- Email: [email protected]
Numerical study of heat transfer in an optically thick semi-transparent spherical porous medium
Journal of Quantitative Spectroscopy & Radiative Transfer 91 (2005) 363 – 372
www.elsevier.com/locate/jqsrt
Numerical study of heat transfer in an optically thick semi-transparent spherical porous medium B. Ch.erif∗ , M.S. Sifaoui Laboratoire de Rayonnement Thermique, Departement de physique, Faculte des Sciences de Tunis, Campus universitaire, Tunis 1060, Tunisia Received 23 May 2003; accepted 17 December 2003
Abstract In this paper, heat transfer by simultaneous convection, conduction and radiation in a semi-transparent spherical porous medium is investigated. The ROSSELAND approximation is adopted to take account of radiation in the heat transfer rate. The routine used here to solve the set of di;erential equations is taken from the IMSL MATH/LIBRARY. Various results are obtained for the dimensionless temperature pro?les in the solid and @uid phases, the radiative, conductive, convective and total heat @uxes. The e;ects of some radiative properties of the medium on the heat transfer rate are examined. ? 2004 Elsevier Ltd. All rights reserved. Keywords: Conduction; Radiation; Convection; Porous medium; Optically thick; Spherical geometry
1. Introduction In recent years, heat transfer by simultaneous radiation, conduction and convection in semitransparent porous media has attracted our attention. So we have studied the problem in an in?nite anisotropic scattering steady state porous medium using the Discrete Ordinates Method combined with an asymptotic analysis [1,2]; in a transient rectangular semi-transparent porous medium [3] and recently in a cylindrical semi-transparent optically thick porous medium [4]. Here, the considered medium consists of a pile of homogeneous spherical particles contained between two isothermal hollow spheres with radii R1 and R2 , respectively. The medium receives a transparent gas @owing through it at a constant velocity as shown in Fig. 1. ∗
Corresponding author. E-mail address: [email protected] (B. Ch.erif).
0022-4073/$ - see front matter ? 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.jqsrt.2003.12.012
364
B. Cherif, M.S. Sifaoui / Journal of Quantitative Spectroscopy & Radiative Transfer 91 (2005) 363 – 372
Fig. 1. Concentric spheres separated by a semi-transparent porous medium.
For the mathematical analysis, the following basic assumptions are made: (a) steady state; (b) the @uid @ow and heat transfer are one dimensional; (c) all the thermophysical properties are constant; (d) the solid is absorbent, participating, scattering and it behaves as an optically dense gray body for radiation; (e) the temperature gradients in the solid particles are negligible; (f) the @uid phase is incompressible and the mass @ow rate at every cross-section of the medium is constant. The optically thick approximation media [5] are used to take an account of radiation e;ect on the heat transfer rate. 2. Formulation The mathematical formulation of the problem is based on the conservative form of @uid and solid energy equations. To normalise the problem, we introduce the reference temperature Tr = T (r)Tf 1 and the space-radial coordinate r = dp r ∗ . Hence, we use Ts (r) = Tf 1 s (r);
Tf (r) = Tf 1 f (r):
These governing equations under consideration are given in [6]. For the solid phase: 2 ds 1 d 2 s + ∗ ∗ = Hs (s − f ) + div(’r (r ∗ )) ∗ 2 dr r dr ks
(1)
(2)
with Hs = 6kr Nu 2r ; kr = kf =ks where kf and ks are the @uid and the solid thermal conductivity, respectively. ’r (r ∗ ) is the radiative heat @ux in the r ∗ direction and Nu is the Nusselt number. r is the optical thickness ratio, r = 2 = d , where 2 = R2 denotes the optical thickness of the second sphere and d = dp of the particles of diameter dp . For the @uid phase: d 2 f 2 df df + ∗ ∗ = A ∗ − Hf (s − f ) (3) ∗ 2 dr r dr dr
B. Cherif, M.S. Sifaoui / Journal of Quantitative Spectroscopy & Radiative Transfer 91 (2005) 363 – 372
365
with Hf = 6Nu[(1 − )=] 2r and A = (Pr Re=) r . is the porosity; Pr and Re are, respectively, the Prandlt and Reynolds numbers. The boundary conditions are de?ned as follows: T s1 and f 1 (r ∗ = r1∗ ) = 1; s1 (r ∗ = r1∗ ) = Tf 1 f 2 (r ∗ = r2∗ ) =
Tf 2 T s2 = s2 (r ∗ = r2∗ ) = = 2 (r2∗ ); Tf 1 Tf 1
(4)
where Ts1 = Ts (r ∗ = r1∗ ), Tf 1 = Tf (r ∗ = r1∗ ); Tf 2 = Tf (r ∗ = r2∗ ), Ts2 = Ts (r ∗ = r2∗ ). The subscripts 1 and 2 refer to the boundaries at r = R1 and R2 . By using the Rosseland approximation [4,6] we take ’r (r ∗ ) = −
16n2 Ts3 (r ∗ ) dTs (r ∗ ) ; 3R2 dr ∗
(5)
where is the Rosseland mean extinction coeKcient, n denotes the refractive index and is the Stefan–Boltzmann constant. The term 16n2 Ts3 (r ∗ )=3R2 can be considered as the ‘radiative thermal conductivity’ [6]. Under the above consideration, the solid conservative energy equation takes the following form: 2 2 d d 2 ds 4 2 d d 2 s s s s ; (6) + = Hs (s − f ) − N 3s2 + s3 − ∗ ∗ dr ∗2 r ∗ dr ∗ 3 dr ∗ dr ∗2 r dr where N = ks =4n2 Tf31 is the Planck number representing the radiation–conduction parameter. The conductive and convective heat @uxes are dTs ’ c = ks ; dr ’cv = h(Ts − Tf );
(7)
where h = kf Nu=dp is the @uid exchange coeKcient. The Nusselt number is given by the following relation for particle diameter such as dp ¿ 10−3 m [7]: Nu = 2:0 + 1:8Pr 0:33 Re0:5 :
(8)
The total dimensionless heat @ux in the medium is determined from ds 4 3 ds T = ’t =ks Tf 1 =dp = − ∗ − + kr r Nu(s − f ): (9) dr 3N s dr ∗ The geometry and coordinates for the concentric spheres separated by a semi-transparent porous medium are shown in Fig. 1. 3. Numerical solution The governing equations for the problem considered here, Eqs. (3)–(6) and the boundary conditions (4) are nonlinear di;erential equations depending on the dimensionless radial coordinate r ∗ , conduction–radiation parameter N , Reynolds number and thermal conductivity ratio kr . In the
366
B. Cherif, M.S. Sifaoui / Journal of Quantitative Spectroscopy & Radiative Transfer 91 (2005) 363 – 372
present analysis, the numerical technique used here namely BVPFD solves a (parameterised) system of di;erential equations with boundary conditions at two points, using a variable order, and a variable step-size ?nite di;erence method with deferred corrections. It is taken from the IMSL MATH/LIBRARY. 4. Results and discussion The numerical results are presented graphically for conduction-to-radiation parameter N , conductivity ratio kr , optical thickness ratio r and the Reynolds number Re. These results are obtained for a heat transfer by simultaneous radiation, conduction and convection in a semi-transparent spherical porous medium. For the calculations we have taken: s1 = 0:25; f 1 = 1; r1∗ = 0:5; and r2∗ = 1. In Tables 1, 2 and 3we give the numerical values of the conduction, radiation and the total heat @uxes at the boundaries for various combination of the Planck number N , the Reynolds number Re and the thermal conductivity ratio kr . In Fig. 2, for kr = 0:02, r = 1 and Re = 100, we demonstrate the e;ect of the Planck number on the solid temperature distribution. This ?gure shows that the temperature in the solid phase increases by decreasing N due to the radiation which becomes important as shown in Fig. 4. Table 1 Planck number e;ect on the boundary @uxes with: kr = 0:02, r = 1 and Re = 100 N
T (r1∗ )
r (r1∗ )
0.01 0.1 1
128.3799 15.1117 4.5235
126.8618 13.8062 2.2610
ds
dr ∗ r ∗ 1
0.9515 1.03555 1.6958
T (r2∗ )
r (r2∗ )
31.4160 3.5086 0.5405
27.6378 2.1929 0.0772
ds
dr ∗ r ∗ 2
1.7783 1.3157 0.4633
Table 2 Reynolds number e;ect on the boundary @uxes with: kr = 0:02, r = 1 and N = 0:1 Re
T (r1∗ )
r (r1∗ )
100 200 300
15.1117 15.3630 15.5549
13.8062 13.9475 14.0551
ds
dr ∗ r ∗ 1
1.0355 1.0461 1.0541
T (r2∗ )
r (r2∗ )
3.5086 3.4407 3.3887
2.1929 2.1504 2.1180
ds
dr ∗ r ∗ 2
1.3157 1.2303 1.2708
Table 3 Thermal conductivity ratio e;ect on the boundary @uxes with: N = 0:1, r = 1 and Re = 100 kr
T (r1∗ )
r (r1∗ )
0.02 0.06 0.1
15.1117 16.3258 17.5279
13.8062 14.4333 15.0491
ds
dr ∗ r ∗ 1
1.0355 1.0825 1.1287
T (r2∗ )
r (r2∗ )
3.5086 3.2808 3.0595
2.1929 2.0505 1.9122
ds
dr ∗ r ∗ 2
1.3157 1.2303 1.1473
B. Cherif, M.S. Sifaoui / Journal of Quantitative Spectroscopy & Radiative Transfer 91 (2005) 363 – 372
367
Dimensionless solid temperature
1
N=0.001 N=0.1 N=10
0.9
0.8
0.7
0.6
0.5 0.5
0.6
0.7
0.8
0.9
1
Dimensionless radial coordinate r*=r/dp
Fig. 2. Planck number e;ect on the dimensionless solid temperature.
Dimensionless fluid temperature
0.5 N=0.001 N=0.01 N=10
0.45
0.4
0.35
0.3
0.25 0.5
0.55
0.6
0.65
0.7
0.75
0.8
0.85
0.9
0.95
1
Dimensionless radial coordinate r*=r/dp
Fig. 3. Planck number e;ect on the dimensionless @uid temperature.
Fig. 3 shows the variation of the @uid temperature in the medium for kr = 0:02, r = 1 and Re = 100, for three values of Planck number. We note that the @uid temperature is a;ected by this number which by decreasing, the radiation becomes relatively important compared to conduction and convection, hence this yields a heating of the medium. Fig. 4 represents the e;ect of the Planck number N on the radiative heat @ux for kr = 0:02, r = 1 and Re = 100. It is clear that on decreasing N , the radiation will be important in the heat transfer rate and reduces, precisely, the conduction in the medium. The e;ect of the thermal conductivity ratio on the radiative heat @ux presented in Fig. 5 for N = 0:1, r = 1 and Re = 100, demonstrates that decreasing kr yields an opposite behaviour at the boundaries: an enhancement of temperature at the cold boundary and a cooling in the hot one.
368
B. Cherif, M.S. Sifaoui / Journal of Quantitative Spectroscopy & Radiative Transfer 91 (2005) 363 – 372 14 N=0.1 N=1 N=10
Dimensionless radiative heat flux
12
10
8
6
4
2
0 0.5
0.55
0.6
0.65
0.7
0.75
0.8
0.85
0.9
0.95
1
Dimensionless radial coordinate r*=r/dp
Fig. 4. Planck number e;ect on the dimensionless radiative heat @ux. 15.5
kr=0.02 kr=0.06
Dimensionless radiative heat flux
13.5
kr=0.1
11.5
9.5
7.5
5.5
3.5
1.5 0.5
0.55
0.6
0.65
0.7
0.75
0.8
0.85
0.9
0.95
1
Dimensionless radial coordinate r*=r/dp
Fig. 5. Thermal conductivity ratio e;ect on the dimensionless radiative heat @ux.
Figs. 6 and 7 display the radiative and conductive heat @uxes, distribution for three values of the Reynolds number for kr = 0:02, r = 1 and N = 0:1. It can be seen that on decreasing the Reynolds number, the heat transfer rate is a;ected and notably the convection which becomes important. For kr = 0:02, r = 1 and N = 0:1 we plot in Fig. 8 the radiative heat @ux for three values of the thermal conductivity ratio. We can observe that the pro?le is a;ected considerably by this grandeur which when increases, reduces the radiation in favour of convection and yields a cooling of the medium. Fig. 9 presents the dimensionless heat @uxes for kr = 0:02, N = 0:1, r = 1 and Re = 100. As it may be seen, the radiation is very important compared to the conduction and convection; it is clear that heat exchange is based essentially on radiation.
B. Cherif, M.S. Sifaoui / Journal of Quantitative Spectroscopy & Radiative Transfer 91 (2005) 363 – 372
369
0.5
Re=100 Re=200 Re=300
Dimensionless radiative heat flux
0.45
0.4
0.35
0.3
0.25 0.5
0.55
0.6
0.65
0.7
0.75
0.8
0.85
0.9
0.95
1
Dimensionless radial coordinate r*=r/dp
Fig. 6. Reynolds number e;ect on the dimensionless radiative heat @ux.
Fig. 7. Reynolds number e;ect on the dimensionless convective heat @ux.
Figs. 10 and 11 represent the dimensionless temperature pro?les in the solid and @uid phases for kr = 0:02 and N = 0:1. We remark that a decrease in the optical thickness ratio yields a heating of the medium, thus an enhancement of the temperature pro?les in the two phases. In fact, this behaviour is due to the easy propagation of radiation in the medium when r decreases. 5. Conclusion The optical thick approximation medium is used to study the combined radiative, conductive and convective heat transfer in a semi-transparent porous medium in a spherical enclosure. The expression
370
B. Cherif, M.S. Sifaoui / Journal of Quantitative Spectroscopy & Radiative Transfer 91 (2005) 363 – 372 1.6
kr=0.06
1.4
Dimensionless convective heat flux
kr=0.02 kr=0.1
1.2
1
0.8
0.6
0.4
0.2
0 0.5
0.6
0.7
0.8
0.9
1
Dimensionless radial coordinate r*=r/dp
Fig. 8. Thermal conductivity ratio e;ect on the dimensionless convective heat @ux.
14
Radiative heat flux 12
Convective heat flux
Dimensionless heat fluxes
Conductive heat flux 10
8
6
4
2
0 0.5
0.55
0.6
0.65
0.7
0.75
0.8
0.85
0.9
0.95
1
Dimensionless radial coordinate r*=r/dp
Fig. 9. Dimensionless heat @ux pro?les.
for the derivative term of the radiative heat @ux appearing in the solid energy equation written in spherical geometry is represented by ROSSELAND approximation. This approach is successfully applied and leads to highly accurate results. The e;ect of the Reynolds and Planck numbers, the conductivity ratio and the optical thickness ratio on the temperature pro?les, on the convective heat @ux and on the radiative heat @ux distributions have been examined. It is shown that the radiative transfer rate and the interaction of di;erent transfer modes are dependent on the conduction–radiation parameter, Reynolds number, the thermal conductivity ratio
B. Cherif, M.S. Sifaoui / Journal of Quantitative Spectroscopy & Radiative Transfer 91 (2005) 363 – 372
371
0.6
Dimensionless solid temperature
0.55
0.5
0.45
0.4
0.35
τ =1 τ =5
0.3
τ = 10 0.25 0.5
0.6
0.7
0.8
0.9
1
Dimensionless radial coordinate r*=r/dp
Fig. 10. Optical thickness ratio e;ect on the dimensionless solid temperature.
1
τ =1 τ =5 τ =10
Dimensionless fluid temperature
0.9
0.8
0.7
0.6
0.5 0.5
0.6
0.7
0.8
0.9
1
Dimensionless radial coordinate r*=r/dp
Fig. 11. Optical thickness ratio e;ect on the dimensionless @uid temperature.
and the optical thickness ratio. We have also demonstrated that, the radiation plays an important role in the heat transfer rate. It is interesting to note that this paper is similar to the one in Ref. [7]; the medium behaviour is the same towards radiative properties and it is independent of the geometry. References [1] Cherif B, Sifaoui MS. Etude du transfert thermique dans un milieu poreux semi-transparent semi-in?ni par la discr.etisation de la luminance associ.ee aR une analyse asymptotique. Rev G.en Therm Fr Tome 1995;34:408.
372
B. Cherif, M.S. Sifaoui / Journal of Quantitative Spectroscopy & Radiative Transfer 91 (2005) 363 – 372
[2] Cherif B, Sifaoui MS. Mod.elisation du transfert de chaleur par rayonnement, conduction et convection dans un milieu poreux semi-in?ni. CongrRes SFT99, Arcachon, Th.eme convection naturelle et forc.ee; Mai 1999. p. 157– 60. [3] Ben Kheder C, Cherif B, Sifaoui MS. Numerical study of transient heat transfer in semitransparent porous medium. Renewable Energy 2002;27:543–60. [4] Cherif B, Sifaoui MS. Theoretical study of heat transfer by radiation conduction and convection in an semi-transparent porous medium in a cylindrical enclosure. J Quant Spectrosco Radiat Transfer 2003;83(3/4):519–527. [5] Modest MF. Radiative heat transfer. New York: McGraw hill; 1993. [6] Cherif B, Sifaoui MS. Modelling heat transfer by conduction radiation and convection in a semitransparent porous medium in a cylindrical enclosure. SFT2001. Comm. no. 48. Nantes, France; Mai 2001. [7] Ranz WE, Levenspiel O. Friction and transfer coeKcients for single particles and packed beds. Chem Eng Progr (USA) 1952;48(5):247–53.
www.elsevier.com/locate/jqsrt
Numerical study of heat transfer in an optically thick semi-transparent spherical porous medium B. Ch.erif∗ , M.S. Sifaoui Laboratoire de Rayonnement Thermique, Departement de physique, Faculte des Sciences de Tunis, Campus universitaire, Tunis 1060, Tunisia Received 23 May 2003; accepted 17 December 2003
Abstract In this paper, heat transfer by simultaneous convection, conduction and radiation in a semi-transparent spherical porous medium is investigated. The ROSSELAND approximation is adopted to take account of radiation in the heat transfer rate. The routine used here to solve the set of di;erential equations is taken from the IMSL MATH/LIBRARY. Various results are obtained for the dimensionless temperature pro?les in the solid and @uid phases, the radiative, conductive, convective and total heat @uxes. The e;ects of some radiative properties of the medium on the heat transfer rate are examined. ? 2004 Elsevier Ltd. All rights reserved. Keywords: Conduction; Radiation; Convection; Porous medium; Optically thick; Spherical geometry
1. Introduction In recent years, heat transfer by simultaneous radiation, conduction and convection in semitransparent porous media has attracted our attention. So we have studied the problem in an in?nite anisotropic scattering steady state porous medium using the Discrete Ordinates Method combined with an asymptotic analysis [1,2]; in a transient rectangular semi-transparent porous medium [3] and recently in a cylindrical semi-transparent optically thick porous medium [4]. Here, the considered medium consists of a pile of homogeneous spherical particles contained between two isothermal hollow spheres with radii R1 and R2 , respectively. The medium receives a transparent gas @owing through it at a constant velocity as shown in Fig. 1. ∗
Corresponding author. E-mail address: [email protected] (B. Ch.erif).
0022-4073/$ - see front matter ? 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.jqsrt.2003.12.012
364
B. Cherif, M.S. Sifaoui / Journal of Quantitative Spectroscopy & Radiative Transfer 91 (2005) 363 – 372
Fig. 1. Concentric spheres separated by a semi-transparent porous medium.
For the mathematical analysis, the following basic assumptions are made: (a) steady state; (b) the @uid @ow and heat transfer are one dimensional; (c) all the thermophysical properties are constant; (d) the solid is absorbent, participating, scattering and it behaves as an optically dense gray body for radiation; (e) the temperature gradients in the solid particles are negligible; (f) the @uid phase is incompressible and the mass @ow rate at every cross-section of the medium is constant. The optically thick approximation media [5] are used to take an account of radiation e;ect on the heat transfer rate. 2. Formulation The mathematical formulation of the problem is based on the conservative form of @uid and solid energy equations. To normalise the problem, we introduce the reference temperature Tr = T (r)Tf 1 and the space-radial coordinate r = dp r ∗ . Hence, we use Ts (r) = Tf 1 s (r);
Tf (r) = Tf 1 f (r):
These governing equations under consideration are given in [6]. For the solid phase: 2 ds 1 d 2 s + ∗ ∗ = Hs (s − f ) + div(’r (r ∗ )) ∗ 2 dr r dr ks
(1)
(2)
with Hs = 6kr Nu 2r ; kr = kf =ks where kf and ks are the @uid and the solid thermal conductivity, respectively. ’r (r ∗ ) is the radiative heat @ux in the r ∗ direction and Nu is the Nusselt number. r is the optical thickness ratio, r = 2 = d , where 2 = R2 denotes the optical thickness of the second sphere and d = dp of the particles of diameter dp . For the @uid phase: d 2 f 2 df df + ∗ ∗ = A ∗ − Hf (s − f ) (3) ∗ 2 dr r dr dr
B. Cherif, M.S. Sifaoui / Journal of Quantitative Spectroscopy & Radiative Transfer 91 (2005) 363 – 372
365
with Hf = 6Nu[(1 − )=] 2r and A = (Pr Re=) r . is the porosity; Pr and Re are, respectively, the Prandlt and Reynolds numbers. The boundary conditions are de?ned as follows: T s1 and f 1 (r ∗ = r1∗ ) = 1; s1 (r ∗ = r1∗ ) = Tf 1 f 2 (r ∗ = r2∗ ) =
Tf 2 T s2 = s2 (r ∗ = r2∗ ) = = 2 (r2∗ ); Tf 1 Tf 1
(4)
where Ts1 = Ts (r ∗ = r1∗ ), Tf 1 = Tf (r ∗ = r1∗ ); Tf 2 = Tf (r ∗ = r2∗ ), Ts2 = Ts (r ∗ = r2∗ ). The subscripts 1 and 2 refer to the boundaries at r = R1 and R2 . By using the Rosseland approximation [4,6] we take ’r (r ∗ ) = −
16n2 Ts3 (r ∗ ) dTs (r ∗ ) ; 3R2 dr ∗
(5)
where is the Rosseland mean extinction coeKcient, n denotes the refractive index and is the Stefan–Boltzmann constant. The term 16n2 Ts3 (r ∗ )=3R2 can be considered as the ‘radiative thermal conductivity’ [6]. Under the above consideration, the solid conservative energy equation takes the following form: 2 2 d d 2 ds 4 2 d d 2 s s s s ; (6) + = Hs (s − f ) − N 3s2 + s3 − ∗ ∗ dr ∗2 r ∗ dr ∗ 3 dr ∗ dr ∗2 r dr where N = ks =4n2 Tf31 is the Planck number representing the radiation–conduction parameter. The conductive and convective heat @uxes are dTs ’ c = ks ; dr ’cv = h(Ts − Tf );
(7)
where h = kf Nu=dp is the @uid exchange coeKcient. The Nusselt number is given by the following relation for particle diameter such as dp ¿ 10−3 m [7]: Nu = 2:0 + 1:8Pr 0:33 Re0:5 :
(8)
The total dimensionless heat @ux in the medium is determined from ds 4 3 ds T = ’t =ks Tf 1 =dp = − ∗ − + kr r Nu(s − f ): (9) dr 3N s dr ∗ The geometry and coordinates for the concentric spheres separated by a semi-transparent porous medium are shown in Fig. 1. 3. Numerical solution The governing equations for the problem considered here, Eqs. (3)–(6) and the boundary conditions (4) are nonlinear di;erential equations depending on the dimensionless radial coordinate r ∗ , conduction–radiation parameter N , Reynolds number and thermal conductivity ratio kr . In the
366
B. Cherif, M.S. Sifaoui / Journal of Quantitative Spectroscopy & Radiative Transfer 91 (2005) 363 – 372
present analysis, the numerical technique used here namely BVPFD solves a (parameterised) system of di;erential equations with boundary conditions at two points, using a variable order, and a variable step-size ?nite di;erence method with deferred corrections. It is taken from the IMSL MATH/LIBRARY. 4. Results and discussion The numerical results are presented graphically for conduction-to-radiation parameter N , conductivity ratio kr , optical thickness ratio r and the Reynolds number Re. These results are obtained for a heat transfer by simultaneous radiation, conduction and convection in a semi-transparent spherical porous medium. For the calculations we have taken: s1 = 0:25; f 1 = 1; r1∗ = 0:5; and r2∗ = 1. In Tables 1, 2 and 3we give the numerical values of the conduction, radiation and the total heat @uxes at the boundaries for various combination of the Planck number N , the Reynolds number Re and the thermal conductivity ratio kr . In Fig. 2, for kr = 0:02, r = 1 and Re = 100, we demonstrate the e;ect of the Planck number on the solid temperature distribution. This ?gure shows that the temperature in the solid phase increases by decreasing N due to the radiation which becomes important as shown in Fig. 4. Table 1 Planck number e;ect on the boundary @uxes with: kr = 0:02, r = 1 and Re = 100 N
T (r1∗ )
r (r1∗ )
0.01 0.1 1
128.3799 15.1117 4.5235
126.8618 13.8062 2.2610
ds
dr ∗ r ∗ 1
0.9515 1.03555 1.6958
T (r2∗ )
r (r2∗ )
31.4160 3.5086 0.5405
27.6378 2.1929 0.0772
ds
dr ∗ r ∗ 2
1.7783 1.3157 0.4633
Table 2 Reynolds number e;ect on the boundary @uxes with: kr = 0:02, r = 1 and N = 0:1 Re
T (r1∗ )
r (r1∗ )
100 200 300
15.1117 15.3630 15.5549
13.8062 13.9475 14.0551
ds
dr ∗ r ∗ 1
1.0355 1.0461 1.0541
T (r2∗ )
r (r2∗ )
3.5086 3.4407 3.3887
2.1929 2.1504 2.1180
ds
dr ∗ r ∗ 2
1.3157 1.2303 1.2708
Table 3 Thermal conductivity ratio e;ect on the boundary @uxes with: N = 0:1, r = 1 and Re = 100 kr
T (r1∗ )
r (r1∗ )
0.02 0.06 0.1
15.1117 16.3258 17.5279
13.8062 14.4333 15.0491
ds
dr ∗ r ∗ 1
1.0355 1.0825 1.1287
T (r2∗ )
r (r2∗ )
3.5086 3.2808 3.0595
2.1929 2.0505 1.9122
ds
dr ∗ r ∗ 2
1.3157 1.2303 1.1473
B. Cherif, M.S. Sifaoui / Journal of Quantitative Spectroscopy & Radiative Transfer 91 (2005) 363 – 372
367
Dimensionless solid temperature
1
N=0.001 N=0.1 N=10
0.9
0.8
0.7
0.6
0.5 0.5
0.6
0.7
0.8
0.9
1
Dimensionless radial coordinate r*=r/dp
Fig. 2. Planck number e;ect on the dimensionless solid temperature.
Dimensionless fluid temperature
0.5 N=0.001 N=0.01 N=10
0.45
0.4
0.35
0.3
0.25 0.5
0.55
0.6
0.65
0.7
0.75
0.8
0.85
0.9
0.95
1
Dimensionless radial coordinate r*=r/dp
Fig. 3. Planck number e;ect on the dimensionless @uid temperature.
Fig. 3 shows the variation of the @uid temperature in the medium for kr = 0:02, r = 1 and Re = 100, for three values of Planck number. We note that the @uid temperature is a;ected by this number which by decreasing, the radiation becomes relatively important compared to conduction and convection, hence this yields a heating of the medium. Fig. 4 represents the e;ect of the Planck number N on the radiative heat @ux for kr = 0:02, r = 1 and Re = 100. It is clear that on decreasing N , the radiation will be important in the heat transfer rate and reduces, precisely, the conduction in the medium. The e;ect of the thermal conductivity ratio on the radiative heat @ux presented in Fig. 5 for N = 0:1, r = 1 and Re = 100, demonstrates that decreasing kr yields an opposite behaviour at the boundaries: an enhancement of temperature at the cold boundary and a cooling in the hot one.
368
B. Cherif, M.S. Sifaoui / Journal of Quantitative Spectroscopy & Radiative Transfer 91 (2005) 363 – 372 14 N=0.1 N=1 N=10
Dimensionless radiative heat flux
12
10
8
6
4
2
0 0.5
0.55
0.6
0.65
0.7
0.75
0.8
0.85
0.9
0.95
1
Dimensionless radial coordinate r*=r/dp
Fig. 4. Planck number e;ect on the dimensionless radiative heat @ux. 15.5
kr=0.02 kr=0.06
Dimensionless radiative heat flux
13.5
kr=0.1
11.5
9.5
7.5
5.5
3.5
1.5 0.5
0.55
0.6
0.65
0.7
0.75
0.8
0.85
0.9
0.95
1
Dimensionless radial coordinate r*=r/dp
Fig. 5. Thermal conductivity ratio e;ect on the dimensionless radiative heat @ux.
Figs. 6 and 7 display the radiative and conductive heat @uxes, distribution for three values of the Reynolds number for kr = 0:02, r = 1 and N = 0:1. It can be seen that on decreasing the Reynolds number, the heat transfer rate is a;ected and notably the convection which becomes important. For kr = 0:02, r = 1 and N = 0:1 we plot in Fig. 8 the radiative heat @ux for three values of the thermal conductivity ratio. We can observe that the pro?le is a;ected considerably by this grandeur which when increases, reduces the radiation in favour of convection and yields a cooling of the medium. Fig. 9 presents the dimensionless heat @uxes for kr = 0:02, N = 0:1, r = 1 and Re = 100. As it may be seen, the radiation is very important compared to the conduction and convection; it is clear that heat exchange is based essentially on radiation.
B. Cherif, M.S. Sifaoui / Journal of Quantitative Spectroscopy & Radiative Transfer 91 (2005) 363 – 372
369
0.5
Re=100 Re=200 Re=300
Dimensionless radiative heat flux
0.45
0.4
0.35
0.3
0.25 0.5
0.55
0.6
0.65
0.7
0.75
0.8
0.85
0.9
0.95
1
Dimensionless radial coordinate r*=r/dp
Fig. 6. Reynolds number e;ect on the dimensionless radiative heat @ux.
Fig. 7. Reynolds number e;ect on the dimensionless convective heat @ux.
Figs. 10 and 11 represent the dimensionless temperature pro?les in the solid and @uid phases for kr = 0:02 and N = 0:1. We remark that a decrease in the optical thickness ratio yields a heating of the medium, thus an enhancement of the temperature pro?les in the two phases. In fact, this behaviour is due to the easy propagation of radiation in the medium when r decreases. 5. Conclusion The optical thick approximation medium is used to study the combined radiative, conductive and convective heat transfer in a semi-transparent porous medium in a spherical enclosure. The expression
370
B. Cherif, M.S. Sifaoui / Journal of Quantitative Spectroscopy & Radiative Transfer 91 (2005) 363 – 372 1.6
kr=0.06
1.4
Dimensionless convective heat flux
kr=0.02 kr=0.1
1.2
1
0.8
0.6
0.4
0.2
0 0.5
0.6
0.7
0.8
0.9
1
Dimensionless radial coordinate r*=r/dp
Fig. 8. Thermal conductivity ratio e;ect on the dimensionless convective heat @ux.
14
Radiative heat flux 12
Convective heat flux
Dimensionless heat fluxes
Conductive heat flux 10
8
6
4
2
0 0.5
0.55
0.6
0.65
0.7
0.75
0.8
0.85
0.9
0.95
1
Dimensionless radial coordinate r*=r/dp
Fig. 9. Dimensionless heat @ux pro?les.
for the derivative term of the radiative heat @ux appearing in the solid energy equation written in spherical geometry is represented by ROSSELAND approximation. This approach is successfully applied and leads to highly accurate results. The e;ect of the Reynolds and Planck numbers, the conductivity ratio and the optical thickness ratio on the temperature pro?les, on the convective heat @ux and on the radiative heat @ux distributions have been examined. It is shown that the radiative transfer rate and the interaction of di;erent transfer modes are dependent on the conduction–radiation parameter, Reynolds number, the thermal conductivity ratio
B. Cherif, M.S. Sifaoui / Journal of Quantitative Spectroscopy & Radiative Transfer 91 (2005) 363 – 372
371
0.6
Dimensionless solid temperature
0.55
0.5
0.45
0.4
0.35
τ =1 τ =5
0.3
τ = 10 0.25 0.5
0.6
0.7
0.8
0.9
1
Dimensionless radial coordinate r*=r/dp
Fig. 10. Optical thickness ratio e;ect on the dimensionless solid temperature.
1
τ =1 τ =5 τ =10
Dimensionless fluid temperature
0.9
0.8
0.7
0.6
0.5 0.5
0.6
0.7
0.8
0.9
1
Dimensionless radial coordinate r*=r/dp
Fig. 11. Optical thickness ratio e;ect on the dimensionless @uid temperature.
and the optical thickness ratio. We have also demonstrated that, the radiation plays an important role in the heat transfer rate. It is interesting to note that this paper is similar to the one in Ref. [7]; the medium behaviour is the same towards radiative properties and it is independent of the geometry. References [1] Cherif B, Sifaoui MS. Etude du transfert thermique dans un milieu poreux semi-transparent semi-in?ni par la discr.etisation de la luminance associ.ee aR une analyse asymptotique. Rev G.en Therm Fr Tome 1995;34:408.
372
B. Cherif, M.S. Sifaoui / Journal of Quantitative Spectroscopy & Radiative Transfer 91 (2005) 363 – 372
[2] Cherif B, Sifaoui MS. Mod.elisation du transfert de chaleur par rayonnement, conduction et convection dans un milieu poreux semi-in?ni. CongrRes SFT99, Arcachon, Th.eme convection naturelle et forc.ee; Mai 1999. p. 157– 60. [3] Ben Kheder C, Cherif B, Sifaoui MS. Numerical study of transient heat transfer in semitransparent porous medium. Renewable Energy 2002;27:543–60. [4] Cherif B, Sifaoui MS. Theoretical study of heat transfer by radiation conduction and convection in an semi-transparent porous medium in a cylindrical enclosure. J Quant Spectrosco Radiat Transfer 2003;83(3/4):519–527. [5] Modest MF. Radiative heat transfer. New York: McGraw hill; 1993. [6] Cherif B, Sifaoui MS. Modelling heat transfer by conduction radiation and convection in a semitransparent porous medium in a cylindrical enclosure. SFT2001. Comm. no. 48. Nantes, France; Mai 2001. [7] Ranz WE, Levenspiel O. Friction and transfer coeKcients for single particles and packed beds. Chem Eng Progr (USA) 1952;48(5):247–53.