# Simulation of Mass and Heat Transfer of Droplets Collision in a Flash Evaporation Pattern

Authors: Facheng Qiu; Xianming Zhang; Xinjie Chai; Yingying Dong; Xingjuan Xie; Zuohua Liu; Renlong Liu; Wensheng Li
Journal: International Journal of Chemical Engineering
 (2023)

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## Abstract

The behavior of droplets collision in a flash evaporation ambient widely exists in various fields. In this work, the deformation analysis and thermal analysis models were established under the condition of flash via a computational fluid dynamics (CFD) method. First, the effects of initial temperature and collision velocity on heat and mass transfer during evaporation were considered. Then, the morphology change of the liquid phase, the mass change, and their influencing factors during the droplet evaporation process were analyzed. A very good agreement is observed between the results of this paper and the published literature. The results show that the interaction between the initial collision velocity and the initial temperature affects the heat and mass transfer performance. The initial collision velocity influences the heat and mass transfer process of the evaporating droplet by affecting the deformation characteristics of the droplet. The collision velocity and the liquid temperature have a competitive relationship with the evaporation process. Under a low-initial temperature, the collision velocity played a leading role in the evaporation of the liquid phase and the mass transfer of steam.


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## Body

## 1. Introduction

The evaporation process of moving droplets is not only often accompanied by relative collisions [1] but also affected by many factors, such as droplet atomization [2], morphological changes, and proximity effects [3]. Meanwhile, the application fields of droplet evaporation are also very wide-ranging, such as internal combustion engine fuel injection combustion [4], contact desulfurization of atomized droplets with flue gas [5–7], seawater desalination [8, 9], particle-enhanced droplet evaporation [10], and so on. Generally, there exist obvious differences in the nonequilibrium evaporation process of droplets under static or flowing conditions [11–13]. When the flow state of droplets is in collision, it not only results in the change of droplet morphology but also affects the heat and mass transfer behavior in the process of droplet evaporation. Thus, it is of great significance to investigate the change characteristics of evaporative droplet collision under different evaporation processes.Recently, because of the wide application of droplet flash mass transfer technology, research on droplet deformation evolution and droplet collision behavior has gradually attracted widespread attention [14, 15]. In particular, the We number factor is often employed to determine the scope of deformation law in droplet collisions process. Moreover, it is found that the evolution laws of coalescence, bouncing, coalescence, stretching separation, and reflexive separation occur successively in the droplet collision process [16–18]. In terms of the formation characteristics of satellite droplets, it is found that the evolution process was affected by liquid viscosity, tension [19], temperature [20], relative collision velocity, and collision form [21]. In terms of heat transfer, Xia et al. [22] obtained the change law of energy in the collision from the energy perspective, and then further explained the change law of droplet interaction. So far, research on droplets collision mainly focused on the morphology analysis, mass, and momentum transfer. However, there were few studies on the coupling of heat and mass transfer in the process of droplets collision.As for the evaporation behavior of droplets, the evaporative mass transfer of micron-scale droplets is strongly limited by heat transfer and molecular collisions [23]. Specifically, when the droplet space size changes in the range of the average free path of steam molecules, its evaporation rate dominates the whole change process, resulting in a significant change in the droplet evaporation rate [24]. Research on droplet evaporation can be roughly divided into two categories. One is to take a single droplet as the main research object of heat and mass transfer. The other is to take the group evaporation considering the interaction of adjacent droplets as the research object. Shin et al. [25] conducted a theoretical study on single droplet flash evaporation under vacuum conditions and then concluded an evaporation model with diffusion as the dominant factor. Castanet et al. [26] and Shahsavan Markadeh et al. [27] studied the group influence effect of droplet evaporation and then proposed that the droplet evaporation process was affected by the evaporation of adjacent droplets. In the comparative study of evaporation of single droplet and group droplet, Wong et al. [28] used experimental methods to compare the evaporation rate under the same conditions and found the evaporation rate of nondiluted group droplets was significantly reduced compared with that of single droplets and even stopped evaporation. In addition, the external conditions in the droplet evaporation process are also crucial to the evaporation process. Based on this, Cossali and Tonini [29] studied the effect of ambient gas density on the heat and mass transfer of adjacent droplets and pointed out that the surrounding gas density was related to the droplet evaporation process. Consolini et al. [30] studied the evaporation process of submicron droplets under subcritical and supercritical conditions in the nitrogen atmosphere and successfully predicted the characteristics of droplet changes during the process.Although the evaporation process of microdroplets has been widely recognized at present, the evaporation process of moving droplets and collisions involved in the process and the changes of droplets or the surrounding environment due to collisions, especially harmful pressure flash evaporation conditions. There are few studies on the influence law of heat and mass transfer under droplet evaporation. Therefore, based on this point, this paper adopts the method of computer software simulation to study the motion analysis and thermal analysis of the moving droplets in the collision process under the condition of flash evaporation and studies the deformation and heat and mass transfer of the droplet collision.The main novelties of the work are as follows: first, the coupling of heat and mass transfer during droplet conditions under negative pressure has been studied. Second, the DO model was chosen to express the radiation heat transfer process. Meanwhile, the interface between the droplet surface and the environment was redefined into the thermal conductivity boundary condition with the user-defined function (UDF) method.


## 2. Problem Description and Model Establishment

### 2.1. Model Simplification Analysis

The movement process of droplets in the actual droplet group is not uniform, resulting in various situations of contact and collision between droplets. For example, the collisions predicted by theory can be divided into opposite collisions according to the direction of movement, chasing, and collision. According to the relative position of the collision, it is divided into eccentric collision and concentric collision, as shown in Figure1. Therefore, two identical droplets are selected with the same momentum to facilitate the expression of the simulated experimental process and the result parameters. The center collision is simulated as a representative study.Figure 1
Diagram of (a) center-to-center collision and (b) off-center collision.
(a)(b)To make the actual model meet the calculation requirements of the simulation theory, the following assumptions are made:(1)
The droplet is spherical when it is in the initial state(2)
In the evaporation under superheated state, the influence of the vaporization core generated inside the droplet is not considered(3)
The diffusion behavior of steam is affected by the external environment and concentration field, regardless of the influence of environmental humidity


### 2.2. Geometric Model

The computational domain and grid division are shown in Figure2(a). The entire computational domain is divided into two parts according to material properties. Inside the cube, two simulated droplets are placed at symmetrical positions in the inner space of the computational domain. During the simulation process, the two droplets move relative to each other. Then, a series of processes such as collision and evaporation are completed at the center. The droplets and surrounding grids are densified to accurately capture the temperature changes, steam movement, and concentration changes in and around the droplets. To ensure the accuracy of the droplet evaporation calculation results, the droplet grid size keeps Dd/Dg = 102 (Dd is the initial droplet diameter and Dg is the grid size). In the calculation process, the overall calculation grid numbers are 902037, 1203184, and 1420583, respectively. Accordingly, taking pressure as the research object, the differences of the above three grids are compared and analyzed. Then, the calculated deviations under the three grid numbers are 8.2%, 3.6%, and 3.4%, respectively. Under the premise of ensuring accuracy, convergence, and computational efficiency, the number of grids selected for the overall calculation is 1203184.Figure 2
Geometric model and mesh generation of simulation experiment.
(a)(b)


### 2.3. Parameter Conditions

In this simulation process, the pure water is used as the simulation medium, the initial diameter is 200μm, the ambient temperature is 293 K and remains unchanged, and the simulated temperature conditions of the droplet are 353 K, 393 K, and 473 K, respectively. Meanwhile, radiant heat transfer is considered. The velocity (absolute velocity) conditions of droplet collision are 0.5 m/s, 1.5 m/s, and 3 m/s, respectively. The property parameters and motion parameters of the simulated medium are imported into each defined area through a patch method, and the variables in the simulation are the droplet collision velocity and the initial droplet temperature.


## 2.1. Model Simplification Analysis

The movement process of droplets in the actual droplet group is not uniform, resulting in various situations of contact and collision between droplets. For example, the collisions predicted by theory can be divided into opposite collisions according to the direction of movement, chasing, and collision. According to the relative position of the collision, it is divided into eccentric collision and concentric collision, as shown in Figure1. Therefore, two identical droplets are selected with the same momentum to facilitate the expression of the simulated experimental process and the result parameters. The center collision is simulated as a representative study.Figure 1
Diagram of (a) center-to-center collision and (b) off-center collision.
(a)(b)To make the actual model meet the calculation requirements of the simulation theory, the following assumptions are made:(1)
The droplet is spherical when it is in the initial state(2)
In the evaporation under superheated state, the influence of the vaporization core generated inside the droplet is not considered(3)
The diffusion behavior of steam is affected by the external environment and concentration field, regardless of the influence of environmental humidity


## 2.2. Geometric Model

The computational domain and grid division are shown in Figure2(a). The entire computational domain is divided into two parts according to material properties. Inside the cube, two simulated droplets are placed at symmetrical positions in the inner space of the computational domain. During the simulation process, the two droplets move relative to each other. Then, a series of processes such as collision and evaporation are completed at the center. The droplets and surrounding grids are densified to accurately capture the temperature changes, steam movement, and concentration changes in and around the droplets. To ensure the accuracy of the droplet evaporation calculation results, the droplet grid size keeps Dd/Dg = 102 (Dd is the initial droplet diameter and Dg is the grid size). In the calculation process, the overall calculation grid numbers are 902037, 1203184, and 1420583, respectively. Accordingly, taking pressure as the research object, the differences of the above three grids are compared and analyzed. Then, the calculated deviations under the three grid numbers are 8.2%, 3.6%, and 3.4%, respectively. Under the premise of ensuring accuracy, convergence, and computational efficiency, the number of grids selected for the overall calculation is 1203184.Figure 2
Geometric model and mesh generation of simulation experiment.
(a)(b)


## 2.3. Parameter Conditions

In this simulation process, the pure water is used as the simulation medium, the initial diameter is 200μm, the ambient temperature is 293 K and remains unchanged, and the simulated temperature conditions of the droplet are 353 K, 393 K, and 473 K, respectively. Meanwhile, radiant heat transfer is considered. The velocity (absolute velocity) conditions of droplet collision are 0.5 m/s, 1.5 m/s, and 3 m/s, respectively. The property parameters and motion parameters of the simulated medium are imported into each defined area through a patch method, and the variables in the simulation are the droplet collision velocity and the initial droplet temperature.


## 3. Mathematical Model and Boundary Condition

### 3.1. Description and Selection of Mathematical Models

The moving microdroplets and the generation and propagation of steam in the gas-phase environment belong to the multiphase flow diffusion process. To accurately simulate the droplet collision and other physical changes of coupling process, the Euler–Euler model is used in this work. The Euler model can be used to simulate the separation flow of multiphase flow and the interaction between phases. Since the multiphase momentum effect is involved in the simulation, it is necessary to introduce the volume fraction of each phase and modify the momentum exchange between the phases. The corresponding continuity equations and momentum equations are established for different phases, and the calculation is performed according to the coupling pressure and the exchange coefficient between the two phases. The transient calculation method is adopted, theκ-ε model is selected for the turbulence model, and the SIMPLE algorithm is used for the solution. The spatial discretization of momentum, energy adds a common discrete coordinates adopts the first-order upwind style. The diffusion equation chooses the mass transfer diffusion without chemical reaction, and the radiation heat transfer model chooses the DO model. The corresponding governing equations are as follows.For liquid and gas phases, the continuity equation is expressed as follows:(1)∂∂t∂iρi+∇∂iρivi⟶=mij•−mji•+Sqm.In the abovementioned formula,∂i, ρi, v⇀i, and m˙ij are the mass fraction of the gas phase or liquid phase, the density, the velocity vector, the mass from the phase transition i to the phase j, and Sqm is the mass source term. In our research system, there is only interphase mass conversion in a closed space and the mass source term in this case is zero.The momentum equation of the gas-liquid two-phase can be expressed as follows:(2)∂∂t∂iρivi⟶+∇∂iρivi⟶vi⟶=−∇P+∇τi⟶+∂iρig⟶+∑mij•vij⟶−mji•vji⟶+F⟶,where P, τi⟶ , and g⟶ are the pressure, the i stress-strain tensor of the gas or liquid phase, and the acceleration of gravity, respectively. F⟶ is the force on the fluid. Combined with the characteristics of this model, for the liquid phase of the droplet, this term is the surface tension, and for gas-liquid two-phase motion, the main force is buoyancy.(3)∂∂t∂iρiE+∇∂iρivi⟶E=−∇∂ivi⟶P+∑Qij+mij•Eij−mji•Eji+∇τi⟶vi⟶−∇EiJi⟶+Sqe,Qij=Kijvi⟶−vj⟶,where K is the interface momentum exchange coefficient. E, Qij, and Ei are the unit energy, the energy exchanged between the phase i to the phase j, and the energy of the component i in the phase, respectively. Sqe is the energy source term.For the process of droplet evaporation to generate steam, the diffusion equation is expressed as follows:(4)∂∂tρXi+∇ρv⟶Xi=−∇Ji⟶+Si,where Ji⟶ is the diffusion flux of the component i, Xi is the mass fraction of the component i, and Si is the source term of the generation rate of the component j.The heat balance equation in the evaporation process:(5)mtCpdTdt−Qh=hAhTl−T∞+Cpmij•−mji•Tl−T∞+Eb.Heat changes in phase transitions and radiative heat transfer are considered in the heat balance equation. The general expression of radiant heat can be expressed as follows:(6)Eb=ε1AbσTl4.The heat flux equation considering only sensible heat is as follows:(7)Q=hTl−T∞−hlat+Eb+qm,where mtCp, Qh, h, Ah, Tl, T∞, and Eb are the droplet mass, specific constant pressure heat capacity, latent heat of phase change, convective heat transfer coefficient, area participating in convective heat transfer, droplet surface temperature, ambient temperature, and radiant heat, respectively. Q, hlat, qm, ε1, Ab, and σ are the static outflow heat flux density, the phase transition latent heat flux density, the heat flux density carried away due to the outflow of steam mass, the radiation emissivity, the emitted radiation area, and the black body radiation constant, respectively.


### 3.2. Initial Boundary Conditions

#### 3.2.1. Initial Conditions

When the droplet is in the initial state, the temperature contained in the interior and its surface is the same, and the temperature and pressure in the ambient space are kept uniform. In the initial state, the droplets did not evaporate.The mathematical expression of the initial state iswhent=0, Tl=T0, Ta=T∞, r=r0, p=p0, Xvap=0, and Xvap=0. Under the initial conditions, the initial temperature of the droplet and the environmental pressure is the effects that need to be simulated parameter the vapor phase volume fraction is 0 at this time.For the thermal conductivity boundary condition for spherical droplets,whenr=0, αXl/αr=0 and αT/αr=0.At the interface between the droplet surface and the environment,whenr=rstλαTl/αr=Q.For the calculated boundary conditions, the calculated boundary is the outflow boundary for the fluid, and the fluid flows out freely without being affected by the boundary after the fluid motion reaches the boundary.For the radiation boundary, the calculation boundary is a nonreflection boundary, and radiation does not absorb and reflect at the calculation boundary.


## 3.1. Description and Selection of Mathematical Models

The moving microdroplets and the generation and propagation of steam in the gas-phase environment belong to the multiphase flow diffusion process. To accurately simulate the droplet collision and other physical changes of coupling process, the Euler–Euler model is used in this work. The Euler model can be used to simulate the separation flow of multiphase flow and the interaction between phases. Since the multiphase momentum effect is involved in the simulation, it is necessary to introduce the volume fraction of each phase and modify the momentum exchange between the phases. The corresponding continuity equations and momentum equations are established for different phases, and the calculation is performed according to the coupling pressure and the exchange coefficient between the two phases. The transient calculation method is adopted, theκ-ε model is selected for the turbulence model, and the SIMPLE algorithm is used for the solution. The spatial discretization of momentum, energy adds a common discrete coordinates adopts the first-order upwind style. The diffusion equation chooses the mass transfer diffusion without chemical reaction, and the radiation heat transfer model chooses the DO model. The corresponding governing equations are as follows.For liquid and gas phases, the continuity equation is expressed as follows:(1)∂∂t∂iρi+∇∂iρivi⟶=mij•−mji•+Sqm.In the abovementioned formula,∂i, ρi, v⇀i, and m˙ij are the mass fraction of the gas phase or liquid phase, the density, the velocity vector, the mass from the phase transition i to the phase j, and Sqm is the mass source term. In our research system, there is only interphase mass conversion in a closed space and the mass source term in this case is zero.The momentum equation of the gas-liquid two-phase can be expressed as follows:(2)∂∂t∂iρivi⟶+∇∂iρivi⟶vi⟶=−∇P+∇τi⟶+∂iρig⟶+∑mij•vij⟶−mji•vji⟶+F⟶,where P, τi⟶ , and g⟶ are the pressure, the i stress-strain tensor of the gas or liquid phase, and the acceleration of gravity, respectively. F⟶ is the force on the fluid. Combined with the characteristics of this model, for the liquid phase of the droplet, this term is the surface tension, and for gas-liquid two-phase motion, the main force is buoyancy.(3)∂∂t∂iρiE+∇∂iρivi⟶E=−∇∂ivi⟶P+∑Qij+mij•Eij−mji•Eji+∇τi⟶vi⟶−∇EiJi⟶+Sqe,Qij=Kijvi⟶−vj⟶,where K is the interface momentum exchange coefficient. E, Qij, and Ei are the unit energy, the energy exchanged between the phase i to the phase j, and the energy of the component i in the phase, respectively. Sqe is the energy source term.For the process of droplet evaporation to generate steam, the diffusion equation is expressed as follows:(4)∂∂tρXi+∇ρv⟶Xi=−∇Ji⟶+Si,where Ji⟶ is the diffusion flux of the component i, Xi is the mass fraction of the component i, and Si is the source term of the generation rate of the component j.The heat balance equation in the evaporation process:(5)mtCpdTdt−Qh=hAhTl−T∞+Cpmij•−mji•Tl−T∞+Eb.Heat changes in phase transitions and radiative heat transfer are considered in the heat balance equation. The general expression of radiant heat can be expressed as follows:(6)Eb=ε1AbσTl4.The heat flux equation considering only sensible heat is as follows:(7)Q=hTl−T∞−hlat+Eb+qm,where mtCp, Qh, h, Ah, Tl, T∞, and Eb are the droplet mass, specific constant pressure heat capacity, latent heat of phase change, convective heat transfer coefficient, area participating in convective heat transfer, droplet surface temperature, ambient temperature, and radiant heat, respectively. Q, hlat, qm, ε1, Ab, and σ are the static outflow heat flux density, the phase transition latent heat flux density, the heat flux density carried away due to the outflow of steam mass, the radiation emissivity, the emitted radiation area, and the black body radiation constant, respectively.


## 3.2. Initial Boundary Conditions

### 3.2.1. Initial Conditions

When the droplet is in the initial state, the temperature contained in the interior and its surface is the same, and the temperature and pressure in the ambient space are kept uniform. In the initial state, the droplets did not evaporate.The mathematical expression of the initial state iswhent=0, Tl=T0, Ta=T∞, r=r0, p=p0, Xvap=0, and Xvap=0. Under the initial conditions, the initial temperature of the droplet and the environmental pressure is the effects that need to be simulated parameter the vapor phase volume fraction is 0 at this time.For the thermal conductivity boundary condition for spherical droplets,whenr=0, αXl/αr=0 and αT/αr=0.At the interface between the droplet surface and the environment,whenr=rstλαTl/αr=Q.For the calculated boundary conditions, the calculated boundary is the outflow boundary for the fluid, and the fluid flows out freely without being affected by the boundary after the fluid motion reaches the boundary.For the radiation boundary, the calculation boundary is a nonreflection boundary, and radiation does not absorb and reflect at the calculation boundary.


## 3.2.1. Initial Conditions

When the droplet is in the initial state, the temperature contained in the interior and its surface is the same, and the temperature and pressure in the ambient space are kept uniform. In the initial state, the droplets did not evaporate.The mathematical expression of the initial state iswhent=0, Tl=T0, Ta=T∞, r=r0, p=p0, Xvap=0, and Xvap=0. Under the initial conditions, the initial temperature of the droplet and the environmental pressure is the effects that need to be simulated parameter the vapor phase volume fraction is 0 at this time.For the thermal conductivity boundary condition for spherical droplets,whenr=0, αXl/αr=0 and αT/αr=0.At the interface between the droplet surface and the environment,whenr=rstλαTl/αr=Q.For the calculated boundary conditions, the calculated boundary is the outflow boundary for the fluid, and the fluid flows out freely without being affected by the boundary after the fluid motion reaches the boundary.For the radiation boundary, the calculation boundary is a nonreflection boundary, and radiation does not absorb and reflect at the calculation boundary.


## 4. Results and Discussion

### 4.1. Analysis of Comparative Collision Deformation

#### 4.1.1. The Verification of Simulation Results in the Collision Process

According to the research provided by Qian and Law [17], there is a direct relationship between collision deformation and the We numbers. Thus, the We numbers was used to summarize the rules of subsequent research for the collision deformation process. Based on this, the We number is used as a variable parameter to study the impact process, and the according results are verified with the literature [31]. Figure 3 shows the evolution of collision contact with a We = 75 and no droplet evaporation. Figure 3 shows the characteristics of collision deformation at different moments after the collision of droplets with a velocity of 2 m/s in the simulation. It can be seen from Figure 3 that the simulation results are basically consistent with the variation characteristics of collisional behavior in the references. Meanwhile, in this collision process, the deformation of the combined droplet morphology is mainly influenced by the liquid surface tension and aerodynamic forces. Therefore, for the form of simulation research, the We number can be used to verify this process more accurately.Figure 3
Verification of simulation results based on literature [31] at We = 75 and no droplet evaporation.


#### 4.1.2. The Analysis of the Collision Deformation

When the flash behavior of droplets is considered at the same time, it can be seen from Figure4 that the deformation of the collision droplet tends to be a more spacious plate from the perspective perpendicular to the collision direction, and the range of stretching and contraction strokes is increased accordingly. After the collision, the droplet deformation is mainly the liquid to overcome the surface tension work process in a series of processes. From the energy perspective, the liquid collision and deformation process are the droplet kinetic energy work to overcome the surface tension and viscous dissipation [32]. Considering the effect of evaporation with mass change on this process, on the one hand, the evaporation process will make the liquid overall mass change, resulting in a droplet morphology change process accompanied by mass loss. On the other hand, evaporation leads to changes in vapor concentration distribution around the droplet, and changes in temperature distribution will also impact the droplet collision process morphology change. This point is discussed in the subsequent temperature and concentration distributions analysis.Figure 4
Simulation of droplet deformation process with a simulated droplet diameter of 200μm and an initial collision velocity of 3 m/s.


### 4.2. Analysis of Velocity and Turbulent Kinetic Energy Variations

For droplet collisions in the evaporating state, the motion collision behavior can affect the movement of the surrounding environment in the evaporation process. Figure5 shows the velocity changes in the whole region of droplet collision process under evaporation in a negative pressure environment. By comparing the initial collision velocity of 0.5 m/s and 1.5 m/s, it is concluded that when the initial contact time is 0.75 ms, the velocity away from the center appears at the symmetrical position of the longitudinal upper and lower sides, which are caused by the squeezing effect caused by the collision. As time goes on, the two velocity cores in Figure 5(a) always exist at the same time point, while the velocity cores in Figure 5(b) gradually diverge. This result proves that the larger the initial kinetic energy is, the faster the velocity core is dispersed.Figure 5
Cloud diagram of the droplet collision velocity (initial temperature 393 K). (a) The initial velocity is 0.5 m/s. (b) The initial velocity is 1.5 m/s.
(a)(b)Generally, the distribution of velocity field in the area around the collision can reveal the evaporation and vapor propagation process. For the composite droplet collision process in the evaporation state, the kinetic energy of droplet collision has a direct effect on the evaporation process and steam movement. On the one hand, the kinetic energy of collision affects the evaporation of droplets. On the other hand, it affects the vapor diffusion behavior that has been produced. As shown in Figure6, the turbulent area caused by collision at low speed is concentrated near the collision area, while at slightly higher collision speed; the distribution range of turbulent kinetic energy is significantly wider. This is because the core area of velocity is obvious at low-collision velocity. At this time, the liquid phase and high-density vapor phase coexist in the core area, while at higher collision velocity; there is no obvious velocity core. Meanwhile, the propagation space of the velocity field is wider, indicating that under the influence of higher collision velocity, the evaporation behavior of liquid phase and the process of vapor diffusion are strengthened.Figure 6
Turbulent kinetic energy variation in perpendicular to collision direction.


### 4.3. Analysis of Temperature Field

There are four ways of releasing heat from inside the liquid droplet to the outside. Namely, vapor-phase heat conduction, radiation heat transfer, convective heat transfer, and vapor-carried heat transfer. The temperature difference is the primary driving force for heat transfer in this process. Figure7 shows the comparison of the temperature change during a collision at different initial temperatures of the droplets. After the droplet collision and merger, the temperature gradually decreases from inside to outside, forming the temperature core and gradient influence zone. The temperature core is located near the center of the collision, and the temperature in this area remains largely uniform. In the core region, the collisional deformation motion process is accompanied by evaporation. Since the vapor wraps the unevaporated liquid phase, a distinct profile is formed, and the surrounding wrapped vapors are subjected to collisional dynamics for mixing. However, in the temperature gradient region formed near the outside of the core region, the temperature changes outward in the form of a gradient decrease. Because the vapor in the gradient change region first propagates outward by concentration diffusion, the vapor carrying effect causes the gradient change in the temperature.Figure 7
Temperature change during a collision at different initial temperatures of the droplets. (a) 353 K and (b) 393 K.
(a)(b)By comparing the temperature diffusion under the collision of two different initial temperatures with the increase of the initial temperature, the temperature diffusion range is more extensive simultaneously, and the temperature gradient boundary during the propagation process appears blurred. This is due to the rapid evaporation of the liquid phase at the high-initial temperature, which causes the volume of the dispersed vapor phase to increase rapidly. Then, the temperature core region increases and the gradient change decreases and blurs under the perturbation of the collision dynamics.As pointed out in the previous analysis, the heat transfer modes that cause temperature changes during the evaporation of liquid droplets are gas-phase heat conduction, radiation heat transfer, convection heat transfer, and diffusion heat transfer with heat carried by steam. Since the radiative heat exchange process is only related to the droplet’s temperature and the spatial environment involved in radiation, the amount of radiative heat exchange is relatively small for this simulation. Convective and vapor-carried heat exchange is used as the focus of analysis.Figure8 compares the total heat transfer coefficients in the XY plane when the droplets collide and evaporate at the initial temperatures of 353 K, 393 K, and 473 K. It can be seen that the overall heat transfer coefficient shows an “hourglass” shape; i.e., the overall heat transfer coefficient is the smallest in the center and gradually increases in the process of deviation from the center. With the increase in the temperature, the minimum heat transfer coefficient difference at the center gradually decreases. The values are δK = 3.75, δK = 1.3, and δK = 0.375. This result indicates that the overall heat transfer coefficient decreases with the increase in the temperature.Figure 8
Heat transfer coefficient in parallel to collision direction on the XY plane at the initial temperatures of 353 K, 393 K, and 473 K.In general, in the evaporation center area, the liquid phase is continuously evaporated, leading to the enrichment of the vapor phase when the vapor layer wraps the vicinity of the center area. The dense vapor outward heat transfer mode is mainly conductive gas heat. As the steam expands and diffuses outward in a manner driven by the concentration gradient, the volume of steam in the environment increases while generating a specific diffusion rate, leading to an enhanced convective heat transfer mode. In the concentration diffusion region influenced by the temperature gradient, the vapor diffusion process is enhanced, resulting in enhanced heat transfer by vapor carryover. These factors enhance the heat transfer enhanced with the evaporative diffusion process. The possible reason for the reduction of the overall heat transfer coefficient at higher initial temperatures is the rapid expansion of the vapor volume generated by the evaporation of the droplets in a short period. Coupled with the disturbance of collision kinetic energy, the diffusion process of concentration law is affected, and finally the overall heat transfer process is hindered.


### 4.4. Analysis of Mass Changes

Figure9 shows the mass change during the collision of droplets in the evaporation state. Figure 9(a) shows that the collision evaporation rate gradually slows down, and the whole process can be divided into a rapid evaporation area and slow evaporation area with 0.2 ms as the boundary. In the rapid evaporation zone, the droplets can basically complete more than 90% of mass evaporation. This is because in the initial state of collision evaporation, the dynamic deformation of droplets caused by collision increases the evaporation mass transfer efficiency. Figure 9(b) compares the influence of low-speed collision difference (1.5 m/s − 0.5 m/s) and high-speed collision difference (2.5 m/s − 1.5 m/s) on the difference of liquid-phase mass change. Through comparison, it is concluded that under the condition of low-speed difference, the higher the initial temperature is, the greater the mass difference is. This shows that at relatively low speed, temperature is the main factor affecting liquid evaporation. However, in the comparison of relatively high-speed difference, the mass difference caused by the high-initial temperature is smaller.Figure 9
Droplet mass change of collision evaporation. (a) Droplet mass change. (b) Droplet mass change difference.
(a)(b)As shown in Figure10, under the condition of dynamic collision, the deformation rate is the largest at 0.2 ms. This shows that the gas-liquid contact surface of the liquid surface is the largest at this time. At the same time, it can be seen that the deformation rate is also increased with the increase of the initial collision kinetic energy. Moreover, larger deformation results in a shorter time for the final droplet to evaporate completely. It shows that the evaporation mass transfer rate of droplets is directly affected by collision deformation.Figure 10
Ratio of the phase surface area change during droplet collision evaporation.Based on the previous discussion, it can be explained that when the initial temperature reaches a certain degree, the liquid phase evaporates in a very short time, weakening the impact of the initial collision velocity on the evaporation rate. Meanwhile, it can be seen that under high temperature conditions, the vapor evaporated instantaneously in the liquid phase is enriched on the surface of the liquid phase, which further hinders the evaporation process. In addition, by comparing the mass changes between different velocity differences at the same temperature, it can be seen that the mass changes caused by high-initial velocity differences at low-initial temperatures are large. The results show that increasing the initial collision velocity can enhance the evaporation of the liquid phase at low-initial temperature. At a higher initial temperature, the influence of initial velocity at the initial stage of collision is dominant. However, with the evaporation, the liquid phase evaporates gradually, and the influence of the initial collision velocity on the evaporation rate is weakened. Therefore, under this comparison condition, the quality changes of the two overlap.Then, the instantaneous flow change when the liquid phase is converted to the gas phase during the collision is considered. As shown in Figure11, the impact of the collision process on the instantaneous evaporation flow is completed in a very short time. In the same deformation time period, the instantaneous flow change process changes violently. Combined with previous research on the deformation of droplets during impact, it is further confirmed that the collision deformation promotes the evaporation of liquid phase. During the impact period, the larger the initial collision velocity and the higher the initial temperature, the more intense the fluctuation of the instantaneous evaporation rate. This is because, on the one hand, the initial collision speed is increased, and the deformation of the liquid interface under dynamic action leads to the expansion of the evaporation interface and surface renewal, which promote the liquid evaporation. On the other hand, as the driving force of evaporation mass transfer, the size of temperature directly affects the speed of droplet evaporation process. Under the condition of static or low-speed collision at 0.5 m/s, the droplet evaporation flow rate is one order of magnitude smaller than the large collision speed. For example, when the initial temperature is 393 K and the collision velocity is 2.5 m/s, the liquid-phase evaporation process is almost completed within the collision influence period. The abovementioned analysis shows that under the condition of small initial collision kinetic energy, the evaporation process of droplets is greatly affected by the initial temperature of droplets. With the increase of initial collision kinetic energy, the contact area and surface renewal of the gas-liquid interface change dramatically, and finally the evaporation process is strengthened.Figure 11
Transient flow rate of the liquid phase during the collision.


### 4.5. Analysis of Droplet Evaporation Model

Based on the abovementioned analysis results of the movement, heat, and mass transfer in the collision process of evaporation droplets, the actual process of collision evaporation can be described, as shown in Figure12. Since the combination of droplet collision and contact, the movement and evaporation behavior are centered on the droplet, and heat and mass transfer are realized through heat and steam dispersion. In this process, temperature influence domain and steam diffusion domain are formed [32]. The collision dynamic action causes the vapor to show spatial anisotropy in the transmission. Specifically, the vapor phase diffusion propagation distance parallel to the collision direction is longer than that perpendicular to the collision direction. Meanwhile, due to the carrying effect of steam, the spatial distribution of temperature also changes.Figure 12
Theoretical droplet evaporation model.


## 4.1. Analysis of Comparative Collision Deformation

### 4.1.1. The Verification of Simulation Results in the Collision Process

According to the research provided by Qian and Law [17], there is a direct relationship between collision deformation and the We numbers. Thus, the We numbers was used to summarize the rules of subsequent research for the collision deformation process. Based on this, the We number is used as a variable parameter to study the impact process, and the according results are verified with the literature [31]. Figure 3 shows the evolution of collision contact with a We = 75 and no droplet evaporation. Figure 3 shows the characteristics of collision deformation at different moments after the collision of droplets with a velocity of 2 m/s in the simulation. It can be seen from Figure 3 that the simulation results are basically consistent with the variation characteristics of collisional behavior in the references. Meanwhile, in this collision process, the deformation of the combined droplet morphology is mainly influenced by the liquid surface tension and aerodynamic forces. Therefore, for the form of simulation research, the We number can be used to verify this process more accurately.Figure 3
Verification of simulation results based on literature [31] at We = 75 and no droplet evaporation.


### 4.1.2. The Analysis of the Collision Deformation

When the flash behavior of droplets is considered at the same time, it can be seen from Figure4 that the deformation of the collision droplet tends to be a more spacious plate from the perspective perpendicular to the collision direction, and the range of stretching and contraction strokes is increased accordingly. After the collision, the droplet deformation is mainly the liquid to overcome the surface tension work process in a series of processes. From the energy perspective, the liquid collision and deformation process are the droplet kinetic energy work to overcome the surface tension and viscous dissipation [32]. Considering the effect of evaporation with mass change on this process, on the one hand, the evaporation process will make the liquid overall mass change, resulting in a droplet morphology change process accompanied by mass loss. On the other hand, evaporation leads to changes in vapor concentration distribution around the droplet, and changes in temperature distribution will also impact the droplet collision process morphology change. This point is discussed in the subsequent temperature and concentration distributions analysis.Figure 4
Simulation of droplet deformation process with a simulated droplet diameter of 200μm and an initial collision velocity of 3 m/s.


## 4.1.1. The Verification of Simulation Results in the Collision Process

According to the research provided by Qian and Law [17], there is a direct relationship between collision deformation and the We numbers. Thus, the We numbers was used to summarize the rules of subsequent research for the collision deformation process. Based on this, the We number is used as a variable parameter to study the impact process, and the according results are verified with the literature [31]. Figure 3 shows the evolution of collision contact with a We = 75 and no droplet evaporation. Figure 3 shows the characteristics of collision deformation at different moments after the collision of droplets with a velocity of 2 m/s in the simulation. It can be seen from Figure 3 that the simulation results are basically consistent with the variation characteristics of collisional behavior in the references. Meanwhile, in this collision process, the deformation of the combined droplet morphology is mainly influenced by the liquid surface tension and aerodynamic forces. Therefore, for the form of simulation research, the We number can be used to verify this process more accurately.Figure 3
Verification of simulation results based on literature [31] at We = 75 and no droplet evaporation.


## 4.1.2. The Analysis of the Collision Deformation

When the flash behavior of droplets is considered at the same time, it can be seen from Figure4 that the deformation of the collision droplet tends to be a more spacious plate from the perspective perpendicular to the collision direction, and the range of stretching and contraction strokes is increased accordingly. After the collision, the droplet deformation is mainly the liquid to overcome the surface tension work process in a series of processes. From the energy perspective, the liquid collision and deformation process are the droplet kinetic energy work to overcome the surface tension and viscous dissipation [32]. Considering the effect of evaporation with mass change on this process, on the one hand, the evaporation process will make the liquid overall mass change, resulting in a droplet morphology change process accompanied by mass loss. On the other hand, evaporation leads to changes in vapor concentration distribution around the droplet, and changes in temperature distribution will also impact the droplet collision process morphology change. This point is discussed in the subsequent temperature and concentration distributions analysis.Figure 4
Simulation of droplet deformation process with a simulated droplet diameter of 200μm and an initial collision velocity of 3 m/s.


## 4.2. Analysis of Velocity and Turbulent Kinetic Energy Variations

For droplet collisions in the evaporating state, the motion collision behavior can affect the movement of the surrounding environment in the evaporation process. Figure5 shows the velocity changes in the whole region of droplet collision process under evaporation in a negative pressure environment. By comparing the initial collision velocity of 0.5 m/s and 1.5 m/s, it is concluded that when the initial contact time is 0.75 ms, the velocity away from the center appears at the symmetrical position of the longitudinal upper and lower sides, which are caused by the squeezing effect caused by the collision. As time goes on, the two velocity cores in Figure 5(a) always exist at the same time point, while the velocity cores in Figure 5(b) gradually diverge. This result proves that the larger the initial kinetic energy is, the faster the velocity core is dispersed.Figure 5
Cloud diagram of the droplet collision velocity (initial temperature 393 K). (a) The initial velocity is 0.5 m/s. (b) The initial velocity is 1.5 m/s.
(a)(b)Generally, the distribution of velocity field in the area around the collision can reveal the evaporation and vapor propagation process. For the composite droplet collision process in the evaporation state, the kinetic energy of droplet collision has a direct effect on the evaporation process and steam movement. On the one hand, the kinetic energy of collision affects the evaporation of droplets. On the other hand, it affects the vapor diffusion behavior that has been produced. As shown in Figure6, the turbulent area caused by collision at low speed is concentrated near the collision area, while at slightly higher collision speed; the distribution range of turbulent kinetic energy is significantly wider. This is because the core area of velocity is obvious at low-collision velocity. At this time, the liquid phase and high-density vapor phase coexist in the core area, while at higher collision velocity; there is no obvious velocity core. Meanwhile, the propagation space of the velocity field is wider, indicating that under the influence of higher collision velocity, the evaporation behavior of liquid phase and the process of vapor diffusion are strengthened.Figure 6
Turbulent kinetic energy variation in perpendicular to collision direction.


## 4.3. Analysis of Temperature Field

There are four ways of releasing heat from inside the liquid droplet to the outside. Namely, vapor-phase heat conduction, radiation heat transfer, convective heat transfer, and vapor-carried heat transfer. The temperature difference is the primary driving force for heat transfer in this process. Figure7 shows the comparison of the temperature change during a collision at different initial temperatures of the droplets. After the droplet collision and merger, the temperature gradually decreases from inside to outside, forming the temperature core and gradient influence zone. The temperature core is located near the center of the collision, and the temperature in this area remains largely uniform. In the core region, the collisional deformation motion process is accompanied by evaporation. Since the vapor wraps the unevaporated liquid phase, a distinct profile is formed, and the surrounding wrapped vapors are subjected to collisional dynamics for mixing. However, in the temperature gradient region formed near the outside of the core region, the temperature changes outward in the form of a gradient decrease. Because the vapor in the gradient change region first propagates outward by concentration diffusion, the vapor carrying effect causes the gradient change in the temperature.Figure 7
Temperature change during a collision at different initial temperatures of the droplets. (a) 353 K and (b) 393 K.
(a)(b)By comparing the temperature diffusion under the collision of two different initial temperatures with the increase of the initial temperature, the temperature diffusion range is more extensive simultaneously, and the temperature gradient boundary during the propagation process appears blurred. This is due to the rapid evaporation of the liquid phase at the high-initial temperature, which causes the volume of the dispersed vapor phase to increase rapidly. Then, the temperature core region increases and the gradient change decreases and blurs under the perturbation of the collision dynamics.As pointed out in the previous analysis, the heat transfer modes that cause temperature changes during the evaporation of liquid droplets are gas-phase heat conduction, radiation heat transfer, convection heat transfer, and diffusion heat transfer with heat carried by steam. Since the radiative heat exchange process is only related to the droplet’s temperature and the spatial environment involved in radiation, the amount of radiative heat exchange is relatively small for this simulation. Convective and vapor-carried heat exchange is used as the focus of analysis.Figure8 compares the total heat transfer coefficients in the XY plane when the droplets collide and evaporate at the initial temperatures of 353 K, 393 K, and 473 K. It can be seen that the overall heat transfer coefficient shows an “hourglass” shape; i.e., the overall heat transfer coefficient is the smallest in the center and gradually increases in the process of deviation from the center. With the increase in the temperature, the minimum heat transfer coefficient difference at the center gradually decreases. The values are δK = 3.75, δK = 1.3, and δK = 0.375. This result indicates that the overall heat transfer coefficient decreases with the increase in the temperature.Figure 8
Heat transfer coefficient in parallel to collision direction on the XY plane at the initial temperatures of 353 K, 393 K, and 473 K.In general, in the evaporation center area, the liquid phase is continuously evaporated, leading to the enrichment of the vapor phase when the vapor layer wraps the vicinity of the center area. The dense vapor outward heat transfer mode is mainly conductive gas heat. As the steam expands and diffuses outward in a manner driven by the concentration gradient, the volume of steam in the environment increases while generating a specific diffusion rate, leading to an enhanced convective heat transfer mode. In the concentration diffusion region influenced by the temperature gradient, the vapor diffusion process is enhanced, resulting in enhanced heat transfer by vapor carryover. These factors enhance the heat transfer enhanced with the evaporative diffusion process. The possible reason for the reduction of the overall heat transfer coefficient at higher initial temperatures is the rapid expansion of the vapor volume generated by the evaporation of the droplets in a short period. Coupled with the disturbance of collision kinetic energy, the diffusion process of concentration law is affected, and finally the overall heat transfer process is hindered.


## 4.4. Analysis of Mass Changes

Figure9 shows the mass change during the collision of droplets in the evaporation state. Figure 9(a) shows that the collision evaporation rate gradually slows down, and the whole process can be divided into a rapid evaporation area and slow evaporation area with 0.2 ms as the boundary. In the rapid evaporation zone, the droplets can basically complete more than 90% of mass evaporation. This is because in the initial state of collision evaporation, the dynamic deformation of droplets caused by collision increases the evaporation mass transfer efficiency. Figure 9(b) compares the influence of low-speed collision difference (1.5 m/s − 0.5 m/s) and high-speed collision difference (2.5 m/s − 1.5 m/s) on the difference of liquid-phase mass change. Through comparison, it is concluded that under the condition of low-speed difference, the higher the initial temperature is, the greater the mass difference is. This shows that at relatively low speed, temperature is the main factor affecting liquid evaporation. However, in the comparison of relatively high-speed difference, the mass difference caused by the high-initial temperature is smaller.Figure 9
Droplet mass change of collision evaporation. (a) Droplet mass change. (b) Droplet mass change difference.
(a)(b)As shown in Figure10, under the condition of dynamic collision, the deformation rate is the largest at 0.2 ms. This shows that the gas-liquid contact surface of the liquid surface is the largest at this time. At the same time, it can be seen that the deformation rate is also increased with the increase of the initial collision kinetic energy. Moreover, larger deformation results in a shorter time for the final droplet to evaporate completely. It shows that the evaporation mass transfer rate of droplets is directly affected by collision deformation.Figure 10
Ratio of the phase surface area change during droplet collision evaporation.Based on the previous discussion, it can be explained that when the initial temperature reaches a certain degree, the liquid phase evaporates in a very short time, weakening the impact of the initial collision velocity on the evaporation rate. Meanwhile, it can be seen that under high temperature conditions, the vapor evaporated instantaneously in the liquid phase is enriched on the surface of the liquid phase, which further hinders the evaporation process. In addition, by comparing the mass changes between different velocity differences at the same temperature, it can be seen that the mass changes caused by high-initial velocity differences at low-initial temperatures are large. The results show that increasing the initial collision velocity can enhance the evaporation of the liquid phase at low-initial temperature. At a higher initial temperature, the influence of initial velocity at the initial stage of collision is dominant. However, with the evaporation, the liquid phase evaporates gradually, and the influence of the initial collision velocity on the evaporation rate is weakened. Therefore, under this comparison condition, the quality changes of the two overlap.Then, the instantaneous flow change when the liquid phase is converted to the gas phase during the collision is considered. As shown in Figure11, the impact of the collision process on the instantaneous evaporation flow is completed in a very short time. In the same deformation time period, the instantaneous flow change process changes violently. Combined with previous research on the deformation of droplets during impact, it is further confirmed that the collision deformation promotes the evaporation of liquid phase. During the impact period, the larger the initial collision velocity and the higher the initial temperature, the more intense the fluctuation of the instantaneous evaporation rate. This is because, on the one hand, the initial collision speed is increased, and the deformation of the liquid interface under dynamic action leads to the expansion of the evaporation interface and surface renewal, which promote the liquid evaporation. On the other hand, as the driving force of evaporation mass transfer, the size of temperature directly affects the speed of droplet evaporation process. Under the condition of static or low-speed collision at 0.5 m/s, the droplet evaporation flow rate is one order of magnitude smaller than the large collision speed. For example, when the initial temperature is 393 K and the collision velocity is 2.5 m/s, the liquid-phase evaporation process is almost completed within the collision influence period. The abovementioned analysis shows that under the condition of small initial collision kinetic energy, the evaporation process of droplets is greatly affected by the initial temperature of droplets. With the increase of initial collision kinetic energy, the contact area and surface renewal of the gas-liquid interface change dramatically, and finally the evaporation process is strengthened.Figure 11
Transient flow rate of the liquid phase during the collision.


## 4.5. Analysis of Droplet Evaporation Model

Based on the abovementioned analysis results of the movement, heat, and mass transfer in the collision process of evaporation droplets, the actual process of collision evaporation can be described, as shown in Figure12. Since the combination of droplet collision and contact, the movement and evaporation behavior are centered on the droplet, and heat and mass transfer are realized through heat and steam dispersion. In this process, temperature influence domain and steam diffusion domain are formed [32]. The collision dynamic action causes the vapor to show spatial anisotropy in the transmission. Specifically, the vapor phase diffusion propagation distance parallel to the collision direction is longer than that perpendicular to the collision direction. Meanwhile, due to the carrying effect of steam, the spatial distribution of temperature also changes.Figure 12
Theoretical droplet evaporation model.


## 5. Conclusion

The thermodynamics and evaporation mass transfer process of droplet collision coupling evaporation behavior in low-pressure environment were simulated by numerical calculation. By analyzing the changes of liquid phase morphology during the collision process, the change characteristics of mass transfer during the evaporation process of collision droplets, and the change characteristics of velocity field, temperature field, and concentration field during the evaporation process, the following conclusions are obtained:(1)
The liquid phase deformation process of collision produces area changes and tensile extrusion disturbs the surrounding environment, resulting in different temperature and concentration distributions in the horizontal and vertical directions.(2)
The collision kinetic energy affects the deformation of the liquid phase, resulting in a sharp change in the instantaneous flow of liquid evaporation during the collision process. The greater the collision kinetic energy, the higher the liquid relative surface area, and the greater the evaporation flow at this time.(3)
The temperature promotes the evaporation of the liquid phase, which leads to the increase of vapor concentration gradient, thus accelerating the temperature propagation carried by vapor diffusion. Therefore, the higher the initial temperature is, the faster the temperature diffusion is. However, the higher the initial temperature is, the steam produced will cover the liquid surface instantaneously, which will reduce the evaporation rate of droplets.



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*Source: 3574285-2023-02-10.xml*