# Thermal Lattice Boltzmann Simulation of Entropy Generation within a Square Enclosure for Sensible and Latent Heat Transfers

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## Abstract

**:**

^{4}and 10

^{5}. The results are presented in terms of the total entropy generation, average Bejan number and average Nusselt number. Within the range considered for the key parameters, the entropy generation is found to be controlled by the heat transfer loss for low Prandtl numbers. However, for the large Prandtl numbers, its variation is dominated by shearing losses. Moreover, the presence of the latent heat state decreases the overall thermodynamic losses while increasing the quantity of heat transferred.

## 1. Introduction

## 2. Problem Description and Formulation

**Figure 1.**Schematic view of the differentially-heated square cavity with initial and boundary conditions (

**a**) Natural convection (no melting); (

**b**) melting with convection.

#### 2.1. Mathematical Model

#### 2.2. Entropy Generation

## 3. Numerical Model

#### 3.1. Thermal Lattice Boltzmann Equations

_{b}is incorporated in the model by shifting the velocity field by a term of ${F}_{b}{\tau}_{f}/\mathsf{\rho}$, as proposed by Shan and Chen [20], where ${F}_{b}$ $\left(=-\mathsf{\beta}g\left(T-{T}_{0}\right)/\left({T}_{h}-{T}_{c}\right)\right)$. By this treatment, there is no need to add a force term to the collision operator. On the other hand, in the g distribution function, the source term is treated as per the method proposed by Luo [21]. Hence, the resulting force in the LBM frame will be: ${S}_{h}^{g}=-{\mathsf{\omega}}_{i}{S}_{h}$ with ${S}_{h}$$\left(=St{e}^{-1}\partial \mathsf{\epsilon}/\partial t\right)$ being the source (or sink) term that handles the phase change.

#### 3.2. Boundary Conditions

#### 3.3. Macroscopic Quantities

## 4. Model Validation

#### 4.1. Melting with Convection

#### 4.1.1. Comparison with Similar LBM Schemes

Parameter | Pr | Ra | Ste | Grid |
---|---|---|---|---|

Value | 1 | 1.7 × 10^{5} | 10 | 150 × 150 |

**Figure 3.**Position and shape of the melting front at different dimensionless times $\mathsf{\zeta}$.

#### 4.1.2. Comparison with Experimental Results and the Finite Volume Method

Parameter | Pr | Ra | Ste | Aspect Ratio |
---|---|---|---|---|

Value | 0.021 | 6 × 10^{5} | 0.039 | H/L = 1.4 |

#### 4.2. Entropy Generation

**Table 3.**The entropy generation parameters as in [6].

Parameter | Pr | Ra | φ | ΔT |
---|---|---|---|---|

Value | 0.7 | 10^{4} | 10^{−3} | 1.0 |

**Figure 5.**Total entropy production vs. dimensionless time $\mathsf{\theta}$ ($=\mathsf{\alpha}t/{L}^{2}$).

## 5. Results and Analysis

**Figure 7.**(

**a**) ${S}_{T}$, ${S}_{l,h}$, ${S}_{l,d}$ and $\text{Nu}$ vs. $\text{Pr}$; (

**b**) zoom on ${S}_{T}$, ${S}_{l,h}$, ${S}_{l,d}$ and $\text{Nu}$ vs. $\text{Pr}$ (= 0.02 till 10).

#### Single-Phase and Solid-Liquid Phase Change with High Prandtl Number

Parameter | Pr | Ra | φ | Ste (Figure 1b) |
---|---|---|---|---|

Value | 50 | 10^{4} − 10^{5} | 10^{−5} | 1.0 |

**Figure 9.**Entropy generations with no phase change (problem in Figure 1a) for $\text{Pr}=50$ and $\text{Ra}={10}^{5}$.

**Figure 10.**Entropy generations with no phase change (problem in Figure 1a) for $\text{Pr}=50$ and $\text{Ra}={10}^{5}$.

**Figure 11.**Nu versus dimensionless time (see Figure 1a) for $\text{Pr}=50$ and $\text{Ra}={10}^{5}$.

**Figure 14.**Entropy generation for melting with convection (problem in Figure 1b) for $\text{Pr}=50$ and $\text{Ra}={10}^{5}$.

**Figure 15.**Nu versus dimensionless time (problem in Figure 1b) for $\text{Pr}=50$ and $\text{Ra}={10}^{5}$.

**Figure 16.**Entropy generation versus ${X}_{m}$ during latent heat transfer with $\text{Pr}=50$ and Ra = 10

^{5}.

## 6. Conclusions

## Author Contributions

## Conflicts of Interest

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Yehya, A.; Naji, H.
Thermal Lattice Boltzmann Simulation of Entropy Generation within a Square Enclosure for Sensible and Latent Heat Transfers. *Appl. Sci.* **2015**, *5*, 1904-1921.
https://doi.org/10.3390/app5041904

**AMA Style**

Yehya A, Naji H.
Thermal Lattice Boltzmann Simulation of Entropy Generation within a Square Enclosure for Sensible and Latent Heat Transfers. *Applied Sciences*. 2015; 5(4):1904-1921.
https://doi.org/10.3390/app5041904

**Chicago/Turabian Style**

Yehya, Alissar, and Hassane Naji.
2015. "Thermal Lattice Boltzmann Simulation of Entropy Generation within a Square Enclosure for Sensible and Latent Heat Transfers" *Applied Sciences* 5, no. 4: 1904-1921.
https://doi.org/10.3390/app5041904