# Effects of temperature-dependent viscosity variation on entropy generation, heat and fluid flow through a porous-saturated duct of rectangular cross-section

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## Abstract

Effect of temperature-dependent viscosity on fully developed forced convection in a duct of rectangular cross-section occupied by a fluid-saturated porous medium is investigated analytically. The Darcy flow model is applied and the viscosity-temperature relation is assumed to be an inverse-linear one. The case of uniform heat flux on the walls, i.e. the H boundary condition in the terminology of Kays and Crawford [12], is treated. For the case of a fluid whose viscosity decreases with temperature, it is found that the effect of the variation is to increase the Nusselt number for heated walls. Having found the velocity and the temperature distribution, the second law of thermodynamics is invoked to find the local and average entropy generation rate. Expressions for the entropy generation rate, the Bejan number, the heat transfer irreversibility, and the fluid flow irreversibility are presented in terms of the Brinkman number, the Péclet number, the viscosity variation number, the dimensionless wall heat flux, and the aspect ratio (width to height ratio). These expressions let a parametric study of the problem based on which it is observed that the entropy generated due to flow in a duct of square cross-section is more than those of rectangular counterparts while increasing the aspect ratio decreases the entropy generation rate similar to what previously reported for the clear flow case by Ratts and Raut [14].

## Key words

entropy generation rate forced convection porous medium rectangular duct temperature-dependent viscosity## Nomenclature

*a*aspect ratio

*A*coefficient defined by Eq. (17)

*Be*Bejan number defined by Eq.(32)

*Br*Brinkman number defined by Eq.(29)

*c*_{P}specific heat at constant pressure

*D*_{H}hydraulic diameter

*D*_{n}coefficient defined by Eq.(14)

*G*Negative of the applied pressure gradient

*H*duct height

*k*thermal conductivity

*K*permeability

*m*coefficient defined by Eq.(14)

*S*_{1},*S*_{2}series defined by Eq.(19b,c)

*Ṡ*_{gen}entropy generation rate per unit volume

*T*^{*}temperature

*T*_{w}wall temperature

*T*_{m}bulk mean temperature

*u*^{*}filtration velocity

*ū*mean velocity

*û*normalized velocity

*u/ū**x,y,z*dimensionless coordinates

*N*viscosity variation number

*N*_{FFI}fluid friction irreversibility

*N*_{HTI}heat transfer irreversibility

*N*_{S}dimensionless entropy generation number defined by Eq.(31)

*N*_{u}Nusselt number defined by Eq.(20)

*P*^{2}viscosity variation parameter defined by Eq.(11)

*P*_{e}Péclet number defined by Eq.(27a)

*q*dimensionless wall heat flux by Eq.(27b)

*q*″wall heat flux

*R*dimensionless parameter defined by Eq.(10)

*x*^{*},*y*^{*},*z*^{*}Cartesian coordinates

## Greek symbols

*ϑ**k**T*_{ w }−*T** /*q*″*H**μ*fluid viscosity

*λ*_{n}Eigenvalues of the problem

*ρ*fluid density

## Subscripts

- cp
constant property

- w
wall

## Chinese Library Classification

O357.3## 2000 Mathematics Subject Classification

76S05## Preview

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## References

- [1]Nield D A, Bejan A. Convection in porous media[M]. New York: Springer-Verlag, 1999.Google Scholar
- [2]Lauriat G, Ghafir R. Forced convective heat transfer in porous media[M]. In: Vafai K (ed).
*Handbook of Porous Media*, New York: Dekker, 2000, 201.Google Scholar - [3]Haji-Sheikh A, Vafai K. Analysis of flow and heat transfer in porous media imbedded inside various-shaped ducts[J].
*Internet J Heat Mass Transfer*, 2004,**47**(8/9):1889–1905.CrossRefzbMATHGoogle Scholar - [4]Haji-Sheikh A. Heat transfer to fluid flow in rectangular passages filled with porous materials[J].
*ASME J Heat Transfer*, 2006,**128**(6):550–556.CrossRefGoogle Scholar - [5]Haji-Sheikh A, Nield D A, Hooman K. Heat transfer in the thermal entrance region for flow through rectangular porous passages[J].
*Internet J Heat Mass Transfer*, 2006,**49**(17/18):3004–3015.CrossRefGoogle Scholar - [6]Hooman K, Merrikh A A. Forced convection in a duct of rectangular cross-section saturated by a porous medium[J].
*ASME J Heat Transfer*, 2006,**128**(6):596–600.CrossRefGoogle Scholar - [7]Hooman K. Analysis of entropy generation in porous media imbedded inside elliptical passages[J].
*Internat J Heat Technology*, 2005,**23**(2):145–149.MathSciNetGoogle Scholar - [8]Hooman K. Fully developed temperature distribution in a porous saturated duct of elliptical cross-section, with viscous dissipation effects and entropy generation analysis[J].
*Heat Transfer Research*, 2005,**36**(3):237–245.CrossRefGoogle Scholar - [9]Hooman K. Entropy-energy analysis of forced convection in a porous-saturated circular tube considering temperature-dependent viscosity effects[J].
*Internet J Exergy*, 2006,**3**(4):436–451.CrossRefGoogle Scholar - [10]Harms T M, Jog M A, Manglik R M. Effects of temperature-dependent viscosity variations and boundary conditions on fully developed laminar steady forced convection in a semicircular duct[J].
*ASME J Heat Transfer*, 1998,**120**(3):600–605.Google Scholar - [11]Ling J X, Dybbs A. The effect of variable viscosity on forced convection over a flat plate submerged in a porous medium[J].
*ASME J Heat Transfer*, 1992,**114**(4):1063–1065.CrossRefGoogle Scholar - [12]Kays W M, Crawford M E. Convective heat and mass transfer[M]. New York: McGraw-Hill, 1993.Google Scholar
- [13]Bejan A. Entropy generation through heat and fluid flow[M]. New York: Wiley, 1982.Google Scholar
- [14]Ratts E B, Raut A G. Entropy generation minimization of fully developed internal flow with constant heat flux[J].
*ASME J Heat Transfer*, 2004,**126**(4):656–659.CrossRefGoogle Scholar - [15]Sahin A Z. Second law analysis of laminar viscous flow through a duct subjected to a constant wall temperature[J].
*ASME J Heat Transfer*, 1998,**120**(1):76–83.MathSciNetGoogle Scholar - [16]Al-Zahranah I T, Yilbas B S. Thermal analysis in pipes: influence of variable viscosity on entropy generation[J].
*Entropy*, 2004,**6**(3):344–363.CrossRefGoogle Scholar - [17]Nield D A, Porneala D C, Lage J L. A theoretical study, with experimental verification of the viscosity effect on the forced convection through a porous medium channel[J].
*ASME J Heat Transfer*, 1999,**121**(2):500–503.Google Scholar - [18]Hooman K. Effects of temperature-dependent viscosity on thermally developing forced convection through a porous medium[J].
*Heat Trans Res*, 2005,**36**(1/2):132–140.CrossRefGoogle Scholar - [19]Narasimhan A, Lage J L. Modified Hazen-Dupuit-Darcy model for forced convection of a fluid with temperature-dependent viscosity[J].
*Internat J Heat Mass Transfer*, 2001,**123**(1):31–38.Google Scholar - [20]Narasimhan A, Lage J L, Nield D A. New theory for forced convection through porous media by fluids with temperature-dependent viscosity[J].
*ASME J Heat Transfer*, 2001,**123**(6):1045–1051.CrossRefGoogle Scholar - [21]Narasimhan A, Lage J L. Variable viscosity forced convection in porous medium channels[M]. In: Vafai K (ed),
*Handbook of Porous Media*, 2nd Ed, Taylor and Francis: Boca Raton, 2005, 195.Google Scholar - [22]Nield D A, Kuznetsov A V. Effects of temperature dependent viscosity in forced convection in a porous medium: layered medium analysis[J].
*J Porous Media*, 2003,**6**(3):213–222.CrossRefMathSciNetzbMATHGoogle Scholar - [23]Mahmud S, Fraser R A. Flow, heat transfer, and entropy generation characteristics inside a porous channel with viscous dissipation[J].
*Internat J Therm Sci*, 2005,**44**(1):21–32.CrossRefGoogle Scholar - [24]Bejan A. Convection heat transfer[M]. Hoboken: Wiley, 1984.Google Scholar