Free and Forced Convective Flow in a Vertical Channel Filled with Composite Porous Medium Using Robin Boundary Conditions
Mixed convection flow and heat transfer in a vertical channel filled with composite porous medium using Robin boundary conditions is analyzed. The flow is modeled using the Darcy-Lapwood-Brinkman model. The viscous and Darcy dissipation terms are included in energy equation. The plate exchanges heat with an external fluid. Both the conditions of equal and different reference temperature of the external fluid are considered. The governing equations are coupled and non-linear because of inclusion of dissipation terms and buoyancy forces. The equations are solved using perturbation method valid for small values of perturbation parameter. However, the restriction on the perturbation parameter is relaxed by finding the solutions of governing equations by using Differential Transform Method. The effects of various parameters such as mixed convection parameter, porous parameter, viscosity ratio, width ratio, conductivity ratio and the Biot numbers on the flow are discussed. The percentage of error between perturbation method and differential transformation method increases as the perturbation parameter increases for both equal and unequal Biot numbers.
Jada Prathap Kumar,
Jawali Channabasappa Umavathi,
Free and Forced Convective Flow in a Vertical Channel Filled with Composite Porous Medium Using Robin Boundary Conditions, American Journal of Applied Mathematics.
Vol. 2, No. 4,
2014, pp. 96-110.
G. Beavers, and D.D. Joseph, “Boundary conditions at a naturally permeable wall,” J. Fluid Mech, vol. 30: pp. 197-207, 1967.
G. Neale, and W. Nadar, “Practical significance of Brinkman’s extension of Darcy’s law: Coupled parallel flows within a channel and a bounding porous medium,” Can. J. Chem. Engrg, vol. 52, pp. 475- 478, 1974.
K. Vafai, and S.J. Kim, “Fluid mechanics of the inter-phase region between a porous medium and a fluid layer-an exact solution,” Int. J. Heat Fluid Flow, vol. 11, pp.254-256, 1990.
K. Vafai, and R. Thiyagaraja, “Analysis of flow and heat transfer at the interface region of a porous medium,” Int. J. Heat Fluid Flow, vol. 30, pp. 1391-1405, 1987.
K. Vafai, and S.J. Kim, “Analysis of surface enhancement by a porous substrate,” J. Heat Transfer, vol. 112, pp. 700-706, 1990.
S.J. Kim, and C.Y. Choi, “Convection heat transfer in porous and overlaying layers heated from below,” Int. J. Heat Mass Transfer, vol. 39, pp. 319-329, 1996.
D. Poulikakos, and M. Kazmierczak, “Forced convection in a duct partially filled with a porous material,” J. Heat Transfer, vol. 109, pp.653-662, 1987.
J.A. Ochoa-Tapie, and S. Whitaker, “Heat transfer at the boundary between a porous medium and a homogeneous fluid,” Int. J. Heat Mass Transfer, vol. 40, pp. 2691-2707, 1997.
J.W. Paek, B.H. Kang, S.Y. Kim, and J.M. Hyan, “Effective thermal conductivity and permeability of aluminum foam materials,” Int. J. Thermophys, vol. 21, pp. 453-464, 2000.
A.V. Kuznetsov, “Analytical study of fluid flow and heat transfer during forced convection in a composite channel partly filled with a Brinkman-Forchheimer porous medium,” Flow Turbulence and Combustion, vol. 60, pp. 173-192, 1998.
D.A. Nield, and A.V. Kuznetsov, “The effect of heterogeneity in forced convection in a porous medium: parallel plate channel or circular duct,” Int. J. Heat Mass Transfer, vol. 43, pp. 4119-4134, 2000.
M.S. Malashetty, J.C. Umavathi, and J. Prathap Kumar, “Convective Flow and Heat Transfer in an Inclined Composite Porous Medium,” J. Porous Media, vol. 4(1), pp. 15-22, 2001.
M.S. Malashetty, J.C. Umavathi, and J. Prathap Kumar, “Two fluid flow and heat transfer in an inclined channel containing porous and fluid layer,” Heat and Mass Transfer, vol. 40, pp. 871–876, 2004.
J.C. Umavathi, Ali J Chamkha, Abdul Mateen, and Al-Mudhaf A, “Oscillatory flow and heat transfer in a horizontal composite porous medium channel,” Int. J. Heat and Tech, vol. 25(2), pp. 75-86, 2006.
J.C. Umavathi, M.S. Malashetty, and J. Prathap Kumar, “Flow and heat transfer in an inclined channel containing a porous layer sandwiched between two fluid layers,” ASME, Modelling Measurement and Controlling, vol. 74, pp. 19-35, 2005.
J.C. Umavathi, J. Prathap Kumar, and K.S.R. Sridhar, “Flow and heat transfer of poiseuille.couette flow in an inclined channel for composite porous medium,” Int. J. Appl. Mech. Engg, vol. 15(1), pp. 249-266, 2010.
J. Prathap Kumar, J.C. Umavathi, and Basavaraj M Biradar, “Mixed convection of composite potous medium in a vertical channel with asymmetrical wall heating conditions,” J. Porous Media, vol. 13(3), pp. 271-285, 2010.
J. Prathap Kumar, J.C. Umavathi, I. Pop, and Basavaraj M Biradar, “Fully developed mixed convection flow in a vertical channel containing porous and fluid layer with isothermal or isoflux boundaries,” Trans. Porous Med, vol. 80, pp. 117–135, 2009.
P. Wibulswas, “Laminar flow heat transfer in non circular ducts,” Ph.D. Thesis, London University, 1966 (as reported by Shah and London in 1971)
R.W. Lyczkowski, C.W. Solbrig, and D. Gidaspow, Forced convective heat transfer in rectangular ducts general case of wall resistance and peripheral conduction. 969. Institute of Gas Technology Tech. Info, Center File 3229, 3424S, State, Street, Chicago, Ill 60616 (as reported by Shah and London in 1971).
V. Javeri, “Analysis of laminar thermal entrance region of elliptical and rectangular channels with Kantorowich method,” Warme- und Stoffuberragung, vol. 9, pp. 85–98, 1976.
E. Hicken, and Das, Temperatur field in laminar durchstromten Kanalen mitechteckquerschnitt bei unterschiedlicher Beheizung der Kanalwade,” Warme- und Stoffubertragung, vol. 1, pp. 98–104, 1968.
E.M. Sparrow, R. Siegal, “Application of variational methods to the thermal entrance region of ducts,” Int. J. Heat Mass Transfer, vol. 1, pp. 161–172, 1960.
V. Javeri, Heat transfer in laminar entrance region of a flat channel for the temperature boundary condition of the third kind,” Warne-und Stoffubertragung Thermo-and Fluid Dynamics, vol. 10, pp. 137-144, 1977.
V. Javeri, “Laminar heat transfer in a rectangular channel for the temperature boundary conditions of third kind,” Int. J, Heat Mass Transfer, vol. 10, pp. 1029-1034, 1978.
E. Zanchini, Effect of viscous dissipation on mixed convection in a vertical channel with boundary conditions of the third kind,” Int. J Heat and Mass Transfer, vol. 41, pp. 3949 –3959, 1998.
M. Kumari, and G. Nath, “Mixed convection boundary layer flow over a thin vertical cylinder with localized injection/suction and cooling/heating,” Int. Heat mass Transfer, vol. 47, pp. 969-976, 2004.
J.K. Zhou, “Differential transformation and its applications for electrical circuits,” Huarjung University Press, 1986. (in Chinese)
M.J. Jang, C.L. Chen, and Y.C. Liu, “Two-dimensional differential transform for partial differential equations,” Appl. Math. Comput, vol 21, pp. 261–270, 2001.
A. Kurnaz, and G. Oturanç, “The differential transform approximation for the system of ordinary differential equations,” Int. J. Comput. Math, vol. 82, pp. 709–719, 2005.
A. Arikoglu, and I. Ozkol, “Solution of differential-difference equations by using differential transform method,” Appl. Math. Comput, vol. 181, pp. 153–62, 2006.
A.S.V. Ravi Kanth, and K. Aruna, “Solution of singular two-point boundary value problems using differential transformation method,” Physics Letters A, vol. 372, pp. 4671–4673, 2008.
M.J. Jang, Y.L. Yeh, C.L. Chen, and W.C. Yeh, “Differential transformation approach to thermal conductive problems with discontinuous boundary condition,” Appl., Math. Comput, vol. 216, pp. 2339–2350, 2010.
M.M. Rashidi, N. Laraqi, and S. M. Sadri, “A novel analytical solution of mixed convection about an inclined flat plate embedded in a porous medium using the DTM-Padé,” Int. J. Thermal Sci, vol. 49, pp. 2405–2412, 2010.
H. Yaghoobi, and M. Torabi, “The application of differential transformation method to nonlinear equations arising in heat transfer,” Int. Commu. Heat and Mass Transfer, vol. 38, pp. 815–820, 2011.
M.M. Rashidi, O. Anwar Beg, and N. Rahimzadeh, “A generalized differential transform method for combined free and forced convection flow about inclined surfaces in porous media,” Chem. Eng. Comm, vol. 199, pp. 257–282, 2012.
A. Barletta, “Laminar mixed convection with viscous dissipation in a vertical channel,” Int. J. Heat Mass Transfm, vol. 41, pp. 3501-3513, 1998.
J.C. Umavathi, and V. Santosh, “Non-Darcy mixed convection in a vertical porous channel with boundary conditions of third kind,” Trans. Porous Media, vol. 95, pp. 111-131, 2012.