Second Law Analysis of Mixed Convection in a Laminar, Non-Newtonian Fluid Flow through a Vertical Channel

The fully developed mixed convection of non-Newtonian laminar flow through a vertical channel is investigated. The boundary conditions of uniform and unequal temperature prescribed at the channel walls are considered. The velocity and temperature fields are obtained by analytically solving the momentum and energy balance equations. The velocity and temperature distributions are used to calculate the entropy generation number NS , the irreversibility ratio Φ , and the Bejan number Be for several values of the viscous dissipation parameter BrΩ−1 , the viscosity index n , and the appropriate dimensionless coordinates. The results show us the regions of high entropy generation.


Introduction
The study of second law analysis of a laminar non-Newtonian, power-law fluid flowing between two heated plates has many significant applications in thermal engineering and industries.The heat transfer of non-Newtonian fluids in ducts is a subject which has received much attention in the last decades.The interest in this field is due, for instance, to many industrial applications which involve polymeric materials.Starting from petroleum industry to various heat exchanger systems, this type of geometry can be observed.Meanwhile, the improvement in thermal systems as well as energy utilization during the convection in any fluid is one of the fundamental problems of the engineering processes, since improved thermal systems will provide better material processing, energy conservation, and environmental effects.One of the methods used for predicting the performance of the engineering processes is the second law analysis.The second law of thermodynamics is applied to investigate the irreversibility in terms of the entropy generation rate.Since the where T is temperature, ρ is the mass density, β is the thermal expansion coefficient, ρ 0 is the mass density at T T 0 , and T 0 is the mean temperature in a channel section, that is, Let us denote by U and V the X-component and the Y -component of the velocity field, respectively.The mass balance implies that the velocity field is solenoidal, while the conditions of fully developed flow imply that ∂U ∂X 0.

2.3
Therefore one can conclude that U depends only on Y and that V is zero.On account of the Ostwald-de Waele constitutive equation, the components where η is the consistency factor and m is the inverse of the power-law index.The case m < 1 corresponds to dilatant fluid behavior, while the case m > 1 occurs for pseudo plastic fluids.Let us assume that the thermophysical properties of the fluid ρ 0 , β, η and the thermal conductivity k are independent of temperature.Then, 2.4 implies that τ depends only on Y .The momentum balance along Y -direction yields ∂P/∂X 0, where P p ρ 0 g X is the difference between the pressure p and the hydrostatic pressure.The momentum balance along the X-direction can be written as By deriving both sides of 2.5 with respect to X, one obtains By integrating both sides of 2.6 with respect to Y in the interval −L < Y < L and by employing 2.2 , one is led to the following conclusion: As a consequence of 2.7 , one infers that dP/dX is a constant.Since the channel walls are isothermal and since 2.7 implies that ∂T/∂X does not depend on Y , one can deduce that ∂T/∂X is zero; that is, T depends only on Y .Therefore the energy balance equation yields The analysis presented by Barletta 11 has been used to compute the entropy generation terms, and therefore a summary of his analysis is shown below.Equation 2.4 can be easily inverted to obtain dU/dY , namely,

2.9
On account of no-slip boundary condition for velocity field, 2.9 allows one to obtain the expression Let us choose a reference velocity as follows: where D 4L is the hydraulic diameter.If one defines the dimensionless quantities,

2.12
Equations 2.5 , 2.8 , and 2.9 can be written in the dimensionless form:

2.15
The boundary conditions for the dimensionless fields can be expressed as

2.16
A further constrain on θ is induced by 2.2 , namely, θ y dy 0. 2.17 Equations 2.2 -2.17 determine uniquely the functions u y , θ y , σ y and the parameter ω, provided that the inverse of the power-law index and the parameter are prescribed.On account of 2.13 , the integration of both sides of 2.15 with respect to y in the interval −1/4, 1/4 yields the following constrain on σ y : σ y σ y m−1 dy 0.

2.19
Equation 2.14 implies that θ y is a linear function of y.Then, the boundary conditions expressed by 2.27 and the additional constraint given by 2.17 yield

2.20
On account of 2.1 and 2.12 , σ y can be expressed as where C m, λ is an integration constant which can be determined by employing the constraint given by 2.19 .Therefore, C m, λ is the solution of the equation On account of 2.15 , 2.16 , and 2.21 , the dimensionless velocity can be evaluated as On account of 2.15 and 2.21 ,

2.25
Since ∂T/∂X 0, the entropy generation rate equation becomes .

2.26
In the previous equation, the superscript indicates per unit volume.Writing the entropy generation in nondimensional form by defining the entropy number N s ,

Irreversibility Ratio
In convection problems, both fluid friction and the heat transfer contribute to the rate of entropy generation.In order to assess which one among the fluid friction and heat transfer dominates, a criterion known as irreversibility ratio is defined by the following equation.Irreversibility ratio Φ is the ratio of the entropy generation due to the fluid friction to the total entropy generation due to heat transfer.
Φ is irreversibility due to fluid friction/irreversibility due to heat transfer 2.30

The Bejan Number
Bejan number is the ratio of heat transfer irreversibility to the total irreversibility due to heat transfer and fluid friction.
Be is irreversibility due to heat transfer/total irreversibility Be 4 ∂θ/∂y

Results and Discussion
The previous mathematical analysis is valid for the second law analysis of a laminar non-Newtonian, power-law fluid flowing between two parallel heated plates.The velocity and temperature distributions are used to calculate the entropy generation number, irreversibility ratio, and the Bejan number for the case of a laminar, non-Newtonian, power-law fluid.These are presented graphically for various values of the viscosity index n , the viscous dissipation parameter BrΩ −1 , and the dimensionless axial distance y .

Viscous Dissipation Parameter (BrΩ −1 )
The viscous dissipation parameter BrΩ −1 is defined as the product of the Brinkman number and the inverse of dimensionless temperature difference.We have the following: i the Brinkman number Br The viscous dissipation parameter is an important dimensionless number for the irreversibility analysis.It determines the relative importance of the viscous effects for the entropy generation.

Entropy Generation Number
The spatial distribution of the entropy generation number is plotted in Figures 2, 3 where N S entropy generation number, N F is BrΩ −1 ∂u/∂y n 1 entropy generation due to the fluid friction, and N H ∂θ/∂y 2 is entropy generation due to heat transfer.Therefore, We note that the entropy generation rate is highest at the isothermal walls and gradually decreases as we move towards the center of the channel.This is because the rate of change of velocity ∂u/∂y is highest at the channel walls and decreases as we move towards the center.
The entropy generation number increases as the viscous dissipation parameter BrΩ −1 increases because an increased viscous dissipation parameter increases the entropy generation due to fluid friction.
The entropy generation number increases as the viscosity index n increases because the more the viscousity the fluid, the higher the entropy generation rate due to fluid friction which eventually increases the entropy generation number.

Irreversibility Ratio
In Figures: 5, 6, and 7 irreversibility ratio is plotted as a function of the transverse distance for different values of the viscous dissipation parameter BrΩ −1 and viscosity index n

3.3
The irreversibility ratio is highest at the channel walls and decreases as we move towards the center.Irreversibility ratio increases as the viscous dissipation parameter BrΩ −1 and the viscosity index n increase.

The Bejan Number (Be)
The Bejan number is the ratio of irreversibility due to heat transfer to the total irreversibility due to heat transfer and fluid friction In Figures 8, 9, and 10, the Bejan number profiles are shown as functions of the transverse distance for different values of the viscous dissipation parameter BrΩ −1 , viscosity index n , and constant λ .
The Bejan number is highest at the center of the channel and decreases as we move towards the channel walls on either direction.
The Bejan number decreases as the viscous dissipation parameter BrΩ −1 and the viscosity index n increase.

Concluding Remarks
This paper presents the application of the second law of thermodynamics to mixed convection in a laminar, non-Newtonian, power-law fluid flowing between two parallel isothermal vertical plates.The velocity and the temperature profiles are obtained analytically and used to compute the entropy generation number, irreversibility ratio, and the Bejan number for several values of the viscous dissipation parameter BrΩ −1 , the viscosity index n , and the dimensionless axial distance y .The numerical results show that the nondimensional entropy number is least at the center of the channel and increases in the transverse direction on either side owing to an increased velocity gradient near the walls.The entropy generation number increases with increase in the viscosity index and increase in the viscous dissipation parameter BrΩ −1 .
Irreversibility ratio is highest at the channel walls and decreases as we move towards the center.Irreversibility ratio increases as the viscous dissipation parameter BrΩ −1 and the viscosity index n increase.The Bejan number is least at the channel walls and increases in the transverse direction as we move towards the center.The numerical results show that the Bejan number decreases as the viscous dissipation parameter BrΩ −1 and the viscosity index n increase.

2 2LFigure 1 :
Figure 1: The local flow model and coordinate system.