Analysis of Heat and Mass Transfer in a Vertical Annular Porous Cylinder Using FEM

Transcription

1 Transp Porous Med DOI.7/s x Analysis of Heat and Mass Transfer in a Vertical Annular Porous Cylinder Using FEM Irfan Anjum Badruddin N. J. Salman Ahmed Abdullah A. A. A. Al-Rashed Jeevan Kanesan Sarfaraz Kamangar H. M. T. Khaleed Received: 8 April 2 / Accepted: 27 August 2 Springer Science+Business Media B.V. 2 Abstract The present study is intended to study heat and mass transfer in a vertical annular cylinder embedded with saturated porous medium. The inner surface of cylinder is maintained at uniform wall temperature and uniform wall concentration. The governing partial differential equations are non-dimensionalised and solved by using finite element method (FEM). The porous medium is discritised using triangular elements with uneven element size. Large number of smaller-sized elements are placed near the walls of the annulus to capture the smallest variation in solution parameters. The results are reported for both aiding and opposing flows. The effects of various non-dimensional numbers such as buoyancy ratio, Lewis number, Rayleigh number, aspect ratio, etc on heat and mass transfer are discussed. The temperature and concentration profiles are presented. Keywords Heat and mass transfer Porous medium Darcy flow Vertical annulus FEM I. A. Badruddin (B) N. J. Salman Ahmed S. Kamangar Department of Mechanical Engineering, Faculty of Engineering, University of Malaya, 563 Kuala Lumpur, Malaysia Irfan_magami@Rediffmail.com A. A. A. A. Al-Rashed Public Authority for Applied Education and Training, Industrial Training Institute, 392 Adiliya, Kuwait J. Kanesan Department of Electrical Engineering, Faculty of Engineering, University of Malaya, 563 Kuala Lumpur, Malaysia H. M. T. Khaleed Department of Mechanical Engineering, Anjuman Engineering College, Bhatkal 5832, India 23

2 I. A. Badruddin et al. List of symbols A Area of an element A r Aspect ratio C p Specific heat C, C Species concentration (dimensional and non-dimensional, respectively) D Mass diffusivity g Gravitational acceleration H Height of cylinder k Thermal conductivity K Permeability of porous media L = r o r i Le Lewis number M Shape function n Refractive index N Buoyancy ratio Nu z, Nu Local and average Nusselt number p Pressure q r Radiation flux r, z Cylindrical coordinates r, z Non-dimensional coordinates R Mean radial distance of an element R Radius ratio Ra Rayleigh number Rd Radiation parameter Sh z, Sh Local and average Sherwood number T, T Dimensional and non-dimensional temperature, respectively u, w Velocity in r and z directions Greek symbols α Thermal diffusivity β C Coefficient of concentration expansion β T Coefficient of thermal expansion β R Rosseland extinction coefficient ρ Density v Coefficient of kinematic viscosity μ Coefficient of dynamic viscosity σ Stefan Boltzmann s constant ψ Stream function ψ Non-dimensional stream function Subscripts w Wall Conditions at infinity i Inner o Outer 23

3 Analysis of Heat and Mass Transfer Introduction The study of heat and mass transfer in a saturated porous medium has received a noticeable attention from the researchers during the last few decades. This is supported by the fact that the heat and mass transfer in porous media involves plenty of applications such as migration of moisture through the air contained in fibrous insulations, the contamination of chemicals in the soil, grain storage installations, cryogenic containers, etc. A deep insight into the various aspects of convection in PM is provided by Nield and Bejan (26), Ingham and Pop (998), Pop and Ingham (2), Bejan and Kraus (23) and edited book by Vafai (2). The literature reveals an extensive work in heat and mass transfer in porous media. Yih (999) has studied the coupled heat and mass transfer by natural convection adjacent to a horizontal cylinder in a saturated porous medium by using Keller box method. Cheng (2) has investigated the heat and mass transfer near a wavy surface and shown that the increasing amplitude wavelength ratio tends to increase the amplitude of the local Sherwood number and local Nusselt number. The investigation into heat and mass transfer in an unsaturated soil was carried out by Thomas and Ferguson (999). The study by Hossain et al. (999) explained that the Nusselt number and Sherwood number were found to decrease with the increased curvature parameter when the fluid was injected through the permeable surface of the cylinder. Trevisan and Bejan (987) have reported the results for heat and mass transfer in porous medium subjected to high Rayleigh number flow. Yang (27) investigated the coupled heat and moisture in double-layer hollow cylinder. Singh and Chandarki (29) studied the effects of the governing parameters such as buoyancy ratio (N) and Lewis number (Le) on the local Nusselt and local Sherwood Numbers for the integral treatment of coupled heat and mass transfer by natural convection from a cylinder in porous media. El-Amin (24) studied the double dispersion effects on natural convection heat and mass transfer in non-darcy porous medium by employing fourth-order Runge Kutta method with shooting technique. Angirasa et al. (997) investigated the heat and mass transfer with opposing buoyancy effects. The mixed convention heat and mass transfer was considered by Yih 998. The analysis of heat transfer through natural convection in case of the vertical annular cylinder was reported by Badruddin et al. (26a, 26b, 27) by considering radiation parameter effect using thermal equilibrium model, whereas the thermal non-equilibrium model was applied for the similar geometry in another study by Badruddin et al. (26a). Cheng (26a) reported the double-diffusive natural convection in a vertical annular porous medium to demonstrate the effect of modified Darcy number on the volume flow rate. In another case of his research, Cheng (26a) investigated the heat and mass transfer rates of the elliptical cylinder with slender orientation and blunt orientations. Cheng (29) also studied the Soret and Dufour effects on the boundary layer flow due to natural convection heat and mass transfer over a downward pointing vertical cone in a porous medium, whereas Kairi and Murthy (2) have investigated the effect of viscous dissipation on natural convection heat and mass transfer from vertical cone. The radiation effect in case of the natural convection was reported by Salman et al. (29) using the thermal equilibrium approach, whereas thermal non-equilibrium approach was adopted to study the mixed convection in an annular vertical cylinder by the same author Salman et al. (2). The present study is aimed to investigate numerically the heat and mass transfer characteristics in a vertical annular cylinder embedded in porous media with constant wall temperature and concentration. The heat and mass transfer in an annular porous body is an important area of interest as it involves practical applications such as insulation of pipe lines, high-temperature gas-cooled reactor vessels, biomass converters, etc. The fluid flow is assumed to be governed by Darcy law. 23

4 I. A. Badruddin et al. Fig. Annular cylinder 2Analysis Consider the flow of fluid under the influence of natural convection in a saturated porous medium embedded in a vertical annular cylinder. Figure depicts the geometry of problem under consideration. Since the geometry is axisymmetric in nature, only two dimensions, i.e., r and z can describe the flow behaviour completely in the medium. The cylinder has inner radiusr i and outer radius r o,. where r and z-axes point towards the width and height of the medium, respectively. Following assumptions are made: Porous medium is saturated with fluid. The fluid and medium are in local thermal equilibrium everywhere inside the medium. The porous medium is isotropic and homogeneous. Fluid properties are constant except the variation in density. With the above assumptions, the governing equations are given by: Continuity equation (ru) r Using Darcy law for flow in porous media Energy equation u T r + w T z = α Concentration equation 23 ( r r u = K μ w = K μ + (rw) z p r ( ) p z + ρg ( r T ) ) + 2 T r z 2 ( ρc p r u C r + w C z = D ( r r = () ( r C ) ) + 2 C r z 2 r (rq r) + z (q r) ρ = ρ [ β T (T T ) β C (C C )] (6) ) (2) (3) (4) (5)

5 Analysis of Heat and Mass Transfer Corresponding boundary conditions are: at r = r i, T = T w, C = C w, u =, (7a) at r = r o, T = T, C = C, u =, (7b) at z = and z = H T z = C z = (7c) where u and w are Darcy velocities in the r and z directions, respectively. The continuity Eq. () can be satisfied by introducing the stream function ψ as: u = ψ (8a) r r w = ψ (8b) r r Invoking Rosseland approximation for radiation Yih (999), Hossain and Alim (997) q r = 4n2 σ T 4 3β R r (9) Expanding T 4 about T in Taylor series, Raptis (998) T 4 4TT 3 3T 4 () The following parameters have been used for non-dimensionalising the governing equations. r = r L, z = z L, ψ = ψ αl, T = (T T ) (T w T ), C = (C C ) (C w C ), Rd = 4n2 σt 3, Ra = gβ T T KL, Le = α β R k να D, N = βc(c () w C ) β T (T w T ) Making use of Eqs. (6 8)and()into(2)and(3) and performing mathematical operations yield 2 ψ z 2 + r ( ) [ ] ψ T = rra r r r r + N C (2) r Replacing u and w by ψ in Eq. (4) and substituting terms from Eqs. (9 ) results [ ] ψ T r r z ψ ( T = + 4Rd ) ( ( ) ) r T + 2 T z r 3 r r r z 2 After substituting the ψ and non-dimensional parameters, Eq. (5) takes the form: [ ] ( ( ) ) ψ C r r z ψ C = r C + 2 C z r Le r r r z 2 (3) (4) when Rd =, the above equations reduce to the case of pure natural convection flow over an isothermal vertical annular cylinder embedded in saturated porous medium. 23

6 I. A. Badruddin et al. Table Nu variation with mesh size Sl. No. No of elements Nu Sh Time (s), , , , ,26.42 The corresponding boundary conditions are at r = r i, T =, C = ψ = (5a) at r = r o, T =, C = ψ = (5b) at z = and z = A r T z = C z = (5c) 3 Numerical Method Equations (2 4) are coupled partial differential equations that govern the heat and mass transfer behaviour. These equations are solved by using finite element method using Galerkin approach. A simple three-noded triangular element is considered. T, C and ψ vary inside the element and can be expressed as: where M, M 2, M 3 are the shape functions given as: T = T M + T 2 M 2 + T 3 M 3 (6) ψ = ψ M + ψ 2 M 2 + ψ 3 M 3 (7) C = C M + C 2 M 2 + C 3 M 3 (8) M i = a i + b i x + c i y, i =, 2, 3 (9) 2A and a i,b i,c i are matrix coefficients Details of FEM formulations can be obtained from Segerland (982), Lewis et al. (24). Integrating Eqs. (2 4) using Galerkin method yield a set of coupled matrix equations. These matrix equations for elements are assembled to get the global matrix equation for the whole domain, which is solved iteratively to get T, C and ψ in the porous medium. For higher accuracy in the results, the tolerance level of solution for T, C and ψ is set at 5, 5 and 9, respectively. Element size in the domain varies, having large number of elements located near the wall where large variations in T, C and ψ are expected. Sufficiently dense mesh is chosen to make the solution mesh invariant. Table shows the average Nusselt and Sherwood numbers for different mesh sizes, at hot surface with A r =, R =, Rd =, N =, Le = andra =. The local and average Nusselt and Sherwood numbers at hot and cold surface of the annulus are evaluated using following relations: 23

7 Analysis of Heat and Mass Transfer ( ) + 4Rd T Nu z = 3 r Ar ( + 4Rd 3 Nu = r= ri, r o ) T r r= ri, r o d z A r (2a) (2b) Sh z = C r Sh = Ar r=ri,r o (2) C r r= ri, r o d z A r (22) The computations are carried out on high-end computer. It can be seen from Table that the variation in Nu and Sh is very small when element size is changed from,8 to 5,. The variation in Nu and Sh is found to be.4 and.8%, respectively, when mesh size is increased from,8 to 5, elements. However, the time required for a mesh of 5, elements is considerably higher than that of,8 elements. Thus, the mesh size of,8 elements is selected for the present study. 4 Results and Discussion The inner surface of cylinder is maintained at constant temperatures T i = and constant species concentration C i =. The outer surface remains at T o = and C o =. Aspect ratio and Radius ratio are defined as A r = H/Land R = (r r i )/r i, respectively. The parameter N indicates the relative importance of concentration and thermal buoyancy force. It may be noted that N is positive for thermally assisting flow and negative for thermally opposing flow. N = indicates the absence of concentration buoyant force, and flow is driven by thermal buoyancy only. Results are obtained in terms of Nusselt number and Sherwood number at hot and cold surface of annulus for various parameters such as aspect ratio, radius ratio, buoyancy ratio, Lewis number, Rayleigh number and radiation parameter. The temperature and concentration gradient in Eqs. (2 22) are evaluated using 4-point polynomial along the nodes near the surface of cylinder. The results are validated for the present method by comparing Nu and Sh with the available literature, i.e., El-Amin (24), Yih (998, 999), Hossain and Alim (997). For the comparison, r i is extended to large value, so that the radial effect becomes negligible and the domain behaves as a vertical plate embedded in porous medium. Figure 2 explicates the comparison of the present method with that of (El-Amin 24). It is evident from this figure that the present method follows the similar trend as the established results by the previous researcher El-Amin (24). The comparison with that of Yih (998)andNakayama and Hossain (995) is also shown in the Table 2. It can be inferred from Table 2 that the present method has good accuracy in predicting the heat and mass transfer characteristics. Furthermore, the results are compared with convective heat transfer in axisymmetric bodies (Rajamani et al. 995) and presented in Table 3. In this case also the prediction of heat transfer behaviour is found to be similar. Thus, it is clear from, Table 2, Fig. 2 and Table 3 that the present method provides sufficient accuracy to simulate the heat and mass transfer behaviour in porous annulus. Figure 3 shows the Sh variation with aspect ratio of annular cylinder for different values of Le. Lewis number Le has pronounced effect on Sherwood number. The Sh increases initially when aspect ratio is increased, reaches a maximum value and then starts decreasing. 23

8 I. A. Badruddin et al. / Sh z / Raz 2 [] / z Nu z Ra / z Sh z Ra / 2 /2 /2 Nu z / Raz Fig. 2 Comparison of results with the previously published work Le Table 2 Comparison of the present method with available literature N Le Nu z /Raz /2 Sh z /Raz /2 Yih (998) Nakayama and Hossain (995) Present Yih (998) Nakayama and Hossain (995) Present The maxima for Sh at Le = occur at an aspect ratio around.25, but it keeps shifting towards lower aspect ratio when Le is increased. The effect of Lewis number on average Nusselt number Nuvariation is comparatively small, due to which it has not been explicated 23

9 Analysis of Heat and Mass Transfer Table 3 Comparison of results for annular body R Nu (Rajamani et al. 995) Nu present Fig. 3 Sh versus aspect ratio Rd = 4, Ra =, R =,N =.5 2.9x -3 K = 2.9x x -5 Fig. 4 Nu and Sh versus aspect ratio Rd =, R =, N =, Le= 5 in the figure. Sh increases with increase in Le. The concentration boundary layer becomes thinner as Le increases, which in turn increases the concentration gradient and thus the Sh. Figure 4 shows the effect of permeability on the heat and mass transfer characteristics of porous medium. The permeability range corresponds to the well-packed soil. It is obvious from Fig. 4 that the permeability has greater effect on mass transfer than the heat transfer. The Nusselt and Sherwood numbers increase with increase in the permeability of the 23

10 I. A. Badruddin et al (a) (b) (c) Fig. 5 a Isotherms, b Isoconcentration, c Streamlines LeftA r =.5, RightA r = atra =,R =, N =.5,Le = 5andRd = 23

11 Analysis of Heat and Mass Transfer Fig. 6 Sh versus aspect ratio at Rd =, Ra = 25, R =, Le = 2 Fig. 7 Sh variations with radius ratio at hot surface for Rd = 2, Ra = 5, A r = 5, N = medium. This could be attributed to decrease in the resistance for fluid flow with increase in permeability which in turn transports the heat and mass from inner surface to the porous medium. It is observed that the Nusselt number is not much affected at low values of permeability ( ) which is shown as single line in Fig. 4. The effect of aspect ratio variation is negligible at low permeability. Figure 5 depicts the isotherms, isoconcentration lines and streamlines for different values of aspect ratio. It is evident from the figure that when aspect ratio is increased from.5 to, the isothermal lines tend to move towards lower corner of inner surface of cylinder, which indicates that the convective heat transfer has increased. However, at the same time, isoconcentration lines also move towards the lower and upper corner of inner and outer surfaces of annular cylinder, respectively. The fluid activity increases at lower and upper corner of inner and outer surfaces, respectively, as indicated by streamlines. 23

12 I. A. Badruddin et al. Fig. 8 Sh variations with radius ratio at cold surface for different values on Le at Rd = 2, Ra = 5, A r = 5, N = Fig. 9 Nu and Shvariations with radius ratio at hot surface for different values on N at Rd = 2, Ra = 5, A r = 5, Le = 2 Figure 6 explains the Sh as a function of aspect ratio for both aiding and opposing flows. Nu is not much affected by buoyancy ratio; thus, it has not been shown in the figure. Sh increases with increase of N. The maximum Sh shifts to lower aspect ratio when N is increased. Figure 7 illustrates the effect of radius ratio on the Sherwood numbers Sh, with respect to Lewis number at hot surface of the annulus. It can be noticed that the Sherwood 23

13 Analysis of Heat and Mass Transfer Fig. Nu and Sh variations with radius ratio at cold surface for different values on N at Rd = 2, Ra= 5, A r = 5, Le= 2 number at hot surface increases with increase in radius ratio. The increase of Sherwood number is found to be almost linear with respect to radius ratio. Figure 8 demonstrates the effect of radius ratio on the Sherwood number with respect to Lewis number at cold surface of the annulus. At cold surface of annulus, the Sh decreases with increase in radius ratio. It is found that the rate of decrease in Sh is higher at lower values of radius ratio. Figure 9 reveals the effect of radius ratio on Nusselt and Sherwood numbers with respect to buoyancy ratio at hot surface. Sherwood number increased by 2.44 times when radius ratio is varied from.5 to at N =.5. The corresponding increase in Sherwood number at N = 2 is found to be.28 times. The Nusselt number increased by 2.47 times and.85 times at N =.5 andn = 2, respectively, when radius ratio is changed from.5 to. The effect of radius ratio on cold surface for various values of buoyancy ratio has been shown in Fig..Thetrendof Nu and Sh variation is found to be similar to that of Fig. 8. The Nusselt and Sherwood numbers are smaller for opposing flow as compared to that of aiding flow. The Sherwood number at R =.5 increased by 2.65 times when N is changed from.5 to 2, and corresponding increase in Nusselt number is found to be.67 times. It is noticed that the difference in Nusselt and Sherwood numbers at hot and cold surface decreases with increase in buoyancy ratio at R =.5. However, this difference increases at R = when buoyancy ratio is increased. Figure shows the isothermal lines, isoconcentration lines and streamlines for two different values of radius ratio. It can be seen that the isothermal lines move towards the hot surface and away from cold surface when radius ratio is increased. This leads Nusselt number at hot surface to increase, whereas it decreases at cold surface. Similar behaviour for isoconcentration lines is observed as shown in Fig. b. It is interesting to note that the fluid flow regime gets divided into two distinct regions at R = as shown by streamlines in Fig. c. 23

14 I. A. Badruddin et al. (a) (b) (c) Fig. a Isotherms, b Isoconcentration, c Streamlines Left R =.5, Right R = 2atRa = 5, A r =, N =.5, Le = and Rd = 2 23

15 Analysis of Heat and Mass Transfer (a) (b) (c) Fig. 2 a Isotherms, b Isoconcentration, c Streamlines, Left N =.5, Right N = 2atRa = 25, A r =, R =, Le = 2andRd = 23

16 I. A. Badruddin et al. (a) (b) (c) Fig. 3 a Isotherms, b Isoconcentration, c Streamlines, Left Rd =, Right Rd = at Ra =, A r =, R =, Le =. and N =.5 23

17 Analysis of Heat and Mass Transfer Fig. 4 Temperature and concentration profile for various values of N at Le = 2, A r =, R =, Rd =, and Ra = 25 Fig. 5 Temperature and concentration profile for various values of Ra at Le = 2, A r =, R =, Rd = 2 and N =.5 Figure 2 shows the isothermal lines, isoconcentration lines and streamlines for different values of N. The isothermal lines are almost parallel to vertical surface of cylinder at N =.5, which indicates that the heat transfer is dominated by conduction. When N is 23

18 I. A. Badruddin et al. increased from.5 to 2, then the isothermal lines and isoconcentration lines concentrate at the lower and upper corner of inner and outer surfaces, respectively. The streamlines are distorted and crowded near the surface of cylinder when N is changed from.5to2(fig.2c), indicating the increased fluid velocity near the surface of cylinder. This increased velocity helps in carrying more amount of heat and mass from the surface to the porous medium, thus increasing the Nusselt and Sherwood numbers. Figure 3 depicts the isothermal lines, isoconcentration lines and streamlines for two different values of Rd, i.e., Rd = (pure natural convection) and Rd = (convection + radiation). The value of Rd corresponds to the ambient temperature of 3 C for water as fluid in the medium. It is obvious from this figure that the conduction mode of heat transfer dominates when radiation effect increases. There is slight change in isoconcentration lines that move towards the hot wall at bottom portion of the annulus with increase in Rd. This indicates that the mass transfer rate increases slightly at bottom portion of annulus with increase in radiation parameter. The increase in mass transfer at bottom portion is counterbalanced by decrease at upper portion; thus, there is negligible effect of radiation parameter on average Sherwood number (figure not shown to conserve the space). Figure 4 shows the temperature and concentration profile for different values of N. For agivenvalueofra, Rd, A r,leand R, temperature and concentration gradient increases with increase in N. The concentration profile is affected to a greater degree due to changes in N, as compared to that of temperature profile. At little distance from the inner surface of cylinder, the variation in concentration profile with respect to N ceases for aiding flow. The effect of Rayleigh number on temperature and concentration profile is shown in Fig. 5. The thermal and concentration boundary layer becomes thinner with increase in Rayleigh number as thermal energy and species concentration diffuses quickly leading to increased heat and mass transfer rate. When Ra is varied, then the temperature profile is affected to a greater extent as compared to concentration profile. 5Conclusion Heat and mass transfer in a vertical annular cylinder embedded with saturated porous medium is studied. The governing equations are solved using finite element method. Effects of various parameters such as buoyancy ratio, Lewis number, Rayleigh number, aspect ratio, and radius ratio on Nu and Sh are investigated. It is found that Sh increases initially when aspect ratio is increased, reaches a maximum value and then starts decreasing. The maxima for Sh keep shifting towards lower aspect ratio when Le is increased. Similar effect on Sh with respect to aspect ratio is observed due to changes in N. In general, it is found that Sh increases with increase in Le. Temperature and concentration gradient increases with increase in buoyancy ratio or Rayleigh number. References Angirasa, D., Peterson, G.P., Pop, I.: Combined heat and mass transfer by natural convection with opposing buoyancy effects in a saturated porous medium. Int. J. Heat Mass Trans. 4, (997) Badruddin, I.A., Zainal, Z.A., Aswatha Narayana, P.A., Seetharamu, K.N.: Heat transfer by radiation and natural convection through a vertical annulus embedded in porous medium. Int. Comm. Heat Mass Trans. 33, 5 57 (26) 23

19 Analysis of Heat and Mass Transfer Badruddin, I.A., Zainal, Z.A., Aswatha Narayana, P.A., Seetharamu, K.N.: Thermal non-equilibrium modeling of heat transfer through vertical annulus embedded with porous medium. Int. J. Heat Mass Trans. 49, (26) Badruddin, I.A., Zainal, Z.A., Zahid, A.K., Zulquernain, M.: Effect of viscous dissipation and radiation on natural convection in a porous medium embedded within vertical annulus. Int. J. Thermal Sci. 46(3), (27) Bejan, A.D.K. (eds.): Heat Transfer Handbook. Wiley, New York. (23) Cheng, C.-Y.: Free convection heat and mass transfer from a horizontal cylinder of elliptic cross section in micropolar fluids. Int. Comm. Heat Mass Trans. 33, 3 38 (26) Cheng, C.-Y.: Fully developed natural convection heat and mass transfer in a vertical annular porous medium with asymmetric wall temperatures and concentrations. Appl. Thermal Eng. 26, (26) Cheng, C.-Y.: Natural convection heat and mass transfer near a vertical wavy surface with constant wall temperature and concentration in a porous medium. Int. Comm. Heat Mass Trans. 26, (2) Cheng, C.-Y.: Soret and Dufour effects on natural convection heat and mass transfer from a vertical cone in a porous medium. Int. Comm. Heat Mass Trans. 36, 2 24 (29) El-Amin, M.F.: Double dispersion effects on natural convection heat and mass transfer in non-darcy porous medium. Appl. Math. Comput. 56, 7 (24) Hossain, M.A., Alim, M.A.: Natural convection radiation interaction on boundary layer flow along a thin vertical cylinder. Heat Mass Trans. 32, (997) Hossain, M.A., Vafai, K., Khanafer, K.M.N.: Non-Darcy natural convection heat and mass transfer along a vertical permeable cylinder embedded in a porous medium. Int. J. Thermal Sci. 38, (999) Ingham, D.B., Pop, I. (eds.): Transport Phenomena in Porous Media. Pergamon, Oxford (998) Kairi, R.R., Murthy, P.V.S.N.: Effect of viscous dissipation on natural convection heat and mass transfer from vertical cone in a non-newtonian fluid saturated non-darcy porous medium. Appl. Math. Comput. 27, 8 84 (2) Lewis, R.W., Nithiarasu, P., Seetharamu, K.N.: Fundamentals of the Finite Element Method for Heat and Fluid Flow. Wiley, Chichester (24) Nakayama, A., Hossain, M.A.: An integral treatment for combined heat and mass transfer by natural convection in a porous medium. Int. J. Heat Mass Trans. 38, (995) Nield, D., Bejan, A.: Convection in Porous Media. 3rd edn. Springer, New York (26) Pop, I., Ingham, D.B.: Convective Heat Transfer: Mathematical and Computational Modeling of Viscous Fluids and Porous Media. Pergamon, Oxford (2) Rajamani, R., Srinivasa, C., Nithiarasu, P., Seetharamu, K.N.: Convective heat transfer in axisymmetric porous bodies. Int. J. Numer. Methods Heat Fluid Flow 5, (995) Raptis, A.: Radiation and free convection flow through a porous medium. Int. Comm. Heat Mass Trans. 25, (998) Salman Ahmed, N.J., Badruddin, I.A., Jeevan, K., Zainal, Z.A., Nazim Ahamed, K.S.: Study of mixed convection in an annular vertical cylinder filled with saturated porous medium, using thermal non-equilibrium model. Int. J. Heat Mass Trans. 54(7-8), (2) Salman Ahmed, N.J., Badruddin, I.A., Zainal, Z.A., Khaleed, H.M.T., Jeevan, K.: Heat transfer in a conical cylinder with porous medium. Int. J. Heat Mass Trans. 52(3 4), (29) Segerland, L.J.: Applied Finite Element Analysis. Wiley, New York (982) Singh, B.B., Chandarki, I.M.: Integral treatment of coupled heat and mass transfer by natural convection from a cylinder in porous media. Int. Comm. Heat Mass Trans. 36, (29) Thomas, H.R., Ferguson, W.J.: A fully coupled heat and mass transfer model incorporating contaminant gas transfer in an unsaturated porous medium. Comput. Geotech. 24, (999) Trevisan, O.V., Bejan, A.: Mass and heat transfer by high Rayleigh number convection in a porous medium heated from below. Int. J. Heat Mass Trans. 3, (987) Vafai, K. (ed.): Hand Book of Porous Media. Marcel Dekker, New York (2) Yang, Y.-C.: Simultaneously estimating the contact heat and mass transfer coefficients in a double-layer hollow cylinder with interface resistance. Appl. Thermal Eng. 27, 5 58 (27) Yih, K.A.: Coupled heat and mass transfer in mixed convection about a vertical cylinder in a porous medium: The entire regime. Mech. Rea. Commun. 25, (998) Yih, K.A.: Coupled heat and mass transfer by natural convection adjacent to a permeable horizontal cylinder in a saturated porous medium. Int. Comm. Heat Mass Trans. 26, (999) Yih, K.A.: Radiation effect on natural convection over a vertical cylinder embedded in a porous media. Int. Comm. Heat Mass Trans. 26, (999) 23