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In this paper, we have considered a fully developed flow of a viscous incompressible fluid in a rectangular porous duct saturated with the same fluid. The duct is heated from the bottom for forced and mixed convection. The Brinkman model is used to simulate the momentum transfer in the porous duct. Using the momentum and thermal energy equations, the entropy generation has been obtained due to the heat transfer, viscous and Darcy dissipations. It is found from the mathematical analysis that the entropy generation is double when the viscous as well as the Darcy dissipations terms are taken in the thermal energy equation in comparison when the viscous as well as the Darcy dissipations terms are not taken in the thermal energy equation. This result clearly shows that there is no need of taking the viscous and Darcy dissipations terms in the thermal energy equation to obtain the entropy generation.

The study of transport phenomena in the channel filled with saturated porous media has attracted considerable attention of scientists, engineers and experimentalists in present time. This attention is mainly due to the applications of this phenomenon in the field of electronics cooling system, geothermal system, storage of nuclear waste materials, microelectronics heat transfer equipment, coal and grain storage, crude oil production, catalytic converters, ground water pollution, fiber and granular insulations, solidification of castings, etc. The advancement in the thermal systems as well as the energy utilization during the convection in any fluid is one of the fundamental problems of the technological processes, because the improved thermal systems will provide better material processing, energy conservation and environmental effects. Also, because of applications to the cooling of electronic equipment there has been an increased interest in the forced convections in the channels and ducts filled with the porous media. One of the important viscous fluid flow situations in the porous media is the Poiseuille flow in a rectangular duct. In addition, convection through the porous medium may be found in the fiber and granular insulation, including structures for high power density, electric machines and nuclear reactors.

Viscosity is the measure of a fluid’s resistance to the flow and it describes the internal friction of a moving fluid. Due to the added resistance of the porous structure, the effective viscosity

In addition to the analysis based on the basic conservation laws, the analysis of second-law of thermodynamics is important in understanding the entropy generation, which is attributed to the thermodynamic irreversibility. This kind of thermodynamic analysis is significant for studying the optimum operating conditions, which helps in designing a system with less entropy and destruction of available work (energy). According to the Gouy-Stodola theorem, the lost available work is directly proportional to the entropy generation. The utilization of the second law of thermodynamics in convective heat transfer is very well presented by Bejan [

Sauoli and Sauoli [

Heat transfer from solid walls to flowing fluids is an area of extreme scientific interest as well as of immense practical importance. Certain flows, passing through the bodies with high porosity, do not follow the Darcy’s law and Brinkman’s model is applicable for that type of flows. Neale and Nader [

Consider the steady, laminar, two dimensional incompressible fluid flow in a saturated porous medium bounded by a rectangular duct of width, 2W and height, H with the origin of a coordinate system located at the corner of the rectangular duct as sketched in

main flow (z) direction and a cross sectional variation in the x-y plane; i.e.

We also assume that the fluid motion can adequately be described by Boussinesq approximation. Under the assumptions of constant thermo-physical properties and linear Boussinesq approximations, the governing conservation equations, namely the equation of continuity, momentum and thermal energy for the isotropic and homogeneous porous medium may be written as:

For the fully developed flow

The first term on the right hand side of above equation is due to the heat transfer

where

The subscript,

For the rate of entropy generation due to the heat transfer, we obtain, after applying integration by parts as well as the adiabatic side wall conditions,

The second term on the right hand side is zero as we apply conservation of energy, Equation (5), the hydrodynamic boundary conditions and integration by parts; i.e.

Also by integrating conservation of energy, Equation (5), over a cross section it may be shown that heat trans- fer at the top boundary is balanced by heat transfer at the bottom boundary; i.e.

Therefore,

with

where

with

The rate of entropy generation due to viscous and Darcy dissipations, upon application of the hydrodynamic conditions as well as continuity, becomes

where

Therefore,

The first term on the right hand side is due to the forced convection (buoyancy induced flow). To represent the first term in Equation (17) in terms of non-dimensional parameters relevant to forced convection, we introduce

where

The rate of entropy generation due to the viscous dissipation associated with the buoyancy-induced flow in porous medium, on the other hand, may be expressed as

where

Therefore,

The rate of non-dimensional entropy generation over a cross section for fully developed mixed convection in porous medium,

The right hand side of Equation (23) consists of contributions from heat transfer, cross-sectional (buoyancy-induced) flow and longitudinal main flow.

Now if Equation (5) includes the viscous as well as Darcy dissipations, i.e.

then Equation. (23) becomes

Entropy is a thermo-dynamical property that is a measure of the energy unavailable for useful work in a thermodynamic process, such as in energy conversion devices, engines, or machines, which can only be driven by convertible energy. When a substance is heated or cooled, there is a change in the entropy and has a theoretical entropy minimization (maximum efficiency) while converting the energy to useful work. From Equation (23) and Equation (25), we see that, if the thermal energy equation includes the viscous as well as Darcy dissipations then the second term (due to viscous dissipation) on the right hand side of the entropy generation rate is double and there is no change in first term (due to heat transfer).

This paper presents the analytical calculation for the non-dimensional entropy generation and the obtained result shows that there is no need to include the viscous and Darcy dissipations in the energy equation.

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.

Shyam Lal Yadav,Ashok Kumar Singh, (2016) Analysis of Entropy Generation in a Rectangular Porous Duct. Journal of Applied Mathematics and Physics,04,1336-1343. doi: 10.4236/jamp.2016.47143