Introduction

3. Introduction 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 reactions

4. Entropy Generation Function In 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.

5. Problem Geometry

6. Nomenclature

8. Analyzing problem 1-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.

9. 2-Temperature field: Energy conservation eq. in the fluid: dimensionless energy conservation equation in the fluid:

10. Walls conduction equation: one dimensional steady state conduction dimensionless heat equation for the walls: i=1 : lower wall & i=2 : upper wall

11. Boundary conditions: continuity conditions for temperature across the fluid-wall interface continuity conditions for heat flux across the fluid-wall interface

12. Boundary conditions: convection in outer walls surfaces using newton’s cooling law

13. Solving problem Thermally fully developed region:

14. Solving problem

15. Results In 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.

16. Results We get the global entropy generation rate per unit length , which only depends on these dimensionless parameters: Z= or ororor . In all cases fixed to 5. Used the physical properties of engine oil at an ambient temperature .

17. Results:

18. Fig. 1 the global entropy generation rate is reported as a function of dimensionless lower wall thickness that optimal δ1 values can be determined. It was found that entropy generation produced by heat transfer in the fluid and the upper wall always increases with δ1. however, the terms associated to viscous dissipation in the fluid and heat transfer in the lower wall always decrease with δ1 due to the increment in the temperature of the system. For small values of δ1, 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.

19. the global entropy generation rate is reported as a function of δ1 for three different values of the wall to fluid thermal conductivity ratio , that optimal δ1 values can be determined.

20. We 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.

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

22. References G. 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) 587–592. A. Bejan, Minimization of Entropy Generation, CRC Press, Boca Raton, 1996.