Journal ArticleDOI

# Effect of aspect ratio on entropy generation in a rectangular cavity with differentially heated vertical walls

01 Jul 2008--Vol. 35, Iss: 6, pp 696-703
TL;DR: In this article, the entropy generation in rectangular cavities with the same area but different aspect ratios is numerically investigated and the variation of the total entropy generation and average Bejan number for the whole cavity volume at different aspects ratios for different values of the Rayleigh number and irreversibility distribution ratio are also evaluated.
About: This article is published in International Communications in Heat and Mass Transfer.The article was published on 2008-07-01 and is currently open access. It has received 291 citations till now. The article focuses on the topics: Bejan number & Rayleigh number.

### 1. Introduction

• Natural convection heat transfer in enclosures has recently been an important topic due to its wide applications in energy storage systems, electronic cooling devices, heating and cooling of buildings etc.
• The second law of thermodynamics has been applied to cavity problems to determine entropy generations due to heat and flow transport in the cavity and consequently minimize the entropy generation.
• Erbay et al. [4, 5] studied entropy generation during transient laminar natural convection in a square enclosure being heated either completely or partially from the left side wall and cooled from the opposite side wall.
• The variation of the total entropy generation due to heat transfer and fluid friction irreversibilities and total entropy generation versus aspect ratio are presented in graphical forms.

### 2. The considered problem

• The horizontal walls are insulated and the vertical walls of the cavity are maintained at different Th (left wall) and Tc (right wall) constant temperatures such that ThNTc.
• The aspect ratio (AR=H /L) for the problem is changed from 1 to 16, while the dimensionless area for all cavities is one.
• The heat transfer in the cavity occurs by laminar natural convection and the effect of radiation is neglected.

### 3. Mathematical modeling

• The governing equations for the problem are the continuity, momentum and energy equations.
• The pressure term in the momentum equation can be eliminated using the vorticitystream function approach.
• The boundary conditions for the problem are:.
• The local Bejan number indicates the strength of the entropy generation due to heat transfer irreversibility [2]: Bel ¼ Sl:h:tSl ð14Þ.
• For any location inside the cavity, the condition of BelN1/2 implies that the heat transfer irreversibility is dominating, while if Belb1/2 the fluid friction irreversibility is dominating.

### 4. Solution procedure

• The set of governing equations is numerically solved by the finite difference method.
• The energy and vorticity equations are solved line by line by employing the ADI method, whereas the stream function equation is solved point by point.
• The finite difference form of diffusion and convection terms are written based on three points central difference which has second order accuracy.
• The detailed information about the numerical procedure and convergence criteria can be found in Ref. [14].

### 4.1. Validation of numerical results

• In order to validate the obtained numerical results, total entropy generation for square cavities with different ϕ and Ra values were obtained and compared with the results of Magherbi et al. [2].
• Fig. 2 depicts the comparison of the present solutions with the reported results.
• Excellent agreement between the two studies is observed.

### 5. Results and discussion

• The comparison of Sl.h.t and Sl.f.f entropy generation due to fluid friction is considerably greater than the heat transfer local entropy generation.
• The variation of the total entropy generation in a cavity with Φ=10−4 versus aspect ratio for different Rayleigh numbers is shown in Fig.
• For the cavity with high Rayleigh number (i.e., Ra=105), Beavb1/2 which indicates the domination of fluid friction irreversibilities.
• As seen from Fig. 7(d), the variation of the average Bejan number with aspect ratio is different from Fig. 6(d), since the Bejan number is the ratio of Sl.h.t to Sl.f.f.

### 6. Conclusion

• For the rectangular cavity with unit area but different aspect ratio, the variation of local and total heat transfer and fluid friction irreversibilities and entropy generation were investigated.
• For cavities with high Ra values (i.e., Ra=105) and φ=10−4, fluid friction irreversibility is dominant and total entropy generation increases with aspect ratio, attains a maximum and then it decreases.
• A peak point for the maximum total entropy generation exists.
• The average Bejan number is a proper criterion to predict the domination of heat transfer or fluid friction irreversibilities for the entire domain.
• The total entropy generation in a cavity increases with Rayleigh number, however, the rate of increase depends on the aspect ratio.

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##### References
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TL;DR: In this paper, the entropy generation due to heat transfer and friction has been determined in transient state for laminar natural convection by solving numerically the mass, momentum and energy balance equations, using a control volume finite-element method.

200 citations

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TL;DR: In this paper, the authors investigate numerically the heat transfer and fluid flow characteristics inside a wavy walled enclosure and apply second law of thermodynamics to predict the nature of irreversibility in terms of entropy generation.

106 citations

### "Effect of aspect ratio on entropy g..." refers background in this paper

• ...Mahmud and Islam [8] numerically investigated laminar free convection and entropy generation Nomenclature AR Aspect ratio Be Bejan number k Thermal conductivity, W/m/K L Length, m H Height of cavity, m Pr Prandtl number Ra Rayleigh number based on cavity length S Dimensionless entropy generation T Temperature, K To Bulk temperature (Th+Tc) / 2, K u, v Velocity components in x and y directions, m/s U, V Dimensionless velocity components in X and Y directions x, y Dimensional Cartesian coordinates, m X, Y Dimensionless Cartesian coordinates Greek symbols α Thermal diffusivity, m2/s β Thermal expansion coefficient, K−1 ϕ Irreversibility distribution ratio ν Kinematic viscosity, m2/s Ω Dimensionless vorticity θ Dimensionless temperature τ Dimensionless time Ψ Dimensionless stream function μ Viscosity (Ns/m2) Subscripts av Average c Cold f Fluid f.f Fluid friction h Hot h.t Heat transfer l Local t Total inside a wavy walled enclosure....

[...]

• ...Mahmud and Islam [8] numerically investigated laminar free convection and entropy generation...

[...]

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TL;DR: In this paper, the authors dealt with the prediction of entropy generation for natural convection in a Γ-shaped enclosure, where heat removal is achieved from cooler vertical sides while the horizontal walls are insulated perfectly.

80 citations

### "Effect of aspect ratio on entropy g..." refers background in this paper

• ...[1] dealt with the prediction of entropy generation of natural convection in a Γ-shaped enclosure....

[...]

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TL;DR: A review of the diverse roles of entropy and the second law in computational thermo-fluid dynamics is presented, and sample numerical results for a multitude of applications in compressible flows, as well as problems with phase change heat transfer.
Abstract: A review of the diverse roles of entropy and the second law in computational thermo-fluid dynamics is presented. Entropy computations are related to numerical error, convergence criteria, time-step limitations, and other significant aspects of computational fluid flow and heat transfer. The importance of the second law as a tool for estimating error bounds and the overall scheme's robustness is described. As computational methods become more reliable and accurate, emerging applications involving the second law in the design of engineering thermal fluid systems are described. Sample numerical results are presented and discussed for a multitude of applications in compressible flows, as well as problems with phase change heat transfer. Advantages and disadvantages of different entropy-based methods are discussed, as well as areas of importance suggested for future research

70 citations

### "Effect of aspect ratio on entropy g..." refers methods in this paper

• ...[13] presented a review of past advances in numerical analysis using the second law of thermodynamics....

[...]

Journal ArticleDOI
TL;DR: In this article, the effect of conduction of horizontal walls on natural convection heat transfer in a square cavity is numerically investigated, and a code based on vorticity-stream function is written to solve the governing equations simultaneously over the entire computational domain.

63 citations

###### Q1. What are the contributions mentioned in the paper "Effect of aspect ratio on entropy generation in a rectangular cavity with differentially heated vertical walls" ?

In the present study, entropy generation in rectangular cavities with the same area but different aspect ratios is numerically investigated. The present results are compared with reported solutions and excellent agreement is observed. The study is performed for 10bRab10, 10bφb10, and Pr=0.

For low Rayleigh numbers (i.e., Ra=102, 103), St.h.t almost varies linearly with aspect ratio; though a non-linear behavior appears for high Rayleigh number due to development of the fluid flow.

The energy and vorticity equations are solved line by line by employing the ADI method, whereas the stream function equation is solved point by point.

The second law of thermodynamics has been applied to cavity problems to determine entropy generations due to heat and flow transport in the cavity and consequently minimize the entropy generation.

An increase of the irreversibility ratio pulls down the average Bejan number curves since the fluid friction irreversibility is increased.

Based on that study, the local heat transfer irreversibility and the local fluid friction irreversibility change by the inclination angle and the minimum entropy generation depends considerably on the inclination.

For low Ra numbers (i.e., Rab104), the value of the average Bejan number is greater than 1/2 (BeavN1/2), which shows the strong heat transfer irreversibility in the major part of the cavity.

Themaximum local entropy generation due to heat transfer is (Sl.h.t)max=2.27 while the maximum value of local fluid friction entropy generation is (Sl.f.f)max=0.25.

8. For Ra=100, the total entropy generationincreases from theminimum aspect ratio (AR=1) to the maximum aspect ratio (AR=16) with the same order.

The isotherms, streamlines and their corresponding local entropy generation maps are plotted to supply useful information for minimizing the entropy generation in rectangular cavities.

The transient state of entropy generation for laminar natural convection in a square cavity with heated vertical walls was numerically solved by Magherbi et al.☆