# Numerical Investigation on Two-Phase Flow Heat Transfer Performance and Instability with Discrete Heat Sources in Parallel Channels

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

^{2}; mass flux G = 50–400 kg/m

^{2}s), R12 (heat flux q = 7.5–129 kW/m

^{2}; mass flux G = 44–832 kg/m

^{2}s), and R134a (heat flux q = 2.2–49.5 kW/m

^{2}; mass flux G = 33–502 kg/m

^{2}s). Meng et al. [6] studied the flow boiling of R141b in vertical tubes and serpentine tubes through experiment and numerical simulation, and the results showed that the heat transfer can be enhanced by inclining the pipe. Huang et al. [7] studied numerically the flow pattern and heat transfer characteristics of flow boiling in the minichannels. They pointed out that the smaller the pipe diameter, the better the heat transfer. Catherine et al. [8] used a reduced-order model to simulate the flow boiling of R245fa and compared it experimentally. They focused on the effect of the time relaxation constant and r on flow and heat transfer, where they defined r as a function of the streamwise location. Daniel et al. [9] analyzed flow boiling in a microgap with circular pin fins using the Lee model and found that the evolution of two-phase flow is accelerated with the time relaxation constant increasing, which causes more uniform distribution of temperature. Similarly, Lee et al. [10] used the Lee model for analyzing thermal performance of the microcooler module in GaN HEMTs. The results showed that the heat transfer coefficient, temperature, and pressure drop depend on different mesh sizes and the time relaxation constant. Konstantinos et al. [11] adopted a numerical method to study the effect of surface wettability on flow boiling characteristics within microchannels. The results showed indeed that surface wettability plays a significant role on the HTC, and the hydrophilic and hydrophobic conditions are about 43.9% and 17.8% higher than the single-phase reference simulation, respectively. Li et al. [12] simulated the bubble departure characteristics and found that the surface tension is vital to simulate bubble departure diameter and active core density. Ali et al. [13] studied the subcooled flow boiling of different concentrations of alumina nanoparticles in a microchannel heat sink. Zhuan et al. [14] investigated the nucleate boiling of water in microchannels by using the VOF model. They displayed the whole process of bubbles changing in the microchannel, such as generating, growing, departing, and so on. Liu et al. [15] discussed the bubble growth and merger of refrigerant in a microchannel by using the coupled level set and volume of fluid method (CLSVOF). Yu et al. [16] investigated heat transfer and pressure drop of mixture refrigerant in a vertical rectangular minichannel, pointing out that as for heat transfer and pressure drop, the influence of heat flux was slight. Yang et al. [17] studied flow boiling of R141b in horizontal coiled tube, based on the Lee model, and the results showed that the temperature was significantly affected by the phase distribution. Using a numerical method, Steffen et al. [18] designed a channel in which vapor bubbles grew only along a predefined direction. Based numerical simulation results, Ma et al. [19] proposed a method to estimate the mean diameter of dispersed phase in saturated boiling.

## 2. Study on Heat Transfer Characteristics

#### 2.1. Physical Model

#### 2.2. Mathematical Model

#### 2.2.1. Governing Equations

_{l}and S

_{v}are the volumetric mass source terms; S

_{ml}and S

_{mv}are the mass transfer between the liquid phase and the vapor phase, which can be obtained in the phase-change model. Subscript “l” and “v” mean liquid phase and vapor phase.

_{lv}is surface tension coefficient, and ${\kappa}_{v}$ and ${\kappa}_{l}$ is surface curvature, which is defined as follows:

_{e}is the volumetric energy source term.

#### 2.2.2. Phase Change Model

_{ml}and S

_{mv}and the energy source term S

_{e}between the liquid phase and the vapor phase due to the boiling. In this paper, the evaporation–condensation phase change model proposed by Lee [34] was used.

_{sat}means saturation temperature. In order to maintain equilibrium between the liquid and gas phases, the chemical potential energy should be equal, so the equation between pressure and temperature can be obtained using the Clausius–Clapeyron equation:

_{lv}is latent heat and P

_{sat}means saturation pressure. Assuming that the bubbles have the same diameter d

_{p}, the area density of the phase interface is given as:

_{l}and λ

_{v}are defined as:

_{ml}and S

_{mv}, can be written as follows:

_{l}and λ

_{v}should have appropriate values. If the value is too large, the calculation diverges easily and if the value is too small, the interface temperature deviates too much from the saturation temperature. Actually, the time relaxation factor is usually set as 0.1–100, referring to the studies by Lee [34] and Yang [17]. In this paper, λ

_{l}and λ

_{v}was set as 100 s

^{−1}in order to maintain numerically the interface temperature within ±1 K compared with saturation temperature.

#### 2.2.3. Solution Methods

^{−4}, the simulation is considered as convergence and the implicit body force is opened to ensure the convergence of numerical simulation.

#### 2.3. Validation of the Simulated Method

^{2}. According to the work of Lin, the boundary conditions of inlet and outlet were velocity boundary (0.42 m/s) and pressure boundary, and the number of cells is 160,979. Based on simulation and experimental data, the variation of heat transfer coefficient along the channel is shown in Figure 2, and the definition of the heat transfer coefficient is written as follows:

#### 2.4. Results and Discussion

## 3. Analysis of Flow Instability

#### 3.1. Mathematical Model

- It is simplified to one-dimension, and only the variation of the parameters in the axial direction is considered;
- The entire tube is in thermal equilibrium;
- The fluid at the inlet of the pipeline is always in a state of supercooling;
- Subcooled boiling is not taken into account in the subcooling zone of the pipeline. The simplified physical model is shown in Figure 6.

_{e}, k, and Δp

_{L}are thermal power per unit volume, gravitational acceleration, Darcy friction coefficient, hydraulic diameter, the throttling coefficient at the inlet or outlet, and local resistance pressure drop, respectively.

^{n}− p is the pressure difference between the next and the current time level. In addition, A

_{j}(j = 1,2,3) and B are constants.

^{n}, h

^{n}, ρ

^{n}, can be determined by substituting δp back into Equations (24) and (28)–(30), respectively. The numerical computations noted are implemented by FORTRAN codes.

#### 3.2. Solution Method and Model Validation

#### 3.3. Result and Discussion

_{pch}and N

_{sub}, is generally used to characterize the stability boundary. N

_{pch}means the phase change number and N

_{sub}represents the subcooling number. Their definitions follow:

_{fg}refer to heating power, mass flow rate, and latent heat of vaporization. Additionally, Δh

_{in}= h

_{l}− h

_{in}is undercooling enthalpy of the inlet. The physical properties and geometric parameters as shown in Table 3 are used for calculation, and the flowing medium is water.

## 4. Conclusions

- (1)
- The relative positions of discrete heat sources affect the heat transfer effect of two-phase flow, and the heat transfer performance can be improved by reasonably designing the relative position of heat sources.
- (2)
- The closer the distribution of discrete heat sources, the worse the heat transfer effect; for the working condition designed in this paper, the heat transfer effect is best when the distance between the centers of the two discrete heat sources on the same branch channel are within a range of 7–9 cm.
- (3)
- Compared with the continuous heat source, the discrete heat source with uniform distribution can enhance the flow stability under low and high inlet undercooling.
- (4)
- When the high-power discrete heat source is closer to the flow outlet and the low-power discrete heat source is closer to the flow inlet, the flow stability is improved.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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Parameter | Value |
---|---|

Volume of aluminum plate (c × a × b)/cm^{3} | 14 × 7 × 1 |

Volume of discrete heat sources (c2 × a2 × b2)/cm^{3} | 2 × 2 × 0.3 |

Volume of single continuous heat source/cm^{3} | 14 × 7 × 0.3 |

Cross section area of inlet (a1 × b1)/cm^{2} | 1 × 0.5 |

Cross section area of outlet (a1 × b1)/cm^{2} | 1 × 0.5 |

Cross section area of branch (a3 × a3)/cm^{2} | 0.5 × 0.5 |

Axis distance of parallel channel (d1)/cm | 4 |

Properties | Liquid | Vapor |
---|---|---|

Density ρ/(kg/m^{3}) | 1182.2 | 39.025 |

Viscosity μ/(kg/(m s)) | 0.00018011 | 0.000011965 |

Coefficient of thermal conductivity k/(W/(m K)) | 0.078424 | 0.014497 |

Specific heat capacity C_{P}/(J/(kg K)) | 1452.7 | 39.025 |

Latent heat h/(kJ/kg) | 171.81 | |

Surface tension $\sigma $/(N/m) | 0.0072429 | |

Saturation temperature T_{sat}/K | 304.48 |

Case | Grid Number 1 | Average Temperature of 4 Discrete Heat Sources T_{h}/K | $\left|({\mathit{T}}_{\mathit{h}}^{\mathit{j}}-{\mathit{T}}_{\mathit{h}}^{\mathit{j}})/{\mathit{T}}_{\mathit{h}}^{\mathit{j}}\right|$ |
---|---|---|---|

Mesh1 | 361061 | 337.0 | 2.37 × 10^{−3} |

Mesh2 | 392049 | 337.8 | 2.07 × 10^{−3} |

Mesh3 | 432322 | 338.5 | 5.91 × 10^{−4} |

Mesh4 | 545087 | 338.7 | - |

Parameter | Value |
---|---|

Heated length/(mm) | 140 |

Cross section of channel/(mm^{2}) | 5 × 10 |

System pressure/(MPa) | 14 |

Total mass flow/(kg/s) | 0.015 |

Inlet subcooling/(K) | 10–40 |

Channel inclination/(°) | 90 |

Discrete Heat Source | Proportion of Total Heating Power | ||
---|---|---|---|

Mode 1 | Mode 2 | Mode 3 | |

Source 1 | 0.5 | 0.1 | 0.1 |

Source 2 | 0.5 | 0.1 | 0.9 |

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**MDPI and ACS Style**

Hu, C.; Wang, R.; Yang, P.; Ling, W.; Zeng, M.; Qian, J.; Wang, Q.
Numerical Investigation on Two-Phase Flow Heat Transfer Performance and Instability with Discrete Heat Sources in Parallel Channels. *Energies* **2021**, *14*, 4408.
https://doi.org/10.3390/en14154408

**AMA Style**

Hu C, Wang R, Yang P, Ling W, Zeng M, Qian J, Wang Q.
Numerical Investigation on Two-Phase Flow Heat Transfer Performance and Instability with Discrete Heat Sources in Parallel Channels. *Energies*. 2021; 14(15):4408.
https://doi.org/10.3390/en14154408

**Chicago/Turabian Style**

Hu, Changming, Rui Wang, Ping Yang, Weihao Ling, Min Zeng, Jiyu Qian, and Qiuwang Wang.
2021. "Numerical Investigation on Two-Phase Flow Heat Transfer Performance and Instability with Discrete Heat Sources in Parallel Channels" *Energies* 14, no. 15: 4408.
https://doi.org/10.3390/en14154408