influence of thermal radiation on free convection inside a porous

Emirates Journal for Engineering Research, 12 (2), 47-52 (2007) (Regular Paper)

INFLUENCE OF THERMAL RADIATION ON FREE CONVECTION INSIDE A POROUS ENCLOSURE I. Zahmatkesh Department of Mechanical Engineering, Shiraz University, Shiraz, Iran [email protected] (Received January 2007 and accepted June 2007)

‫يھدف البحث إلى دراسة أثر انتقال الحرارة باإلشعاع على مجال الحركة بالحمل داخل حيز مملوء بمادة مسامية‬ ‫( للمادة المائعة‬Laminar) ‫ تم عمل الحسابات النتقال الحرارة تحت ظروف جريان منتظم‬.‫مشبعة بالمادة المائعة‬ ‫ وتقريب‬Darcy ‫ في الحل تم استخدام كالً من قانون‬.‫ والطاقة‬،‫ الزخم‬،‫عن طريق الحل العددي لمعادالت حفظ المادة‬ ‫( لكن‬gray) ‫ اما من حيث اإلشعاع الحراري فقد تم اعتبار المائع من النوع الرمادي‬.Oberbeck-Bousinesg ‫ الضوابط‬.‫ لالنتشار لحساب مقدار اإلشعاع في معادلة حفظ الطاقة‬Rosseland ‫ ومع استخدام تقريب‬،‫بدون تشتيت‬ ‫ بينما الحائطين‬،‫ إثنان من الحائطين المتقابلين لھما درجات حرارة ثابتة لكن مختلفة‬:‫الحدية المستخدمة في الحل ھي‬ ‫( وبمقارنة نتائج ھذا البحث مع نتائج منشورة سابقا ً فقد لوحظ تطابق‬Adiabatic) ً ‫اآلخرين تم عزلھما حراريا‬ ‫( والمجال‬Flow) ‫ وبھذا فإن الموديل المستخدم تم التحقق من صحة أدائه للتنبؤ بمجال الحركة‬.‫ممتاز بينھم‬ ‫ على الحائطين غير المعزولين حراريا ً والتوزيع الحراري‬Nusselt ‫ إضافة لذلك فإن التغيرات في عدد‬.‫الحراري‬ .‫على الحائطين المعزولين تم دراستھم تحت ظروف إشعاع مختلفة‬ This investigation is undertaken to study the influence of thermal radiation on the development of free convective flow inside an enclosure filled with a fluid-saturated porous medium. The laminar free convective heat transfer is calculated by solving numerically the mass, momentum, and energy balance equations, invoking Darcy's law and the Oberbeck-Boussinesq approximation. In addition, the fluid is considered gray, absorbing-emitting, but non-scattering and the Rosseland diffusion approximation is used to describe the radiative heat flux in the energy equation. As thermal boundary conditions of the enclosure, two opposite walls are kept at constant but different temperatures while the other two are maintained adiabatic. Comparison between the results of the present numerical model with previously published works provides excellent agreement. The validated model is used to study the influence of thermal radiation on the development of flow as well as thermal fields. In addition, variations of Nusselt number on the non-adiabatic walls and temperature distribution on the adiabatic walls are investigated under different values of radiation parameter. Keywords: Porous medium; Thermal radiation; Free convection; Enclosure

1. INTRODUCTION Due to wide range of industrial applications of porous medium in the construction of different devices ranging from electrical heaters[1] to solar collectors[2], considerable interest has been drawn to study flow as well as thermal fields inside these materials. One of the issues in this field goes back to the development of free convective flows inside enclosures filled with fluid-saturated porous medium. This problem, finds its application in diverse practical situations including cooling of radioactive waste containers, grain storage, terrestrial heat flow through aquifer, and heat exchange in granular insulating materials. The presence of such a wide application has made the problem of free convection inside porous enclosures as one of the mostly emphasized subjects in the literature concerning porous medium. Among others, Moya et al.[3], Baytas and Pop[4], Saeid and. Pop[5],

Misirlioglu et al.[6], and Oztop[7] have contributed with some important theoretical contributions to this topic. All of the aforementioned studies have concerned two-dimensional rectangular enclosures in which, free convection is the only heat transfer mode and the literature is less abundant for the cases wherein thermal radiation is present. In this paper, is studied the influence of thermal radiation on the development of free convective flow inside an enclosure filled with a fluid-saturated porous medium. The laminar flow and heat transfer is calculated by solving the balance equations for mass, momentum, and energy. Here, Darcy's law and the Oberbeck-Boussinesq approximation are invoked. In addition, the Rosseland diffusion approximation is used for thermal radiation inside the gray, absorbingemitting, but non-scattering medium.

47

I. Zahmatkesh

2. MATHEMATICAL FORMULATION Free convective flow in the presence of thermal radiation inside an enclosure filled with fluid-saturated porous medium is concerned here. Physical model of the 2D porous enclosure is depicted in Figure. 1. Here, the left wall of the enclosure is maintained at temperature TH which is warmer than the right wall with the temperature of TC . Additionally, the top and bottom walls of the enclosure are maintained adiabatic. In order to facilitate the solution of the governing equations, several assumptions are adopted. These assumptions include: 1. The developed flow is laminar. 2. Darcy’s model is appropriate for flow prediction inside the porous medium. 3. There is local thermal equilibrium between the medium and the fluid. 4. The porous medium has isotropic and homogenous permeability. 5. The fluid physical properties are constant, except the density in the body force term in the momentum equation for which the OberbeckBoussinesq approximation is invoked. 6. The fluid is a gray, emitting-absorbing, but nonscattering medium. 7. The optically thick radiation limit is appropriate for the thermal radiation where the radiative flux term can be simplified by using the Rosseland diffusion approximation. 8. The flux of thermal radiation in y-direction is negligible compared with x-direction. Under these assumptions and by using the cold wall temperature ( TC ) as the reference temperature, the conservation equations for the analysis are written as: Continuity equation

∂u ∂v + =0 ∂x ∂y

(1)

K ∂p u=− μ ∂x

(2)

K ∂p T − TC + Kgβ μ ∂y υ

(3)

(6)

It is assumed that the temperature differences within the flow are such that the term T 4 can be considered as a linear function of temperature. This is accomplished by expanding T 4 in a Taylor series about TC4 and neglecting higher order terms. Thus[9];

T 4 ≅ 4TTC3 − 3TC4

(7)

In the view of (7), the energy equation becomes:

u

⎛ ∂ 2T ∂ 2T ⎞ 16σ 3 ∂ 2T ∂T ∂T +v = α ⎜⎜ 2 + 2 ⎟⎟ + Tc ∂y ∂x ∂y ⎠ 3ρac p ∂x 2 ⎝ ∂x

The boundary conditions to be satisfied for the problem are expressed as:

2

⎞ 1 ∂qr ⎟− ⎟ ρc ∂x p ⎠

(4)

Eliminating pressure terms in the momentum equations by applying cross-differentiation gives:

∂u ∂v Kgβ ∂T − =− ∂y ∂x υ ∂x

u = 0 , v = 0 , and T = TH

(9a)

At the left wall ( x = L and 0 < y < L ):

⎛∂ T ∂ T ∂T ∂T +v = α ⎜⎜ 2 + 2 ∂y ∂x ∂y ⎝ ∂x 2

48

4σ ∂T 4 qr = − 3a ∂x

At the right wall ( x = 0 and 0 < y < L ):

Energy equation

u

The quantity qr on the right hand side of Eq. (4) represents the radiative flux in y-direction. By utilizing the Rosseland diffusion approximation, the flux inside the optically thick medium is calculated as[8]:

(8)

Momentum equation

v=−

Figure 1. Physical model of the 2D porous enclosure.

(5)

u = 0 , v = 0 , and T = TC

(9b)

At the bottom wall ( 0 < x < L and y = 0 ): u = 0 , v = 0 , and ∂T / ∂y = 0

(9c)

At the top wall ( 0 < x < L and y = L ): u = 0 , v = 0 , and ∂T / ∂y = 0

(9d)

Introducing streamfunction (ψ ) as;

Emirates Journal for Engineering Research, Vol. 12, No.2, 2007

Influence of Thermal Radiation on Free Convection Inside A Porous Enclosure

u=

∂ψ ∂ψ and v = − ∂x ∂y

(10)

by which the continuity equation is automatically satisfied, and the dimensionless parameters in the form of;

x Dimensionless width: X = L y Dimensionless height: Y = L

(11a) (11b)

Dimensionless temperature: Θ =

4σ

4 R ∂Θ ⎤ ⎡ Nu = − ⎢(1 + d ) 3 ∂X ⎥⎦ X =0,1 ⎣ ___

Dimensionless streamfunction: Ψ =

Radiation parameter: Rd =

The quantities of physical significance are the local and the average Nusselt numbers at the non-adiabatic walls. As the flow and thermal fields are developed, these quantities are calculated from the following expressions:

1

∫

Nu = Nu dY

(16)

0

ψ α

(11c)

T − TC TH − TC

(11d)

Tc3

(11e)

ak

Darcy-modified Rayleigh number: Ra =

(15)

gβ ΔTKL

να

(11f) the governing equations, Eqs. (5) and (8), reduces to the following dimensionless form:

∂ 2Ψ ∂ 2Ψ ∂Θ + = − Ra 2 2 ∂ X ∂X ∂Y

(12)

4R ⎞ ∂ 2Θ ∂ 2Θ ∂Ψ ∂Θ ∂Ψ ∂Θ ⎛ − = ⎜1 + d ⎟ 2 + 3 ⎠ ∂X ∂Y ∂X ∂X ∂Y ⎝ ∂Y 2

(13)

Also, the corresponding boundary conditions become: At the right wall ( X = 0 and 0 < Y < 1 ):

Ψ = 0 and Θ = 1 At the left wall ( X = 1 and 0 < Y < 1 ):

(14a)

Ψ = 0 and Θ = 0

(14b)

Ψ = 0 and ∂Θ / ∂Y = 0 At the top wall ( 0 < X < 1 and Y = 1 ):

(14c)

Ψ = 0 and ∂Θ / ∂Y = 0

(14d)

At the bottom wall ( 0 < X < 1 and Y = 0 ):

4. GRID INDEPENDENCE AND VALIDATION To obtain a grid suitable for the range of Darcymodified Rayleigh number studied here ( Ra ~ 10 − 100 ), a grid independence test is performed and the results are listed in Table 1. Clearly, a 100 × 100 grid can be expected to provide acceptable accurate results for the whole range of the flows considered here. In order to verify the accuracy of the present numerical investigation, the results of the developed model are compared with the previously published works under two different Darcy-modified Rayleigh numbers in Table 2. Here, the role of radiation is not taken into account in the model to be similar with the previous investigations. As can be seen, under the both cases, the present model provides acceptable results. Table 1. Grid independence test. Grid size

Ra = 10

Ra = 100

200x200

1.069

3.085

100x100

1.067

3.026

50x50

1.055

2.950

25x25

1.032

2.812

Table 2. Comparison of the mean Nusselt number with previously published works. Author

3. SOLUTION PROCEDURE The resulting dimensionless coupled differential equations (Eqs. (12) and (13)) are solved simultaneously along with the corresponding boundary conditions (Eqs. (14)). Here, a control volume, finite-difference approach [10] is applied. The governing equations are converted into a system of algebraic equations through integration over each control volume. The algebraic equations are solved by a line-by-line iterative method. The method sweeps the domain of integration along the x and y axes and uses Tri-Diagonal Matrix Algorithm (TDMA) to solve the system of equations. The convergence criterion employed is the maximum residuals of all variables which must be less than 1× 10 −5 .


Said and Pop Oztop

[5]

Ra = 10

Ra = 100

-

3.002

-

2.980

1.065

2.801

[4]

1.079

3.16

Misirlioglu et al. [6]

1.119

3.05

Present simulation

1.067

3.026

[7]

Moya et al. [3] Baytas and Pop

5. RESULTS AND DISCUSSION The predicted streamlines as well as isothermal lines for the case wherein the value of radiation parameter is taken as 8.0 are depicted in Figure 2. By referring to the figure, the physics of the problem at hand can be understood. As can be seen, close to the hot isothermal wall (the right wall), the fluid is heated and expands, thus giving rise to an ascending motion. The fluid

49

I. Zahmatkesh

changes its direction when reaching the neighborhood of the top horizontal wall, proceeds nearly in the horizontal direction from left to right, and changes direction when reaching the neighborhood of the cold vertical wall. As the fluid releases heat there, its temperature decreases, it becomes denser and sinks down. Close to the lower horizontal wall the flow is essentially horizontal. These results in the establishment of a closed-loop for fluid flow which transfers heat from the hot wall to the cold wall[11]. It is obvious that the value of local Nusselt number is high in the lower and upper corners for the left and right walls, respectively. This corresponds to the presence of higher temperature gradient there. To investigate the influence of thermal radiation on the development of the free-convective flow, distribution of the predicted streamlines and isotherms in the absence of thermal radiation are shown in Figure 4. Comparison of this figure with Figure 2, wherein thermal radiation is taken into account, reveals that thermal radiation is of profound effect on the establishment of the flow as well as thermal fields inside the enclosure. In fact, it makes temperature distribution nearly uniform in vertical sections inside the enclosure. This in turn causes the streamlines to be nearly parallel with the vertical walls. Figure 5 indicates the variations of the average Nusselt number with the radiation parameter under different Darcy-modified Rayleigh numbers. As can be seen, under a constant radiation parameter, increase in the Darcy-modified Rayleigh number always raises the value of the average Nusselt number. This is obvious due to enhanced free-convective flow inside the enclosure. In addition, it is observed from the figure that, due to augmented heat transfer by radiation, the value of the average Nusselt number rises by increasing the radiation parameter. As can be seen the increase is almost linearly while the difference between the results of the various Darcymodified Rayleigh numbers decreases as the radiation parameter increases. This occurs since the role of radiation becomes the dominant heat transfer mode under high values of the radiation parameter. Variations of the adiabatic wall temperatures along the width of the enclosure under two different radiation parameters can be observed from Figure 6. As it is clear from the figure, temperature gradient is high in the right and left sides for the top and bottom walls, respectively. As seen previously, presence of thermal radiation may completely change the developed flow and thermal fields inside the enclosure. Thus, although in this study the flux of thermal radiation is considered just in the x-direction, it can influence adiabatic temperatures of the horizontal walls. This effect can be seen from Figure 6. Indeed, thermal radiation makes the distribution of the adiabatic wall temperature, nearly uniform.

50

a) Streamlines

b) Isothermal lines Figure 2. Distributions of streamlines as well as isothermal lines inside the enclosure. (Ra=100 and Rd=8)

6. CONCLUDING REMARKS The influence of thermal radiation on the development of free convective flow inside an enclosure filled with fluid-saturated porous medium was investigated here. It was shown that, thermal radiation is of profound effect on the establishment of the flow as well as thermal fields inside the enclosure. The results indicated that, presence of thermal radiation makes temperature distribution nearly uniform in the vertical sections inside the enclosure and causes the streamlines to be nearly parallel with the vertical walls. In addition, it may lead to nearly uniform adiabatic wall temperatures. It was also found that, average Nusselt number rises almost linearly with increasing the radiation parameter.


Influence of Thermal Radiation on Free Convection Inside A Porous Enclosure

15

10

Nu

5

1

1

0.8

0.8

0.6

0.6

0.4

0.4

0.2

0.2

0

0 0

0

5

Nu

10

15

Figure 3. Variations of the local Nusselt number along the height of the enclosure

Average Nusselt number

20 16

Ra=100 Ra=50 Ra=10

12 8 4 0 0

a) Streamlines

2

4 6 Radiation parameter

8

10

Figure 5. Variations of the average Nusselt number with radiation parameter under different Darcy-modified Rayleigh numbers.

Adiabatic wall temperatures

1

0.8 0.6

Top wall, Rd = 0 Top wall, Rd = 8 Bottom wall, Rd = 0 Bottom wall, Rd = 8

0.4

0.2

0 0

0.2

0.4

X

0.6

0.8

1

Figure 6. Variations of the adiabatic wall temperature along the width of the enclosure with radiation parameter (Ra=100). b) Isotherms Figure 4. Predicted streamlines and isothermal lines in the absence of thermal radiation. (Ra=100 and Rd=0)


51

I. Zahmatkesh

NOMENCLATURE 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18.

a cp g k K L

Nu ___ y Nu p Ra Rd T TC TH ΔT u x, y X ,Y

mean absorption coefficient constant pressure specific heat (J/Kg.K) gravitational acceleration (m/s2) thermal conductivity (W/m.K) permeability (m2) enclosure height (m) local Nusselt number average Nusselt number pressure (Pa) Darcy-modified Rayleigh Number radiation parameter temperature at any point (K) temperature of the cold wall (K) temperature of the hot wall (K) temperature difference, ΔT = TH − TC (K) velocity components in x- and y-direction (m/s) Cartesian coordinates (m) dimensionless coordinates

Greek symbols 19. 20. 21. 22. 23. 24.

ρ μ υ α σ ψ

density (kg/m3) dynamic viscosity (kg/m.s) kinematic viscosity (m2/s) effective thermal diffusivity (m2/s) Stefan-Boltzman constant (KW/m2 K4) streamfunction (m2/s)

25. 26.

Ψ Θ

dimensionless streamfunction dimensionless temperature

52

REFERENCES 1. Zahmatkesh, I. and Yaghoubi, M., 2006. Studies on Thermal Performance of Electrical Heaters by Using Porous Materials, International Communications in Heat and Mass transfer, Vol. 33, 259-267. 2. Chen, W. and Liu, W., 2004. Numerical analysis of heat transfer in a composite wall solar-collector system with a porous absorber, Applied Energy, vol. 78, 137-149. 3. Moya, S.L., Ramos, E., and Sen, M., 1987. Numerical study of natural convection in a tilted rectangular porous material, International Journal of Heat and Mass Transfer, Vol. 30, 741–756. 4. Baytas, A.C. and Pop, I., 1999. Free convection in oblique enclosures filled with porous medium, International Journal of Heat and Mass Transfer, Vol. 42, 1047-1057. 5. Saeid, N.H. and Pop, I., 2004. Transient free convection in a square cavity filled with a porous medium, International Journal of Heat and Mass Transfer, Vol. 47, 1917–1924. 6. Misirlioglu, A., Baytas, A.C. and Pop, I., 2005. Free convection in a wavy cavity filled with a porous medium, International Journal of Heat and Mass Transfer, Vol. 48, 1840-1850. 7. Oztop, H.F., 2007. Natural convection in partially cooled and inclined porous rectangular Enclosures, International Journal of Thermal Sciences, Vol. 46, 149-156. 8. Sigel, R. and Howell, J.R., 1992. Thermal radiation heat transfer, 3rd ed., Hemisphere, New York. 9. Quaf, M.E.M., 2005. Exact solution of thermal radiation on MHD flow over a stretching porous sheet, Applied Mathematics and Computation, Vol. 170, pp. 1117-1125. 10. Patankar, S.V., 1980. Numerical heat transfer and fluid flow, Hemisphere/ McGrawHill, Washington DC. 11. Bejan, A., 1984. Convection heat transfer, Wiley, New York, 110-111.
