Pore-scale Simulation of Two-phase Flow in Foamed Porous Materials

This paper establishes a pore cell model of foamed silicon carbide material based on a tetradecahedron structure, and uses FLUENT software to numerically study the flow rate and pressure drop changes in the porous structure at different flow rates and the gas-liquid two-phase flow state of the channel. The research results show that the porous skeleton structure has a great influence on the velocity and pressure distribution in the calculation domain; the inlet flow velocity has a certain impact on the gas-liquid mixing zone and gas-liquid interface area in two-phase flow. The application of tetradecahedral porous media increases the thickness of the gas-liquid mixing zone and the phase interface area are increased, with a maximum increase of 11.5%, thus promoting gas-liquid heat and mass transfer.


Introduction
Foamed silicon carbide materials have controllable pore structure, high open porosity, long service life and good product regeneration performance.[1][2].Based on the above properties, it has been widely used in various industrial fields, such as liquid/gas filters, catalytic carriers, membrane separation and pollution control [3][4][5]; but its complex pore structure has caused many inconveniences in exploring its internal structure.With the development of computers, numerical simulations are applied to foam ceramics based on topological structures, it is a relatively simpler and more efficient research method [6][7][8][9][10][11].
In summary, the flow mechanism of gas-liquid two-phase flow in complex porous media needs to be deepened, which is of great significance for the application of foam ceramics.This paper establishes a topological model based on the tetradecahedron and performs three-dimensional numerical simulation.Fluent simulation is used to obtain the velocity field and pressure field distribution cloud diagrams in the calculation domain, and explores the changes in the flow field under different flow rates and the morphological distribution of the gas-liquid two-phase interface.The fluid flow mechanism in the porous medium was further investigated.In this paper, the Kelvin tetradecahedral is used to generate the topology, and the structural parameters are measured from the real foam silicon carbide structure.Figure 2 shows the pore cell structure, with a cell body diameter of , a rib band length of   , a rib diameter of   , and a porosity of ε.A porous structure with a porosity of 0.735, a pore size of 3.626 mm and a pore density of 10 PPI was established for numerical simulation.The relationship between the relevant parameters in geometric modeling refers to the formula proposed by Wu [12], the corresponding formula is as follows:

Geometry
In addition, for porous media, saturation  is defined as a percentage of the pore volume occupied by a certain fluid in the medium: where,   is the pore volume of the porous medium occupied by the fluid,   is the total pore volume of the porous medium.For gas-liquid two-phase flows, there are: where,  , and  , are the percentage of pore volume occupied by liquids and gases in the porous medium.In this paper,  , is considered to be 1 when the liquid film completely covers the surface of the porous skeleton, at which point the porous medium reaches saturation.In order to avoid the influence of outlet reflux, the outlet section is set with length d in the empty fluid domain, the calculated domain of the Kelvin columnar structure is set in the z direction length according to 5d, the y direction is 2d, the size of the air domain and the porous area are the same, and the three-dimensional model is established as shown in figure 3. The liquid flows in from the upper part of the porous structure area, forms a liquid film within the porous structure, and then discharged from the bottom end.

Mathematical Models
2.2.1.Governing equations.The basic control equations followed by the fluid flow process, including the continuity equation and momentum equation, are as follows: where   ,   , and  are the volume fraction, density, and velocity vectors of phase q.The units are / 3 and /.
where is mixture phase density,/ 3 ;  is pressure, ; is dynamic viscosity,  •;  is gravitational acceleration, / 2 ;  is momentum source term, the surface tension of the porous medium is considered and calculated by the CSF model.
The VOF method requires additional phase equations to calculate the gas-liquid interface:

Boundary condition and solution method.
The two-phase flow VOF model was used to trace the gas-liquid two-phase interface by introducing the phase volume fraction variable.The turbulence model selects the standard k-ε turbulence model.The inlet boundary is given an average flow velocity, the outlet is the pressure outlet, and the gauge pressure is 0. For porous media, the wall contact angle is set to 45°, and the empty fluid domain of the outlet section is not included in the data processing.In the calculation process, the convergence residuals are set to 10 −4 for each equation.The PISO algorithm is adopted, the pressure term is PRESTO!, and the rest are in the second-order upwind.
When the pressure and flow rate of the outlet fluid basically reach a certain value, the calculation is stopped.

Meshing And Irrelevance Verification
In order to ensure the reliability and accuracy of the simulation results, two inlet flow rates with apparent flow rates of 0.01m/s and 0.1m/s were selected, and the pressure gradient of water flowing through the porous medium was used as a reference, and three sets of grids were set for comparison of the geometric model.The number of meshes and the corresponding calculation results and change rates were shown in Table 1.Finally, Mesh2 is selected for calculation.

Model Validation
Figure 4. Verification of porous media model.
In order to verify the accuracy of the established idealized geometric model and numerical simulation solution method, the experimental process of air flowing through porous medium as working fluid was compared.The Fluent software was used to establish a porous media model with the same parameters for the pressure drop experiment of Maxime Lacroix et al. [13], the same test boundary was used as that of the predecessors, and the single-phase flow simulation was used for verification.The results are shown in figure 4. The maximum error of the tetradecahedral model in this experiment is about 9.96% compared with the experimental data in the literature.Therefore, the model in this paper can accurately reproduce the results obtained under the actual test conditions.Figure 5 is the velocity vector diagram of the fluid velocity v=0.1m/s in the middle plane of the xdirection of the porous medium region of the model.It can be seen that the velocity of the fluid is uniform at the inlet, but the velocity changes after entering the porous medium, and the fluid forms a high-velocity region after the skeleton splits the beam.The streamlines in the high-velocity region are densely distributed, while the velocity in the direction of the inlet is the lowest at the connection of the skeleton.At the same time, most of the liquid flows downward along the porous skeleton, but there is a transverse velocity between different cell bodies at some locations, which is due to the hydrophilicity of the porous skeleton wall and the gas-liquid interphase force, which makes the liquid flow along the pores to the area with more liquid distribution.This position accelerates the liquid disturbance, which will have a certain promoting effect on gas-liquid heat and mass transfer.Figure 6 shows the distribution of fluid flow pressure in the middle plane in the x direction of the porous medium region of the model, taking the flow velocity of the fluid inlet v=0.1m/s as an example.Due to the periodic modeling of the tetrahedral structure, the pressure drop is relatively uniform, and the pressure increases first and then decreases at each cell, finally, the pressure returns to 0 at the fluid outlet.Combined with the velocity vector distribution, the skeleton structure has a greater influence on the pressure in the calculation domain, and the fluid causes greater pressure loss and flow velocity at the skeleton structure.At the middle axis, the flow channel is narrow, the flow velocity and pressure drop is large, and the flow rate of the leeward side of the skeleton is small, so the pressure drop is relatively slow.Observing the flow field at different flow velocities, it is found that the larger the flow velocity, the greater the flow resistance in the calculation domain, and the greater the pressure loss.Take the plane in the middle of the x direction of the model, and plot the volume fraction distribution of the liquid phase at different liquid inlet velocities, as shown in Figure 7.It can be seen that there is a liquid film of a certain thickness in the porous structure area, the liquid phase volume fraction   is 1, and there is a layer of gas-liquid mixing region at the junction of liquid and gas, in this area,   gradually transitions from 1 to 0. In the porous structure area, the thickness of this layer is larger inside the porous medium, but the thickness at the inlet is small.With the increase of the flow, the liquid phase occupies the porous structure, and this thickness decreases significantly.This is because the liquid flow rate in the porous medium is smaller than the surface, the air in the air domain is easier to enter, and the gas-liquid mixture is more sufficient, which is conducive to the diffusion of air into the porous medium and the heat and mass exchange with the liquid.Inside the hydrophilic porous structure, the liquid and solid structures interact to form a curved liquid-gas interface, an isotopic surface with a liquid volume fraction   of 0.5 is used as the gas-liquid interface.The existence of porous structure makes the free surface have certain fluctuations, and the surface tension and the force between the gas-liquid phase also have a certain influence on the shape of the gas-liquid two-phase interface.As shown in Figure 8, the left side of the gas-liquid interface is liquid, the right side is gas, with the different inlet velocity of the gas-liquid phase, the shape of the gas-liquid interface is also significantly different, and with the increase of the flow rate, the gas-liquid interface gradually approaches the surface of the porous area.

Liquid Film Distribution
With the help of CFD-Post software, the contact area between the gas-liquid phase can be obtained by integral, and the gas-liquid interface area provided by the liquid distribution state in the porous medium is obtained as shown in Table 2. Besides, the vertical plate area of the model is 65.7m 2 .Considering that when the flow rate is greater than 0.1m/s, the porous medium area reaches saturation and the liquid film completely covers the porous skeleton surface, so its gas-liquid interface area remains unchanged.It can be seen that with the increase of the inlet flow rate, the gas-liquid interface area in the porous medium increases first and then decreases, and there are small fluctuations within a certain range.Although the occupation of the porous medium skeleton will reduce the original area, its area is still larger than the original vertical plate area in the unsaturated state, with a maximum increase of 11.5%.Therefore, the use of porous media will provide a larger contact area for gas-liquid mass transfer, which will be more significant under macroscopic conditions considering the size of the model in this paper.

Conclusion
In this paper, the topology of silicon carbide ceramic tetradecahedral was constructed, and the following conclusions can be drawn through numerical calculation and comparison with experimental results reported in the literature.
(1) The porous skeleton makes the fluid velocity unevenly distributed, the velocity of the fluid is the largest at the throat of the small-sized hole, and lowest at the skeleton in the direction of the back to the inlet, and the size of the flow rate affects the pressure drop of the fluid, the larger the flow rate, the greater the flow resistance, and the greater the pressure loss.
(2) With the increase of flow rate, the porous area reaches saturation, the gas-liquid interface gradually approaches the surface of the porous area, and the thickness of the gas-liquid mixing zone decreases.The gas-liquid interface area in the porous medium increases and then decreases with the inlet flow rate, the maximum increase is 11.5% compared with the original vertical plate area.The application of the porous medium will provide a larger contact area for gas-liquid mass transfer.

Figure 3 .
Figure 3. Overall structure of the model.

Figure 6 .
Figure 6.pressure distribution in porous region at v=0.1m/s.

Figure 7 .
Figure 7. Volume fraction distribution of liquid phase at different liquid inlet velocities: v=0.03m/s, 0.05m/s and 0.1m/s.

Table 2 .
Gas-liquid contact area at different liquid inlet speeds.