# Numerical Study on Entropy Generation in Thermal Convection with Differentially Discrete Heat Boundary Conditions

^{1}

^{2}

^{3}

^{*}

^{†}

## Abstract

**:**

_{u}) generally increases in magnitude at the central region of the channel with increasing Ra. Total entropy generation rate (S) and thermal entropy generation rate (S

_{θ}) are larger in magnitude in the region with the largest temperature gradient in the channel. Our results also indicated that the thermal entropy generation, viscous entropy generation, and total entropy generation increase exponentially with the increase of Rayleigh number. It is noted that lower percentage of single heat source area in the bottom boundary increases the intensities of viscous entropy generation, thermal entropy generation and total entropy generation. Comparing with the classical homogeneous thermal convection, the thermal entropy generation, viscous entropy generation, and total entropy generation are improved by the presence of discrete heat sources at the bottom boundary.

## 1. Introduction

## 2. Thermal Fluid Dynamics Equation and Numerical Method

#### 2.1. Thermal Fluid Dynamics Equation

#### 2.2. Viscosity and Thermal Entropy Generation

#### 2.3. Numerical Method for Thermal Fluid Dynamics Equation

_{1}= εx) and two macroscopic time scales (t

_{1}= εt, t

_{2}= εt) are employed to obtain the classical Oberbeck–Boussinesq equations (Equations (1)–(3)). As two time scales ∂t = ε∂

_{t}

_{1}+ ε

^{2}∂

_{t}

_{2}and one spatial scale ∂

_{x}= ε∂

_{α}are carried out for the Frisch, Hasslacher and Pomeau (FHP) model. The inertial terms in the classical Oberbeck–Boussinesq equations (Equation (7)) can be reproduced by executing the streaming step using the above Chapman–Enskog expansion.

_{down}, which is denoted by black regions, or adiabatic boundary condition, ∂

_{y}θ = 0. The upper boundary (y = H) is kept at constant temperature, θ = θ

_{up}. In this simplified geometry, the width to height ratio of the cavity, ξ = H/L, and other two new dimensionless parameters are used to define the geometrical configuration of the heating boundary, namely the percentage of single heat source area λ = l/L, and the total percentage of heating source area, η = nl/L, in Figure 1, where n is the number of heat source and l is the length of single heat source region. In the limiting case of η = 1, the model recovers to the usual homogeneous RB convection. The quantities λ and η are important factors for discrete heat source boundary conditions. It is easy to understand the changes with different λ at fixed η from an applied point of view.

**L**denotes the periodicity direction and length of the flow pattern.

## 3. Results and Discussion

_{c})

^{0.296}, and the circle symbols denote the results of the present simulations using LBM, and the plus symbols represent the numerical results obtained by Clever and Busse [41]. It is seen that the value of the Nusselt number obtained using the present LBM is quite consistent with the theoretical value and benchmark solutions, which validates the accuracy of the numerical simulations by the LBM.

#### 3.1. Analysis of Flow and Temperature Field

^{3}, 10

^{4}, 10

^{5}and 10

^{6}. As shown in Figure 4, the local hot fluid close to each heat source moves upward in an independent way and the isotherms merge at the top region of the channel at Ra = 10

^{3}. However, the cold fluid close to the cold sources moves upward and increases the temperature near every cold source. It is seen that with increasing Rayleigh number, the heat sources effectively heat the fluid from the central portion of the channel to the top wall. The hot fluid flow moves upward and the cold fluid moves downward near the top wall flow, while the temperature decreases near the side boundaries. Two trends with increasing Rayleigh number are found for the isotherm distribution. The mixing of the cold and hot fluids is enhanced, and an increased temperature gradient is found in the region close to the bottom and top boundaries. It is concluded that the presence of the discrete heat sources leads to the enhanced heat transfer in the channel.

^{3}, two large symmetrical vortices appear in the central region of channel as the typical behaviors of the heated flow; two symmetrical secondary vortices appear in the central region of the bottom wall, and small vortices appear between the heat sources on the bottom wall. The vortices gradually expand to be of elliptic shape with increasing Ra, and move towards both vertical sides of the channel. As Ra reaches to 10

^{6}, the two vortices break up into multiple vortices.

#### 3.2. Analysis of S_{u} and S_{θ}

^{3}, 10

^{4}, 10

^{5}and 10

^{6}. The viscous entropy generation rate at four Ra ispresented in Figure 6a–d. As shown in Figure 6 and Figure 7, the significant S

_{u}mainly clusters in the region with steepest velocity gradient. It is concluded that with increasing Ra, the significant S

_{u}gradually propagates to the central region of the channel, which mainly occurs in the region with largest velocity gradient in the majority portion of the channel.

_{θ}at four Ra. From Figure 4 and Figure 8, it is seen that the significant S

_{θ}clusters on the interfaces between hot and cold fluids, which is mainly originated from the largest temperature gradient near wall regions. As shown in Figure 8b–d, the significant S

_{θ}constantly propagates to the central region of the channel, which is closely related with the largest temperature gradient in the majority of the channel.

_{θ}at the same Ra. It is shown that the heat transfer plays a leading role on the flow in the channel. Comparing Figure 7 and Figure 8, it is seen that S

_{θ}is much larger in magnitude than S

_{u}. It is also indicated that the heat transfer irreversibility plays a leading role in the entropy generation of thermal convection.

## 4. Conclusiuons

## Author Contributions

## Funding

## Conflicts of Interest

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**Figure 2.**Nusselt number Nu as a function of Rayleigh number in traditional Rayleigh–Bénard (RB) convection for homogeneous boundary conditions (η = 1).

**Figure 3.**Nusselt number with changing Rayleigh number and with discrete heat boundary conditions for different η.

**Figure 4.**Temperature distributions (isotherms) at various Rayleigh number from (

**a**) to (

**d**). (

**a**) Ra = 10

^{3}; (

**b**) Ra = 10

^{4}; (

**c**) Ra = 10

^{5}; (

**d**) Ra = 10

^{6}.

**Figure 5.**Streamlines of the thermal convection flow at various Rayleigh number from (

**a**) to (

**d**). (

**a**) Ra = 10

^{3}; (

**b**) Ra = 10

^{4}; (

**c**) Ra = 10

^{5}; (

**d**) Ra = 10

^{6}.

**Figure 6.**Velocity distribution of the thermal convection flow at various Rayleigh number from (

**a**) to (

**d**). (

**a**) Ra = 10

^{3}; (

**b**) Ra = 10

^{4}; (

**c**) Ra = 10

^{5}; (

**d**) Ra = 10

^{6}.

**Figure 7.**Viscous entropy generation rate at various Rayleigh number. (

**a**) Ra = 10

^{3}; (

**b**) Ra = 10

^{4}; (

**c**) Ra = 10

^{5}; (

**d**) Ra = 10

^{6}.

**Figure 8.**Thermal entropy generation rate at various Rayleigh number from (

**a**) to (

**d**). (

**a**) Ra = 10

^{3}; (

**b**) Ra = 10

^{4}; (

**c**) Ra = 10

^{5}; (

**d**) Ra = 10

^{6}.

**Figure 9.**Total entropy generation rate at various Rayleigh number from (

**a**) to (

**d**). (

**a**) Ra = 10

^{3}; (

**b**) Ra = 10

^{4}; (

**c**) Ra = 10

^{5}; (

**d**) Ra = 10

^{6}.

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Wang, Z.; Wei, Y.; Qian, Y. Numerical Study on Entropy Generation in Thermal Convection with Differentially Discrete Heat Boundary Conditions. *Entropy* **2018**, *20*, 351.
https://doi.org/10.3390/e20050351

**AMA Style**

Wang Z, Wei Y, Qian Y. Numerical Study on Entropy Generation in Thermal Convection with Differentially Discrete Heat Boundary Conditions. *Entropy*. 2018; 20(5):351.
https://doi.org/10.3390/e20050351

**Chicago/Turabian Style**

Wang, Zhengdao, Yikun Wei, and Yuehong Qian. 2018. "Numerical Study on Entropy Generation in Thermal Convection with Differentially Discrete Heat Boundary Conditions" *Entropy* 20, no. 5: 351.
https://doi.org/10.3390/e20050351