# Entropy Generation in Pressure Gradient Assisted Couette Flow with Different Thermal Boundary Conditions

## Abstract

**:**

## Introduction

## Hydrodynamic and Thermal Analysis

^{2}is the dimensionless temperature and k is the thermal conductivity of the fluid. The use of dimensionless temperature in this form eliminates the Eckert number Ec and Prandtl number Pr which appear in traditional analyses. The solution of equation (5) depends on the thermal boundary conditions imposed at the plates. We consider the solutions for four combinations of boundary conditions.

## Constant Plates Temperatures

_{1}and the moving plate at a constant temperature T

_{2}. In terms of θ, the boundary conditions become

_{1}

_{2}

## Constant Temperature at Stationary Plate and Constant Heat flux at the Moving Plate

^{2}is the dimensionless heat flux. q″ is the heat flux at the moving plate.

## Constant Temperature at Stationary Plate and Convection at the Moving Plate

_{a}= kT

_{a}/μV

^{2}, T

_{a}is the convection environment temperature.

## Convection at both plates

_{1}= h

_{1}a/k, Bi

_{2}= h

_{2}a/k, θ

_{a}

_{,1}= kT

_{a}

_{,1}/μV

^{2}, θ

_{a}

_{,2}= kT

_{a}

_{,2}/μV

^{2}and h

_{1},T

_{a}

_{,1}and h

_{2},T

_{a}

_{,2}are the heat transfer coefficients and convection environment temperatures at the stationary and moving plates, respectively.

## Entropy Generation Rate

^{3}K) provided by Bejan [1,2] to the present convective flow situation, we have

## Results and Discussion

## Constant Plates Temperatures

_{1}= 20,θ

_{2}= 5). In the range from P = 0-10, the temperature distribution changes only slightly. Since the temperature profiles do not exhibit any maxima, the entropy generation due to heat conduction occurs at all locations across the gap. However, a distinct maximum in temperature is observed at P = 50 and indicates zero entropy generation due to heat conduction at that location.

_{1}= 20 and P = 2, the effect of varying θ

_{2}on the entropy generation rate is depicted in Fig.4. Fig. 4a shows the entropy generated due to heat conduction i.e. ${\dot{S}}_{gen,h}^{m}{a}^{2}/k$ (denoted by S

_{h}) while Fig. 4b shows that due to fluid friction i.e ${\dot{S}}_{gen,f}^{m}{a}^{2}/k$ (denoted by S

_{f}). The temperature is varied from θ

_{2}= 20 (red curve) in increment of 10 to 70 (aqua curve). For θ

_{2}= 20, the local entropy generation due to heat conduction (Fig 4a) is negligible. However, the entropy generation due heat conduction increases sharply as θ

_{2}is increased particularly at the stationary plate where the largest temperature gradients occur. The effect of θ

_{2}on the local entropy generation at the moving plate is rather small because the changes in θ

_{2}result in comparatively smaller changes in the temperature gradients at the moving plate. The local entropy generation due to fluid friction (Fig. 4b) is maximum at the stationary plate and minimum at the moving plate. The highest entropy generation due to fluid friction occurs at the lowest value of θ

_{2}(red curve). This is due to the presence of θ in the denominator of the fluid friction term in equation (19). A comparison of the results in Fig. 4a and Fig. 4b and the corresponding results for other pressure gradients reveals that the entropy generation due to fluid friction is much smaller than that due to heat conduction for the range of P investigated i.e. from 0 to 50.

_{h}+ S

_{f}for convenience, as a function of the pressure gradient P for θ

_{2}= 20, 40, 60, and 80 with θ

_{1}fixed at a value of 20. The contribution of S

_{f}is much smaller than S

_{h}. The total entropy generation rate increases as P increases and/or θ

_{2}increases. This is a consequence of the higher velocity and temperature gradients caused by the increase in P and θ

_{2}, respectively. From a system design perspective, the total entropy generation is minimized only when the plates are held at identical temperatures and the pressure gradient is zero, that is, when the flow is driven solely by the motion of the upper plate.

## Constant Temperature at Stationary Plate and Constant Heat flux at the Moving Plate

_{1}was fixed at a value of 20. The effect of varying the pressure gradient P and the heat flux Q at the moving plate is illustrated in Fig. 6. The total entropy generation rate increases significantly with increase in pressure gradient but is comparatively less affected by the changes in heat flux at the moving plate. The increase in entropy generation rate is due to the increase in velocity and temperature gradients that accompany the increase in P and Q. It may be noted that in this case the moving plate is colder than the stationary plate and extracts energy from the fluid. The entropy generation is minimized only when the moving plate is insulated, the pressure gradient is zero and the flow is induced solely by the motion of the plate. A comparison of Fig.5 and Fig.6 shows that by extracting heat at a constant rate from the moving plate, the entropy generation in the process can be significantly reduced.

**Fig.6.**Effect of pressure gradient P and heat flux Q on total entropy generation when the moving plate delivers heat to the fluid.

**Fig.7.**Effect of pressure gradient P and heat flux Q on total entropy generation rate when the moving plate extracts heat from the fluid.

## Constant Temperature at the Stationary Plate and Convection at the Moving Plate

_{1}= 20 and θ

_{a}= 5 and illustrate in Fig.8 the effect of Biot number on the total entropy generation rate for selected values of the pressure gradient. For each value of P, the total entropy generation rate increases as Bi increases i.e. the convection at the moving plate gets stronger. The increase in heat removal from the fluid by the moving plate results in enhanced entropy generation rate. For a fixed Bi, the higher the pressure gradient and hence the larger the velocity gradients, the higher the entropy generation rate.

_{a}on the entropy generation rate is depicted in Fig. 9 for selected values of the pressure gradient P. This figure was generated by fixing θ

_{1}= 20 and Bi = 2. As θ

_{a}increases, the convective heat removal from the moving plate decreases which leads to a reduction in entropy generation rate.

## Convection at both plates

_{a}

_{,1},θ

_{a}

_{,2}, Bi

_{1}and Bi

_{2}. We fix the first three at P = 2, θ

_{a}

_{,1}= 20, θ

_{a}

_{,2}= 10 and study the effect of varying Bi

_{1}and Bi

_{2}. In Fig. 10, the entropy generation rate is plotted as a function of Bi

_{2}for parametric values of Bi

_{1}. At Bi

_{1}= 1, the entropy generation increases sharply as Bi

_{2}increases. However at Bi

_{1}= 2, 3, 4 and 5, the increase in entropy generation with Bi

_{2}is moderate. Also the effect of Bi

_{1}is significantly attenuated beyond Bi

_{1}= 3.

**Fig.10.**Entropy generation rate as a function of Bi

_{2}and Bi

_{1}with P = 2, θ

_{a}

_{,1}= 20, θ

_{a}

_{,2}= 10.

_{1}and Bi

_{2}, the entropy generation attains a minimum. A close examination of their Fig. 3 for G=2 (P = 2 in present work) indicates the variation in entropy generation as Bi

_{1}(Bi

_{2}in present work) increases from 0 to 6 is only about 0.2 percent for the three values of Bi

_{2}(Bi

_{1}in present work ) used namely 20, 25 and 30 with the minimum entropy generation occurring at Bi

_{1}(Bi

_{2}in present work) = 0.5. Because of the variation of 0.2 percent, the minimum could not be identified graphically without a large magnification of S axis. We use the values of P,θ

_{a}

_{,1}, θ

_{a}

_{,2}, Bi

_{1}and Bi

_{2}on which their Fig. 3 is based and present our results in Fig.11. It can be observed that in the range of Bi

_{2}from 0.5-3.0 and Bi

_{1}≥ 2, the entropy generation rate is virtually a minimum. A distinct minimum can be identified by magnifying the S axis or examining the numerical results but the exact determination of the minimum would appear to be of little practical use. The same conclusion was reached with the results for other values of P,θ

_{a}

_{,1}, and θ

_{a}

_{,2}.

_{a}

_{,1}= 20, θ

_{a}

_{,2}= 10.The designer then has the flexibility of providing cooling at the stationary plate within Biot numbers in the range 0.5-3.

**Fig.11.**Entropy generation rate as a function of Bi

_{2}and Bi

_{1}with P = 2, θ

_{a}

_{,1}= 10, θ

_{a}

_{,2}= 10.

## Conclusions

## References

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**MDPI and ACS Style**

Aziz, A. Entropy Generation in Pressure Gradient Assisted Couette Flow with Different Thermal Boundary Conditions. *Entropy* **2006**, *8*, 50-62.
https://doi.org/10.3390/e8020050

**AMA Style**

Aziz A. Entropy Generation in Pressure Gradient Assisted Couette Flow with Different Thermal Boundary Conditions. *Entropy*. 2006; 8(2):50-62.
https://doi.org/10.3390/e8020050

**Chicago/Turabian Style**

Aziz, Abdul. 2006. "Entropy Generation in Pressure Gradient Assisted Couette Flow with Different Thermal Boundary Conditions" *Entropy* 8, no. 2: 50-62.
https://doi.org/10.3390/e8020050