Study on the Flow Boiling Heat Transfer Characteristics of the Liquid Film in a Rotating Pipe

Abstract

A three-dimensional numerical model is established to study the flow boiling heat transfer characteristics of the liquid film in a rotating pipe, and the effectiveness of the model is verified by a comparison between the numerical results and the experimental results. The effects of rotational speed, heat flux, and Coriolis force on the characteristics of heat transfer of the rotating liquid film are investigated. The conclusions are drawn as follows: (1) The convection of the rotating liquid film is enhanced while the nucleate boiling in the rotating liquid film is inhibited by the increase in the rotational speed; (2) With the influence of these two factors, the heat transfer coefficient increases with centrifugal acceleration increasing from 20 g to 40 g, then decreases with centrifugal acceleration increasing from 40 g to 120 g; (3) The turbulent intensity of the flow with Coriolis force is obviously increased compared to that without Coriolis force when the centrifugal acceleration ranges from 20 g to 80 g, which shows no increase at higher centrifugal accelerations when the turbulence is sufficiently strong. The Coriolis force also has an impact on the heat transfer coefficient of the liquid film, which should not be ignored when studying the boiling heat transfer of a rotating liquid film.

1. Introduction

Since the concept of the rotating heat pipe (RHP) was proposed by Gray [1] in 1969, much research [2,3,4,5,6] has been conducted to study the heat transfer characteristics of the RHPs. An outstanding heat transfer capacity is obtained by vaporizing the working medium in the heating section and condensing the medium in the condensing section of a RHP, in which the condensate is driven by the centrifugal force due to the rotating heat pipe. Generally, the boiling heat transfer in the liquid film at the heating section has significant impact on the performance of a RHP [7].

There has been considerable interest in developing models for the flow and heat transfer characteristics of the heating section in a RHP in order to optimize its design. Bertossi et al. [8] took the natural convection caused by temperature gradient and centrifugal acceleration into consideration to study the flow pattern and heat transfer characteristics of a heated rotating liquid film. Hassan et al. [9] studied the flow and heat transfer characteristics of the nanofluid film in a rotating pipe (a < 160 g). Only the heat conduction of the evaporative liquid film was taken into consideration in their theoretical calculation. A 2D model was developed by Lian [10,11] for further research into the flow and heat transfer characteristics inside an RHP, which predicted the two-phase flow and heat transfer phenomena of a rotating pipe, and the results agree well with the experiment. Machado [12] adopted a 2D model to simulate the flow pattern of the vapor and the liquid in a homogenous wick, where the phase-changing process occurs at the interface between the liquids. However, previous research did not capture all of the flow and heat transfer characteristics inside a rotating heat pipe with the limitations of the simplified two-dimensional model.

On the other hand, different from the boiling procedure in the liquid in a static container, boiling suppression has been found that nucleate boiling can be suppressed within the liquid film in the evaporator as the centrifugal acceleration increases [13,14,15,16]. However, the detailed boiling characteristics (including the boiling suppression) inside an axially rotating liquid film is still not clear, and better physical understanding of the heat transfer mechanism is expected.

Many models were built for describing a nucleate boiling phenomenon, among which two models have been widely used: (1) RPI-DNB model [17], based on an empirical formula obtained from experiments which is often combined with the Euler model; (2) the Lee model [18], based on the bubble dynamics, which is often combined with VOF model to describe the boiling characteristics. The accuracy of the RPI-DNB model depends on having enough experimental databases, and the Lee model employed an empirical coefficient which is not very well-known [19,20,21,22,23]. For further improving the accuracy, a phase-change model was developed by Sun et al. [24] without empirical coefficient. The accuracy and feasibility of this phase-change model was verified by classical evaporation problems. Therefore, the Sun model is adopted here to study the phase-change phenomenon in a rotating liquid film.

The objective of this paper is to reveal the boiling heat transfer characteristics of a thin liquid film inside an axially rotating pipe. A three-dimensional two-phase model is established using a phase-change model developed by Sun. The volume of fluid (VOF) method is adopted to trace the phase interface with the purpose of studying the bubble distribution. The impact of Coriolis force on the flow and heat transfer characteristics is also discussed. It is expected that the present work may provide a deeper understanding of the flow boiling of the liquid film in a rotating pipe, and help to improve the design of the rotating heat pipes.

2. Simulation Model and Method

2.1. Simulation Model

When a circular pipe rotates about its own axis of rotation (axially rotating pipe), the liquid flow becomes fully annular at centrifugal accelerations greater than about 20 g [25]. In this case, a thin liquid film is formed near the inner wall of the pipe, as shown in Figure 1.

The full discretized Navier–Stokes equations are employed here and the VOF model is adopted to simulate the two-phase flow in the rotating pipe, which causes a considerable computing cost. Therefore, a part of the liquid film is focused on for the three-dimensional simulation of the film boiling. However, before that, the thickness and axial velocity of the liquid film should be determined, which depends on the rotational speed of the liquid film. To obtain the characteristic parameters in a 3-D model, a two-dimensional simulation of the flow in the whole rotating pipe is carried out first. Therefore, the scheme of simulation is combined with a two-dimensional model of a whole rotating liquid film and a three-dimensional model of flow boiling in a part of the liquid film, as shown in Figure 2. Here the tapered section of the pipe provides a centrifugal force acting on the liquid film to maintain an axial flow. An expansion space is introduced at the end of the rotating pipe to simulate the phenomenon of the rotating liquid film as flowing out of the rotating pipe. Since the 3-D flow field is consistent in the circumferential direction, the 3-D model in circumferential direction is selected as a 1/30 periodic segment. A part of the flow field is intercepted along the axial direction and the partial 3-D flow field is set by the liquid film thickness and pressure gradient obtained from 2-D numerical simulation.

According to the previous study [8], the thickness of the liquid film in the pipe with a high rotational speed is less than 5 mm. In the present study, the vapor domain far away from the rotating wall is not considered in the 2D model since it has little influence on the flow and heat transfer in the liquid film, and the geometry and mesh structure of the computational domain is shown in Figure 3.

The distribution of the water film gained in the 2D model is shown in Figure 4. The vapor region is represented by red color while the water region is represented by the blue color, and can provide the thickness of liquid film in 3-D simulation. As expected, the computational domain covers all the liquid film and a large aera of vapor near the liquid.

Through 2D simulation of the two-phase flow, the thickness and velocity distribution of the liquid film at the developed region is obtained. Based on the results, the 3D model is built at a periodic part of the fluid with a 0.75 cm length in the axial direction and a 12° angle in the circumferential direction. The outer radius of the model is 36 mm, and the thickness of the model is 2 mm. The boundary condition used for the three-dimensional model is shown in Figure 5.

The outlet of the vapor domain is set as a pressure outlet, with the outlet temperature of 373.15 K and the pressure of 101,325 Pa. The wall is set as a moving wall, and the constant heat flux is set on the moving wall. The angular velocity is set according to the rotational speed. The contact angle between vapor and water is set as 65°. To avoid the back flow during the simulation, the periodic boundary is adopted to replace the inlet and outlet of the liquid film, and a momentum source derived from the axial pressure gradient is added to drive the axial flow in the liquid film.

The initial temperature of the liquid is 372.15 K while the temperature of vapor is 373.15 K. The initial phase distribution is shown in Figure 6. The k-Epsilon model is adopted to simulate the turbulent flow in the liquid film and the Enhanced wall function is selected. The variable time stepping method is adopted in the unsteady solver and the Courant number is limited to 0.25 to ensure the accuracy.

In this study, based on the requirement of y + (1~5), the grid independence for the 2-D model and 3-D model is verified in Figure 7.

The working conditions, which are used to study the flow boiling heat transfer characteristics of the rotating liquid film in 2-D and 3-D numerical simulation, is shown in

Table 1. Working conditions used for numerical simulation.

Volume Flow Rate (V)	0.18 m3/h
Heat flux (q)	44,000 W/m2
	51,000 W/m2
	66,000 W/m2
Rotational speed (ω)	73.7 rad/s (a = 20 g)
	104.34 rad/s (a = 40 g)
	127.8 rad/s (a = 60 g)
	147.57 rad/s (a = 80 g)
	164.99 rad/s (a = 100 g)
	180.74 rad/s (a = 120 g)
Working medium	water

. Here g is the gravity acceleration, equals to 9.8 m/s².

2.2. Simulation Method [10]

The numerical method can be divided into 2-D numerical calculation and 3-D numerical simulation, based on different simulation models. Since the advantage in capturing free phase interface accurately, the VOF model is used to obtain overall flow field characteristics of rotating pipe section in 2-D simulation, such as free interface and flow field axial pressure gradient.

The continuity equation for one phase volume fraction, momentum equation and energy equation can be seen as Equations (1), (2) and (4).

$$\frac{\partial {\rho }_1{\alpha }_1}{\partial t}+\nabla \cdot \left({\rho }_1{\alpha }_{1}V\right)={\dot{m}}_{vl}-{\dot{m}}_{lv}$$

(1)

where ${\alpha }_1$ including liquid volume fraction ${\alpha }_l$ and gas volume fraction ${\alpha }_v$, ${\alpha }_l+{\alpha }_v=1$; $\rho$ is density; $V$ is velocity vector; ${\dot{m}}_{lv}$ is the mass transfer rate from liquid phase to vapor phase; ${\dot{m}}_{vl}$ is the mass transfer rate from vapor phase to liquid phase.

$$\rho \frac{dV}{dt}=-\nabla p+f_{\eta }+f_c+f_{st}$$

(2)

$$f_c=-2mV\times \omega$$

(3)

where $p$ is mixed phase pressure; $f_{\eta }$ is viscous force; $f_c$ is Coriolis force and the calculation formula is shown in Equation (3); $f_{st}$ is surface tension; $m$ is mixed phase mass; $V$ is velocity vector; $\omega$ is rotation angular velocity vector.

$$\frac{\partial }{\partial t}\left(\rho E\right)+\nabla \cdot \left(V\left(\rho E+p\right)\right)=\nabla \cdot \left(\lambda \nabla T\right)+S_E$$

(4)

where $T$ is mixed phase temperature; $\lambda$ is the thermal conductivity of mixed phase; $S_E$ is energy source term; $E$ is internal energy.

The model of gas-liquid phase-change in rotating pipe proposed by Sun [24] in 2014 is adopted in the 3-D numerical simulation. In the boiling model, work medium can be divided into saturated and unsaturated phases, which need to be marked respectively, as the interface and mesh of saturated phase and unsaturated phase shown in Figure 8.

The mass transfer rate at the phase interface in evaporation can be expressed as follows [10].

$${\dot{m}}_{lv}=-{\dot{m}}_{vl}=\frac{Q_{C_I,in}}{h_{fg}{V}_{C_I}}+\frac{Q_{C_{Is},in}}{h_{fg}{V}_{C_{Is}}}$$

(5)

The mass transfer rate at the phase interface in condensation can be expressed as follows [10].

$${\dot{m}}_{lv}=-{\dot{m}}_{vl}=\frac{Q_{C_I,out}}{h_{fg}{V}_{C_I}}+\frac{Q_{C_{Is},out}}{h_{fg}{V}_{C_{Is}}}$$

(6)

where $lv$ indicates from liquid phase to vapor phase; $vl$ indicates from vapor phase to liquid phase; $Q_{C_I,in}$ is the energy transfer from nearby grid cell to saturated phase grid; $Q_{C_I,out}$ is the energy transfer from saturated phase grid to nearby grid cell; $h_{fg}$ is boiling latent heat; $V$ is volume.

The energy transfer in the gasification process at the phase interface can be expressed as follows.

$$S_E=\frac{-Q_{C_I,in}}{V_{C_I}}$$

(7)

The energy transfer during liquefaction at the phase interface can be expressed as follows.

$$S_E=\frac{Q_{C_I,out}}{V_{C_I}}$$

(8)

The boiling model above is suitable for the general evaporation and condensation system and the reliability has been verified by classical experiments. The realization method of the mass-energy coupling transfer model can be seen in Figure 9. The Fluent solver is used for solving the vapor-liquid phase volume fraction continuity equation, momentum conservation equation, and energy conservation equation. In addition, the mass, momentum, and energy source terms solver needed in calculation are provided by UDF. The working medium physical parameters and boundary condition of heat flux in moving wall and pressure outlet is defined by UDF. In the UDF phase interface tracking module, the grid data are obtained from the solver at the beginning of each iteration step, which is used to determine the vapor-liquid phase interface and the heat and mass transfer capacity through the phase interface, and then calculate the source term through UDF. In the UDF momentum source term module, the Coriolis force is converted to be the source term of the momentum equation and transfer it to solver. In the UDF mass source term module, the vapor source or liquid phase source term in boiling or condensation process is calculated, transferred to solver mass equation, and the pressure correction equation is solved. In the UDF energy source term module, the energy exchange in the evaporation or condensation process is calculated by UDF and transferred the solved energy equation.

Compared with the previous model [19,20,21], numerical method and boiling model mentioned above are not necessary to determine the empirical coefficient in the heat and mass transfer process, which is difficult to obtain, and match the heat and mass transfer in the actual evaporation and condensation process. For detailed model introduction, please refer to Lian [10].

The heat transfer coefficient of the liquid film in simulation or experiment can be obtained from Newton Cooling Theory refer to the paper from Zhao [26], and the equation can be seen below:

$$h=\frac{q}{T_w-T_s}$$

(9)

where $q$ is the heat flux; $T_w$ is the temperature of pipe wall, which can be obtained from the temperature of moving wall in numerical simulation and thermocouple in experiment measurement; $T_s$ is the saturation temperature under local atmospheric pressure.

3. Model Validation

A set of experiments is conducted on a visual experimental platform [11], which consists of (1) a testing pipe; (2) a mechanical support system and a driving motor; (3) a working medium circulation system; (4) a data acquisition system. A set of sharp shot of the liquid and bubbles is obtained by a high-speed camera (Photron Fastcam Mini UX100, 200,000 fps, 1024 × 1280) through the transparent testing pipe made of quartz. The distribution of bubbles obtained by simulations are compared with that obtained by experiments, as shown in Figure 10, where the pictures are taken from the bottom of the liquid layer.

The boiling inhibition is observed both in simulations and experiments, in which the bubble concentration decreases with the centrifugal acceleration. The tendency of the heat transfer coefficient of the liquid film under different working conditions obtained in the experiments are taken into comparison with the theoretical predictions as shown in Figure 11.

Since the heat transfer coefficient is influenced by the convective and the boiling heat transfer, it shows an increase with the centrifugal acceleration under low rotational speeds due to the enhanced convection, and then a decrease under higher rotational speeds due to the boiling suppression, which is consistent with the phenomenon expressed in Ponnappan’s paper [2]. It is found that the predictions agree well with the experimental results with the errors ranging from 1.2% to 32%. This model can be used to predict the flow boiling heat transfer of a liquid film in a rotating pipe within limits.

4. Results and Discussions

In this paper, the boiling heat transfer characteristics of the rotating liquid film under different heat fluxes and rotational speeds are obtained, and the influence of the Coriolis force on the flow and heat transfer process of the liquid film are also discussed.

4.1. Effect of Heat Fluxes and Rotational Speeds on the Heat Transfer Characteristics

The heat transfer ability of the liquid film is determined by the convective and the boiling heat transfer processes, both of which are impacted by the rotational speed, since it affects the thickness of the liquid film, the flow velocity, and the boiling characteristic, etc. To analyze the characteristics of the convective heat transfer of the liquid film, a simulation without boiling is carried out with a heat flux of 44,000 W/m². The variations of the thickness and the axial velocity distribution of the liquid film are examined when the centrifugal acceleration increases from 20 g to 120 g, which are shown in Figure 12.

The decrease in the thickness of the liquid film, accomplished with a significant increase in the axial velocity, is observed with the centrifugal acceleration increasing from 20 g to 120 g. Moreover the axial flow, driven by the pressure gradient, the circumferential velocity of different layers, driven by the viscous force in the rotating liquid film, are also analyzed. Compared with the absolute circumferential velocity, it is more appropriate to take the relative circumferential velocity (the difference of the absolute circumferential velocity between the moving wall and the liquid film) into analysis for the characteristics of the rotating motion. The distribution of the relative circumferential velocities of different flow layers is shown in Figure 13.

It can be inferred from Figure 12 and Figure 13 that the axial velocity and the relative circumferential velocity inside the liquid film have a tendency to increase with the rise of the rotational speed. The convective heat transfer of the liquid film can be enhanced with the increase of the flow velocity of the liquid film, as shown in Figure 14.

It is obvious that the convective heat transfer coefficient has a tendency to rise with the increase of the centrifugal acceleration due to larger axial velocity and relative tangential velocity. On the other hand, the boiling of the liquid film is suppressed at higher rotational speeds, even if imposed on larger heating power, as shown in Figure 15.

It can be concluded that both the size and the number of the bubbles have a tendency to decline with the increase of the rotational speed. Since the material parameter, the roughness of the wall, and heat flux are kept as constant during the simulation under certain conditions, the boiling inhibition may be resulted from the evolution of the flow pattern with the increase of rotational speed.

To obtain a further understanding of the influence of rotational speed on the liquid flow pattern, the distribution of the streamline in the liquid film at a certain circumferential section under different working conditions are examined and shown in Figure 16. The outward flow is positive, the inward flow is negative, and the thickness of the liquid film decreases with the increase in centrifugal acceleration.

It can be seen that with the increase in centrifugal acceleration, the fluctuation of radial velocity could be stronger in the liquid film, and the direction of the streamline is changed more frequently. To be more precise, the radial velocity of the intermediate flow layer (half height) of the liquid film along the axial direction is shown in Figure 17.

This means the energy transfer between different layers is enhanced as the rotational speed increases, which may lead the fluid near the heating wall to being driven away before absorbing enough heat to induce nucleate boiling. As is known, the nucleate boiling has an outstanding heat transfer capacity for the huge latent heat during the process of the phase-changing, affecting the heat transfer capacity of the rotating liquid film in a large degree. Under the influence of the boiling inhibition and the characteristics of convective heat transfer, the heat transfer coefficient has a tendency to behave as shown in Figure 11.

With the centrifugal acceleration increasing from 20 g to 40 g, the boiling is not obviously weakened while the flow velocity is increased, enhancing the heat transfer between the liquid film and the wall. As the centrifugal acceleration ranges from 60 g to 120 g, the boiling inhibition overcomes the enhancement of convection, resulting in the decrease in the heat transfer coefficient.

4.2. Effect of Coriolis Force on the Heat Transfer Characteristics

Coriolis force is induced by the inertia of the motion of an object, which is a kind of inertia force set for the conversion of reference frame and only deflect the direction of the flow velocity, without changing the velocity value. In most of the previous numerical studies on a rotating liquid film, the influence of Coriolis force cannot be explored based on the two-dimensional theoretical models. In this paper, the influence of Coriolis force is considered through the source term of the momentum equation.

It is found that the Coriolis force has an obvious effect on the flow of the liquid film. Taking the centrifugal acceleration of 20 g and 120 g as examples, the radial velocity distribution along the axial orientation of the intermediate liquid layer under the influence of Coriolis force is shown in Figure 18 and Figure 19.

It can be seen that the radial velocity distribution is obviously changed when considering the effect of Coriolis force, even if the average of it shows little difference, which is more obvious at a low rotational speed. That is to say, the turbulence of the flow may be enhanced by Coriolis force under relatively low centrifugal accelerations. Figure 20 shows the trend of the turbulence intensity with the increase in the centrifugal acceleration of the liquid film.

It can be seen that the turbulent intensity of the flow with Coriolis force is obviously increased compared to that without Coriolis force when the centrifugal acceleration ranges from 20 g to 80 g, which shows no increase at higher centrifugal accelerations when the turbulence is sufficiently strong. The contrast of the heat transfer coefficient between the liquid film with Coriolis force and the liquid film without Coriolis force is shown in Figure 21. It can be found that the Coriolis force has a significant influence on the heat transfer ability of the rotating liquid film.

5. Conclusions

A numerical model was developed to predict the boiling heat transfer process of the liquid film in a rotating pipe. A phase-change model developed by Sun is adopted in a 3D model to simulate the boiling process in the rotating liquid film. The boiling phenomenon and heat transfer coefficient predicted by the numerical model are in satisfactory agreement with the experimental results.

The main achievements can be concluded as follows:

(1)

Nuclear boiling is developed in the liquid film under low rotational speeds, and gradually suppressed as the rotational speed increases. Since the heat transfer coefficient is influenced by the convective and the boiling heat transfer, it shows an increase with the centrifugal acceleration under low rotational speeds due to the enhanced convection, and then a decrease under higher rotational speeds due to the boiling suppression.

(2)

The turbulent intensity of the flow with Coriolis force is obviously increased compared to that without Coriolis force when the centrifugal acceleration ranges from 20 g to 80 g, and a larger heat transfer coefficient is obtained when considering this force, which should not be ignored when studying the boiling heat transfer of a rotating liquid film.