# Darcy-Forchheimer natural, forced and mixed convection heat transfer in non-Newtonian power-law fluid-saturated porous media

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## Abstract

The governing equation for Darcy-Forchheimer flow of non-Newtonian inelastic power-law fluid through porous media has been derived from first principles. Using this equation, the problem of Darcy-Forchheimer natural, forced, and mixed convection within the porous media saturated with a power-law fluid has been solved using the approximate integral method. It is observed that a similarity solution exists specifically for only the case of an isothermal vertical flat plate embedded in the porous media. The results based on the approximate method, when compared with existing exact solutions show an agreement of within a maximum error bound of 2.5%.

## Key words

non-Newtonian fluids power-law fluids Darcy-Forchheimer flow natural convection forced convection mixed convection## Nomenclature

*A*cross-sectional area

*b*_{i}coefficient in the chosen temperature profile

*B*_{1}coefficient in the profile for the dimensionless boundary layer thickness

*C*coefficient in the modified Forchheimer term for power-law fluids

*C*_{1}coefficient in the Oseen approximation which depends essentially on pore geometry

*C*_{i}coefficient depending essentially on pore geometry

*C*_{D}drag coefficient

*C*_{t}coefficient in the expression for

*K*^{*}*d*particle diameter (for irregular shaped particles, it is characteristic length for average-size particle)

*f*_{p}resistance or drag on a single particle

*F*_{R}total resistance to flow offered by

*N*particles in the porous media*g*acceleration due to gravity

*g*_{x}component of the acceleration due to gravity in the

*x*-direction- \(Gr_{K^* }\)
Grashof number based on permeability for power-law fluids

*K*intrinsic permeability of the porous media

*K*^{*}modified permeability of the porous media for flow of power-law fluids

*l*_{c}characteristic length

*m*exponent in the gravity field

*n*power-law index of the inelastic non-Newtonian fluid

*N*total number of particles

- Nu
*x*,_{D,F} local Nusselt number for Darcy forced convection flow

- Nu
*x*,_{D-F,F} local Nusselt number for Darcy-Forchheimer forced convection flow

- Nu
*x*,_{D,M} local Nusselt number for Darcy mixed convection flow

- Nu
*x*,_{D-F,M} local Nusselt number for Darcy-Forchheimer mixed convection flow

- Nu
*x*,_{D,N} local Nusselt number for Darcy natural convection flow

- Nu
*x*,_{D-F,N} local Nusselt number for Darcy-Forchheimer natural convection flow

- \(\bar p\)
pressure

*p*exponent in the wall temperature variation

_{Pe}*c*characteristic Péclet number

_{Pe}*x*local Péclet number for forced convection flow

_{Pe′}*x*modified local Péclet number for mixed convection flow

_{Ra}*c*characteristic Rayleigh number

_{Ra}*x*local Rayleigh number for Darcy natural convection flow

_{Ra′}*x*local Rayleigh number for Darcy-Forchheimer natural convection flow

- Re
convectional Reynolds number for power-law fluids

- \(\operatorname{Re} _{K^* }\)
Reynolds number based on permeability for power-law fluids

*T*temperature

*T*_{e}ambient constant temperature

*T**w*,_{ref}constant reference wall surface temperature

*T*_{w}(X)variable wall surface temperature

- δ
*T*_{w} temperature difference equal to

*T**w*,_{ref}−*T*_{e}*T*_{1}term in the Darcy-Forchheimer natural convection regime for Newtonian fluids

*T*_{2}term in the Darcy-Forchheimer natural convection regime for non-Newtonian fluids (first approximation)

*T*_{N}term in the Darcy/Forchheimer natural convection regime for non-Newtonian fluids (second approximation)

*u*Darcian or superficial velocity

*u*_{1}dimensionless velocity profile

*u*_{e}external forced convection flow velocity

*u*_{s}seepage velocity (local average velocity of flow around the particle)

*u*_{w}wall slip velocity

*U**c*_{M}characteristic velocity for mixed convection

*U**c*_{N}characteristic velocity for natural convection

*x, y*boundary-layer coordinates

*x*_{1},*y*_{1}dimensionless boundary layer coordinates

*X*coefficient which is a function of flow behaviour index

*n*for power-law fluids*α*effective thermal diffusivity of the porous medium

*α*′shape factor which takes a value of

*Μ*/4 for spheres*Β*′shape factor which takes a value of

*Μ*/6 for spheres*Β*_{0}expansion coefficient of the fluid

*δ*_{T}boundary-layer thickness

*δ**T*_{1}dimensionless boundary layer thickness

*ε*porosity of the medium

*η*similarity variable

*θ*dimensionless temperature difference

*λ*′coefficient which is a function of the geometry of the porous media (it takes a value of 3

*Μ*for a single sphere in an infinite fluid)*Μ*_{0}viscosity of Newtonian fluid

*Μ*^{*}fluid consistency of the inelastic non-Newtonian power-law fluid

*ξ*constant equal to

*X*(2*ε*^{2−n}*λ*′)/*α*′*ρ*density of the fluid

*Φ*dimensionless wall temperature difference

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## References

- Bejan, A., 1987, Convective heat transfer in porous media, in S. Kakac, R. K. Shah, and W. Aung (eds),
*Handbook of Single-Phase Convective Heat Transfer*, Wiley, New York, Chapt. 16.Google Scholar - Bejan, A. and Poulikakos, D., 1984, The non-Darcy regime for vertical boundary layer natural convection in a porous medium,
*Int. J. Heat Mass Transfer***27**, 717–722.Google Scholar - Brinkman, H. C., 1947, A calculation of the viscous force exerted by a flowing fluid on a dense swarm of particles,
*Appl. Sci. Res.***A1**, 27–34.Google Scholar - Chhabra, R. P., 1986, Steady non-Newtonian flow about a rigid sphere,
*Encyclopedia of Fluid Mechanics Vol 1*, Gulf Publishing Co., Ch. 30, pp. 983–1033.Google Scholar - Chen, H-T. and Chen, C-K., 1988a, Free convection flow of non-Newtonian fluids along a vertical plate embedded in a porous medium,
*Trans ASME, J. Heat Transfer***110**, 257–260.Google Scholar - Chen, H-T. and Chen, C-K., 1988b, Natural convection of a non-Newtonian fluid about a horizontal cylinder and a sphere in a porous medium,
*Int. Comm. Heat Mass Transfer***15**, 605–614.Google Scholar - Christopher, R. H. and Middleman, S., 1965, Power-law flow through a packed tube,
*I & EC Fundamentals***4**, 422–426.Google Scholar - Churchill, S. W., 1977, A comprehensive correlating equation for laminar, assisting, forced and free convection,
*AIChE J.***23**, 16.Google Scholar - Combarnous, M. A. and Bories, S. A., 1975, Hydrothermal convection in saturated porous media,
*Adv. ydrosci.***10**, 231–307.Google Scholar - Dharmadhikari, R. V. and Kale, D. D., 1985, Flow of non-Newtonian fluids through porous media,
*Chem. Engg. Sci.***40**, 527–529.Google Scholar - Fand, R. M., Steinberger, T. E., and Cheng, P., 1986, Natural convection heat transfer from a horizontal cylinder embedded in a porous medium,
*Int. J. Heat Mass Transfer***29**, 119–133.Google Scholar - Forchheimer, P., 1901, Wasserbewegung durch boden,
*Zeitschrift Ver. D. Ing.***45**, 1782–1788.Google Scholar - Goldstein, S., 1938,
*Modern Developments in Fluid Dynamics*, Vol. II, Oxford University Press, London.Google Scholar - Hellums, J. D. and Churchill, S. W., 1964, Simplification of the mathematical description of boundary and initial value problems,
*AIChE J.***10**, 110–114.Google Scholar - Herschel, W. H. and Bulkley, R., 1926, Konsistenzmessungen von gummi-benzollosungen,
*Koll. Z.***39**, 291–295.Google Scholar - Ingham, D. B., 1986, The non-Darcy free convection boundary layer on axisymmetric and two-dimensional bodies of arbitrary shape,
*Int. J. Heat Mass Transfer***29**, 1759–1763.Google Scholar - Kafoussias, N. G., 1990, Principles of flow through porous media with heat transfer,
*Encyclopedia of Fluid Mechanics*Vol. 10, Gulf Publishing Co., Ch. 20, pp. 663–686.Google Scholar - Kakac, S.
*et al.*(eds), 1991,*Convective Heat and Mass Transfer in Porous Media*, Kluwer Academic Pubs, Dordrecht.Google Scholar - Kemblowski, Z. and Michniewicz, M., 1979, A new look at the laminar flow of power-law fluids through granular beds,
*Rheol. Acta***18**, 730–739.Google Scholar - Lauriat, G. and Prasad, V., 1987, Natural convection in a vertical porous cavity: A numerical study for Brinkman-extended Darcy formulation,
*Trans. ASME, J. Heat Transfer***109**, 688–696.Google Scholar - Muskat, M., 1946,
*The Flow of Homogeneous Fluids through Porous Media*, Edwards, MI.Google Scholar - Nakayama, A. and Koyama, H., 1991, Buoyancy-induced flow of non-Newtonian fluids over a nonisothermal body of arbitrary shape in a fluid-saturated porous medium,
*Appl. Sci. Res.***48**, 55–70.Google Scholar - Nakayama, A. and Pop, I., 1991, A unified similarity transformation for free, forced and mixed convection in Darcy and non-Darcy porous media,
*Int. J. Heat Mass Transfer***34**, 357–367.Google Scholar - Nakayama, A. and Shenoy, A. V., 1993, Combined forced and free convection heat transfer in non-Newtonian fluid saturated porous medium,
*Appl. Sci. Res.*(to appear).Google Scholar - Nakayama, A., Kokudai, T., and Koyama, H., 1988, An integral treatment for non-Darcy free convection over a vertical flat plate and cone embedded in a fluid-saturated porous medium,
*WÄrme-und Stoffubertragung***23**, 337–341.Google Scholar - Nakayama, A., Koyama, H. and Kuwahara, F., 1989, Similarity solution for non-Darcy free convection from a non-isothermal curved surface in a fluid-saturated porous medium,
*Trans. ASME, J. Heat Transfer***111**, 807–811.Google Scholar - Nakayama, A., Kokudai, T., and Koyama, H., 1990, Forchheimer free convection over a nonisothermal body of arbitrary shape in a saturated porous medium,
*Trans. ASME, J. Heat Transfer***112**, 511–515.Google Scholar - Nield, D. A. and Joseph, D. D., 1985, Effects of quadratic drag on convection in a saturated porous medium,
*Phys. Fluids***28**, 995–997.Google Scholar - Oseen, C. W., 1927,
*Neuere Methoden und Ergebnisse in der Hydrodynamik*, Akademische Verlagsgesellschaft, Leipzig.Google Scholar - Pascal, H. and Pascal, J. P., 1989, Nonlinear effects of non-Newtonian fluids on natural convection in a porous medium,
*Physica D***40**, 393–402.Google Scholar - Pascal, H., 1990a, Nonisothermal flow of non-Newtonian fluids through a porous medium,
*Int. J. Heat Mass Transfer***33**, 1937–1944.Google Scholar - Pascal, H., 1990b, Some self-similar two-phase flows of non-Newtonian fluids through a porous medium,
*Studies in Applied Math.***82**, 305–318.Google Scholar - Plumb, O. A. and Huenefeld, J. C., 1981, Non-Darcy natural convection from heated surfaces in saturated porous media,
*Int. J. Heat Mass Transfer***24**, 765–768.Google Scholar - Poulikakos, D. and Bejan, A., 1985, The departure from Darcy flow on natural convection in vertical porous layer,
*Phys. Fluids***28**, 3477–3484.Google Scholar - Ruckenstein, E., 1978, Interpolating equations between two limiting cases for the heat transfer coefficient,
*AIChE J.***24**, 940.Google Scholar - Rumer, R. R. Jr., 1969, Resistance to flow through porous media, in R. J. M. De Wiest (ed.),
*Flow Through Porous Media*, Academic Press, New York, pp. 91–108.Google Scholar - Shenoy, A. V., 1980a, A correlating equation for combined laminar forced and free convection heat transfer to power-law fluids,
*AIChE J.***26**, 505–507.Google Scholar - Shenoy, A. V., 1980b, Combined laminar forced and free convection heat transfer to viscoelastic fluids,
*AIChE J.***26**, 683–685.Google Scholar - Shenoy, A. V., 1986, Natural convection heat transfer to power-law fluids,
*Handbook of Heat Mass Transfer*Vol. 1, Gulf Publishing Co. Ch. 5, pp. 183–210.Google Scholar - Shenoy, A. V., 1988, Natural convection heat transfer to viscoelastic fluids,
*Encyclopedia of Fluid Mechanics*Vol. 7, Gulf Publishing Co. Ch. 10, pp. 287–304.Google Scholar - Tien, C. L. and Vafai, K., 1990, Convective and radiative heat transfer in porous media,
*Adv. Appl. Mech.***27**, 225–281.Google Scholar - Tong, T. W. and Subramanian, E., 1985, A boundary-layer analysis for natural convection in vertical porous enclosures — use of the Brinkman-extended Darcy-model,
*Int. J. Heat Mass Transfer***28**, 563–571.Google Scholar - Trevisan, O. V. and Bejan, A., 1990, Combined heat and mass transfer by natural convection in a porous medium,
*Adv. Heat Transfer***20**, 315–352.Google Scholar - Vasantha, R., Pop, I., and Nath, G., 1986, Non-Darcy natural convection over a slender vertical frustum of a cone in a saturated porous medium,
*Int. J. Heat Mass Transfer***29**, 153–156.Google Scholar - Wang, C. and Tu, C., 1989, Boundary-layer flow and heat transfer of non-Newtonian fluids in porous media,
*Int. J. Heat Fluid Flow***10**, 160–165.Google Scholar - Wang, C., Tu, C. and Zhang, X., 1990, Mixed convection of non-Newtonian fluids from a vertical plate embedded in a porous medium,
*Acta Mechanica Sinica***6**, 214–220.Google Scholar - Ward, J. C., 1969, Turbulent flow in porous medium,
*Proc. Am. Soc. Civil Engng. No. HY5*,**90**, 1–12.Google Scholar