Numerical simulation of condensation in microchannels between spherical particles

Abstract

In this article, a numerical investigation of vapor condensation in a two-dimensional ordered microchannel was conducted with computational fluid dynamics software Fluent. A simplified physical model was built up to simulate a rectangular channel filled with particles. A constant wall heat flux was added to the side walls of the rectangular channel. Volume of fluid model was adopted to pursue the interface of the gas and liquid. The results showed that a better heat transfer performance could be obtained with the porous structure. The local heat transfer coefficient obtained from simulation was in good accordance with the former experimental data, which increased with the increase in fluid velocity and decreased along the flow direction. Parametric analyses were conducted concerning the effects of initial vapor velocity u₀, initial temperature T₀, and wall heat flux q_w on local heat transfer coefficient. The velocity u₀ played a significant role during the process of condensation. Temperature distributions along the porous channel and side walls were also analyzed. The results showed that the temperature decreased along the flow direction and increased with the increase in fluid velocity.

Keywords

Phase change heat transfer condensation microchannel porous channel computational fluid dynamics method

Introduction

The porous structure can intensify the heat transfer process effectively. With this property, it has been widely used in many fields such as chemical engineering, oil recovery, and nuclear energy. But the most common applications are packed beds in cooling systems, heat exchangers, and regenerators. Numerous studies have been carried out to understand the mechanism of the heat and mass transfer in it.

Although theoretical analyses and experimental investigations were performed extensively on heat transfer process in the porous media, the problem concerning the phase change is still a difficult area that people rarely involved with. Several studies have been carried out by researchers such as Chang and colleagues^1,2 and Link and Weigand³ in this field, but it is still far from enough. In Chang and colleagues’^1,2 study, film wise condensation on the surface of horizontal disk embedded in the porous medium was investigated, and the results revealed that the average Nusselt number was enhanced by 75% approximately by the capillary force. After that, Link and Weigand³ investigated the quasi-steady solidification within two parallel plates filled with the porous media and found that Darcy number and Rayleigh number affected the fluid velocity field in the porous media; the thickness of the frozen crust was also obtained by solving the equation of interface energy. Agwu Nnanna et al.⁴ also performed a series of experiments to study the phase change phenomenon in the porous media, and the result showed that the effective latent heat and the heat transfer coefficient played an important role in the process of the phase change. The same conclusion was also achieved by Harris et al.⁵ with numerical method. Akbari et al.⁶ studied the heat and mass transfer during the freezing process in the capillary porous media experimentally and numerically; Luikov’s equations were used in their research and the phase transformation number was also taken into account.

Except the condensation and solidification, evaporation also gained much attention. Nuske et al.⁷ studied the evaporation of multicomponent fluids in the porous medium; based on Ahrenholz et al.’s⁸ work, a refined mass exchange term was added to the numerical model. The results showed that the scaling factor f was a key parameter in the process of evaporation. Leu et al.⁹ studied film evaporation in a non-Darcy porous media; the effects of porosity ε, liquid Reynolds number Re_l, and vapor Reynolds number Re_v were investigated; the research showed that a higher Re_l and Re_v lead to a better heat transfer performance in non-Darcian porous media. Hong et al.¹⁰ conducted an experiment to investigate the boiling heat transfer in porous wick; the effects of the particle size as well as particle type and wick thickness on the effective heat transfer coefficient were obtained from their works. All these researches provide us an important theoretical foundation to understand the phase change in the channels between the porous structures.

In the porous media with the phase change, heat transfer between the fluids and solid particles is quite complicated and important. The local thermal equilibrium (LTE) model and local thermal non-equilibrium (LTNE) model are two kinds of methods commonly used to study the heat transfer in the porous media. The LTE method assumes that temperature difference between the solid and fluids is small and can be neglected; thus, one energy equation is shared by both the solid and the fluid. Although this method has been reported to be accurate for some certain circumstance,^11–13 the LTNE model with two energy equations for fluid and solid is more applicable when substantial temperature difference exists between the solid and the fluid. This method was adopted by many researchers such as Mahmoudi and Maerefat,¹⁴ Dehghan et al.,¹⁵ and Buonomo et al.¹⁶ and has been proved to be accurate. Moreover, a more recent research carried out by Mahmoudi¹⁷ shows that the thermal radiation from solid phase in a porous media also played an important role in the temperature field within a porous/fluid system; the ignorance of the thermal radiation will lead to a substantial error in temperature prediction. In Dehghan et al.’s¹⁸ work, the effect of thermal radiation was also taken into account when studying the heat transfer in the microchannel filled with the porous material.

This work analyzes the condensation in micro/minichannels between the particles; all particles are filled in a rectangular channel and used as a pack bed in chemical engineering; a constant wall heat flux is added to the side walls of the rectangular channel. The purpose of this study is to determine the local heat transfer coefficient and temperature distribution at the wall as well as the temperature of the fluids in the mini/microchannel in the porous media. The influence of the inlet temperature, wall heat flux, and inlet velocity on the process of condensation is also analyzed.

Physical models, governing equations, and methods

Physical models

The pore structure in the porous media is always complicated and heterogeneous. Many physical and chemical properties are highly associated with the pore structure such as strength¹⁹ and permeability.²⁰ In the former literature, some pore structure models were built in order to simplify the research of heat and mass transfer in the porous media. One of the most famous pore structure models is the bundle of capillary tube model which was built up by Chatzis and Dullien²¹ in 1975 and assumed that the pores of the porous media consisted of a bundle of capillary tube with different diameters. It is mainly used for calculating the capillary pressure in the porous media packed with spherical particles. Then the pore network model was introduced as a more effective and accurate model to simulate the pore structure. Ryazanov et al.²² used this model to study the residual oil in a water-wet system, and the results showed that the predictions described in their work were in good accordance with the experiment. In recent years, many fractal pore structure models were established on the basis of fractal theory. This kind of method was first applied by Katz and Thompson²³ to study the crystal growth in rock formation, and this model showed a great accuracy in porosity prediction. From the report of Zhang et al.²⁴ and Chen et al.,²⁵ the fractal pore model was also accurate when summarizing the pore distribution in shale and coal. In their works, a two-dimensional ordered physical model was established to describe the distribution of the particles and minichannels. This physical model was also used by Florez et al.²⁶ to investigate the sintered porous media, and their result showed that the effective thermal conductivity obtained from the numerical results agreed well with the experimental data. Differences can be controlled within 6.7%. So in our work, a simplified model is also involved to simulate the complicated pore structure.

On the other hand, it is very difficult to get the full-scale simulation of the whole porous channel due to the huge amount of particles. Thus, only a small part of the channel is involved in this work. In order to simplify the research, several hypotheses are established as follows:

The analysis assumes that the mass transfer is from vapor to liquid; evaporation of the liquid will not take place.

The thickness of the wall is not taken into account; the temperature gradient in the direction of thickness is neglected.

The effective diameter of the pore is much larger than the mean free path of the gas molecule, and the gas is not the rare gas so the continuity hypothesis can be satisfied.

The surface of the particles is smooth, and there is no adsorption between the solid particles and the fluids during the condensation process.

The analysis also assumes that the porous media are homogeneous and isotropic. The particles in the channel are uniformly distributed, the surface of which is regarded as adiabatic. Based on these hypotheses, Figure 1(a) and (b) shows a schematic diagram of the physical models and coordinate system used in this work. As shown in Figure 1(a), the fluid enters the channel from the inlet and leaves from the outlet. A phase change phenomenon occurs in the mini/microchannel between the porous media. The vapor enters the channel with a constant velocity u₀ and a constant temperature T₀. The diameter of the particles d_p is 2 mm. The minimum width of the pores between the particles is 0.2 mm (Figure 1(b)). No wall-slip assumption is also included. The wall of the channel is aluminum and a constant heat flux is added to the side walls of the channel.

Figure 1.

Physical model and coordinate system: (a) Computational domain and boundary conditions and (b) Geometry size of the particles and channels used in the simulation.

Governing equations

The two-phase flow in the study is modeled by volume of fluid (VOF) approach. The VOF model is suitable for simulating multiphase flow by solving a single set of momentum equations and tracking the volume fraction of each fluid throughout the domain. It can also trace the interface of the gas and liquid and resolves the transient motion between them. It was first put forward by Hirt and Nichols.²⁷ Then the energy equations were involved to make the model more accurate in different situations. The volume fraction of the vapor and liquids in a computational cell were represented by α_v and α_l, respectively, which are determined by equation (1)

α_{v} + α_{l} = 1

(1)

Based on equation (1), the governing equations were implemented in a commercial computational fluid dynamics (CFD) code Fluent version 16.1 (Ansys, Inc.). The numerical model for multiphase flow includes continuity, momentum conservation, and energy conservation equations, as given by equations (2)–(12), respectively:

Equation of continuity

\frac{\partial α_{i}}{\partial t} + \nabla (α_{i} V) = \frac{S_{i}}{ρ_{i}}

(2)

The subscript i represents the liquid or vapor phase; S_i is the mass source term due to the phase change which can be depicted by Lee’s²⁸ model

\begin{matrix} S_{v} = λ_{v} α_{l} ρ_{l} | \frac{T_{l} - T_{sat}}{T_{sat}} |, & T_{l} > T_{sat} \\ S_{v} = 0, & T_{l} \leq T_{sat} \end{matrix}}

(3)

\begin{matrix} S_{l} = λ_{l} α_{v} ρ_{v} | \frac{T_{sat} - T_{v}}{T_{sat}} |, & T_{v} < T_{sat} \\ S_{l} = 0, & T_{v} \geq T_{sat} \end{matrix}}

(4)

λ_v and λ_l are the time relaxation parameters for evaporation and condensation with a unit of 1/s, which are used to represent the significance of the evaporation and condensation processes, respectively. These parameters could be given by equations (3) and (4) that are defined in the Hertz–Knudsen equation. T is the fluid temperature

λ_{v} = \frac{6}{d} δ \sqrt{\frac{M}{2 π R T_{sat}}} H \frac{ρ_{v}}{ρ_{l} - ρ_{v}}

(5)

λ_{l} = \frac{6}{d} δ \sqrt{\frac{M}{2 π R T_{sat}}} H \frac{ρ_{l}}{ρ_{l} - ρ_{v}}

(6)

In equations (5) and (6), δ is the accommodating coefficient that shows the portion of the vapor molecules going into the liquid surface and adsorbed by this surface, R is the universal gas constant, and H is the latent heat.

Equation of momentum

\frac{\partial (ρ V)}{\partial t} + \nabla (ρ VV) = \nabla p + \nabla [μ (\nabla V + \nabla V^{T})] + ρ g + F

(7)

where ρ and µ are the mixture density and mixture viscosity, respectively, defined as

\begin{matrix} ρ = α_{l} ρ_{l} + α_{v} ρ_{v} \\ μ = α_{l} μ_{l} + α_{v} μ_{v} \end{matrix}}

(8)

F is the surface tension which can be obtained via continuum surface force (CSF) model as follows

F = σ \frac{2 ρ (\nabla n) \nabla α_{i}}{ρ_{l} + ρ_{v}}

(9)

where σ is the surface tension coefficient and n is the unit normal vector defined by

n = \frac{\nabla α_{l}}{| \nabla α_{l} |}

(10)

Energy equation

\frac{\partial}{\partial t} (ρ E) + \nabla (V (ρ E + p)) = \nabla (k_{eff} \nabla T) + S

(11)

E is the energy, p is the static pressure, and S is the energy source term. The effective thermal conductivity k_eff is calculated as the average of thermal conductivities of the liquid and vapor phases, weighted by their respective volume fractions. The mixture thermal conductivity is defined as

k_{eff} = α_{l} k_{l} + α_{v} k_{v}

(12)

CFD methods and boundary conditions

In this study, toluene is chosen as the working fluid. The toluene vapor enters the channel at a fixed mass flow rate, a constant heat flux is acting on the channel wall when the fluid is passing through the mini/microchannel between the particles, the wall was impermeable, and all the physical properties of the fluid at the inlet were assumed to be constant. Since the superficial velocities of droplet and gas are low, the flow pattern in the channel is treated as laminar flow. Thus, the boundary conditions can be defined as follows:

At the inlet, the saturated vapor enters the channel at a fixed mass flow rate; therefore, the boundary conditions are

y = 0 : v = u_{0}, T = T_{0}, α_{v} = 1, α_{l} = 0

(13)

At the outlet, the flow and heat transfer are assumed to be fully developed so the derivatives of all the scalars along the flow direction are set to 0

y = L : \frac{\partial V}{\partial y} = 0, \frac{\partial α_{v}}{\partial y} = \frac{\partial α_{l}}{\partial y} = 0

(14)

A uniformed wall heat flux is assumed at the side walls, thus

x = 0 and x = w : k \frac{\partial T}{\partial x} = q_{w}

(15)

The SIMPLE algorithm was used to couple the pressure and velocities. The second-order upwind advection model was used in the momentum and energy equations. The flow in the minichannel was treated as laminar flow when the boundary conditions at the inlet were set to velocity inlet and outflow for the outlet. The pressure staggering option (PRESTO!) scheme was chosen for the interpolation, and the geometric reconstruction scheme was used for the interface interpolation.

Figure 2 demonstrates the calculation region and the grids used for the simulations. The computational domain in the simulation is 9.5 mm wide with a length of 15.5 mm. The grid system is established and discretized with the GAMBIT mesh generator using quadrilateral meshes.

Figure 2.

Geometry mesh.

The grid independence is checked before extensive simulations were carried out by increasing the amount of meshes from 2000 to more than 120,000, with different mesh sizes; the average heat transfer coefficient can be obtained from the simulation results; it is defined as

h_{av} = \frac{q_{w}}{T_{w} - T_{b}}

(16)

where q_w is the wall heat flux, T_w is the mean wall temperature, and T_b is the bulk mean temperature. Figure 3 shows the effect of mesh sizes on heat transfer coefficient under the conditions of q_w = 5000 W/m², u₀ = 0.05 m/s, and T₀ = 380 K. As shown in Figure 3, the adopted grid has a number around 11,000 cells and the time step varied from 10⁻⁴ to 10⁻⁶ s to ensure the convergence.

Figure 3.

Effect of mesh size on heat transfer coefficient.

Result and discussion

The condensation process in the channel

Figure 4 shows the variation in the liquid volume fraction during the condensation process. The values of the constant parameters are given as q_w = 3500 W/m², u₀ = 0.05 m/s, and T₀ = 380 K. As can be seen from Figure 4(a)–(d), the droplets first appear at the bottom of the channel when t = 0.3 s; the condensate could be seen on the surface of the particle and driven downward by gravity. This can be explained by the temperature distribution in the rectangular channel along the flow direction. At the beginning of the condensation, the toluene vapor enters the channel with a temperature higher than the saturate temperature, and when it passes through the microchannels between the particles, the temperature of the fluids begins to decrease due to the cooling heat flux from the side walls, so the fluid temperature near the inlet is much higher than the temperatures at the outlet. Thus, the droplets first appear at the bottom of the channel. Figure 5 illustrates the volume fraction of the liquid in the channel from 0.5 to 9.75 s during condensation. The constant parameters were q_w = 2000 W/m², u₀ = 0.05 m/s, and T₀ = 400 K. It shows that as the condensation process continues, more droplets appear on the particle surface and pores between the particles. After t = 5 s, liquid film could be seen in the pores between the particles. Most of the droplets are distributed uniformly around the inlet and side walls; only a small part of the droplets can be found around the center of the porous channel. This may be caused by a stronger heat transfer process at the inlet and side walls. The temperature of the vapor near the walls is much lower than that near the central of the porous channel, so the local heat transfer coefficient is much higher in these zones and more condensate can be seen in these zones.

Figure 4.

Volume fraction of liquid phase in condensation: (a) t = 0.3 s, (b) t = 0.35 s, (c) t = 0.4 s, and (d) t = 0.45 s.

Figure 5.

Volume fraction of liquid phase in the first 5 s: (a) t = 0.5 s, (b) t = 1 s, (c) t = 3 s, (d) t = 3.75 s, (e) t = 5.5 s, and (f) t = 9.75 s.

Influence of porous structure on local heat transfer coefficient

The local heat transfer coefficient was defined as

h_{y} = \frac{q_{w}}{T_{w} - T_{b, y}}

(17)

where h_y is the local heat transfer coefficient, q_w is the heat flux at the wall, T_w is the local temperature of the wall, and T_b,y is the local bulk mean temperature. Figure 6 shows the local heat transfer coefficient of condensation in an empty channel and that in the porous channel. Figure 6(a) and (b) shows the distributions of the local heat transfer coefficient with the y-coordinate. The boundary conditions are q_w = 2000 W/m², T₀ = 400 K, u₀ = 0.07 m/s and q_w = 3500 W/m², T₀ = 420 K, u₀ = 0.05 m/s, respectively. The result shows that the local heat transfer coefficient in the porous media is much higher than that in the empty channel. This enhancement can reach 1.5–2 times compared with that in the empty channel. It is because the velocity fields of the fluids in the porous channel are more complicated than the empty channel, and this is helpful for the heat transfer enhancement. From the curves in Figure 6, we can also find that h_y decreases with the increase in the y-coordinate. That means the local heat transfer coefficient is higher at the inlet than the outlet. This is another reason why there are more droplets around the inlet mentioned above.

Figure 6.

Comparison between the fluid flow in porous media and non-porous media: (a) q_w = 2000 W/m², T₀ = 400 K, u₀ = 0.07 m/s and (b) q_w = 3500 W/m², T₀ = 420 K, u₀ = 0.05 m/s.

Influence of the velocity

In Figure 7, the local heat transfer coefficients at different inlet velocities are listed for comparison. For Figure 7(a) and (b), the inlet temperature of the fluid T₀ is fixed at 400 K and the wall heat fluxes are set to q_w = 2000 W/m² and q_w = 3500 W/m², respectively. In these two cases, the magnitude of the velocities at the entrance varied from u₀ = 0.03 to 0.07 m/s. The results indicate that u₀ has an obvious influence on the local heat transfer coefficient when the fluid was laminar flow; the local heat transfer coefficient increases with the increase in u₀ which is coincident with the experiments and numerical analyses carried out by Jiang and colleagues.^29,30 In Jiang’s experiments, the test channel was filled with both the sintered and non-sintered particles with different diameters; a constant wall heat flux was added to the channel wall. The results all showed that the local heat transfer coefficient would decrease along the axial direction and increase with the increase in superficial velocity. Along with the increase in u₀, the Reynolds number will also increase when other conditions are fixed, and a higher Reynolds number will lead to an enhancement in heat transfer process.

Figure 7.

Comparison between the fluid flow in porous media with different velocities: (a) q_w = 2000 W/m², T₀ = 400 K and (b) q_w = 3500 W/m², T₀ = 400 K.

Influence of the inlet temperature

Figure 8 presents the distribution of the local heat transfer coefficient h_y in the porous media under different inlet temperatures T₀ = 380 K, T₀ = 400 K, and T₀ = 420 K; the wall heat flux and vapor velocity are set to q_w = 3500 W/m² and u₀ = 0.05 m/s, respectively. The results show that the inlet temperature has much fewer effects on the local heat transfer coefficient. It is possible that the vapor receives a constant heat flux from two side walls during the condensation and the heat flux can be divided into two parts. One is used to consume the latent heat of condensation and the other is used to reduce the fluid temperature. Compared to the latent heat, the sensitive heat is quite small, so the vapor temperature in the channel will not increase or decrease dramatically due to the different inlet temperatures. Thus, the vapor inlet temperature does not have much influence on local heat transfer coefficient.

Figure 8.

Comparison between the fluid flow in porous media with different inlet temperatures.

Influence of the wall heat flux

Another factor which may affect the heat transfer process during condensation in the porous media is the wall heat flux. In Figure 9, the distributions of the local heat transfer coefficient under three different wall heat fluxes q_w = 2000 W/m², q_w = 3500 W/m², and q_w = 5000 W/m² are demonstrated. However, the values of the local heat transfer coefficient under the three different wall heat fluxes are very close to each other; there is no obvious difference between the three curves. So compared to velocity u₀, the wall heat flux also has much less influence on the heat transfer coefficient. Because the wall temperature will decrease remarkably with the increase in the wall heat flux while the vapor temperature will not decrease so dramatically due to the lower effective conductivity. The difference between the wall temperature and the vapor temperature will also increase with the increase in the wall heat flux.

Figure 9.

Comparison between the fluid flows in porous media with different heat fluxes.

Temperature distribution

The temperature distribution during condensation has attracted a lot of interests in the past several decades. Many researchers have been working on it experimentally and numerically. Lu and Shen³¹ investigated the heat and mass transfer with the phase change during the process of paper drying; distributions of the wall temperature and fluid temperature were obtained from their experiment. In Agwu Nnanna et al.’s⁴ research, an experiment with the phase change in the porous media under a constant wall heat flux was conducted for which the temperature distributions were also tested by the thermocouples. All these reports proved the importance of the temperature distributions.

Figure 10 presents the wall temperature distribution along the channel in our simulation. As depicted in the figure, the channel wall temperature decreases with the increase in inlet vapor velocity. Along the flow direction, the wall temperature decreases in a wavy manner which is mainly caused by the phase change in the vapor. When the condensation occurred, the latent heat of the vapor is released and the temperature increases slightly. This is in agreement with the experimental data from Li and Leong.³² In Li and Leong’s research, a temperature jump could be observed on the heated surface indicating the transaction of the phase change phenomenon. Another experiment carried out by Chen et al.³³ in 2014 on condensation in microchannel indicates that the wall temperature increased with the increase in the Reynolds number and decreased with the axial coordinate during the process of condensation.

Figure 10.

Wall temperature distribution along the channel.

In Figure 11, the temperature of the fluids is listed when flow time is t = 3 s along the line of x = 1.2 mm. As shown in Figure 11, the fluid temperature decreases along the flow direction. In Barletta and Nield’s³⁴ research of the effect of viscous dissipation on forced and free convection, the temperature distribution of the fluid was obtained by solving the equations with mathematical method, and the results suggested that in the channel filled with the particles and under a constant heat flux acting on the wall, one has a decreasing temperature and a decreasing temperature gradient in forced convection along the flow direction. And the results from our simulations are in accordance with this conclusion.

Figure 11.

Fluid temperature distribution along the channel.

Conclusion

The understanding of heat transfer in the porous media with the phase change is quite difficult and a further study is still required. Following the description of the physical process, this work presents an approximate method to understand the extent of condensation in the micro porous channels, and some conclusions could be described as follows:

For forced convection heat transfer with condensation, the porous structure has an obvious effect on the process of heat transfer. When the fluid is laminar flow, the porous structure can enhance the heat and mass transfer effectively.

The fluid velocity plays a more important role than other factors do during the heat transfer in porous media, when the local heat transfer coefficient increases with the increase in fluid velocity u₀ and decreases along the flow direction.

The condensation process first appeared on the surface of the particles at the end of the channel. As the process continues, more droplets could be seen near the wall and around the surface of the particles.

The wall temperature distribution increases with the fluid velocity and decreases along the flow direction. A temperature jump can be observed on the surface of the wall which is resulted from the phase change phenomenon. The fluid temperature distribution decreases with the increase in the y-coordinate.

Footnotes

Appendix 1

Academic Editor: Oronzio Manca

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: The authors acknowledge the financial support provided by National Natural Science Foundation of China (grant no. 51576095 and no. 51406079) and Natural Science Foundation of Jiangsu Province (grant no. BK20151539).

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