# Entropy Generation Analysis of the Flow Boiling in Microgravity Field

^{*}

## Abstract

**:**

^{2}to 50,000 W/m

^{2}, so as to investigate their influence on irreversibility during flow boiling in the tunnel. A phase–change model verified by the Stefan problem is employed in this paper to simulate the phase–change process in boiling. The numerical simulations are carried out on ANSYS-FLUENT. The entropy generation produced by the heat transfer, viscous dissipation, turbulent dissipation, and phase change are observed at different working conditions. Moreover, the Be number and a new evaluation number, E

_{P}, are introduced in this paper to investigate the performance of the boiling phenomenon. The following conclusions are obtained: (1) a high local entropy generation will be obtained when only heat conduction in vapor occurs near the hot wall, whereas a low local entropy generation will be obtained when heat conduction in water or evaporation occurs near the hot wall; (2) the entropy generation and the Be number are positively correlated with the heat flux, which indicates that the heat transfer entropy generation becomes the major contributor of the total entropy generation with the increase of the heat flux; (3) the transition of the boiling status shows different trends at different velocities, which affects the irreversibility in the tunnel; (4) the critical heat flux (CHF) is the optimal choice under the comprehensive consideration of the first law and the second law of the thermodynamics.

## 1. Introduction

^{2}), extreme high temperature environments, and so on [2]. An outstanding heat transfer capacity can be obtained through the huge latent heat during the boiling phase change process at a constant temperature. The unique thermodynamic characteristics of the boiling process make it a good choice for the designs of the thermal control system. The thermal control systems based on boiling phenomenon have successfully been utilized [3,4,5] in industrial manufacturing with strict requirements for temperature. It provides a promising prospect in the design of spacecraft cooling systems, compared with the thermal control systems based on the single phase [6]. The cooling system in a spacecraft operates under a microgravity field, which is totally different from the Earth. The absence of gravity contributes to a different boiling phenomenon in space compared with that on earth. Bubbles generated when boiling tend to adhere on the hot wall under the microgravity field, distinguishing the heat and mass transfer characteristics from that under the gravity field [7].

_{P}, which is a criterion combining the first law and the second law of the thermodynamics, is introduced in this paper to evaluate the performance of the boiling process.

## 2. Mathematical Model

#### 2.1. Governing Equations of CFD Calculation

**Continuity Equation**

**Momentum Equation**

**Energy Equation**

**Evaporation**

**Condensation**

#### 2.2. Simulation Model

^{2}to 50,000 W/m

^{2}. The impact of heat flux on entropy production characteristics in the flow boiling is studied every 10,000 W/m

^{2}. A set of velocities, ranging from 1 m/s to 4 m/s every 1 m/s, is taken into consideration to observe the tendency of the entropy generation in flow boiling with different flow patterns. The velocity acceleration in the y direction is set as −0.0005 m/s

^{2}. The water vapor is employed as the primary phase whereas the liquid water is employed as the secondary phase. The transient simulation is adopted in this paper with the courant number smaller than 0.25 to ensure a set of more accurate results.

^{2}. The time-average temperatures of the hot wall from 0 s to 0.05 s is chosen as the criteria. The results obtained in the simulations are displayed in Figure 2.

## 3. Entropy Generation Model

^{th}orientation, and the ${S}_{gen}^{\u2034}$ is the entropy generation rate. The governing entropy equation is considered to be able to deduce the entropy generation model from the simulation region as follows [35]:

**Convective Terms**

**Entropy Generation by Dissipation**

**Entropy Generation by Heat Transfer**

**Entropy Generation by Phase Change**

- (1)
- The entropy generation caused by turbulent dissipation can be expressed as:

- (2)
- Utilizing the Boussinesque-like approach [36], the entropy generation caused by fluctuating temperature gradients is described as follows:

## 4. Results and Discussion

#### 4.1. Validation of the Simulation Model

#### 4.2. Influence of the Heat Flux

^{3}·K when the heat flux is 10,000 W/m

^{2}, whereas the peak of the heat transfer entropy generation is 0.0505 W/m

^{3}·K when the heat flux is 50,000 W/m

^{2}. It can be seen that the peak of the heat transfer entropy differs by two orders of magnitude under the above two conditions; however, the heat flux has no influence on the bottom of the heat transfer entropy generation curves. The bottom values of the heat transfer entropy generation under all conditions are all of the order of 10

^{−5}. The fluctuations of turbulent dissipation entropy generation seem to have no correlation with the heat flux, which are different from those of the heat transfer entropy generation. It can be concluded from Figure 6f that the tendencies of the phase–change entropy generation and the viscous dissipation entropy generation are not affected by the heating distance. The difference between the two entropy generations is the influence of heat flux. The heat flux has influence on the phase change entropy generation, but not on the viscous dissipation entropy generation. The phase–change entropy generation increases from 0.000004 W/m

^{3}·K to 0.000012 W/m

^{3}·K when the heat flux increases from 10,000 W/m

^{2}to 50,000 W/m

^{2}. The total entropy generation tends to be proportional to the heat flux with the contributions of all kinds of entropy generation.

#### 4.3. Influence of the Velocity

^{2}and 20,000 W/m

^{2}, respectively; however, nucleate boiling translates to film boiling when the heat flux is 30,000 W/m

^{2}, 40,000 W/m

^{2}, and 50,000 W/m

^{2}, respectively. This tendency concerning water vapor distribution diverging when the heat flux differs, still exists when the velocity is the 4 m/s. Few bubbles can be found near the hot wall when the heat flux is 10,000 W/m

^{2}and 20,000 W/m

^{2}, respectively. A vapor film that doesn’t cover the entire hot wall can be observed when the heat flux is 30,000 W/m

^{2}, 40,000 W/m

^{2}, and 50,000 W/m

^{2}, respectively. A common rule is shared amongst the working conditions, that the size of bubbles (films) tend to be smaller with the increase in velocity. The distribution of the water vapor phase determines the characteristics of irreversibility. The dimensionless entropy generation number is adopted to take a comparison between the characteristics of the local irreversibility under a different heat flux. The entropy generation number is defined as follows.

_{b}stands for the bulk temperature, and $Q$ represents the heat input to the domain. The entropy generation numbers under a different heat flux are presented in Figure 10, as are a set of representative water vapor distributions, respectively.

^{2}and 20,000 W/m

^{2}. The irreversibility at working conditions with high heat fluxes tends to increase then decrease as shown in Figure 10b, which is totally different from the case with low heat flux.

^{2}and 20,000 W/m

^{2}do not interact with each other, whereas the bigger bubbles tend to emerge near each other at the working conditions of higher heat fluxes. The vapor film formed by the merging between the bubbles actually leads to a high temperature gradient in the domain, contributing to a higher irreversibility in the fluid region.

#### 4.4. Performance Evaluation

_{P}, which takes the first and the second law of the thermodynamics into consideration, is described as:

_{P}can characterize the boiling heat transfer capacity per unit of entropy generation. The E

_{P}criterion is adopted to analyze the influence of velocity and heat flux on the performance of the boiling phenomenon in the tunnel. It is noticeable that not all the operating conditions in Figure 9 are taken into the analysis for the occurrence of the film boiling in some working conditions. The occurrence of the film boiling may lead to the melting of the hot wall, which is unacceptable in industrial facilities. The performances of the flow boiling in the tunnel are presented in Figure 11.

_{P}and the CHF shows the rationality of the criterion developed in this paper.

## 5. Conclusions

- (1)
- The local distribution of the water vapor has a great influence on local entropy generation. A high local entropy generation will be obtained only when heat conduction in vapor occurs near the hot wall, whereas a low local entropy generation will be obtained when heat conduction in water or evaporation occurs near the hot wall. The vapor–liquid distribution near the heating wall changes alternately as the bubbles grow and fall off in nucleate boiling, causing the total entropy generation to fluctuate with the increase of the heating distance. The near-wall region is filled with vapor in film boiling, which causes the total entropy generation to rise continuously with the increase of the heating distance.
- (2)
- The heat transfer entropy generation becomes the major contributor of the total entropy generation with the increase of the heat flux. Unlike the heat transfer entropy generation, the turbulent dissipation entropy generation and the viscous dissipation entropy generation are only influenced by the velocity of the flow. The phase–change entropy generation is more complex and is determined by the boiling state. The boiling state is under the coupled influence of velocity and heat flux. As the result, it is difficult to analyze the effect of a single variable on phase–change entropy generation.
- (3)
- The velocity in the tunnel has a great effect on the boiling status and determines the entropy generation in the tunnel. The increase of the velocity at a low heat flux will restrain the nucleate boiling, reducing the irreversibility in the tunnel; however, the increase of the velocity at a high flux will promote the boiling status transition from nucleate boiling to film boiling, creating more irreversibility in the tunnel.
- (4)
- The optimal operating condition can be achieved through the introduction of the evaluation number E
_{P.}A positive correlation between the heat flux and the E_{P}can be observed when the velocity keeps constant. As a result, the CHF is the optimal choice under the first law and the second law of the thermodynamics.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Nomenclature

${\alpha}_{l}$ | Volume fraction of liquid |

${\alpha}_{v}$ | Volume fraction of vapor |

Be | Bejan number |

${\mathrm{C}}_{D}$ | Bubbles-liquid drag force coefficient |

${h}_{fg}$ | Latent heat (kJ/kg) |

$J$ | Flow |

$Ja$ | Jakob number |

${J}_{i}^{s}$ | Flow |

l | Length of the fluid domain (cm) |

${\stackrel{\u2022}{m}}_{vl}$ | Condensation mass (kg/m^{3}) |

${\stackrel{\u2022}{m}}_{lv}$ | Evaporation mass (kg/m^{3}) |

${N}_{s}$ | Entropy generation number |

$p$ | Pressure (Pa) |

${Q}_{{C}_{Is},in}$ | Heat into the interface cell (J) |

${Q}_{{C}_{Is},out}$ | Heat out the interface cell (J) |

$R{e}_{b}$ | Reynolds number of the bubbles |

${S}_{}$ | Source term |

${S}_{T}$ | Energy source term generated by viscosity (W/m^{3}) |

${S}_{B}$ | Energy source term generated by boiling (W/m^{3}) |

${\dot{S}}_{gen}^{\u2034}$ | Entropy generation rate (J/(K·s·m^{3})) |

${\dot{S}}_{gen}^{}$ | Entropy generation (J/(K·m^{3})) |

${s}_{m}$ | Entropy variable (J/(kg·K) |

$\overline{{s}_{m}}$ | Time-averaged entropy variable (J/(kg·K) |

${s}_{m}^{\prime}$ | Fluctuating entropy variable (J/(kg·K) |

$T$ | Temperature (K) |

$\overline{{T}_{m}}$ | Time-averaged Temperature (K) |

${T}_{m}^{\prime}$ | Fluctuating Temperature (K) |

${u}_{i}$ | x components of velocity (m/s) |

${u}_{j}$ | y components of velocity (m/s) |

$\overline{{u}_{i}}$ | Time-averaged x components of velocity (m/s) |

$\overline{{u}_{j}}$ | Time-averaged y components of velocity (m/s) |

${u}_{i}^{\prime}$ | Fluctuating x components of velocity (m/s) |

${u}_{j}^{\prime}$ | Fluctuating y components of velocity (m/s) |

VOF | Volume of fluid |

Greek symbols | |

$\rho $ | Density (kg/m^{3}) |

$\mu $ | Viscosity (kg/(m·s)) |

$\epsilon $ | Turbulent dissipation rate |

${\kappa}_{}$ | Thermal conductivity (W/(m·K)) |

${\kappa}_{eff}$ | Effective thermal conductivity (W/(m·K)) |

Subscript | |

${}^{\prime}$ | Fluctuating |

$\overline{}$ | Time-averaged |

${}_{m}$ | Mixture phase |

${}_{l}$ | Liquid phase |

${}_{v}$ | Vapor phase |

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**Figure 4.**The comparison between the bubbles obtained in simulations and the experiments. (

**a**) Bubbles obtained in simulation. (

**b**) Bubbles obtained in experiments. Reprinted with permission from Ref. [36]. Copyright 2005, Elsevier.

**Figure 6.**Distribution of the local entropy generation with different heating distance when the velocity is 1 m/s; (

**a**) q = 10,000 W/m

^{2}; (

**b**) q = 20,000 W/m

^{2}; (

**c**) q = 30,000 W/m

^{2}; (

**d**) q = 40,000 W/m

^{2}; (

**e**) q = 50,000 W/m

^{2}; (

**f**) The tendency of the average entropy generation of the entire tunnel.

**Figure 7.**The behavior of bubbles. (

**a**) The bubble at the induced status. (

**b**) The bubble at the growing status. (

**c**) The exfoliated bubble.

**Figure 10.**The entropy generation number vs. velocity. (

**a**) N

_{S}vs. velocity when q is 10,000 W/m

^{2}and 20,000 W/m

^{2}. (

**b**) N

_{S}vs. velocity when q is 30,000 W/m

^{2}, 40,000 W/m

^{2}and 50,000 W/m

^{2}.

**Figure 11.**Performance of the boiling process with different working conditions. (

**a**) E

_{P}vs. Velocity. (

**b**) E

_{P}vs. heat flux.

Simulation | Experiment [33] | |
---|---|---|

medium | water | FC 72 |

${\rho}_{l}$ (kg/m^{3}) | 998 | 1680 |

V (m/s) | 1 | 0.14 |

$\mu $ (kg/(m·s)) | 0.001003 | 0.00064 |

L (m) | 0.02 | 0.005 |

${\mathrm{C}}_{D}$ | 5,416,700.436 | 5,416,700.012 |

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**MDPI and ACS Style**

Sun, Z.; Zhang, H.; Wang, Q.; Sun, W.
Entropy Generation Analysis of the Flow Boiling in Microgravity Field. *Entropy* **2022**, *24*, 569.
https://doi.org/10.3390/e24040569

**AMA Style**

Sun Z, Zhang H, Wang Q, Sun W.
Entropy Generation Analysis of the Flow Boiling in Microgravity Field. *Entropy*. 2022; 24(4):569.
https://doi.org/10.3390/e24040569

**Chicago/Turabian Style**

Sun, Zijian, Haochun Zhang, Qi Wang, and Wenbo Sun.
2022. "Entropy Generation Analysis of the Flow Boiling in Microgravity Field" *Entropy* 24, no. 4: 569.
https://doi.org/10.3390/e24040569