There are two methods that have been developed for multi-dimensional simulation of droplet evaporation, one-fluid method and two-fluid method. For one-fluid method, a set of governing equations are solved, while for two-fluid method, the governing equations for each phases are solved.
In the last decades, two-fluid method has been widely used to investigate the problem of droplet evaporation. The two-fluid method usually assumes that the droplet maintains the rigid spherical shape during the process of droplet evaporation. Masoudi and Sirignano[1997 The influence of an advecting vortex on the heat transfer to a liquid droplet] investigated the three-dimensional interaction between the advecting vortex and an evaporating droplet. The droplet convective heat transfer number is sensitive to the initial position and structure of the advecting vortex. Furthermore, the vertical structures effect the mass and heat transfer mechanisms in spray combustion systems. The study by Zhang used two-fluid model to simulate a evaporating fuel droplet in a forced convective environment. Good agreement was observed between numerical predictions and experimental data.
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There have been a few studies using one-fluid method for analysis of droplet evaporation. The main advantage of one-fluid method is it can consider the deformation of evaporating droplet due to interface tracking and interface capturing methods.
In order to take into account the deformation of evaporating droplet, Deng et al. developed a method for droplet dynamics and evaporation using Arbitrary Lagrangian-Eulerian (ALE) method. They found that the droplet could be deformed under a small relative velocity between gas phase and liquid phase. The ALE method has a low efficiency and the regrid is needed when a large deformation of the interface occurs, so it is not widely used.
The Volume-Of-Fluid (VOF) method has been introduced in the studies of droplet evaporation in recent years. The VOF method is one of the interface capturing methods. Nikolopoulos et al. presented the VOF approach for the computation of the evaporation process of a liquid droplet impinging onto a hot substrate using an adaptive local grid refinement technique at the liquidï¿½Cgas interface. They focus on mainly on the droplet shape evolution during droplet impingement on a heated surface. The calculations have been done in both two-dimensional axisymmetric and fully three-dimensional domains. The algorithm of evaporated mass flux considers the kinetic model. Strotos et al. [2008 Numerical investigation on the evaporation of droplets depositing on heated surfaces at low Weber numbers] made a numerical comparison against Fickï¿½ï¿½s law, kinetic model and conventional hydrodynamic model for evaporation of droplets impinging and depositing on heated solid walls using the VOF method. A coupled model based on kinetic model and conventional hydrodynamic model was formulated for the computation of droplet evaporation in contact with the hot wall. A method for three-dimensional direct numerical simulation of evaporating droplets based on VOF approach was presented by Schlottke and Weigand[7, 8]. They used the PLIC method to capture the interface of droplet and the evaporation rate was computed based on Fickï¿½ï¿½s law.
This paper is focused on the study of droplet evaporation process using VOF method. Firstly, modelling of droplet heating and evaporation using VOF model is described. Secondly, the comparison of simulation results of different models is studied. Thirdly, the comparison of simulation results between droplets array and a single droplet is discussed. Fourthly, the effects of different droplets spacing on droplet evaporation is showed. Finally, we compare the droplet deformation in different ambient pressures.
2. Modelling of droplet heating and evaporation using VOF method
The VOF (Volume of Fluid) method is proposed on the basis of the MAC method, which can be used to deal with the issue of any free interface. The VOF describes two or more immiscible fluid (phase). The main idea of VOF method is: The volume fraction of the fluid is set to a function in a cell and the phases interface is captured through the calculation of the function. The phase interface capturing problem is transformed into the simulation of flow field transport through constructing the equations of the fluid volume fraction. The equations determine the position of the moving interface, shape and deformation of the fluid. The position and shape of the interface is determined by the ratio of volume of fluid and cell.
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2.1 Volume of Fluid equation
This study uses VOF method to investigate the droplet evaporation and two phases are set, liquid phase and gas phase. The gas phase includes the vapour and the air.
If the value is 0, it indicated that the cell is full of gas phase. If the value is 1, it indicated that the cell is full of liquid phase. Furthermore, if the value is 0 < ï¿½ï¿½ < 1, it indicated that the cell is filled with gas phase and liquid phase. The slope and curvature in the phase interface is calculated by the value of volume fraction of the adjacent cells, thereby the discrete equations can be solved.
The fluid flow must obey the law of conservation, including the law of conservation of mass, the law of conservation of momentum and the law of conservation of energy and so on. The basic equation of VOF method is the conservation equation of volume of fluid. The VOF equation has the following form,
Where is the mass source for the qth phase. The source terms for liquid phase and gas phase are,
For the two-phase cell, the transport properties are determined by the volume fraction average method of liquid phase and gas phase. All the properties are computed in this manner. For example, the density is given as,
2.2 Momentum equation
One set of momentum equation for both phases is solved in the computational domain. The velocity field is shared in each phase and the momentum equation is calculated based on the volume fraction of each phase. Due to the effect of evaporation, the liquid phase lost the momentum and the gas phase gain the momentum. The momentum equation and source term is expressed as,
2.3 Energy equation
The energy equation is shared between the liquid phase and the gas phase. The energy equation is similar to the momentum equation, the source term caused by the evaporation can be obtained by the by the evaporation rate and latent heat of evaporation. The radiation effects are ignored. The energy equation and source term is expressed as,
2.4 Species transport equation
Species conservation law represents that the time rate of change of the chemical composition in the cell is equal to the net diffusion flux in the interface and the generation rate due to the chemical reaction. In this study, because the gas phase is composed of air and n-heptane vapour, the transport equation of the n-heptane vapour is solved. In the liquid phase, the transport equation for n-heptane is solved. The species equation and source term is given by
2.5 Turbulence equation
Large eddy simulation (LES) was first proposed by Smagorinsky in 1963. In LES model, large eddies are resolved by the N-S equations directly. However, small eddies cannot be resolved directly, Subgrid-Scale Models are introduced to model the small eddies. In this study, Smagorinsky-Lilly Subgrid-Scale Model is used for the calculations of droplet evaporation in forced convection.
2.6 Volumetric mass source term
There are a large number of mathematical models describing droplet evaporation process, such as empirical models, a pure diffusion model and molecular dynamics model. In this study, we use pure diffusion controlled model to conduct the study of the droplet evaporation process. This model assumes that the vapour-liquid interface is in phase equilibrium state and the evaporation rate is calculated by Fick's law. The volumetric mass source of vapour is calculated as 
Where is the local interface density.
2.7 Saturate vapour pressure
At the interface of two phases, the droplet evaporation rate can be calculated using this equation.
There are many methods to calculate the saturate vapour pressure, which are based on integral of the Clausius-Clapeyron equation. This equation presents gas phase and liquid phase is in the status of phase equilibrium; therefore the temperatures, pressures and chemical potential of two phases are equivalent:
Clapeyron equation used for calculating saturate vapour pressure of fuel droplets:
2.8 Geometric Reconstruction Scheme
PLIC-VOF (piecewise linear interface construction) method is second order. The two-phase interface is composed of some straight lines using PLIC-VOF. (Fig. 1)
Figure 1. Comparison
3. ETC model
Effective thermal conductivity (ETC) model takes into account finite liquid thermal conductivity [12, 17]. The model considers Hill's type of internal circulation inside the droplets, so it leads to an increasing of the sphere thermal conductivity by a correction factor . The correction factor was approximated as
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The correction factor varies over the range of 1-2.72. The transient heat conduction equation is described as
The thermal radiation effects are neglected. where is the liquid thermal diffusivity, is the liquid thermal conductivity, is specific heat capacity, is liquid density, is the droplet temperature, R is the distance from the centre of the droplet.
The boundary condition of the droplet is presented in the following form.
For the ETC model, the liquid thermal conductivity is replaced by the so-called effective thermal conductivity :
3. Two-way coupling method
The two-way coupling simulation is based on a coupling between the external flow computations with the CFD software OPENFOAM, and an in-house code solving the time dependent Navier-Stokes equations for the internal fields of the droplet[19, 20]. The two-way coupling approach adopts two-fluid method and two sets of conservative equations are solved for liquid phase and gas phase. The energy equation of liquid phase takes the following form:
The effective heat flux resulting from the external flow computation are used as non-uniform boundary conditions at the droplet surface.
The two-way coupling method adopts quasi-static evaporation hypothesis.
4. Results and discussion
The evaporation of droplets tandem is investigated using ANSYS FLUENT 12.0. The calculation of droplets evaporation performed using a two-dimensional structured mesh. The computational domain and boundary conditions of numerical simulation under this study are sketched in Fig. 2, which is a velocity inlet condition at the left side, free slip wall conditions on lateral boundaries, a outflow condition at the right side, a axis symmetry condition on the axis line. The size of computational domain is 4.66mm (20d) ï¿½ï¿½ 2.33mm (10d).
Figure 2. Computational domain and boundary conditions configuration for the numerical simulation of droplets tandem
Table 1 summarize the values of experimental parameters. A linear monodisperse droplet stream is generated by Rayleigh disintegration of a liquid jet with the use of a mechanical vibration obtained by a piezoceramic.
Table 1 Values of experimental parameters
Injection temperature 297K
Droplets diameter 233ï¿½ï¿½m
Droplets spacing 4.0d
Ambient temperature 1150K
Droplet velocity 10.1m/s
Surface tension coefficient 0.02N/m
3.1 Grid independence
The size of the grid has a major impact on the simulation results. In order to obtain the trusted simulation results, it is necessary to carry out the grid independence study. When grid size decreases, the simulation results will be approaching a limit gradually. When the simulation results close to the limit, the influence of the grid size weakened rapidly, and then the simulation results are independent of the grid size. Grid independence study is the important preparatory work of CFD.
In order to verify the independence of the grid, three different grids numbers are set. The grids numbers of the droplet occupied are 80, 316 and 716 respectively. The computational time of the three kinds of grids is 3.3 hours, 14.6 hours and 24.5 hours using parallel computing. Figure 3 shows a comparison of the droplets surface temperature of the three kinds of grids. The maximum error between coarse grid and fine grid is 0.55%, the maximum error between medium grid and fine mesh is 0.28%. In addition, in the calculation cycle, the average error between coarse grid and fine grid is 0.18%, while the average error between medium grid and fine grid is 0.04%. As the error of the calculation result increases with the grid size decreases, the calculation result is in a gradual convergence. Considering the calculation time and errors, the medium grid is selected as the computational grid paper and the size of medium grid is 11.65ï¿½ï¿½m ï¿½ï¿½ 11.65ï¿½ï¿½m.
Figure 3. Comparison of the droplets surface temperature of three kinds of grids
3.2 Comparison of different models
The section is focused on the comparison of the effects of different models on droplet heating. The plots of droplet surface temperature versus time for different models are shown in Fig.4. At the initial stage of droplet evaporation, the effect of different models on time evolution of surface temperature is relatively small. However, the effect becomes clearly visible and the difference in surface temperature predicted by the models can reach almost 4% at t = 2ms. The ETC model predicts lowest droplet surface temperatures compared with other approaches, while the two-way coupling approach predicts highest droplet surface temperature.
An explanation to this may be provided as follows. In the ETC model, the droplet heating is controlled by the
In the VOF model,
Figure 4. Comparison of different models on droplets surface temperature
The relation between surface temperature and average temperature in ETC model can be presented in the following.
Figure 5. Comparison of the droplets average temperature of different models
Figure 6. Comparison of experimental values & simulation values of n-heptane droplet evaporation at 550K
The temperature distributions of liquid phase
Figure 7. Comparison of experimental values & simulation values of n-heptane droplet evaporation at 550K
55 52 48 45 41 38 34 31 27 24 20
Figure 4 shows the temperature distributions and streamline fields of liquid phase. The difference of temperature distribution between lead droplet and middle droplet or downstream droplet is relatively large, while the difference of temperature distribution between middle droplet and downstream droplet is relatively small.
Figure 8. The temperature distributions and streamline fields of liquid phase at time =1ms
3.3 Comparison between droplets array and isolated droplet
The temporal evolution of droplets surface temperature, shown in Fig.9, indicates notable differences between the middle one of droplets tandem and isolated droplet. At the very initial stage of droplet heating and evaporation, the values of droplet surface temperature are relatively insensitive for the interaction of droplets. At intermediate time, the surface temperature of droplets tandem is lower than the isolated droplet. The difference between both predictions is due to the effect of the vapour wake of the lead droplet. Because the temperature of vapour is lower than that of air and the middle droplet is at the position of vapour wake of lead droplet, the vapour wake affects the surface temperature of the middle droplet.
Figure 9. Comparison of the droplets surface temperature between droplet tandem and isolated droplet
Figure 10. Comparison of the droplets average temperature between droplet tandem and isolated droplet
3.4 The effects of different droplets spacing
For different spacing of the droplets, the downstream droplet is located at different positions of the lead droplet vapour wake, therefore different droplets spacing will affect the evaporation process of downstream droplet. This section will focus on this topic and the droplets spacing is from 2d to 8d.
Vorticity magnitude is the magnitude of the vorticity vector. Vorticity is a measure of the rotation of a fluid element as it moves in the flow field, and is defined as the curl of the velocity vector [FLUENT 12 Ëµï¿½ï¿½ï¿½ï¿½]:
It can be seen from Fig.9, the distribution of gas phase vorticity at the front sides of lead droplets is same with different droplets spacing. However the distribution of vorticity at the front sides of downstream droplets is different with different droplets spacing. At a smaller droplets spacing of 2d, there is a little vorticity at the front side of the downstream droplet. With the increase of droplets spacing, the distribution area of vorticity at front side of downstream droplet become larger. There is a little effect of distribution of vorticity at front side of downstream droplet with a droplets spacing 6d and 8d. Furthermore, the distribution area of vorticity at front side of downstream droplet is close to that of lead droplet with a droplets spacing 6d and 8d.
Figure 11. Vorticity magnitude of different droplets spacing at
Fig.10 shows the surface temperatures of downstream droplet predicted by different droplets spacing. An obvious increase in surface temperature due to an increase in droplets spacing, is attributed to the positions of downstream droplet located. The larger the droplets spacing is, the smaller the effect of lead droplet is. The surface temperature is lowest with the droplets spacing of 2d while the surface temperatures are almost same with the droplets spacing of 6d and 8d. This can be explained as follows. The surface temperature is mainly controlled by the interface temperature of gas-liquid phase, which is effected by the vapour wake of lead droplet. The temperature of vapour wake is lower than ambient temperature and the temperature of vapour wake is proportional to droplets spacing.
Figure 12. Vorticity magnititude of different downstream droplets spacing
The drag coefficient Cd for the droplet is determined from the aerodynamic force and a dynamic pressure based on a relative velocity. The drag force Fd is given by
Fig. 13 compares the drag coefficient b the downstream droplet with isolate droplet. As can be seen from Figure, the different droplet spacing, when the droplet spacing more hours, after the droplets are greater the resistance; when the droplets the greater the spacing, the smaller the resistance after droplets are.
The drag coefficients of different droplets spacing is shown in Fig.13. As follows from this figure,
Figure 13. Droplet coefficients of the downstream droplets tandem
3.5 Droplets deformation
An important difference between the droplets and solid particles is that the droplet will significantly deform in the course of movement. Furthermore, the droplets also occur in the airflow under rotation. Droplet evaporation, the theoretical model assumes that the droplet is rigid spherical shape and deformation of the droplets is ignored. However, in actual droplet evaporation process, the area of evaporating droplets surface is changing. In the case of the same volume, the surface area of the spherical droplets is minimal.
The axis length of droplet perpendicular to the airflow direction is defined as a, and the axis length of droplet parallel to the airflow direction is defined as b (Fig. 12).
Figure 14. Comparison of experimental values & simulation values of n-heptane droplet evaporation at 550K
The deformation factor of the droplets is defined as follows:
The deformation factor characterizes the change in shape of the evaporating droplets and reflects the change of the droplets shape in different directions. When >1, it indicates that the deformation of droplets in the perpendicular to the airflow direction becomes larger; When = 1, it indicates that the droplet doesnï¿½ï¿½t deform and keeps spherical; when <1, it indicates that the deformation of droplets parallel to the airflow direction becomes larger.
Fig.13 shows the shape of droplets in different ambient pressures at 2 ms. Droplet is substantially spherical when the ambient pressure 1bar. As the ambient pressure increases, the eccentricity of the ellipse becomes larger and the droplet shape gradually becomes flat, which shows the influence of the ambient pressure on the droplet shape is more obvious.
Figure 15. Comparison of experimental values & simulation values of n-heptane droplet evaporation at 550K
It can be seen from Fig.16, the droplet deformation factors increase with increasing environmental pressure. The deformation factors of the droplets vary periodically. At a higher ambient pressure of , the maximum deformation factors of droplets is 2.35. The shape of the droplet is relatively long length spherical from the axial direction. However at the ambient pressure of , the maximum deformation factors of droplets is 1.15. As follows from this figure, at higher ambient pressures, the deformation factors of the droplets are larger. The ambient pressure affects the speed of the droplet. It can be seen that the greater the ambient pressure is, the shorter the time of the droplet escape the computational domain is.
Figure 16. Comparison of experimental values & simulation values of n-heptane droplet evaporation at 550K
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