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Analysis of Entropy Generation in Annular Porous Duct

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Abstract

This paper presents an analytical solution of the velocity and temperature fields in terms of the modified Bessel functions by including the viscous as well as the Darcy dissipations of a fully developed viscous and incompressible fluid, flowing between horizontal concentric annular ducts filled with a saturated porous medium. Both the circular boundaries are assumed to have constant heat fluxes. Using the obtained expressions of the velocity and temperature fields, the entropy generation rate and irreversibility ratio have been also obtained. The effects of the Brinkman number, Darcy number, Péclet number and viscosity ratio parameter on the temperature and entropy generation rate are given by using the graphs and tables. The numerical computation of the analytical solution shows that the heat flux applied at the outer cylindrical surface is more effective in comparison with the inner cylindrical surface on the temperature, entropy generation rate and irreversibility ratio.

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Abbreviations

a :

Dimensionless radius of inner cylinder

\({\bar{a}}\) :

Radius of inner cylinder

Br :

Brinkman number

\(C_p \) :

Specific heat at constant pressure

Da :

Darcy number

G :

Applied pressure gradient \((-{\partial p}/{\partial z})\)

\(I_n (r)\) :

Modified Bessel function of first kind of order n

\(K_n (r)\) :

Modified Bessel function of second kind of order n

K :

Permeability

L :

Radius of outer cylinder

M :

Viscosity ratio parameter, \({\mu _e }/\mu \)

p :

Pressure

Pe :

Péclet number

q :

Constant heat flux

r :

Radial coordinate in non-dimensional form

\({\bar{r}}\) :

Radial coordinate

s :

\(1/{\sqrt{\textit{MDa}}}\)

\(T_0 \) :

Inlet wall temperature

T :

Temperature of the fluid

U :

Characteristic fluid velocity in axial direction

\({\bar{U}}\) :

Fluid velocity

z :

Dimensionless axial coordinate

\({\bar{z}}\) :

Axial coordinate

\(\alpha \) :

Heat flux coefficient

\(\kappa \) :

Fluid thermal conductivity

\(\mu \) :

Fluid viscosity

\(\mu _e \) :

Effective viscosity in Brinkman term

\(\theta \) :

Dimensionless temperature

\(\varOmega \) :

Dimensionless constant heat flux \({qL}/{\kappa T_0 }\)

\(\rho \) :

Fluid density

References

  • Adler, P.M., Mills, P.M.: Motion and rupture of a porous sphere in a linear flow field. J. Rheol. 23, 25–37 (1979)

    Article  Google Scholar 

  • Bejan, A.: Convection Heat Transfer, 4th edn. Wiley, New York (2013)

    Book  Google Scholar 

  • Bejan, A.: Second law analysis in heat transfer and thermal design. Adv. Heat Transf. 15, 1–58 (1982)

    Article  Google Scholar 

  • Breugem, W.P.: The effective viscosity of channel-type porous medium. Phys. Fluids 19, 1–16 (2007). doi:10.1063/1.2792323

    Article  Google Scholar 

  • Brinkman, H.C.: On the permeability of media consisting of closely packed porous particles. Appl. Sci. Res. A1, 81–86 (1947)

    Google Scholar 

  • Demirel, Y., Kahraman, R.: Thermodynamic analysis of convective heat transfer in an annular packed bed. Int. J. Heat Fluid Flow 21, 442–448 (2000)

    Article  Google Scholar 

  • Jha, B.K.: Free-convection flow through an annular porous medium. Heat Mass Transf. 41, 675–679 (2005)

    Article  Google Scholar 

  • Kamish, F.: Analysis of laminar flow and forced convection heat transfer in a porous medium. Transp. Porous Media 80, 345–371 (2009)

    Article  Google Scholar 

  • Kim, S., Russell, W.B.: Modelling in porous media by renormalization of Stokes equations. J. Fluid Mech. 154, 269–286 (1985)

    Article  Google Scholar 

  • Kolodziej, J.A.: Influence of the porosity of a porous medium on the effective viscosity in Brinkman’s filtration equation. Acta Mech. 75, 241–254 (1988)

    Article  Google Scholar 

  • Koplik, J., Levine, H., Zee, A.: Viscosity renormalization in the Brinkman equation. Phys. Fluids 26, 2864–2870 (1983)

    Article  Google Scholar 

  • Kuznotsov, A.V., Xiong, M., Nield, D.A.: Thermally developing forced convection in a porous medium: circular duct with walls at constant temperature, with longitudinal conduction and viscous dissipation effects. Transp. Porous Media 53, 331–345 (2003)

    Article  Google Scholar 

  • Lundgren, T.S.: Slow flow through stationary random beds and suspensions of spheres. J. Fluid Mech. 51, 273–299 (1972)

    Article  Google Scholar 

  • Mahmud, S., Fraser, R.A.: The second law analysis in fundamental convective heat transfer problems. Int. J. Therm. Sci. 42, 177–186 (2003)

    Article  Google Scholar 

  • Makinde, O.D., Osalusi, E.: Second law analysis of laminar flow in a channel field with saturated porous media. Entropy 7, 148–16 (2005)

    Article  Google Scholar 

  • Mohammad, A.A.: Heat transfer enhancement in heat exchanger fitted with porous media. Part I: constant wall temperature. Int. J. Therm. Sci. 42, 385–395 (2003)

    Article  Google Scholar 

  • Narusawa, U.: The second law analysis of mixed convection in rectangular ducts. Heat Mass Transf. 37, 197–203 (2001)

    Article  Google Scholar 

  • Nield, D.A., Bejan, A.: Convection in Porous Media. Springer, New York (2013)

    Book  Google Scholar 

  • Peng, Y.Z.: The variable separation method and exact Jacobi elliptic function solutions for the Nizhnik–Novikov–Veselov equation. Acta Phys. Pol. A 110, 3–9 (2006)

    Google Scholar 

  • Polyanin, A.D., Zhurov, A.I., Vyazmin, A.V.: Generalized separation of variables in nonlinear heat and mass transfer equations. J. Non-Equilib. Thermodyn. 5, 251–267 (2000)

    Google Scholar 

  • Polyanin, A.D.: Handbook of Partial Differential Equations for Engineers and Scientists. CRC Press, Boca Raton (2001)

    Book  Google Scholar 

  • Rott, N.: Thermoacoustics. Adv. Appl. Mech. 20, 135–175 (1980)

    Article  Google Scholar 

  • Sahin, A.Z.: A second law comparison for optimum shape of duct subjected to constant wall temperature and laminar flow. Heat Mass Transf. 33, 425–430 (1998)

    Article  Google Scholar 

  • Sauoli, S., Aiboud-Sauoli, S.: Second law analysis of laminar falling liquid film along an inclined heated plate. Int. Commun. Heat Mass Transf. 31, 879–886 (2004)

    Article  Google Scholar 

  • Singh, A.K., Thorpe, G.R.: Natural convection in a confined fluid overlying a porous layer—a comparison study of different models. Indian J. Pure Appl. Math. 26, 81–95 (1995)

    Google Scholar 

  • Song, L.L., Shang, Y.D.: Variable separation and exact solutions for the Kadomtsev–Petviashvili equation. Adv. Pure Math. 5, 121–126 (2015)

    Article  Google Scholar 

  • Swift, G.W.: Thermoacoustics: A Unifying Perspective for Some Engines and Refrigerators. ASA Publication, New York (2002)

    Google Scholar 

  • Umavathi, J.C., Chamkha, A.J., Sridhar, K.S.R.: Generalized plain Couette flow and heat transfer in a composite channel. Transp. Porous Media 85, 157–169 (2010)

    Article  Google Scholar 

  • Yadav, S.L., Singh, A.K.: Effects of viscous and Darcy dissipations on entropy generation rate of flow through a horizontal porous channel. Int. J. Energy Technol. 6(17), 1–7 (2014)

    Google Scholar 

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Acknowledgments

The author (SLY) would like to thank the Council of Scientific and Industrial Research, New Delhi, India, for financial support in the form of a Junior Research Fellowship.

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  1. Department of Mathematics, Banaras Hindu University, Varanasi, 221005, India

    Shyam Lal Yadav & A. K. Singh

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  1. Shyam Lal Yadav
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Correspondence to Shyam Lal Yadav.

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Yadav, S.L., Singh, A.K. Analysis of Entropy Generation in Annular Porous Duct. Transp Porous Med 111, 425–440 (2016). https://doi.org/10.1007/s11242-015-0602-x

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