A numerical study on bubble dynamics and heat transfer of flow boiling in an inclined microchannel

Flow boiling, as an important heat transfer method, is widely applied in many devices. In this study, a gas-liquid thermal lattice Boltzmann model is adopted to study the flow boiling in an inclined microchannel. Specifically, the bubble growth processes and heat transfer efficiency are studied for different inclined angles. The numerical results show that the liquids occur phase change almost at the same time while the inclined angles are different. As the inclined angle increases, the bubble departure time decreases. Also, the heater temperatures change periodically for the studied cases. The maximum heater temperature is not affected by the inclined angle while the minimum heater temperature decreases when the inclined angle of the microchannel increases. Therefore, the larger inclined angle increases the flow boiling heat transfer efficiency.


Introduction
In recent years, the issue of equipment heat receives more and more attention.It should be noted that flow boiling is widely used in many devices, such as microthermo-fluidics devices, nuclear reactors, and heat pipes.Therefore, many researchers study the flow boiling and try to improve the flow boiling heat transfer.However, the flow boiling mechanism is still unclear because the boiling process is affected by many factors, such as phase change, force segregation, and vapor-liquid interface.Recently, many influential papers have been conducted to study flow boiling [1][2][3].Chinnov et al. [1] found that in flow boiling, the flow patterns of gas and liquid were affected by two-phase flow.Later, Fu et al. [2] found the flow pattern maps of flow boiling in a diverging microchannel.Similarly, Grzybowski et al. [3] observed four different momentary boiling patterns in a mini-channel with a diameter of 1 mm.
With the development of computational technology and numerical method, numerical simulation is becoming an efficient method of flow boiling study.It is worth mentioning that the lattice Boltzmann method (LBM) has received particular attention in flow boiling studies.The flow boiling at low velocity in a horizontal tube was studied by Gong et al. [4] using LBM.In recent years, Sun et al. [5,6] investigated the boiling heat transfer performance and flow pattern in a vertical tube with fluid flowing, and they also studied the influences of fluid velocity and wall contact angle on vertical flow boiling pattern.In addition, many papers on flow boiling using numerical simulation have been reported [7][8][9].The past flow boiling studies focused on the vertical or horizontal channels, however, the two-phase flow of gas and liquid is affected obviously by the inclined angle of the microchannel.
Based on the above review, we adopt the lattice Boltzmann method i.e.LBM [10] to research the flow boiling in inclined microchannel numerically.According to the LBM, the interface of gas-liquid can be observed, and the heater temperature change trend is analyzed in detail.The growth and departure processes of the bubble in the microchannel are simulated.Besides, the heater's temperature is presented and the influences of the inclined angle on flow boiling in the inclined microchannel are compared.

Lattice Boltzmann method
Gong and Cheng conducted a phase-change lattice Boltzmann model [10], which is widely used in boiling studies in recent years.In this section, a brief introduction to their model is presented.

Density field evolution in LBM
In Gong and Cheng's phase change model, the movements of bubble and liquid are shown by density distribution, and the evolution of that is given by: 1 It should be noted that  is the relaxation time, fi (x, t) is the liquid or gas density distribution function.
In LBM, i e is the discrete speed.Here fi eq (x, t) is such as: where  i is the model parameter of LBM, s c is the lattice sound speed.And  fi (x, t) is given by: ( , ) is the increase of speed, and force F is given by: x e e x g .
In Equation (4), is the effective mass in LBM.It is worth noting that in the selected D2Q9 lattice structure, c0 is fixed at 6.0.In this paper, pressure is given by: 2 1 ( 1) In this model, the fluid physical parameters of density and velocity are: , The real fluid velocity U is defined as: 0.5

Temperature field evolution in LBM
The evolution for the temperature is given as follows: As  is the source term, it is responsible for the occurrence of phase change in the liquid.The source term is given as:  The computation domain in this study is shown in Figure 1.As shown, the upper and bottom boundaries of the calculation domain are adiabatic walls, as well as both sides are periodic boundaries.A heater with 8 lattices is set at the bottom wall, and the fluids are driven by inertial force.In this study, the microchannel is inclined with an angle , and Figure 1 shows the case of =0°.It should be noted that the computation domain is set to be 1000 ×300.Initially, the numerical domain is filled with saturated liquid, and the initial temperature is fixed at 0.9Tc.And Tc is the lattice critical temperature.The adiabatic wall wettability is fixed at neutral, and l =5.426, g =0.8113.

Flow boiling processes for different inclined angles
To compare the bubble growth processes in a microchannel under different inclined angles, here three values of inclined angle =0°, 30°, 60°are examined.The bubble growth processes for the three cases are presented in Figures 2-4.
Figure 2 shows the bubble growth process with an inclined angle =0°.As shown, a bubble nucleation site is observed at t=6000, meaning that the liquid phase changed at this time.As the bubble grows, it slides on the bottom wall and departs from the wall finally as shown at t=18000.Meantime, a new bubble is observed on the heater, indicating that the liquid phase changed again.At t=20000, the second bubble slides from the heater, and the third bubble is observed at this time.The bubble grows and departs periodically, the results are similar to the previous study [5].Figure 3 presents the bubble growth process with an inclined angle =30°.As shown, bubble nucleation appears at t=6000, which is the same as the case of =0° at t=6000.Then, it is observed that the bubble departs at t=16000, which is earlier than the case of =0° at t=18000.Comparing Figure 3 and Figure 2, it can be found that the inclined angle has an obvious effect on flow boiling, and this phenomenon is reasonable.This is because the bubble movement is influenced by gravity, which is conductive to bubble departure.At t=20000, four bubbles appear in the inclined microchannel, while three bubbles are observed in the microchannel for the case of =0°, indicating that the period of bubble growth decreases for the inclined angle of =30°.Based on the above results, it can be obtained that the larger inclined angle is conductive to bubble growth in flow boiling.
Figure 4 shows the bubble growth process with inclined angle =60°.As shown, the first bubble nucleation site also appears at t=6000, which is the same as the previous cases and the bubble nucleation time of the three studied cases are the same.At t=16000, the first bubble departs from the wall, on the other hand, the first bubble moves further than the case of =30° because gravity has a more obvious effect on the vertical direction for the case of =60°.The impact of gravity is more significant when the inclined angle increases.At t=20000, two bubbles have departed from the wall, meaning that the larger inclined angle decreases the bubble departure time.Also, the departure bubble is smaller than the previous two cases.This is because gravity has a more obvious effect on vertical direction.On the other hand, the fourth bubble is larger than the case of =30°, indicating that the larger incline angled is more conductive to phase change.
To further investigate the flow boing heat transfer, the time histories of the temperature on heaters for the studied inclined angles are presented in Figure 5.As shown, the three curves all show periodic trends, which corresponds to the periodic bubble growth as shown in the previous.Therefore, the periodic trend of heater temperature is reasonable.Taking the case of =60° as an example, the temperature increases until t=4000, because the liquid is heated during this time.When the liquid temperature reached a certain value, phase change in the liquid occurs.At t=4000, the temperature reached the maximum because phase change occurs.Then the bubble continues to grow and the heater temperature decreases.This is because the bubble growth needs heat steadily.From the figure, it can be seen that while the bubble grows, it slides gradually.At t=14500, the bubble slides from the heater and the heater is covered by the liquid again, indicating that the heater enters the next boiling cycle.Therefore, the bubble grows and departs cyclically in the inclined microchannel.With the increase of inclined angle, it can be seen that the boiling period decreases, this is because the influence of gravity is more obvious.On the other hand, the minimum temperature in a boiling period decreases when the inclined angle is increased, indicating that the larger inclined angle of the microchannel increases the flow boiling heat transfer performance.

Conclusion
In this paper, flow boiling is explored in an inclined microchannel by LBM.Compared with the horizontal microchannel, the inclined microchannel is more conductive to bubble detachment, and the bubble detachment time decreases as the inclined angle increases.It should be noted that the inclined angle has little effect on bubble nucleation time.On the other hand, the heater's minimum temperature decreases as the inclined angle increases, meaning that the inclined angle is more conductive to flow boiling heat transfer.

Figure 1 .
Figure 1.The computation domain of flow boiling in this paper.