Slide 1

112Introduction Irreversibility Some cases of irreversible processes : -heat transfer through a finite temperature difference -Friction -inelastic deformation -electric flow through a resistance -unrestrained expansion of fluids -spontaneous chemical reactions23Entropy Generation FunctionIn a given process irreversible losses can be characterized and evaluated through the entropy generation function.In this problem irreversibility are due to viscous friction and heat transfer.

34Problem Geometry

45Nomenclature , external ambient temperature [K]a , half separation between the planar walls [m]C, specific heat of the fluid, A, constant axial temperature gradient, , dynamic viscosity of the fluid , external convective heat transfer coefficient, , local entropy generation rate per unit length, , internal convective heat transfer coefficient , ,dimensionless local entropy generation rate per unit length,K , fluid thermal conductivity , dimensionless global entropy generation rateNu , Local Nusselt number,

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7Analyzing problem1-Velocity field:We consider the Steady, fully developed viscous fluid in the longitudinal constant pressure gradient, dp/dx .

No slip conditions on the velocity are applied.

782-Temperature field:Energy conservation eq. in the fluid:

dimensionless energy conservation equation in the fluid:

8Walls conduction equation:

one dimensional steady state conduction

dimensionless heat equation for the walls:

i=1 : lower wall & i=2 : upper wall

9Boundary conditions:

continuity conditions for temperature across the fluid-wall interface

continuity conditions for heat flux across the fluid-wall interface

10Boundary conditions:

convection in outer walls surfaces using newtons cooling law

11Solving problem

Thermally fully developed region:

12Solving problem

13ResultsIn dimensionless term entropy generation per unit length can be :written in the form

That first four terms account for irreversibilities caused by Heat flow in the fluid and walls while the last term considers irreversibilities due to viscous dissipation in the fluid.

14Results

We get the global entropy generation rate per unit length , which only depends on these dimensionless parameters:

Z= or or or or .

In all cases fixed to 5.Used the physical properties of engine oil at an ambient temperature .

15Results:

16Fig. 1

the global entropy generation rate is reported as a function of dimensionless lower wall thickness that optimal 1 values can be determined.It was found that entropy generation produced by heat transfer in the fluid and the upper wall always increases with 1. however, the terms associated to viscous dissipation in the fluid and heat transfer in the lower wall always decrease with 1 due to the increment in the temperature of the system.For small values of 1, this reduction in the irreversibilities associated to viscous dissipation and heat transfer in the lower wall dominates over the increment in the terms associated to heat transfer in the fluid and in the upper wall in such a way that Entropy Generation shows a minimum value.17the global entropy generation rate is reported as a function of 1 for three different values of the wall to fluid thermal conductivity ratio , that optimal 1 values can be determined.

18We can determine the optimal Bi number for lower wall and the wall to fluid thermal conductivity ratio for lower wall values in fig. 3 and 4.

19We can determine the optimal value of the axial temperature gradient, A and peclet number in fig. 5 and 6.

20ReferencesG. Ibanez, A. Lopez, S. Cuevas, Optimum wall thickness ratio based on minimization of entropy generation in a viscous flow between parallel plates, International Journal of Heat and Mass Transfer 39 (2012) 587592.A. Bejan, Minimization of Entropy Generation, CRC Press, Boca Raton, 1996.21