Numerical simulation of prandtl number effect on entropy generation in a square cavity

Abstract

Keywords: Generation Entropy Generation; Prandtl Number; Square Cavity; Finite Difference Method

Introduction

Flow and heat transfer in different cavities have a wide range of applications in the engineering devices and industry, such as: the solar energy system, cooling systems producing nuclear energy, chemical and electronic devices and heat exchangers etc [1-3]. Due to its uttermost importance, many applications, many researches, investigations and studies have been created to raise the performance and quality of those engineering systems. Among these studies is the study of the entropy generation. Entropy generation is a measure of dissipated energy and degradation of the performance of operational devices, such as transport and flow process [1]. The study of entropy generation has been initiated by Bejan [3], which considers that the entropy generation of fluid flow under forced and laminar convection is induced by viscous and thermal mechanisms, predicting (involving) velocity and temperature gradients. We present in the following some studies concerning this subject. Yilbas et al. [4] presented a numerical study of the entropy generation in natural convection and laminar flow in a square cavity with bottom-heated, topdown and vertical walls adiabatic. Nine cases were studied by varying the difference of temperature between the hot wall and the cold wall (the Rayleigh number (Ra) varies between 1.36 10 5 and 3.96 105). They found that entropy generation increases as the velocity of flow in the horizontal direction increases. The minimum of the entropy generation is raised for a particular number of Rayleigh numbers (Ra). Baytas et al. [5] reported a numerical study of the entropy generation in laminar natural convection in an inclined porous square cavity. Simulations are performed for angles of inclination varying between 0 ° and 360 ° and Rayleigh number values (Ra) varying between 102 and 104. They found that thermal irreversibilities dominate viscous irreversibilities as the Rayleigh number (Ra) increases. The Bejan number (Be) increases significantly between φ = 150 ° and φ = 270 ° (angles of inclination). Shuja et al. [6] conducted a numerical study of the entropy generation in laminar mixed convection in a square cavity, containing an obstacle. The influence of the location of the fluid outlet and the aspect ratio (a = 0.25 and 4) of the obstacle were examined. Yilbas et al. [7] have taken over the same study as the previous one with the same geometric configuration. In this second study, the influence of the Stanton number and the aspect ratio (a = 0.25, a = This paper was recommended for publication in revised form by Regional Editor Younes Menni 1 Laboratory of Biotechnology, National Higher School of Biotechnology, Taoufik Khaznadar, University town, Constantine, Algeria 2 Department of Mechanical Engineering, Istanbul Medeniyet University, Istanbul, Turkey 3 Faculty of Process Engineering, University of Salah Boubnider Constantine 3, Algeria 4 Faculty of Sciences and Applied Sciences, University of Larbi BenM'hidi, Oum el Bouaghi, Algeria *E-mail address: ferroudj.nawal@gmail.com, n.ferroudj@ensbiotech.edu.dz Orcid id: 0000-0001-7743-0971, 0000-0002-1907-9420, 0000-0003-4420-4331, 0000-0003-0136-0748 Manuscript Received 28 October 2020, Accepted 12 February 2021

Journal of Thermal Engineering, Research Article, Vol. 7, No. 4, pp. 1016-1029, May, 2021 1, and a = 4) of the obstacle were examined. Magherbi et al. [8] conducted a numerical study on the entropy generation of unsteady natural convection in a square cavity. The Rayleigh number (Ra) varies between 103 and 105, whereas the ratio of the irreversibility distribution (φ) varies between 10-4 and 10-1. The evolution of the maximum entropy generation in this configuration has been reported. The effect of the ratio of irreversibility distribution was examined. Ovando-Chacon et al. [9] studied numerically the mixed convection in a square cavity heated from corners. They found that the entropy generation by heat transfer becomes more intense at the vertical walls and the central low part of the cavity due to the stronger temperature gradients at this region. Bouchoucha et al. [10] presented an investigation of the natural convection and entropy generation of nanofluids in a square cavity. They observed that the average Nusselt number ratio increases with the solid volume fraction, and the maximum total entropy generation ratio occurs at a low Rayleigh number for different values of ϕ. Selimefendigil et al. [11] examined the mixed convection and entropy generation in a vented cavity with inlet and outlet ports under the effects of an inclined magnetic field. The numerical simulations are performed for various values of Reynolds numbers between 100< Ra <500, Hartmann number between 0 and 50, and solid particle volume fractions of CuO nanoparticles between 0 and 4%. They found that the addition of nanoparticles increases the overall entropy generation rate. The literature review shows that the entropy generation in laminar mixed convection in cavities is a subject that remains to be explored. Different studies were studied and reported in the literature [12-16]. The present work investigates the entropy generation of mixed laminar convection in a square cavity. The main objective is to determinate the effect of Prandtl number on the fields of streamlines, isotherm, heat and viscous entropy generation and Bejan number, evolution of heat and viscous entropy generation -profiles, Bejan end Nusselt number and total entropy generation- profiles.

Material And Methods

Problem Statement The physical model considered is shown in Fig.1. It is a square cavity the bottom of which a part is subjected to a constant temperature, of length equal to four fifth of the cavity L. the side walls of this cavity move upwardly with a specified velocity, and the all other parts are considered adiabatic. The entropy of the fluid flow is generated within the cavity created 2 by the irreversibility of the viscous and thermal mechanisms due to velocity and temperature gradients. It is assumed that the fluid in the cavity is Newtonian, incompressible, and has constant properties, for laminar flow, neglecting the viscous dissipation effect, and employing the Boussinesq approximation. Y

Journal of Thermal Engineering, Research Article, Vol. 7, No. 4, pp. 1016-1029, May, 2021 Dimensionless Governing Equations The governing equations for the present problem for steady two-dimensional mixed convection flow in a square cavity are the conservation continuity of mass, momentum and energy equation than can be expressed as follows [17]: • Continuity equation:

  2u  2u  u u u 1 P* +u +v =− +  2 + 2  t x y  x y   x

  2v  2v  v v v 1 P* +u +v = − +   2 + 2  + g  (T − T f ) t x y  y y   x

• Energy equation:   2T  2T  T T T +u +v = 2 + 2  t x y y   x

It is assumed that the fluid in the cavity is Newtonian, incompressible, and has constant properties, for laminar flow, neglecting the viscous dissipation effect, and employing the Boussinesq approximation. The dimensionless form of the governing equations may be written with following dimensionless variables of numbers: T − Tc V x y u v p −  gy X = , Y= , = , U= , V= , P= ,  =t 0 2 L L TH − Tc V0 V0 L V0 The governing equations in dimensionless form are transformed into stream-vorticity function (ω, ψ) and energy equation, are expressed as follows: •

Energy equation    1   2  2  +U +V = +    X Y Re Pr  X 2 Y 2 

   1   2  2   +U +V =  2 + 2  + Ri  X Y Re  X X Y 

Journal of Thermal Engineering, Research Article, Vol. 7, No. 4, pp. 1016-1029, May, 2021 Reynolds (Re), Prandtl (Pr) and Grashof (Gr) are the principal controls parameters; they are defined as follow:

The local Nusselt number Nu(X) is used to characterize the heat flux between the heated part of the bottom wall, and the fluid in the cavity. It is expressed as [18]:

The average Nusselt number is obtained by integration the previous local Nusselt number Nu(X): L

• Entropy generation equation Entropy generation rate for a two-dimensional flow is expressed as follow [19, 20]:

2 2 2 2 2 k  T   T      u   v    u v   + + 2 + + +                 T 2  x   y   T   x   y    y x    

and by using the same dimensionless parameters, the dimensionless local

entropy generation due to heat transfer and fluid friction for a two dimensional heat and fluid flow in Cartesian coordinates can be written as: 2 2 2   •   2    2   U   V     U V   S gen =  + + +  +   2  +      X   Y      X   Y     Y X     S S

where γ is the irreversibility distribution ratio is given as 1019

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The first and the second terms (Sθ, Sψ) are represented the local entropy generation due to the heat transfer and viscous effects, respectively. The total entropy generation is expressed by: S gen =

The Bejan number Be is defined as the ratio between entropy generations due to heat transfer irreversibility to the total entropy generation, it’s written as:

It is appropriate to mention average Bejan number in order to determinate which is the dominant, heat transfer or fluid friction irreversibility. Therefore, Be>0.5 implies dominance of heat transfer irreversibility, and Be<0.5 implies dominance of fluid friction irreversibility. Dimensionless Boundary Conditions The boundary conditions which are adopted for the resolution of the problem are:

and 0  Y  1 1-  1+  and X 2 2 1-  1 +  and 0  X  ,  X 1 2 2 and

Numerical Method

The dimensionless equations are solved by a numerical approach on the finite difference method. The system of equations (5)-(14), together with the initial and boundary conditions, stated above were discredited of the various terms of the governing equations is developed using high order scheme as the third order upwind scheme for the convective terms [21], centred fourth order for the diffusive terms, as well as the partial derivatives of the first order. The Runge-Kutta fourth order used to discredit the temporal term scheme [22]. An iterative procedure based on successive Non Linear Over Relaxation method (NLOR) was used to solve the discredited stream function equation [23]. There are different studies related the applications in the literature [24-29]. A convergence criterion for the stream function was imposed as each time step. The computation is carried out using a developed FORTRAN code. In order to convergence the obtained numerical results, total entropy generation for square cavities with different ϕ and Ra values were obtained and compared with the results of similar studies. Results are agreement with the literature. 1020

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GRID Independence Study

The effect of grid size is examined on four different grids 81x81; 101x101; 161x161 and 201x201. These comparisons allow us the choice of mesh 101x101 for the present work, as it provides a good compromise between the accuracy of these calculations and the duration of computation time. Table 1. Maximum values of Num for different grids, at Ri=17 Grid 81x81 101x101 161x161 201x201

CODE Validation

In order to check on the validity of the computer code developed for the solution of the problem considered in the present study, it was validated with Aydin and Yang [30]. Figure 2 illustrates a comparison between the obtained results and those obtain in the Aydin’s results. The results are presented in terms of dimensionless vertical velocity profiles V, at the horizontal mid-plane. The figures show excellent agreement between the present code and their results. 1.5 Ri=0.1 Ri=0.5

Figure 2. Comparison of the horizontal mid-plane velocity component at Y = 0.5 with the results obtained in Aydin and yang (Lines: Present work, Symbols: Aydin’s results)

Results And Discussion

This investigation is mainly related to entropy generation distribution in a square cavity by using the second law of thermodynamics. Detailed computations have been carried for a fixed Reynolds number (Re=100), the length of heated part (l=0.8), and for various fluids used in this study are (Pr=0.0212, 0.71 and 6.35), within the values of Richardson number and irreversibility distribution ration are Ri=1 and γ=10-4, respectively. For all the studied grid of size (101 × 101) with a time steps Δt = 10−4 are found sufficiently enough to achieve the steady state and to realie the imposed convergence criteria mentioned above.

Journal of Thermal Engineering, Research Article, Vol. 7, No. 4, pp. 1016-1029, May, 2021 Streamlines and Isotherms Figure 3 illustrates isotherms (θ) and streamlines (ψ) for Pr=0.0212, 0.71 and 6.35. The streamlines represent a symmetrical flow, characterized by a pair of counter-rotating and symmetrical cells for all values of Prandtl number. Thus, the streamlines are concentrated near the upper walls. The general structures of these fields are qualitatively similar, regardless of the Prandtl number. For low Prandtl number (Pr=0.0212), the thermal field shows a stratification of temperatures near the heated bottom wall, when Prandtl number increases and reaches the value Pr=0.71 a part of the heat is trained to the upward of the cavity through velocity of the side walls. For high Prandtl number (Pr=6.35), the isotherm shows that the heat recovered from the hot source is transported by convection to the upwards of the cavity by the pair of center cells.

Figure 3. Streamlines a) and isotherms b) for different Prandtl numbers, Ri=1, Re=100 and γ=10-4 Fields of Heat and Viscous Entropy Generation The effect of Prandtl number on the fields of heat and viscous entropy generation present in figure. 4 (a-b), for Ri=1 and γ=10-4. The thermal entropy generation is consistent with the temperature distribution; it follows the propagation of heat, given by Figure 3 (a), from the bottom heated part, it is created all along this wall (especially to the extremities of the heated part for the low Pr=0.0212 and 0.71). When the Prandtl number increases the heat entropy generation increases, until it is concentrated also along the moving sidewalls for the high Prandtl number (Pr=6.35). The viscous entropy generation is created near the two side walls and the four corners of the cavity. Its concentration is decreasing with the increase of the Prandtl number.

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Figure 4. Fields of entropy generation due to heat transfer Sθ (a) and fluid friction Sψ (b) for different Prandtl numbers, Ri=1, Re=100 and γ=10-4 Fields of Bejan Number The fields of Bejan number obtained have been presented at three different values of Prandtl number (Pr = 0.0212, 0.71 and 6.35) for comparison supposes as shown in figure 5. It shows that the Bejan number (development) decreases with the increase in the Prandtl number; it increases until it reaches its maximum value in the lower twothirds and an upper part of the cavity for Pr=6.35. Moreover, the low values of this number are observed on the center of the upper wall and the top of the side walls. These results are confirmed by those obtained in the figures (34).

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Pr=6.35 Figure 5. Fields of Bejan number for different Prandtl numbers, Ri=1, Re=100 and γ=10-4 Profiles of Thermal (Sθ) and Viscous (Sψ) Entropy Generation In order to investigate the influence of Prandtl number on both local thermal (Sθ) and viscous (Sψ) entropy generation, Figure 6 (a-b) illustrates the variation of and with different values of Prandtl number. As it can be seen, for all values of this number, a very rapid decrease then a sudden increase in all the first time, then stabilization towards a constant asymptotic value, has been observed. The increase in the number of Prandtl implies the increase of local thermal entropy generation. It is observed also that, for all values of Prandtl number, the viscous entropy generation decreases by increasing the Prandtl number regardless the values of Ri and Re. From the value of Pr = 0.71, the Sψ varies slightly with the increase of the Prandtl number. We have also observed the low values of Sψ compared to the values of the Sθ, this is confirmed by the results obtained in the figure 4 (a-b).

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a) b) Figure 6. Profiles of entropy generation due to heat transfer Sθ a) and fluid friction Sψ b) for different Prandtl numbers, Ri=1, Re=100 and γ=10-4 Evolution of Bejan Number This evolution increases very fast in the first moments to stabilize towards a constant value. It observed that, regardless of the Prandtl number, the Bejan number is greater than 0.5, which shows that the creation of entropy is essentially due to temperature gradients. It shows the increase of Bejan number into the cavity by augmenting Prandtl number. 1

τ Figure 7. Evolution of Bejan number for different Prandtl numbers, Ri=1, Re=100 and γ=10-4 Evolution of Nusselt Number The heat transfer inside the square enclose is presented by average Nusselt numbers given by equation (12). It is found, for all values of the Prandtl number, a very fast decrease of Num in the first moments, then a tendency towards a limit value when the time increases. The increase in the number of Prandtl results in an increase in the average Nusselt number.

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τ Figure 8. Evolution of average Nusselt number for different Prandtl numbers, Ri=1, Re=100 and γ=10-4 Effect of The Irreversibility Distribution Ratio on The Total Entropy Generation The evolution of the effect of irreversibility distribution ratio (γ=10-2 and 10-4) on the total entropy for different Prandtl number are plotted in figure 9 (a-b-c). It is observed that, for all Prandtl number, the total entropy generation increase by increasing the irreversibility distribution ratio and the increase of Prandtl number regardless the values of Ri=1 and Re=100 because of the increase of the velocity gradients. 4.5

τ c) Pr=6.35 Figure 9. Effect of the irreversibility distribution ratio on the total entropy generation for different Prandtl numbers, Ri=1, Re=100 and 10-2<γ<10-4 1026

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Conclusion

In this numerical simulation of the Prandtl number effect on the entropy generation in laminar mixed convection, within a two dimensional square enclosure with moving cold sidewalls and heated from the bottom wall was investigated. The equations governing this phenomenon were discretized by the finite difference method, and the algebraic systems obtained were solved by appropriate numerical methods. The analysis of the preliminary results shows that the creation of the entropy increases with the increase of Pran dtl number (because of the increase of the velocity gradients) and the irreversibility distribution ratio, when this number increases (Pr) the creation of the total entropy is essentially due to temperature gradients which is confirmed by the values of the Bejan number "Be>0.5". As thermodynamic approach entropy generation in different form such as laminar mixed convection in a square fluid-filled cavity is critical process in many applications. It is discussed in this study to understand the response of the system which has heated wall with a constant temperature Th while the side-walls are maintained at a constant temperature Tc, and a constant velocity. Also it is seen that the irreversibility distribution ratio on the entropy generation in the cavity affected for the Prandtl number =0.0212, 0.71 and 6.35.

Nomenclature

𝐶𝑝 Specific heat, kJ / kg oK 𝑔 Gravitational acceleration, m s−2 k Thermal conductivity, W / m oK 𝑙 Length of the heated part, m 𝐿 Length of the cavity, m 𝑃∗ Pressure, kg m−1 s−2 𝑃 Dimensionless pressure 𝑃𝑟 Prandtl number Ri Richard number 𝑠𝑔𝑒𝑛 Entropy generation rate, J m−3 s−1 K−1 𝑆𝑔𝑒𝑛 Dimensionless local entropy generation 𝑡 Time, sec 𝑇 Temperature, K 𝑇𝑐 Cold temperature, K 𝑇ℎ Hot temperature, K 𝑢, 𝑣 Velocity components along x and y, respectively, m s−1 𝑈, 𝑉 Dimensionless velocity components 𝑥, 𝑦 Cartesian coordinates, m 𝑋, 𝑌 Dimensionless coordinates Greek symbols 𝛽 𝜀 𝜌 𝜃 𝜏

Thermal expansion coefficient, K-1 Dimensionless length of the heated part Density of a fluid. kg/m3 Dimensionless temperature Dimensionless temperature

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