Numerical Study of the Gas-Liquid Two-Phase Flow in a Self-Designed Mixer for a Ga-R113 MHD System

Abstract

Liquid metal MHD (Magneto-Hydro-Dynamic) systems can be employed to produce electricity from a wide range of heat resources. In such a system, a low-boiling organic fluid and a high-temperature liquid metal fluid mix. The former evaporates, and carries the latter to flow through an MHD channel, where the electricity is generated. The mixing process and the gas-liquid flow characteristics will have a significant effect on the power generating efficiency. In the present work, trifluorotrichloroethane (R113) was chosen as the organic fluid, and gallium (Ga) as the liquid metal, respectively. Numerical study was subsequently carried out on the gas-liquid flow and heat transfer in a self-designed spherical mixer. The effects of the main factors, including the inlet velocities and inlet temperatures of Ga and R113, were separately determined, with suggested values or ranges discussed in detail.

1. Introduction

At present, the ever-increasing energy cost, shortage of fossil fuel resources, and environmental pollution have aroused the development of high-efficiency energy conversion technologies [1]. MHD power-generation technology extends Faraday’s law to conductive fluids [2], which can be classified into high-temperature plasma MHD systems and LMMHD (Liquid Metal MHD) systems, based on the sort of working medium.

In high-temperature plasma MHD systems, the working gas has to be ionized at a very high temperature, usually up to over 5000 K [3,4], so as to reach an appropriate electrical conductivity [5]. Although ionization seeds can be used to lower this temperature, it still exceeds 2000~2500 K [6,7]. In addition, there is another crucial technical problem to overcome—the slagging [8].

Compared to high-temperature plasma MHD systems, LMMHD systems have no such problems. There are typically two types of fluids in an LMMHD system: the thermodynamic fluid (low-boiling organic medium) and the power-generating fluid (conductive liquid metal). The thermal efficiency is theoretically close to the Carnot cycle in that the low-boiling organic medium is continuously heated by the high-temperature liquid metal [9]. More significantly, LMMHD systems can be utilized for a wide range of heat resources, including solar, geothermal, fossil fuels, nuclear, and chemical reactions, etc.

The schematic of a two-phase LMMHD system is shown in Figure 1. The liquid metal is first heated up in the heater. In the mixer, the two fluids mix and the low-boiling organic medium evaporates. Subsequently, the gas-liquid two-phase fluids flow through the MHD generator and electricity is created. Finally, the fluids are separated and pumped back to the mixer through the upper and lower loops, respectively.

LMMHD systems are drawing increasing attention on various topics, including flow and heat transfer [10,11,12], magnetic effects [13,14], power-generating characteristics [15], and bubble behaviours [16], etc. For example, Kakarantzas et al. [12] investigated the MHD liquid metal flow and heat transfer in vertical annuli under a horizontal magnetic field by direct numerical simulations. They found that the fluid motion increases with the aspect ratio and annular gap, and the highest spatially averaged heat transfer rates are obtained for aspect ratios equal to one. Bakalis and Hatzikonstantinou [14] studied the MHD flow of a liquid metal in a curved annular channel, so as to examine the effect of curvature and the magnetic field on the velocity distribution. Schwarz and Froehlich [16] numerically simulated the upward motion of a single bubble in liquid metal exposed to an external magnetic field. The results showed that for large bubbles, the rise velocity increases first and then decreases; on the other hand, for small bubbles, the rise velocity decreases when strengthening the magnetic field.

However, there are rather few studies focusing on the mixing process and two-phase flow characteristics inside the mixer which can significantly affect the follow-up power-generation process. In our previous research, with tin and trifluorotrichloroethane (R113) chosen as the liquid metal and low-boiling working medium, respectively, a preliminary two dimensional numerical study was carried out on the two-phase flow characteristics [2]. Further research on the effects of the physical properties of liquid metal indicates that, compared to tin, Gallium (Ga) has a more beneficial influence on the flow and heat transfer process due to its higher heat capacity and conductivity [17]. Therefore, in the present paper, we will advance the research on a self-designed spherical mixer for two-phase LMMHD systems, with Ga and R113 as the working media. Their physical properties are presented in

Table 1. Physical properties of Ga.

Physical Properties	Liquid Gallium
Molar mass (g·mol−1)	69.723
Density (kg·m−3)	5904
Melting point (K)	303
Heat capacity (J·kg−1·K−1)	383.52
Heat conductivity (W·m−1·K−1)	58
Viscosity (kg·m−1·s−1)	1.94 × 10−3

and

Table 2. Physical properties of R113 and R113 gas (R113 g).

Physical Properties	R113	R113 g
Molar mass (g·mol−1)	187.376	187.376
Density (kg·m−3)	1565	7.38
Boiling point (K)	321	—
Latent heat of vaporization (kJ·kg−1)	146.7	—
Heat capacity (J·kg−1·K−1)	912	673
Heat conductivity (W·m−1·K−1)	0.0657	0.0778
Viscosity (kg·m−1·s−1)	4.97 × 10−4	1.08 × 10−5
Standard-state enthalpy (J·kmol−1)	−8 × 108	−6.95 × 108

. A three-dimensional numerical study will be carried out, aiming to determine the effects of the main impacting factors on the gas-liquid two-phase flow characteristics.

2. Modelling

2.1. Model Set-Up

A self-designed mixer modeled by Unigraphics (UG) is shown in Figure 2. High-temperature liquid Ga is initially stored inside this adiabatic spherical mixer. R113 and liquid Ga enter the mixer from the bottom and left inlets, respectively. Once R113 makes contact with liquid Ga, it is heated up and evaporates into gas. The continuous expansion of R113 gas increases the inner pressure, and pushes the two-phase fluids out of the mixer from the right outlet. This mixer features a simple structure, and provides an appropriate route and space for mixing, flow, and heat transfer, especially for such gas-liquid two-phase media. In this way, the flow velocity is accelerated to a much higher value at the outlet, which is of great benefit to the electrical power generating in the follow-up MHD generator.

2.2. Numerical Model

2.2.1. Governing Equations

The multiphase models in FLUENT can be classified into the VOF (Volume of Fluid) model, Mixture model, and Eulerian model. Among them, the Mixture model is relatively suitable for simulating the mixing process of R113 and Ga, in combination with a standard k-ε two-equation turbulent model. In addition, the SIMPLE scheme is applied to resolve the pressure-velocity coupling equation, and the second-order upwind difference scheme is used for discretization, with the convergence precision of 10⁻⁶ to obtain a satisfactory accuracy. The governing equations [18] are as follows.

(1)

Continuity Equation

$$\frac{\partial }{\partial t}\left({\rho }_{\mathrm{m}}\right)+\nabla \cdot \left({\rho }_{\mathrm{m}}{\overrightarrow{v}}_{\mathrm{m}}\right)=0$$

(1)

(2)

Momentum Equation

$$\frac{\partial }{\partial t}\left({\rho }_{\mathrm{m}}{\overrightarrow{v}}_{\mathrm{m}}\right)+\nabla \cdot \left({\rho }_{\mathrm{m}}{\overrightarrow{v}}_{\mathrm{m}}{\overrightarrow{v}}_{\mathrm{m}}\right)=-\nabla p+\nabla \cdot \left[{\mu }_{\mathrm{m}}\left(\nabla {\overrightarrow{v}}_{\mathrm{m}}+\nabla {\overrightarrow{v}}_{\mathrm{m}}^{\mathrm{T}}\right)\right]+{\rho }_{\mathrm{m}}\overrightarrow{g}+\overrightarrow{F}+\nabla \cdot \left({\displaystyle \sum _{k=1}^n{\alpha }_k{\rho }_k{\overrightarrow{v}}_{\mathrm{dr},k}{\overrightarrow{v}}_{\mathrm{dr},k}}\right);$$

(2)

where ${\overrightarrow{v}}_{\mathrm{m}}$ is the mass-averaged velocity $\frac{{\displaystyle \sum _{k=1}^n{{\displaystyle \alpha }}_k{{\displaystyle \rho }}_k{{\displaystyle \overrightarrow{v}}}_k}}{{{\displaystyle \rho }}_{\mathrm{m}}}$, ${\rho }_{\mathrm{m}}$ is the mixture density ${\displaystyle \sum _{k=1}^n{{\displaystyle \alpha }}_k{{\displaystyle \rho }}_k}$, n is the number of phases, $\overrightarrow{F}$ is the body force, α_k is the volume fraction of phase k, μ_m is the viscosity of the mixture ${\displaystyle \sum _{k=1}^n{{\displaystyle \alpha }}_k{{\displaystyle \mu }}_k}$, and ${\overrightarrow{v}}_{\mathrm{dr},k}$ is the drift velocity of phase k.

(3)

Energy Equation

$$\frac{\partial }{\partial t}{\displaystyle \sum _{k=1}^n\left({\alpha }_k{\rho }_k{E}_k\right)}+\nabla \cdot {\displaystyle \sum _{k=1}^n\left[{\alpha }_k{\overrightarrow{v}}_k\left({\rho }_k{E}_k+p\right)\right]}=\nabla \cdot \left(k_{\mathrm{eff}}\nabla T\right)+S\prime ;$$

(3)

E_k is the energy of phase k; S′ is the volumetric heat sources; and k_eff is the effective thermal conductivity ${\displaystyle \sum _{k=1}^n{{\displaystyle \alpha }}_k\left({{\displaystyle k}}_k+{{\displaystyle k}}_t\right)}$, where k_k is the thermal conductivity of phase k and k_t is the turbulent thermal conductivity.

(4)

Volume fraction equation of the second phase

$$\frac{\partial }{\partial t}\left({\alpha }_k{\rho }_k\right)+\nabla \cdot \left({\alpha }_k{\rho }_k{\overrightarrow{\nu }}_k\right)=-\nabla \cdot \left({\alpha }_k{\rho }_k{\overrightarrow{\nu }}_{\mathrm{dr},k}\right).$$

(4)

A simple and accurate model proposed by Lee [19] and widely acknowledged is employed in the present paper to calculate the evaporation process of R113, with the relevant source terms expressed as follows.

R113 source term:

$$S_{\mathrm{M}}=-r{\alpha }_{\mathrm{R}}{\rho }_{\mathrm{R}}\left(\frac{T-T_{\mathrm{b}}}{T_{\mathrm{b}}}\right);$$

(5)

R113 (gas) source term:

$$S_{\mathrm{M}}=r{\alpha }_{\mathrm{R}}{\rho }_{\mathrm{R}}\left(\frac{T-T_{\mathrm{b}}}{T_{\mathrm{b}}}\right);$$

(6)

Energy source term:

$$S_{\mathrm{E}}=-r{\alpha }_{\mathrm{R}}{\rho }_{\mathrm{R}}\left(\frac{T-T_{\mathrm{b}}}{T_{\mathrm{b}}}\right)\mathsf{\Delta }H.$$

(7)

where S_M and S_E are the mass and energy source terms, respectively; α_R and ρ_R are the volume fraction and density of R113; T and T_b are the mixture temperature and boiling point of R113; ΔH is the latent heat; and r is the phase change factor.

2.2.2. Boundary Conditions

The velocity-inlet and pressure-outlet boundary conditions are applied to the computational domain, where the outlet pressure is set as the standard atmosphere. The boundary conditions for the pipe wall are impermeable, non-slip, and adiabatic, with an inner roughness at 5 × 10⁻⁵ m. Gravity acceleration is taken into consideration, whereas the radiant heat exchange between the working media is ignored.

2.2.3. Independence Verification

The 3D computational domain with meshes is depicted in Figure 3. The mesh number has a great influence on the simulation accuracy. Generally, increasing the mesh number will, on the one hand, obtain more accurate simulation results; on the other hand, it will result in a much longer computing time. In order to balance the simulation accuracy and computing time, it is necessary to perform the verification of grid independence.

The mesh number is set as 2.1 × 10⁵, 3.7 × 10⁵, 6.0 × 10⁵, 8.9 × 10⁵, and 12.1 × 10⁵, respectively, and the corresponding outlet velocity (v_lmo) and volume fraction of liquid Ga (VF_o) are plotted in Figure 4. It can be seen that the values fluctuate after the mesh number exceeds 6.0 × 10⁵, which can be consequently recognized as the appropriate mesh number.

The numerical method in this paper is testified by simulating a gas-liquid flow process in the literature [20], as shown in Figure 5. The figure describes the axial pressure distribution at different Reynolds numbers, and the simulated results acquired by the present method are in good agreement with the experiments, with the maximum error of less than 10%.

3. Results and Discussion

3.1. Effects of v_lmi

The volume fraction distribution of liquid Ga (VF) and the flow field across the longitudinal section (Z = 0) under different inlet velocities of liquid Ga (v_lmi) are depicted in Figure 6. As v_lmi increases, the falling position of liquid Ga gradually rises, and the evaporation area (or mixing area) enclosed by the high-temperature Ga enlarges, which is beneficial to R113 evaporation.

The controlling variables method is applied in the present paper. Keeping the inlet temperature of Ga (T_lmi), inlet velocity of R113 (v_Ri), and inlet temperature of R113 (T_Ri) constant at 573 K, 0.6 m·s⁻¹, and 310 K, respectively, the inlet velocity of liquid Ga (v_lmi) varies from 0.6 m·s⁻¹ to 3.0 m·s⁻¹. It can be noticed from Figure 7 that as v_lmi increases, all of the outcomes rise, with the outlet velocity of Ga (v_lmo) increasing from 3.46 m·s⁻¹ to 7.81 m·s⁻¹, outlet volume fraction of Ga (VF_o) from 18.78% to 39.63%, and evaporation rate of R113 (ER) from 35.42% to 64.14%, respectively. It is understandable that more liquid Ga is provided due to the increasing inlet velocity. In addition, the evaporation area increases, as shown in Figure 6. They both facilitate the evaporating process and improve the carrying ability of R113. As a result, v_lmo increases. Since more liquid Ga is supplied and conveyed, VF_o increases accordingly.

Although in general, increasing v_lmi has a positive effect on the outlet flow characteristics, it should be pointed out that there should be a limit on it. Figure 7a shows that the outlet temperature of liquid Ga (T_lmo) is very close to its inlet temperature (573 K) when the velocity exceeds 2.4 m·s⁻¹, suggesting that the isothermal expansion of the two-phase mixture is realized and the optimal thermodynamic efficiency is achieved. A further increase of v_lmi will not obviously improve the evaporation area, as shown in Figure 6d,e. Conversely, it will consume more pump power and reduce the overall efficiency of the system. Taking the above factors into consideration, the suggested range for v_lmi is between 1.8 m·s⁻¹ and 2.4 m·s⁻¹.

3.2. Effects of T_lmi

Keeping the inlet velocity of liquid Ga (v_lmi), inlet velocity of R113 (v_Ri), and inlet temperature of R113 (T_Ri) constant at 2.4 m·s⁻¹, 0.6 m·s⁻¹, and 310 K, respectively, the variations of inlet temperature of liquid Ga (T_lmi) are presented in Figure 8. This figure shows that with T_lmi increasing, the outlet velocity of Ga (v_lmo) rises from 6.48 m·s⁻¹ to 8.59 m·s⁻¹, and the evaporation rate of R113 (ER) rises from 48.73% to 80.20%; whereas the outlet volume fraction of Ga (VF_o) declines from 38.33% to 29.10%. The rising T_lmi is beneficial to R113 evaporation; thus, the carrying-ability of R113 gas is promoted and the outlet velocity of Ga (v_lmo) rises. Nevertheless, the increasing volumetric proportion of R113 gas will adversely reduce VF_o and the electrical conductivity of the fluids. The reason for keeping VF_o above a certain value will be subsequently discussed in further detail.

3.3. Effects of v_Ri

Keeping the inlet velocity of liquid Ga (v_lmi), inlet temperature of liquid Ga (T_lmi), and inlet temperature of R113 (T_Ri) constant at 2.4 m·s⁻¹, 573 K, and 310 K, respectively, the inlet velocity of R113 (v_Ri) ranges from 0.2 m·s⁻¹ to 1.0 m·s⁻¹. A higher v_Ri results in a higher R113 supply. As the thermodynamic fluid, its carrying-ability is enhanced. As a result, a higher outlet velocity of Ga (v_lmo) is obtained (Figure 9a) from 4.44 m·s⁻¹ to 9.43 m·s⁻¹. On the contrary, the outlet volume fraction of Ga (VF_o) declines from 55.61% to 26.81% due to the higher volumetric percentage of R113 gas. Because of the increased average velocity, the heat-transfer time length between the two fluids is reduced; thus, the evaporation rate of R113 (ER) decreases from 77.85% to 51.65% (Figure 9b). Still, it should be pointed out that VF_o should be controlled and this will be discussed in the next section.

3.4. Effects of T_Ri

Keeping the inlet velocity of liquid Ga (v_lmi), inlet temperature of liquid Ga (T_lmi), and inlet velocity of R113 (v_Ri) constant at 2.4 m·s⁻¹, 573 K, and 0.6 m·s⁻¹, respectively, the variations of inlet temperature of R113 (T_Ri) are presented in Figure 10. The temperature difference between the two fluids is reduced with T_Ri, which raises ER, and lowers VF_o slightly from 34.99% to 33.39% accordingly. Besides, since more R113 carrying gas is generated, the outlet velocity of Ga (v_lmo) rises from 7.17 m·s⁻¹ to 7.42 m·s⁻¹. Note that a very low VF_o may result in the R113 gas films covering the pipe walls in the follow-up MHD power-generating channel. In this situation, the annular flow pattern may be formed and leads to a loose contact between Ga and the electrodes. It is advised by Wallis [21] that VF_o be higher than 20% so as to avoid the above problem.

Specifically, Figure 11 provides more detailed distributions for three different volume fractions of Ga along Y = 0 at the outlet cross section. Note that at VF_o = 50% and VF_o = 40%, the volume fraction of liquid Ga (VF) in the vicinity of the pipe wall is much higher than those at other locations, which is beneficial to power generation. However, with VF_o decreasing to 30%, the differences of VF among different locations are reduced, with the VF adjacent to the pipe wall decreasing dramatically from 67% to 37%. This indicates that the proportion of continuous R113 gas increases, which will have a detrimental impact on the power generating process by detaching the liquid metal from the electrodes.

4. Conclusions

Liquid metal MHD systems can be utilized to produce electricity from varieties of heat resources. The mixing process and the gas-liquid flow characteristics will have a significant impact on the power-generating efficiency. In the present work, a three-dimensional numerical study is conducted on the above process in a self-designed spherical mixer, with liquid metal gallium (Ga) selected as the working medium. The impacts of the main parameters are determined, respectively, with the conclusions as follows:

(1)

The evaporation area enlarges with the inlet velocity of liquid Ga (v_lmi), which is beneficial to R113 evaporation.

(2)

Generally, increasing v_lmi plays a positive role in the gas-liquid flow characteristics, because v_lmo, VF_o and ER increase with v_lmi. However, an excessively higher v_lmi will result in an overload on the pump power, and consequently reduce the whole efficiency. The suggested range for v_lmi is 1.8 m·s⁻¹ to 2.4 m·s⁻¹.

(3)

The inlet temperatures of liquid Ga (T_lmi) and R113 (T_Ri) have similar impacts on the gas-liquid mixing and flow characteristics. With T_lmi or T_Ri increasing, v_lmo, T_lmo, and ER rise, while VF_o declines. As the thermodynamic fluid, a higher inlet velocity of R113 (v_Ri) will, on one hand, obtain a higher v_lmo; on the other hand, it will reduce T_lmo, ER, and VF_o.

(4)

It is suggested that VF_o be kept above a certain value. Otherwise an undesirable annular flow pattern may be formed, which has a detrimental impact on the power-generating process by detaching the liquid metal from the electrodes.

(5)

It is advised that future research work be centered on experimental investigations, microscopic bubble motions, economic performance, and commercialized operations in this field.