminar Combined Convection in a Horizontal Annulus Subjected to Constant Heat Flux Inner Wall and Adiabatic Outer Wall", J. Heat Transfer, ASME, Vol.108, PP.392-397. Kotake S. and Hattori N., 1985, "Combined Forced and Free Convection Heat Transfer for Fully –Developed Laminar Flow in Horizontal Annuli", J. Heat and Mass Transfer, Vol.28, No.11, PP.2113-2120. Shu C. and Wu Y. L., 2001, "Domain Free Discretization Method for Doubly Connected Domain and Its Applications to Simulate Natural Convection in Eccentric Annuli", Internet, Science Direct, Vol. 191, Issues 17-18, PP. 1827-1841 . William T. Snyddi, 1963, "An Analysis of Slug Flow Heat Transfer in an Eccentric Annulus", A.I.Ch.E. Journal, Vol. 9, No. 4, PP. 503-506 .


562

Yasin K. Salman, Jalal M. Jalil and Ahmed F. Khudheyer, 2006, "Experimental Study of Mixed Convection Heat Transfer to

Thermally Developing Air Flow in a Horizontal Rectangular Duct", convection heat transfer through concentric annuli"، University of Baghdad .

563



1

1

1

1

1

1

1

2

2

2

2

2

2

3

3

3

3

3

3

4

4

4

4

4

5

5

5

5

5 6

6

6

6

6

7

77

7

8

8

8

8

8

9

9

9

9

10

10

Level TEMP.10 4.29679 4.01038 3.72387 3.43746 3.15095 2.86454 2.57803 2.00512 1.14581 0.5729

1

1

2

2

2

3

3

3

3

4

4

4

4

4

5

5

5

5

5

6

6

6

6

6

6

7

7

7

7

7

8

8

8

8

9

9

9

10

10LevelSTREAM10 1.77919 1.38378 0.98847 0.59306 0.19765 -0.19784 -0.59323 -0.98862 -1.38391 -1.7793

Figure (2) Isotherms & Streamlines for Concentric Annulus (φo=0o,ε=0.0)

Ra=200, Re=50, Pr=0.7, =2.6, N=3, =1


564

12

2

3

3

4

4

5

5

5

6

6

6

7

7

7

7

8

8

8

8

9

9

9

9

10

10

10

10

10

10

11

11

11

11

11

12

12

12

12

13

13

13

13

14

14

14

14

15

15

15

16

1617

17

17

18

18

19

20

LevelSTREAM20 2.937019 2.627818 2.318617 2.009516 1.700315 1.391214 1.082013 0.772912 0.463711 0.154610 -0.15469 -0.46378 -0.77297 -1.08206 -1.39125 -1.70034 -2.00953 -2.31872 -2.62781 -2.9370

1

1

11

1

1

1

2

2

22

2

2

23

3

3

3

3

34

4

4

4

4

4

5

5

5

5

5

55

6

66

6

6

6

7

7

7

7

7

7

8

88

8

8 9

9

9

9

10

1010

10

11

11

11

11

12

12

12

13

13

13

14

1414

15

15

15

16

16

17

17

17 18

1819

19

20

Level TEMP.20 5.652619 5.370018 5.087417 4.804716 4.522115 4.239514 3.956913 3.674212 3.391611 3.109010 2.82639 2.54378 2.26117 1.97846 1.69585 1.41324 1.13053 0.84792 0.56531 0.2826

1

2

3 3

4

4

5

5

6

6

6

7

7

7

78

8

8

9

9

9

9

10

10

10

10

10

11

11

11

11

11

12

12

12

12

13

13

13

1414

14

15

15

15

1616

16

17

17

18

18

19

20

LevelSTREAM20 4.349719 3.891818 3.434017 2.976116 2.518315 2.060414 1.602513 1.144712 0.686811 0.228910 -0.22899 -0.68688 -1.14467 -1.60256 -2.06045 -2.51824 -2.97613 -3.43402 -3.89181 -4.3497

1

1

1

1

1

1

2

2

2

2

2

2 3

3

33

3

3

4

4

4

4

4

5

5

5

5

5

6

6

6

6

6

7

7

7

7

8

8

8

8

8

9

9

9

9

10

10

10

11

11

11

11

12

12

12

13

13

13

14

14

14 15

15

15

16

16

16

17

17

1718

18

19

19

20

Level TEMP.20 6.706319 6.371018 6.035617 5.700316 5.365015 5.029714 4.694413 4.359112 4.023811 3.688510 3.35319 3.01788 2.68257 2.34726 2.01195 1.67664 1.34133 1.00592 0.67061 0.3353

1

2

3

3

4

4

5

5

6

6

67

7

7

8

8

8

8

9

9

9

9

10

10

10

10

10

11

11

11

11

12

12

12

12

13

13

13

14

14

14

15

15

16

16

16

17

17

18

1819

19

20

LevelSTREAM20 5.437919 4.865518 4.293117 3.720716 3.148315 2.575914 2.003513 1.431112 0.858711 0.286210 -0.28629 -0.85868 -1.43107 -2.00346 -2.57585 -3.14824 -3.72063 -4.29302 -4.86551 -5.4379 1

1

1

1

1

1

2

2

2

2

2

2

3

3

3

3

3

3

4

4

4

4

4

5

5

5

5

5 6

6

6

6

6

7

7

7

7

7

8

88

8

9

9

9

9

910

10

10

10

11

11

11

11

12

12

12

12 13

1313

1314

14

14

14 15

15

15

1516

16

16

16

17

17

1718

18

19

19

20

20

Level TEMP.20 7.700619 7.315618 6.930617 6.545516 6.160515 5.775514 5.390413 5.005412 4.620411 4.235410 3.85039 3.46538 3.08037 2.69526 2.31025 1.92524 1.54013 1.15512 0.77011 0.3850

Figure (3)

Isotherms & Streamlines for Eccentric Annulus (φo=0o) Ra=200, Re=50, Pr=0.7, =2.6, N=3, =1

565



1

1

2

2

3

3

4

4

4

5

5

55

6

6

6

6

7

7

7

7

7

8

8

8

8

8

9

9

9

9

10

10

10

10

11

11

11

11

12

12

12

13

13

13

1314

14

14

15

15

1516

16

1617

17

18

18

1919

2020

LevelSTREAM20 3.717319 3.418918 3.120617 2.822216 2.523915 2.225514 1.927113 1.628812 1.330411 1.032010 0.73379 0.43538 0.13707 -0.16146 -0.45985 -0.75814 -1.05653 -1.35492 -1.65321 -1.9516

1

1

1

1

1

11

2

2

2

2

2

2

2

3

3

3

3

3

33

4

4

4

4

4

4 5

55

5

5

56

6

66

6

6

7

7

7

7

7

8

8

8

8

8

8

9

9

9

9

10

10

10

10

11

11

11

11

1212

12

13

13

13

13

14

14

15

15

16

16

16

17

17

18

18

1919 20

20

Level TEMP.20 5.722719 5.436518 5.150417 4.864316 4.578115 4.292014 4.005913 3.719712 3.433611 3.147510 2.86139 2.57528 2.28917 2.00296 1.71685 1.43074 1.14453 0.85842 0.57231 0.2861

23

4

4

45

5

56

6

6

7

7

7

8

8 8

8

9

9

9

9

9

9

9

10

10

10

10

11

11

11

11

12

12

12

13

13

13

1414

14

15

15

15

16

16

17

17

18

18

19

1920

LevelSTREAM20 4.816919 4.377118 3.937317 3.497516 3.057815 2.618014 2.178213 1.738512 1.298711 0.858910 0.41919 -0.02068 -0.46047 -0.90026 -1.34005 -1.77974 -2.21953 -2.65932 -3.09901 -3.5388

1

1

1

1

1

1

12

2

2

2

2

2

2

3

3

3

3

3

3

4

4

4

4

4

4

5

5

5

5

5

6

6

6

6

6

7

7

7

7

7

8

8

8

8

9

9

9

9

10

10

10

11

11

11

11

12

12

12

13

13

13

14

14

14

14

15

15

15

16

16

1617

17

18

18

19 1920

20

Level TEMP.20 6.849219 6.506718 6.164317 5.821816 5.479315 5.136914 4.794413 4.452012 4.109511 3.767010 3.42469 3.08218 2.73977 2.39726 2.05485 1.71234 1.36983 1.02742 0.68491 0.3425

1

2

3

3

4

4

5

5

6 6

7

7

8

8

8

9

9

9

9

10

10

10

10

11

11

11

11

11

12

12

12

12

13

13

13

14

14

14

1415

15

15

16

16

17

17

1818

19

20

LevelSTREAM20 5.217919 4.660618 4.103317 3.546016 2.988615 2.431314 1.874013 1.316712 0.759411 0.202010 -0.35539 -0.91268 -1.46997 -2.02736 -2.58465 -3.14194 -3.69923 -4.25652 -4.81391 -5.3712

1

1

1

1

1

1

2

22

2

2

2

3

3

3

3

3

4

4

4

4

4

5

5

5

5

5

6

6

6

6

6

7

7

7

7

8

88

8

8

9

9

9

9

9

10

10

10

10

1111

11

11

12

12

12

13

13

13

14

14

14

1415

15

15

16

16

16

17

17

17

18

18

18

19

19

19

20

20

Level TEMP.20 7.874219 7.480518 7.086817 6.693016 6.299315 5.905614 5.511913 5.118212 4.724511 4.330810 3.93719 3.54348 3.14977 2.75606 2.36235 1.96854 1.57483 1.18112 0.78741 0.3937

1

2

2

2

2

2

3

3

3

3

3

3

4

4

4

4

4

44

5

5

5

5

5

6

6

6

6

6

7

7

7

7

7

8

8

8

89

9

9

9

10

10

10

1011

11

11

11

12

12

12

12

13

13

14

14

14

15

15

16

1617

17

18

18

19

20

LevelSTREAM20 4.252519 3.982418 3.712417 3.442316 3.172315 2.902214 2.632213 2.362112 2.092011 1.822010 1.55199 1.28198 1.01187 0.74186 0.47175 0.20174 -0.06843 -0.33852 -0.60851 -0.8786

1

1

1

1

1

1

12

2

2

2

2

2

2

3

3

33

3

3

4

4

4

4

4

4

4

5

5

5

5

5

5

5

6

6

66

6

6

7

7

77

7

78

8

8

8

8

8

9

9

9

9

9

10

10

10

11

11

11

11

11

12

12

12

13

13

13

14

14

14

15

15

15

16

16

17

17

18

18

19

20

20

Level TEMP.20 5.876719 5.582918 5.289117 4.995216 4.701415 4.407614 4.113713 3.819912 3.526011 3.232210 2.93849 2.64458 2.35077 2.05696 1.76305 1.46924 1.17533 0.88152 0.58771 0.2938

Figure (4)

Isotherms & Streamlines for

Eccentric Annulus (φo=45o) Ra=200,

Re=50, Pr=0.7, =2.6, N=3, =1


566

1

2 3

3

4

4

5

5

5

6

6

6

6

6

6

6

6

7

7

7

7

7

8

8

8

8

9

9

9

9

10

10

10

10

11

11

11

12

12

12

13

13

14

14

15

15

16

16

17

17

18

19

19

20

LevelSTREAM20 5.350119 4.963418 4.576617 4.189916 3.803215 3.416514 3.029813 2.643112 2.256411 1.869710 1.48309 1.09638 0.70967 0.32296 -0.06385 -0.45054 -0.83723 -1.22392 -1.61061 -1.9973

1

1

1

1

1

1

1

2

2

22

2

2

2

3

3

3

3

3

3

4

4

4

4

4

4

5

55

5

5

6

6

6

6

7

7

7

7

8

8

8

8

8

99

9

9

10

10

10

10

11

11

11

11

12

12

12

13

13

13

14

14

14

15

15

1616

16

1717

1718

18

19 2 0

Level TEMP.20 7.316719 6.950918 6.585017 6.219216 5.853415 5.487514 5.121713 4.755912 4.390011 4.024210 3.65849 3.29258 2.92677 2.56096 2.19505 1.82924 1.46333 1.09752 0.73171 0.365812

3

3

4

4

5

5

6

6

7

7

7

8

8

8

8

9

9

9

9

9910

10

10

10

10

11

11

11

11

11

12

12

12

12

13

13

13

14

14

14

1515

16

1616

17

17 18

18

19

20

LevelSTREAM20 5.357019 4.856518 4.356017 3.855516 3.355115 2.854614 2.354113 1.853712 1.353211 0.852710 0.35229 -0.14828 -0.64877 -1.14926 -1.64975 -2.15014 -2.65063 -3.15112 -3.65151 -4.1520

1

1

1

1

1

1

2

2

2

2

2

33

3

3

3

3

4

4

4

4

5

5

55

5

6

6

6

6

6

7

7

7

7

7

8

8

88

8

9

9

99

9

1010

10

10

11

11

11

11

12

12

12

12

13

13

13

13

14

14

14

14

15

15

15

15

16

16

16

17

17

17

1818

18

19

19

19

20

20

Level TEMP.20 8.474019 8.050318 7.626617 7.202916 6.779215 6.355514 5.931813 5.508112 5.084411 4.660710 4.23709 3.81338 3.38967 2.96596 2.54225 2.11854 1.69483 1.27112 0.84741 0.4237

1

1

2

2

3

3

4

4

4

5

5

5

5

6

6

6

6

6

6

7

7

7

7

7

8

8

8

8

9

9

9

9

10

10

10

10

11

11

11

12

12

12

13

13

13

14

14

14

15

15

16

16

16

17

1718

18

19

19

20

LevelSTREAM20 4.195719 3.890118 3.584517 3.278916 2.973315 2.667714 2.362113 2.056512 1.750911 1.445310 1.13979 0.83418 0.52857 0.22296 -0.08275 -0.38834 -0.69393 -0.99952 -1.30511 -1.6107

1

1

1

1

1

1

1

2

2

2

2

2

2

2

3

3

3

3

3

3

3

4

4

44

4

4

5

5

5

5

5

5

5

6

6

6

6

6

6

7

7

7

7

7

7 8

8

88

8

8 9

9

9

9

9

10

1010

10

10

11

11

11

11

11

12

12

12

12

1213

13

13

13

14

14

14

15

15

15

16

16

17

17

18

18

19

19

20

Level TEMP.20 5.865219 5.572018 5.278717 4.985416 4.692215 4.398914 4.105713 3.812412 3.519111 3.225910 2.93269 2.63938 2.34617 2.05286 1.75965 1.46634 1.17303 0.87982 0.58651 0.2933

1

2

2

3

3

4

44

5

5

5

5

56

6

6

6

6

6

7

7

7

7

7

8

8

8

8

89

9

9

9

10

10

10

11

11

11

12

12

12

13

13

13

14

14

15

15

16

16 17

17 18

18 1920

LevelSTREAM20 5.876119 5.460418 5.044717 4.629116 4.213415 3.797714 3.382013 2.966312 2.550711 2.135010 1.71939 1.30368 0.88807 0.47236 0.05665 -0.35914 -0.77483 -1.19042 -1.60611 -2.0218

1

1

11

1

1

2

2

2

2

2

2

23

3

3

3

3

3

3

4

4

4

4

4

4

5

5

5

5

5

6

6

6

6

6

7

7

7

7

7

8

8

8

8

9

9

9

9

9

10

10

10

10

10

11

11

11

11

12

12

12

12

13

13

13

14

14

14

14

15

15

15

16

16

17

17

18

19

1920

Level TEMP.20 7.707119 7.321718 6.936417 6.551016 6.165715 5.780314 5.395013 5.009612 4.624311 4.238910 3.85369 3.46828 3.08287 2.69756 2.31215 1.92684 1.54143 1.15612 0.77071 0.3854

Figure (5)


567



1

2

2

3

3

4

44

5

5

5

5

6

6

6

6

6

7

7

7

7

7

8

8

8

8

9

9

9

9

10

10

10

11

11

11

11

12

12

12

1313

13

14

14

15

15

15

16

16

17

17

1818

1920

LevelSTREAM20 6.551419 6.066318 5.581117 5.096016 4.610915 4.125714 3.640613 3.155412 2.670311 2.185210 1.70009 1.21498 0.72977 0.24466 -0.24055 -0.72574 -1.21083 -1.69592 -2.18111 -2.6662

1

1

1

1

1

1

2

2

2

2

2

2

3

3

3

3

3

4

4

4

4

4

4

5

5

5

5

5

6

6

6

6

6

6

7

7

7

7

7

7

8

8

8

8

9

9

9

9

9

10

10

10

10

11

11

11

11

12

12

12

1213

13

13

14

14

14

15

15

15

15

16

16

17

17

17

18

18

19

1920

Level TEMP.20 9.288819 8.824318 8.359917 7.895416 7.431015 6.966614 6.502113 6.037712 5.573311 5.108810 4.64449 4.17998 3.71557 3.25116 2.78665 2.32224 1.85783 1.39332 0.92891 0.4644

1

2

23

3

4

4

4

5

5

6

6

6

7

7

7

78

8

8

8

9

9

9

9

9

10

10

10

10

10

11

11

11

11

11

12

12

12

12

13

13

13

13

14

14

14

15

15

15

16

16

16

17

17

18

18

1920

LevelSTREAM20 3.132919 2.802918 2.473017 2.143016 1.813115 1.483114 1.153213 0.823212 0.493211 0.163310 -0.16679 -0.49668 -0.82667 -1.15656 -1.48655 -1.81644 -2.14643 -2.47632 -2.80631 -3.1362

1

1

1

1

1

1

1 2

2

2

2

2

2

23

3

3

3

3

3

3

4

4

44

4

4

5

5

5

5

5

5

5

6

6

6

6

6

67

7

7

7

7

7

8

8

88

8

8 9

9

99

9

910

10

1010

10

10

11

11

11

11

11

12

12

12

12

12

13

13

1313

13

13

14

14

14

14

15

15

15

15

16

16

16

16

17

17

18

18

18

19

19

20

20

Level TEMP.20 5.457419 5.184518 4.911717 4.638816 4.365915 4.093114 3.820213 3.547312 3.274511 3.001610 2.72879 2.45588 2.18307 1.91016 1.63725 1.36444 1.09153 0.81862 0.54571 0.2729

1

2

3

3

4

4

45

5

5

6

6

6

7

7

7

8

8

8

9

9

9

9

10

10

10

10

11

11

11

11

12

12

12

12

13

13

13

14

14

14

15

15

15

1616

17

17

18

18

1919 20

LevelSTREAM20 4.533819 4.056118 3.578417 3.100716 2.623015 2.145314 1.667613 1.189912 0.712211 0.234510 -0.24329 -0.72098 -1.19877 -1.67646 -2.15415 -2.63184 -3.10953 -3.58722 -4.06491 -4.5426

1

1

1

1

1

1

1 2

2

2

2

2

2

2

3

3

33

3

3 4

4

4

4

4

45

5

5

5

5

5

6

6

6

6

6

6

7

77

7

8

8

88

8

8

9

9

9

9

910

10

10

10

10

11

11

11

11

12

12

12

12

13

13

13

13

14

14

14

15

15

15

15

16

16

16

17

17

18

18

18

19

19

19

2020

Level TEMP.20 6.850819 6.508218 6.165717 5.823216 5.480615 5.138114 4.795513 4.453012 4.110511 3.767910 3.42549 3.08288 2.74037 2.39786 2.05525 1.71274 1.37023 1.02762 0.68511 0.3425

Figure (6) Isotherms & Streamlines for Eccentric Annulus (φo=135o)

Ra=200, Re=50, Pr=0.7, =2.6, N=3, =1


568

1

2

2

3

4

4

5

5

5

6

6

6

7

7

7

8

8

8

8

9

9

9

9

10

10

10

10

10

11

11

11

11

12

12

12

12

13

13

13

14

14

14 1515

16

16

16

17

17

18

18

19

19 20

LevelSTREAM20 5.632819 5.039418 4.445917 3.852416 3.258915 2.665414 2.071913 1.478412 0.885011 0.291510 -0.30209 -0.89558 -1.48907 -2.08256 -2.67595 -3.26944 -3.86293 -4.45642 -5.04991 -5.6434

1

1

11

1

1

2

2

2

2

2

23

3

3

3

3

3 4

4

4

4

4

4

5

5

5

5

5

6

6

6

6

6

7

7

7

7

7

8

8

8

8

8

8

9

9

9

9

10

10

10

10

11

11

11

11

12

12

12

12

13

13

13

13

14

14

14

14

15

15

15

15 16

16

16

16 17

17

17

18

18 19

1919

19

20

20

Level TEMP.20 8.160619 7.752518 7.344517 6.936516 6.528515 6.120414 5.712413 5.304412 4.896311 4.488310 4.08039 3.67238 3.26427 2.85626 2.44825 2.04014 1.63213 1.22412 0.81611 0.4080

0 30 60 90 120 150 180φ

4

5

6

7

8

9

10

Nu

ε = 0.25ε = 0.50ε = 0.75

o

Ra=200Re=50Pr=0.7R=2.6N=3 , t=1

Figure (7)


Figure (8) Effect of Eccentricity on

Nu

Present Work

569



θmax.= 0.2942 θmax.= 0.2584 Ψmin.= -25.0405 Ψmin.= -21.548

Figure (9) Comparison of Isotherms & Streamlines for Eccentric Annulus

(φo=180o,ε=0.625), Ra=106, Pr=0.7, =2.6, N=0, =0 NOMENCLATURE: LATIN SYMBOLS:

Symbol Description Unit

Radius Ratio ( ) ---

Optical Thickness ( ) --- A Axial Pressure Gradient N/m3

C Axial Temperature Gradient ( ) K/m De Hydraulic Diameter De = 2(ro - ri ) m e Space Between the Centers of the Inner and Outer Cylinders m g Gravitational Acceleration m/s2 J Jacobean of Direct Transformation --- K Thermal Conductivity of the Air W/m.K Kr Volumetric Absorbtion Coefficient m-1 N Radiation-Conduction Parameter ( ) --- n Dimensionless Outer Normal Direction --- Ni Number of Gridlines in the φ-direction --- Nj Number of Gridlines in the r-direction ---


570

Nub Bulk Nusselt Number --- P Normalized Air Pressure --- p Air Pressure N/m2 Pr Prandtl No. ( ) ---

Ra Rayleigh Nnmber ( ) --- Symbol Description Unit

Re Reynolds Number (

) ---

Ri Dimensionless Inner Cylinder Radius --- ri Inner Cylinder Radius m

RO Dimensionless Outer Cylinder Radius --- ro Outer Cylinder Radius m T Air Temperature K t Time Second

u, v, w Velocity Components in x, y and z Direction Respectively m/s

U, V, W Dimentionless Velocity Components in X, Y and Z Direction Respectively ---

x, y, z The physical Coordinates of The Annulus m X, Y, Z The Dimensionless Physical Coordinates of The Annulus ---

GREAK SYMBOLS:

Symbol Description Unit Angular coordinate around inner cylinder Degree

, , γ Coefficient of Transformation of BFC. ---

o Angular position for the inner cylinder Degree

α Thermal Diffusivity Coefficient of Transformation of BFC

m2/s ---

β Coefficient of Thermal Expansion Coefficient of Transformation of BFC

1/K ---

ε Dimensionless Eccentricity (ε = e / De) --- ζ , η Coordinates in The Transformed Domain m θ Dimensionless Air Temperature ( ) ---

σ Stefan Boltizman Constant Relaxation Parameter

W/m2K4 ---

τ Dimensionless Time ( ) --- υ Kinematic Air Viscosity m2/s Ψ Dimensionless Air Stream Function --- Ω Dimensionless Air Vorticity --- Є Emissivity ---

mixed convective and radiative heat transfer in a...

Documents