The Condensation Characteristics of Propane in Binary and Ternary Mixtures on a Vertical Plate

Abstract

Natural gas is one of the most common forms of energy in our daily life, and it is composed of multicomponent hydrocarbon gas mixtures (mainly of methane, ethane and propane). It is of great significant to reveal the condensation mechanism of multicomponent mixtures for the development and utilization of natural gas. A numerical model was adopted to analyze the heat and mass transfer characteristics of propane condensation in binary and ternary gas mixtures on a vertical cold plate. Multicomponent diffusion equations and the volume of fluid method (VOF) are used to describe the in-phase and inter-phase transportation. The conditions of different wall sub-cooled temperatures (temperature difference between the wall and saturated gas mixture) and the inlet molar fraction of methane/ethane are discussed. The numerical results show that ethane gas is more likely to accumulate near the wall compared with the lighter methane gas. The thermal resistance in the gas boundary layer is one hundred times higher than that of the liquid film, revealing the importance of diffusion resistance. The heat transfer coefficients increased about 11% (at ΔT = 10 K) and 7% (at ΔT = 40 K), as the molar fraction of ethane increased from 0 to 40%. Meanwhile, the condensation heat transfer coefficient decreased by 53~56% as the wall sub-cooled temperature increased from 10 K to 40 K.

1. Introduction

With the increasing energy demand and rising attention to the environment, researchers are widely concerned with natural gas as a clean and efficient fossil energy. Liquefied natural gas (LNG) is an important technological process in the natural gas exploitation, which can improve transportation efficiency and reduce the supply cost. As multicomponent hydrocarbon mixtures, the natural gas liquefies at the boiling point. When propane condenses, methane and ethane become non-condensable gases. According to our previous research [1,2], the condensation heat transfer coefficient of propane decreases dramatically with a high methane molar fraction. However, the effect of the third gas on the propane condensation characteristics is not yet clear. Hence, research on the condensation characteristics of propane in multicomponent mixtures is of great significance to the design and develop of LNG equipment.

The mass transfer in binary gas mixtures is driven by its concentration gradient. While in the ternary gas mixture, the diffusivity of one component is a function of all components in the system. Therefore, the traditional two-component diffusion equation cannot describe the diffusion of ternary gas mixtures. Maxwell and Stefan [3] were the first to research the mass transfer in a multicomponent system and proposed diffusion equations. Toor [4] studied mass transfer in ternary mixtures via solving the Maxwell–Stefan equation and verified it with experimental research. He summarized the phenomena that occur in ternary gas mixture diffusion: (1) the diffusion rate of a component is zero when its concentration gradient is not zero (diffusion barrier); (2) the diffusion rate of a component is not zero when its concentration gradient is zero (osmotic diffusion); and (3) a component diffuses against its concentration gradient (reverse diffusion). According to the research of Krishna and Wesselingh [5], Fick’s law is limited in its description of multicomponent diffusion, while the Maxwell–Stefan equation is the most general and simplest method to describe mass transfer in a multicomponent system, which involves the thermodynamic non-idealities and external force fields. Moreover, the Maxwell–Stefan equation has been adopted in considerable research [6,7,8] to describe the mass transfer process of multicomponent gas mixtures.

Due to the existence of multicomponent gases, the analytical model of vapor condensation (e.g., boundary layer model [9,10], diffusion layer model [11], and heat and mass transfer analogy method [12]) must be adjusted with multicomponent diffusion equations. Taitel and Tamir [13] generalized the Nusselt model [14] for the condensation process of multicomponent gas mixtures and combined it with multicomponent diffusion to solve equations via integral methods. The boundary layer condensation model was adopted by Sage [15] to analyze the natural convection condensation of vapor in ternary mixtures. The results showed that species with a large molecular weight could enhance the heat transfer coefficient due to the free convection sweeping effect. Peterson et al. [16] utilized the diffusion layer model with an effective mass diffusion coefficient to address the problem of vapor condensation with multiple non-condensable gases. They found that heavy species preferentially accumulate near the interface, which deforms the diffusion layer. Ganguli et al. [17] promoted a numerical model via modifying the derivation for the calculation of effective diffusivity and condensation conductivity to solve the steam condensation outside a vertical tube in the presence of air and helium. They found that the accumulation of non-condensable gases could reduce the heat transfer coefficient, and the addition of the third species changes the influence of the regularity of the wall sub-cooled temperature on the heat transfer coefficient. Karkoszka and Anglart [18,19] proposed two methods to solve the problem of vapor condensation in the presence of non-condensable gases. One is an analytical model based on the boundary layer approximation method, which can only solve the vapor condensation of binary mixtures in a simple geometric structure. The other is a numerical model via solving conservation equations, which can be used to calculate both binary and ternary mixture conditions. Under the conditions of the ternary mixture (steam/air/helium), increasing the amount of helium in the main flow will increase the air mass fraction near the interface, which shows the interaction among non-condensable gases.

Experimental research about vapor condensation in multicomponent gas mixtures mainly uses the ternary mixture of steam/air/helium. According to the research of Liu et al. [20], the introduction of helium into the steam/air mixture outside a vertical tube could decrease the condensation heat transfer coefficient evidently, and there was an obvious gas stratification phenomenon when the helium mole fraction was in excess of 60%. Similar results were obtained by Su et al. [21], in which the condensation heat transfer coefficient of steam/air/helium was about 20% lower than that of the steam/air mixture. The effect of wall sub-cooled temperatures on the heat transfer coefficient in a ternary mixture was more significant than that of the pure steam condition. Xu et al. [22] investigated steam/air/helium condensation in a horizontal tube. The steam condensation heat transfer coefficient increased with the increase in the helium volume fraction, showing different profiles with the increase in the wall sub-cooled temperature at different flow patterns. Park et al. [23,24] focused on the degradation effect of light gas (helium) mixed with a steam/air mixture on the condensation heat transfer outside a vertical tube. The results showed that, when the helium molar fraction reached 40%, the condensation heat transfer coefficient decreased by approximately 35%. According to the above research, light gas is more likely to increase the diffusion resistance and decrease the condensation heat transfer coefficient in a ternary gas mixture.

Above all, numerical and experimental studies about the condensation of multicomponent gas mixtures are mainly using steam/air and steam/air/helium mixtures. There are few studies about the condensation of hydrocarbon gas mixtures, especially natural gas, owing to the low condensation temperature, complex multicomponent condition and excellent flammability. However, the fluid dynamic and heat and mass transport mechanism of the condensation of multicomponent hydrocarbon gas mixtures are important to the exploration of natural gas condensation process.

The numerical model proposed in this paper was established using an iterative procedure to solve the mass condensation source terms and diffusion equations in ANSYS 17.1. A user-defined function (UDF) was compiled for the condensation rate, which was calculated based on the conservation of mass and energy at the gas–liquid interface instead of an empirical formula. Meanwhile, Maxwell–Stefan equations were applied to address mass diffusion in the gas mixtures. This numerical model can be applied in multi-working conditions with different types of working fluids via changing the parameters.

2. Model and Methods

2.1. Physical Model

The condensation heat and mass transfer model of a ternary gas mixture (methane, ethane and propane) flowing along a vertical cold wall is shown in Figure 1. When the gas mixture flows down the wall, the propane gas condenses and forms a liquid film that adheres to the wall, while methane and ethane are as non-condensable gases and accumulate near the gas–liquid interface. As propane condenses, the concentration of the non-condensable components near the wall gradually increases, which prohibits the diffusion of propane from the mainstream to the cold wall. According to the research of Toor and Duncan [25], different from the binary diffusion, there are phenomena of reverse diffusion, osmotic diffusion and diffusion barrier in the ternary gas mixture diffusion process. As the green lines show in Figure 1, the concentration of methane and ethane shows a peak or valley in the gas boundary layer, rather than a monotone distribution.

2.2. Governing Equations

The two-dimensional condensation model was solved under the assumption that the flow is steady, incompressible with constant properties. The cold wall is smooth and isothermal, and there is no slip and contact thermal resistance. There is no velocity and temperature jump at the gas–liquid interface. The conservation equations of the liquid film and methane, ethane and propane gas mixture are expressed as follows.

Mass conservation equation:

$$\mathrm{Liquid} \mathrm{phase}: {\rho }_l\left(\frac{\partial u_l}{\partial x}+\frac{\partial v_l}{\partial y}\right)=S_l$$

(1)

$$\mathrm{Gas} \mathrm{phase}: {\rho }_m\left(\frac{\partial u_m}{\partial x}+\frac{\partial v_m}{\partial y}\right)=S_m$$

(2)

where ρ is density; the subscript l and m represent the liquid and gas mixture phase, respectively; and S_l = −S_m is the mass source term due to condensation.

Momentum conservation equation:

$$\mathrm{Liquid} \mathrm{phase}: {\rho }_l\left(u_l\frac{\partial u_l}{\partial x}+v_l\frac{\partial u_l}{\partial y}\right)=\left({\rho }_l-{\rho }_{\infty }\right)g+\mu \frac{{\partial }^2{u}_l}{\partial y^2}+{\overrightarrow{F}}_x$$

(3)

$${\rho }_l\left(u_l\frac{\partial v_l}{\partial x}+v_l\frac{\partial v_l}{\partial y}\right)=\left({\rho }_l-{\rho }_{\infty }\right)g+\mu \frac{{\partial }^2{v}_l}{\partial y^2}+{\overrightarrow{F}}_y$$

(4)

$$\mathrm{Gas} \mathrm{phase}: {\rho }_m\left(u_m\frac{\partial u_m}{\partial x}+v_m\frac{\partial u_m}{\partial y}\right)=\left({\rho }_m-{\rho }_{\infty }\right)g+\mu \frac{{\partial }^2{u}_m}{\partial y^2}+{\overrightarrow{F}}_x$$

(5)

$${\rho }_m\left(u_m\frac{\partial v_m}{\partial x}+v_m\frac{\partial v_m}{\partial y}\right)=\left({\rho }_m-{\rho }_{\infty }\right)g+\mu \frac{{\partial }^2{v}_m}{\partial y^2}+{\overrightarrow{F}}_y$$

(6)

where μ is the dynamic viscosity, the subscript ∞ represents the main flow, $\overrightarrow{F}$ is the body force driven by the surface tension at the gas–liquid interface, and the first term on the right of the equations are buoyancy forces due to the density difference between the local and main flows.

According to the research of Karkoszka and Anglart [18], in a ternary mixture, the buoyancy force on the momentum conservation equation can be expressed as:

$$\frac{{\rho }_m-{\rho }_{\infty }}{{\rho }_m}=\alpha \left(w_{C1}-w_{C1,\infty }\right)+\beta \left(w_{C2}-w_{C2,\infty }\right)$$

(7)

According to the ideal gas law, the constants α and β are expressed as follows.

$$\alpha =\frac{M_{C2}{M}_{C3}-M_{C1}{M}_{C2}}{\left(M_{C1}{M}_{C2}-M_{C2}{M}_{C3}\right)w_{C1,\infty }+\left(M_{C1}{M}_{C2}-M_{C1}{M}_{C3}\right)w_{C2,\infty }-M_{C1}{M}_{C2}}$$

(8)

$$\beta =\frac{M_{C1}{M}_{C3}-M_{C1}{M}_{C2}}{\left(M_{C1}{M}_{C2}-M_{C1}{M}_{C3}\right)w_{C1,\infty }+\left(M_{C1}{M}_{C2}-M_{C1}{M}_{C3}\right)w_{C2,\infty }-M_{C1}{M}_{C2}}$$

(9)

where w is the species mass fraction; M is the molar weight; the subscripts C1 and C2 represent the non-condensable methane and ethane, respectively; and C3 is the condensable propane.

Energy conservation equation:

$$\mathrm{Liquid} \mathrm{phase}: {\rho }_l{c}_{pl}\left(u_l\frac{\partial T}{\partial x}+v_l\frac{\partial T}{\partial y}\right)=k_l\frac{{\partial }^2{T}_l}{\partial y^2}+S_h$$

(10)

$$\mathrm{Gas} \mathrm{phase}: {\rho }_m{c}_{pm}\left(u_m\frac{\partial T}{\partial x}+v_m\frac{\partial T}{\partial y}\right)=k_m\frac{{\partial }^2{T}_l}{\partial y^2}+\left(c_{pC3}-c_{pm}\right)j_{C3}\frac{\partial T}{\partial y}-S_h$$

(11)

where C_p is the specific heat, k is the thermal conductivity, and S_h is the energy source term produced by condensation. The heat transfer mechanism in a liquid film is convection, while in the gas phase, there is also heat diffusion caused by mass transport, which is shown in the second term on the right in Equation (8).

2.3. Phase Change and Mass Transfer Model

The VOF method was adopted to solve the two-phase flow. The sum of volume fraction of the liquid and gas phase are equal to the unit in a single cell, shown as Equation (12). The cell is full in the liquid phase when F_m = 0, and the cell is full in the gas phase when F_m = 1. Equation (13) is solved to trace the gas–liquid interface.

$$F_m+F_l=1$$

(12)

$$\nabla \cdot \left(F_m\rho \mathit{v}\right)={\dot{m}}_{cond}$$

(13)

According to Fick’s law, the diffusion velocity of one species is driven by the concentration difference and depends only on its concentration gradient and temperature gradient. Considering the multicomponent diffusion, Fick’s law cannot describe the interaction between ternary components and the different efficiencies of various species, which may be less than zero or tend to infinity. Thus, the Maxwell–Stefan equation was adopted to solve the mass diffusion of the multicomponent condensation process.

$$\frac{1}{RT}\frac{d{\mu }_i}{dz}={\displaystyle \sum _{\begin{array}{l}i=1\\ i\ne j\end{array}}^n\frac{W_j(u_i-u_j)}{{\overline{D}}_{ij}}}$$

(14)

where R is the ideal gas constant, μ_i is the molar chemical potential, u_i is diffusion velocity, and ${\overline{D}}_{ij}$ is the diffusion coefficient. The mass diffusion flux of each component can be expressed as:

$$j_i=-\rho {\displaystyle \sum _{j=1}^{n-1}{D}_{ij}}\nabla w_j,i=1,\dots ,n-1$$

(15)

$${\displaystyle \sum _{i=1}^n{j}_i}=0$$

(16)

Meanwhile, the mass diffusion equation of species C1 and C2 in the gas boundary layer are expressed as:

$${\rho }_m\left(u_m\frac{\partial w_{C1}}{\partial x}+v_m\frac{\partial w_{C1}}{\partial y}\right)=-\frac{\partial j_{C1}}{\partial y}$$

(17)

$${\rho }_m\left(u_m\frac{\partial w_{C2}}{\partial x}+v_m\frac{\partial w_{C2}}{\partial y}\right)=-\frac{\partial j_{C2}}{\partial y}$$

(18)

In the propane vapor condensation process, the non-condensable gases of methane and ethane accumulate near the liquid film and form a gas boundary layer. The propane vapor must diffuse through the gas boundary layer from the main flow to the gas–liquid interface. When the system reaches equilibrium, the propane vapor flows from the main stream to the gas–liquid interface via diffusion, and the convection force should be equal to the propane vapor condensation at the interface. Thus, according to the mass transfer conservation at the gas–liquid interface, the local propane vapor mass flux due to convection and diffusion can be expressed as:

$${\dot{m}}_{C3}=w_{C3}\dot{m}+j_{C3}$$

(19)

where ${\dot{m}}_{C3}$ is the propane vapor flux from the main stream to the liquid film, $\dot{m}$ is the local mass flux, and $w_{C3}$ is the mass fraction of the propane vapor.

It was assumed that a non-condensable gas cannot permeate into a liquid film, and that the mass flux of methane and ethane is zero at the gas–liquid interface. Accordingly, the local mass flux at the gas–liquid interface is as follows.

$${\dot{m}}_i={\dot{m}}_{C3,i}+{\dot{m}}_{C1,i}+{\dot{m}}_{C2,i}={\dot{m}}_{C3,i}={\dot{m}}_{cond}$$

(20)

where ${\dot{m}}_i$, ${\dot{m}}_{C3,i}$, ${\dot{m}}_{C1,i}$ and ${\dot{m}}_{C2,i}$ are, respectively, the total, propane vapor and non-condensable methane and ethane mass flux at the gas–liquid interface; and ${\dot{m}}_{cond}$ is the condensation flux of propane at the interface.

Equation (15) can be expressed in the following forms by substituting it in Equation (16).

$${\dot{m}}_{cond}=w_{C3,i}{\dot{m}}_{cond}+j_{C3,i}$$

(21)

$${\dot{m}}_{\mathit{cond}}=\frac{j_{C3,i}}{1-w_{C3,i}}$$

(22)

The condensation rate was calculated using Equations (13) and (22), which are expressed in an iterative procedure [26] compiled by UDF in this model, shown in Figure 2. The condensation source terms for mass and energy are defined as:

$$S_m={\dot{m}}_{cond}\frac{A_{eff}}{V_{cell}}$$

(23)

$$S_h={\dot{m}}_{cond}\frac{A_{eff}}{V_{cell}}{h}_{fg}$$

(24)

where A_eff is the effective condensation area, h_fg is the latent heat of propane, and V_cell is the computational cell volume.

The VOF model was adopted to trace the liquid–gas interface, and the UDF was used to calculate the heat and mass transfer between the gas and liquid phase. The iterative procedure can be explained as follows: (1) The pointer scans and finds the cells in first layer near the cold wall (n = 1). (2) The condensation term is calculated according to the judgement of the temperature and liquid volume fraction in the cell. (3) The pointer turns to the next layer (n = n + 1) and repeats step (2) until the entire control volume is completed.

2.4. Numerical Setup and Model Validation

The conservation equations of this model were calculated using a pressure-based solver. The second-order upwind scheme was adopted for the momentum, VOF and energy equations, while the pressure staggering option (PRESTO!) scheme was employed for the spatial discretization of pressure. The pressure–velocity coupling was addressed using the pressure-implicit with splitting of operators (PISO) algorithm. The Green–Gauss cell-based method was adopted to handle the evolution of the gradient. Considering the multicomponent species transport and two-phase flow, the convergent criteria of the residuals were set as 10⁻⁶ for the conservation equations to ensure the accuracy of the simulation. At the same time, the area-weighted average heat transfer coefficient and the liquid-phase volume fraction were both monitored for the judgment of convergence.

To verify the validation of this model, both a mesh independence test and a comparation with analytical results were conducted. Considering the thin liquid film, the mesh was refined near the cold wall as shown in the mesh sample of Figure 1. The distribution of the average condensation heat transfer coefficient with varying grid quantities for propane/methane/ethane condensation at the sub-cooled wall temperature of 10 K is shown in Figure 3. As the grid number reaches 70,000, the h_av remains stable at about 84 W. Hence, a seventy-thousand grid was adopted in this paper to discretize the computational domain, assure an accurate numerical result and reduce the computation cost.

Due to the little data about the condensation of hydrocarbon mixtures, a numerical simulation using a steam/air mixture was conducted to compare with the analytical results obtained by Minkowycz and Sparrow [9]. As shown in Figure 4, the numerical and analytical results show a good agreement and the over-prediction is within 10%. Hence, the numerical model is convincing to describe the condensation of multicomponent gas mixtures.

3. Results

The condensation of propane with methane and ethane acting as non-condensable components was studied in this paper. In the numerical model, the ternary mixture flowed into the control volume with a propane molar fraction (W_C3,∞) of 10%, an ethane molar fraction (W_C2,∞) ranging from 0 to 40% and a methane molar fraction (W_C1,∞) ranging from 50% to 90%. The sub-cooled temperature of the vertical wall varied from 10 K to 40 K. As the ternary mixture flowed downstream, the propane condensed and formed a liquid film that adhered to the cold wall, whose temperature was below the boiling point of propane. While the propane condensed, the methane and ethane acted as non-condensable gases, which accumulated near the liquid film and formed a gas boundary layer. Accordingly, the condensation rate of propane was mainly dependent on the heat and mass transfer in the gas boundary layer, and thus heat transfer in the liquid film occurred.

3.1. Characteristics of the Filmwise Condensation

The distribution of the liquid volume fraction, molar fraction, temperature and velocity on the outlet section are shown in Figure 4 under the conditions of W_C3,∞ = 10% and ΔT = 10 K. As shown in Figure 5a, the liquid volume fraction decreases sharply to zero in the x-direction near the wall, showing that the liquid film that adhered to the wall is extremely thin. Compared with a binary mixture (W_C2,∞ = 0), the liquid volume fraction of the ternary mixture decreases more sharply and the liquid film is thinner. The distribution of the propane molar fraction near the wall is shown in Figure 5b. The propane concentration declines significantly near the wall due to condensation and increases with the increase in inlet ethane molar fraction and decrease in the inlet methane molar fraction. The gradient of the temperature (shown in Figure 5c) and propane concentration is mainly within the range of 0.02 m near the wall. The temperature increases linearly near the wall and then smoothly transits to the main flow. Similarly, the temperature increases with the increase in inlet ethane molar fraction and decrease in the inlet methane molar fraction. The velocity gradient is also within the range of 0.02 m near the wall in Figure 5d, while there is a peak value before approaching the main flow velocity. This is due to the buoyancy, temperature difference and mass diffusion in the gas boundary layer, which could disturb the velocity. In the case of the ternary mixture, the temperature, velocity, propane concentration and liquid volume fraction increase with the inlet ethane molar fraction. However, the case of binary mixture shows different variations compared with that of the ternary mixture. This is because there are different transport mechanisms between the binary and ternary mixtures.

3.2. Distribution of the Non-Condensable Components

Figure 6 shows the distribution of the methane and ethane molar fraction on the sections perpendicular to the flow direction at W_C1,∞ = 0.6, W_C2,∞ = 0.3 and ΔT = 10 K. The distribution of the molar fraction on the inlet section (y = 0) is uniform, subject to the boundary conditions. Methane and ethane accumulate near the wall (x = 0) along the flow direction with propane condensation. Different from the monotonic distribution of the non-condensable gas in the binary mixture, there are peak and valley values in distribution of the methane and ethane concentration along the x-direction. Moreover, ethane is more likely to accumulate near the wall than methane, relative to the concentration of the main flow. Similarly to the research of Sage [15], the carrying effect of the larger-molecular-weight ethane on methane could result in a nonlinear distribution of the non-condensable gas concentration.

The profile of the methane and ethane gas molar fraction at the outlet section with different inlet molar fractions is shown in Figure 7, at W_C3,∞ = 0.3 and ΔT = 10 K. The distribution of methane concentration is monotonous under the condition of W_C2,∞ = 10~20%. As W_C2,∞ is greater than 30%, there is a peak value of methane concentration near the wall. This illustrates that the ethane gas of a certain concentration can accumulate methane gas near the gas–liquid interface. With the increase in W_C2,∞, the concentration gradient of methane decreases, and the concentration gradient of ethane increases near the wall.

To compare the accumulation ability of methane and ethane gas near the wall, a dimensionless molar fraction was introduced as Equation (25). The profile of the dimensionless molar fraction for the non-condensable methane and ethane gas at the outlet is shown in Figure 8. It can be concluded that the ethane gas is more likely to accumulate near the wall, and the increase in the W_C2,∞ lowers the accumulation of methane near the wall. Furthermore, under the condition of W_C2,∞ = 40%, the concentration of methane nearby the wall appears to be lower than the main flow value.

$${\Omega }_{Ci}=\frac{W_{Ci}}{W_{Ci,\infty }}$$

(25)

The influence of the wall sub-cooled temperature on the distribution of methane and ethane gas molar fraction at the outlet at W_C1,∞ = 0.6 and W_C2,∞ = 0.3 is shown in Figure 9. It illustrates that, with the increase in the sub-cooled temperature, the peak value of methane increases, and valley value of ethane decreases slightly, meanwhile the peak and valley values of the non-condensable methane and ethane gas move towards the cold wall. Therefore, the increase in the wall sub-cooled temperature can increase both the concentration gradient of methane and ethane gas.

3.3. The Distribution of the Liquid Film and Boundary Layers

The distribution of the liquid film along the flow direction is shown in Figure 10, under the conditions of W_C3,∞ = 0.3 and ΔT = 10 K. As it can be seen in the figure, the liquid film of the ternary mixture is thinner and more uniform than that of the binary mixture. The results are consistent with the characteristics of the liquid phase volume fraction, shown in Figure 5a. Consequently, the addition of ethane could reduce the liquid film thickness of the methane and ethane mixture. Under the conditions of the ternary mixture, the variation in W_C2,∞ has little effect on the liquid film thickness, as the W_C3,∞ remains unchanged. The liquid volume fraction contours of the binary and ternary mixtures are shown in Figure 11. The liquid film of the binary mixture fluctuates significantly downstream, while the liquid film of the ternary mixture forms earlier at the entrance and remains stable downstream. To reveal the mechanism of the liquid film fluctuations, the velocity distribution on the sections perpendicular to the flow direction at W_C1,∞ = 10% and ΔT = 10 K is shown in Figure 12. There is a peak velocity value near the cold wall, which contributes to the liquid film fluctuation. Meanwhile, the velocity in the x-direction, indicating the mass transportation from the main flow to the liquid film, results in a wavy liquid film.

Figure 13 shows the distribution of the velocity, temperature and gas boundary layer along the flow direction at W_C3,∞ = 0.3 and ΔT = 10 K. The velocity boundary layer and temperature boundary layer are defined as the curves where the velocity or temperature are equal to 99.99% of the main flow value. Similarly, the gas boundary layer is the curve where the molar fraction of the propane vapor is equal to 99.99% of the main flow value. It can be seen from the figure that the addition of ethane gas could thicken the gas boundary layer, while with the increase in W_C2,∞ and decrease in W_C1,∞, the gas boundary layer becomes thinner gradually. The thickness of the temperature boundary layer also decreases with the increase in W_C2,∞. While with the increase in W_C2,∞, the thickness of velocity boundary layer firstly increases and then decreases. In the binary mixture (W_C2,∞ = 0), the gas boundary layer has almost the same thickness with the temperature boundary layer. While with the addition of ethane gas, the gas boundary layer and temperature boundary layer are separated, and the separation distance increases as the increase in W_C2,∞.

3.4. The Condensation Heat Transfer Characteristics

In the process of propane vapor condensation with methane and ethane acting as non-condensable gases, there is heat transfer resistance in the liquid film and gas boundary layer. The heat transfer in the liquid film depends mainly on conduction, while heat transfer in the gas boundary layer is composed of condensation latent heat and convection. Accordingly, the heat transfer resistance in the liquid film and the gas boundary layer are expressed as follows.

$$R_l=\frac{1}{h_l}=\frac{T_i-T_w}{q}$$

(26)

$$R_m=\frac{1}{h_{conv}+h_{cond}}=\frac{T_{\infty }-T_i}{q}$$

(27)

where the subscripts I and w represent the gas–liquid interface and the cold wall, respectively; h_l is the heat transfer coefficient of the liquid film; h_conv and h_cond are, respectively, the convection and condensation heat transfer coefficients; and q is the total heat flux.

To compare the thermal resistance between the liquid film and gas boundary layer, the ratio of resistance is defined as

$$\frac{R_m}{R_l}=\frac{h_l}{h_{conv}+h_{cond}}=\frac{T_{\infty }-T_i}{T_i-T_w}$$

(28)

Figure 14 shows the thermal resistance distribution of the liquid film and gas boundary layer under different wall sub-cooled temperatures, at W_C3,∞ = 10%. The thermal resistance of both the liquid film and gas boundary layer increase with the increase in the wall sub-cooled temperature. As shown in Figure 14a, the liquid film thermal resistance of the binary mixture is 1.3~1.7 times that of the ternary mixture, showing that the addition of ethane reduces the thermal resistance of the liquid film. Considering the liquid film distribution shown in Figure 10, the reason for this is that the addition of ethane reduces the liquid film thickness. Additionally, for ternary mixtures, the liquid film thermal resistance decreases with the increase in W_C2,∞. From Figure 14b, the gas boundary layer thermal resistance of the binary mixture is higher than that of the ternary mixture and it decrease with the increase in W_C2,∞. Therefore, the increase in W_C2,∞ can both reduce the liquid film resistance and gas boundary layer resistance.

The thermal resistance ratio of the gas boundary layer and liquid film under different wall sub-cooled temperatures is shown in Figure 15. The thermal ratio of the ternary mixture is considerably higher than that of the binary mixture. In the ternary mixture, the thermal ratio increases with the increase in W_C2,∞. This shows that the addition of ethane improves the proportion of the gas boundary layer thermal resistance compared with the liquid film thermal resistance. Under the calculated conditions, the thermal resistance of the gas boundary layer is one-hundred times higher than that of liquid film. Thus, in the condensation process of propane vapor with a high non-condensable gas concentration, the heat transfer resistance is mainly in the gas boundary layer.

The average condensation heat transfer coefficient (defined as Equation (29)) with different wall sub-cooled temperatures and inlet molar fractions of methane and ethane is shown in Figure 16. The condensation heat transfer coefficient decreases by 53~56% with the increase in the wall sub-cooled temperature from 10 K to 40 K. This is because the thermal resistance of both the liquid film and gas boundary layer increase with the wall sub-cooled temperature. As the W_C3,∞ remains constant, the heat transfer coefficient increases by about 11% (at ΔT = 10 K) and 7% (at ΔT = 40 K), with the W_C2,∞ increase from 0 to 40% and W_C1,∞ decrease from 90% to 50%. The addition of ethane could increase the heat transfer coefficient via reducing the thermal resistance of both the liquid film and gas boundary layer. The correlation formula for predicting the condensation heat transfer coefficient of propane/methane/ethane gas mixtures (at W_C3,∞ = 0.1 and ΔT = 10~40 K) was obtained by the least squares method. As shown in Equation (30), the R-squared value of the fitting formula was 0.99205.

$$h=\frac{q_w}{T_{sat}-T_w}$$

(29)

$$h_{av}=\frac{301.97584}{W^{0.14204}\Delta T^{0.60487}}$$

(30)

4. Conclusions

A numerical model was adopted in this paper to study the heat and mass transfer characteristics of the condensation of binary and ternary gas mixtures (propane/ethane/methane) along a vertical cold wall. The distribution of the liquid film, gas boundary layer, velocity and temperature boundary layer were analyzed, meanwhile the effect of non-condensable gases (methane and ethane) and the wall sub-cooled temperature on the condensation heat transfer coefficient were discussed in detail.

(1)

Different from the monotonic distribution of the non-condensable gas concentration in a binary mixture, there were peak and valley values, respectively, in the profile of the methane and ethane concentration along the x-direction in the ternary mixture. The increase in the wall subcooled temperature promotes the accumulation of methane and ethane gas near the gas–liquid interface.

(2)

The addition of ethane to the binary mixture (methane/propane) separated the temperature boundary layer and gas boundary layer. The separation distance becomes larger with the increase in inlet molar fraction of ethane. The ethane gas is more likely to accumulate near the wall compared with the lighter methane. Meanwhile, the increase in ethane concentration lowers the accumulation of methane near the wall.

(3)

In the condensation process of the propane vapor with a high non-condensable gas molar fraction of 90%, the thermal resistance of the gas boundary layer is one hundred times higher than that of the liquid film. The effect of the wall sub-cooled temperature is more significant on the gas boundary layer than that of the liquid film.

(4)

The addition of ethane to the propane and methane mixture increases the heat transfer coefficient by about 11% (at ΔT = 10 K) and 7% (at ΔT = 40 K), as the molar fraction of ethane increases from 0 to 40%. Under the calculated conditions of W_C2,∞ = 10%, the condensation heat transfer coefficient decreases by 53~56% as the wall sub-cooled temperature increases from 10 K to 40 K.