Abstract
The flow of an incompressible Newtonian fluid confined in a planar geometry with different wall temperatures filled with a homogenous and isotropic porous medium is analyzed in terms of determining the unsteady state and steady state velocities, the temperature and the entropy generation rate as function of the pressure drop, the Darcy number, and the Brinkman number. The one-dimensional approximate equation in the rectangular Cartesian coordinates governing the flow of a Newtonian fluid through porous medium is derived by accounting for the order of magnitude of terms as well as accompanying approximations to the full-blown three-dimensional equations by using scaling arguments. The one-dimensional approximate energy and the entropy equations with the viscous dissipation consisting of the velocity gradient and the square of velocity are derived by following the same procedure used in the derivation of velocity expressions. The one-dimensional approximate equations for the velocity, the temperature, and the entropy generation rate are analytically solved to determine the velocity, the temperature, and the entropy distributions in the saturated porous medium as functions of the effective process parameters. It is found that the pressure drop, the Darcy number, and the Brinkman number affect the temperature distribution in the similar way, and besides the above parameters, the irreversibility distribution ratio also affects the entropy generation rate in the similar way.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
Abbreviations
- Br :
-
Brinkman number=Ec Pr
- Cp :
-
Specific heat (J kg−1 · K−1)
- Pr :
-
Prandtl number=C p /k
- Ec :
-
Eckert number = \({u_{m}^{2}}/{Cp\Delta T}\)
- Da :
-
Darcy number K/h 2
- F :
-
Empirical coefficient
- Fr i :
-
Froude number \({{u_{0}^{1}}/{hg_i}}\)
- h :
-
Half transverse distance (m)
- g i :
-
Gravity in i-direction
- k f :
-
Thermal conductivity (W · m−1 · K−1)
- K :
-
Permeability (m 2)
- ℓ :
-
Characteristic length in the x-direction (m)
- N DB :
-
Entropy generation number, Darcy–Brinkman dissipation = Bru 2/ΩDa
- N F :
-
Entropy generation number, velocity gradient dissipation = Br(∂u/∂y)2/Ω
- N S :
-
Entropy generation number, total
- N Y :
-
Entropy generation number, conduction = (∂θ/∂ y)2
- P :
-
Dimensionless pressure \({={\tilde{P}}/ \left({\mu u_0}\ell/{h^2}\right)}\)
- P 0 :
-
Dimensionless pressure gradient dP/dx
- Re :
-
Cross-flow Reynolds number = ρu 0 h/μ
- S G :
-
Entropy generation rate (W · m−3 · K−1)
- S G,C :
-
Characteristic entropy transfer rate
- t :
-
Dimensionless time \({={\tilde {t}}/{\left({{h^{2}\rho}/\mu}\right)}}\)
- T :
-
Temperature (K)
- T 0 :
-
Reference temperature (K)
- T 1 :
-
Temperature at the lower plate (K)
- T 2 :
-
Temperature at the upper plate (K)
- \({\tilde {u}}\) :
-
Velocity in x-direction (m · s−1)
- u 0 :
-
Characteristic velocity (m · s−1)
- \({\tilde{v}}\) :
-
Velocity in y-direction (m · s−1)
- v 0 :
-
Characteristic velocity in y-direction (m · s−1)
- \({\tilde {w}}\) :
-
Velocity in z-direction (m · s−1)
- u, v, w:
-
Dimensionless velocities
- \({\tilde{x}}\) :
-
Axial direction
- \({\tilde{y}}\) :
-
Transverse direction
- \({\tilde{z}}\) :
-
Normal to the x-direction
- x, y, z:
-
Dimensionless directions
- θ :
-
Dimensionless temperature
- Ω:
-
Dimensionless temperature difference = ΔT/T 0
- \({\Phi}\) :
-
Total viscous dissipation
- μ :
-
Dynamic viscosity (Pa · s)
- ρ :
-
Density of the fluid (kg · m−3)
References
Al-Hadhrami A.K., Elliot L., Ingham D.B.: A new model for viscous dissipation in porous media across a range of permeability values. Transp. Porous Media 53, 117–122 (2003). doi:10.1023/A:1023557332542
Bejan A.: A study of entropy generation in fundamental convective heat transfer. J. Heat Transf. 101, 718–725 (1979)
Bejan A.: Convective Heat Transfer. Wiley, New York (1984)
Bejan A., Dincer I., Lorente S., Miguel A.F., Reis A.H.: Porous and Complex Flow Structures in Modern Technologies. Springer, New York (2004)
Denn M.M.: Process Fluid Mechanics. Prentice-Hall, New Jersey (1980)
Fang T.: Further discussion on the incompressible pressure-driven flow in a channel with porous walls. Int. Comm. Heat Mass Transf. 31, 487–500 (2004). doi:10.1016/S0735-1933(04)00030-2
Haji-Sheikh A., Vafai K.: Analysis of flow and heat transfer in porous media imbedded inside various shaped ducts. Int. J. Heat Mass Transf. 47, 1889–1905 (2004). doi:10.1016/j.ijheatmasstransfer.2003.09.030
Haji-Sheikh A., Nield D.A., Hooman K.: Heat transfer in the thermal entrance region for flow through rectangular porous passages. Int. J. Heat Mass Transf. 49, 3004–3015 (2006). doi:10.1016/j.ijheatmasstransfer.2006.01.040
Hooman K., Haji-Sheikh A.: Analysis of heat transfer and entropy generation for a thermally developing Brinkman-forced convection problem in rectangular duct with isoflux walls. Int. J. Heat Mass Transf. 50, 4180–4194 (2007). doi:10.1016/j.ijheatmasstransfer.2007.02.036
Hooman K., Gurgenci H., Merrikh A.A.: Heat transfer and entropy generation optimization of forced convection in porous-saturated ducts of rectangular cross-section. Int. J. Heat Mass Transf. 50, 2051–2059 (2007). doi:10.1016/j.ijheatmasstransfer.2006.11.015
Ingham D.B., Pop I.: Transport Phenomena in Porous Media III. Elsevier, Oxford (2005)
Kamışlı F.: Pressure-driven laminar flow of a non-Newtonian fluid in a slit with wall suction or injection. Chem. Eng. Process. 47, 585–595 (2008). doi:10.1016/j.cep.2006.11.012
Kaviany M.: Laminar flow through a porous channel bounded by isothermal parallel plates. Int. J. Heat Mass Transf. 28, 851–858 (1985)
Mahmud S., Fraser R.A.: Flow, thermal and entropy generation characteristics inside a porous channel with viscous dissipation. Int. J. Therm. Sci. 44, 21–32 (2005). doi:10.1016/j.ijthermalsci.2004.05.001
Mitrovic J., Maletic B.: Heat transfer with laminar flow forced convection in a porous channel exposed to a thermal asymmetry. Int. J. Heat Mass Transf. 50, 1106–1121 (2007). doi:10.1016/j.ijheatmasstransfer.2006.06.046
Mohammad A.A.: Heat transfer enhancement in heat exchangers fitted with porous media. Part, I.: constant wall temperature. Int. J. Therm. Sci. 42, 385–395 (2003). doi:10.1016/S1290-0729(02)00039-X
Morosuk T.V.: Entropy generation in conduits filled with porous medium totally and partially. Int. J. Heat Mass Transf. 48, 2548–2560 (2005). doi:10.1016/j.ijheatmasstransfer.2005.01.018
Nield D.A.: Resolution of a paradox involving viscous dissipation and nonlinear drag in a porous medium. Transp. Porous Media 41, 349–357 (2000)
Nield D.A., Bejan A.: Convection in Porous Media, 3rd edn. Springer, New York (2006)
Nield D.A., Junqueira S.L.M., Lage J.L.: Forced convection in a fluid-saturated porousmedium channel with isothermal or isoflux boundaries. J. Fluid Mech. 322, 201–214 (1996). doi:10.1017/S0022112096002765
Nield D.A., Kuznetsov A.V., Xiong M.: Thermally developing forced convection in a porous medium: parallel plate channel with walls at uniform temperature, with axial conduction and viscous dissipation effects. Int. J. Heat Mass Transf. 46, 643–651 (2003)
Pop I., Ingham D.B.: Convective Heat Transfer: Mathematical and Computational Modeling of Viscous Fluids and Porous Media. Pergamon, Oxford (2001)
Schlicting, H., Gersten, K.: Boundary Layer Theory. 8th Revised and Enlarged edn. (English). Springer-Verlag, New York (2000)
Vafai K.: Handbook of Porous Media, 2nd edn. Taylor & Francis, New York (2005)
Vafai K., Kim S.J.: Forced convection in a channel filled with a porous medium: an exact solution. ASME J. Heat Transf. 111, 1103–1106 (1989)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Kamışlı, F. Analysis of Laminar Flow and Forced Convection Heat Transfer in a Porous Medium. Transp Porous Med 80, 345–371 (2009). https://doi.org/10.1007/s11242-009-9364-7
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11242-009-9364-7