Abstract

Evaporation of a droplet in hotter environment for the same fluid in subcritical condition has been studied in this work by analyzing the thermodynamic properties. The physical model used is based on Navier-Stokes equations. To simplify the computations, we assumed that the droplet has a spherical symmetry and evolves in quasi steady case and laminar conditions. The study is mainly characterized by the fact that the equation of conservation of momentum is effectively taken into account and that the velocity of the droplet is not always uniform. The analysis of the evolution of temperature, mass flow and velocity in the droplet and in the gaseous phase reveals the presence of discontinuities in the drop towards its boundary and of an unbalanced energy layer attached to the interface when the speed is not uniform in the drop.

1. Introduction

Droplet evaporation is an important process for the performance of fuel combustion systems, such as diesel, gas turbines and rocket engines. In these systems, fuel is injected as a cloud of droplets into the combustion chamber which vaporize and oxidize to release heat ¹. At the moment of injection, the temperature and the pressure in the combustion chamber exceed the critical point of the fuel thus creating subcritical/supercritical conditions ². According to the classical fuel injection theory ³ when the fuel is injected into the subcritical environment, a large number of droplets will be separated from the jet surface with the effects of surface tension and ambient gas disturbance. Many scientists and researchers have carried out work in the sense of knowing the behavior of fluid drops in subcritical environments. Sazhin ⁴ recent review summarizes the development of various droplet evaporation and heating models with varying levels of complexity. Maxwell ⁵, as early as 1877, provided the simplest model predicting the steady evaporation rate for a spherical drop in infinite and uniform medium. He assumed that the driving force for liquid evaporation is the difference in vapor concentration between the surface of the drop and the free stream ⁶. Similarly, in previous work on the subcritical regime, Spalding ⁷ and Godsave ⁸ simultaneously proposed a first quasi-steady model on the evaporation of isolated drops, the law of d². This modeling shows that an isolated drop evaporating reaches a state of equilibrium during which the square of the diameter of the drop decreases linearly with time and the average temperature of the drop remains constant. Although this model is very successful in describing the vaporization process of fuel drop, the assumptions upon which the model has been developed are subjected to several experimental and numerical analyses ^{9, 10}. Amongst the most controversial assumptions are those of constancy and uniformity of the macroscopic variables and the thermodynamic coefficients in the cold drop, the absence of convection and the steadiness of the gaseous phase which fits instantaneously the changes in the boundary conditions and the variation of the size of the drop. Sagna et al ¹¹ studied the behavior of drop in the gas phase. It turns out that the square of the diameter of a combustion drop is no longer a linear function of time. The same results showed that a transition region generally separates two regions of the fluid under subcritical conditions, thus creating thermal and mass shock waves characterized by discontinuities. These precursor discontinuities of incomplete combustion in the engine combustion chamber ¹² are partly responsible for air pollution.

Despite these criticisms ¹³ which they suffered, especially after experimental results invalidated some of the hypotheses on which they are based, the results of these theories remain valid and are still widely disseminated.

The main objective of the present study is to deepen the examination of the law of recession and the thermodynamic properties of the drop in the flow and in the gaseous phase by the analytical resolution of the asymptomatic method of order 0 and 1 using the models of the Navier-Stokes equations used in the study of droplet evaporation.

2. Mathematical Model

A drop of pure liquid of temperature uniformly equal to the temperature of saturated vapor at the subcritical pressure considered is abruptly introduced into a hotter infinite atmosphere initially in the rest. The radius, the density and the temperature of the drop at the initial moment are respectively and . The gaseous flow surrounding the drop is in a steady state characterized by a temperature , a pressure and a density . We suppose that there are neither external volume forces nor a forced convection, the initial spherical symmetry of the drop is thus preserved in the time ¹¹.

We denote by the distance from the center of the drop and by the radius of the drop. A law of state (which will never be specified in the study because of the uniformity of the pressure) gives the pressure in the flow as function of the density and the temperature ; is the velocity of the flow. The fluid is supposed perfect and we obtain the following system of conservation equations for the problem:

(1)

We rewrite these equations in a reference moving with the boundary of the drop and we introduce the variable . We retain two characteristic times in the description of the phenomenon: the characteristic time of thermal diffusion and the drop’s lifetime . To take into account the gradient of density in the environment, we use two characteristic densities: in the drop and in the surrounding gas. We introduce the nondimensional numbers: and and the following nondimensional variables: ; ; ; ; ; ; and . Then the dimensionless equations of the problem are given by (2):

(2)

with the initial conditions:

(3)

(4)

(5)

the boundary conditions:

(6)

(7)

(8)

and the jump relations at the interface take the dimensionless form:

(9)

is the total enthalpy corresponding to the relative motion.

Particular importance is given to the quasi-steady case in the study of drop evaporation and combustion. In the classical dimensional analyzes of the literature ^{14, 15} only one of the parameters or is used. The quasi-steady state being reached for small values of the chosen parameter. The form of system (2) of dimensionless equations shows that the unsteady phenomena in the mass and energy conservation equations are governed by the ratio . In the conservation of momentum equation, controls the terms related to the variation of the radius of the drop.

Thus, when tends to zero, the system of equations (2) tends to a quasi-steady equation. We obtain to the first approximation the system of equations (10) ^{11, 16}.

(10)

And at second approximation of equations, we obtain (11):

(11)

3. Analytical Resolution

In the following, we assume that the heat capacity and the thermal conductivity are uniform and therefore do not depend on the distance

The fact that the pressure is uniform makes it possible to decouple the system from the conservation equations and to treat separately the system formed by the mass conservation equation and the momentum conservation equation and the conservation equation of energy.

The integration of the second equation of (10) and of (11) gives (12):

(12)

where are constants. The solution (12) makes it possible to take into account the variation of the relative velocity as a function of the drop and/or the gaseous phase.

According to Koffi Sagna et al. works ¹¹, the constants are determined by the boundary conditions on the interface.

From the first equation of relations (10) and (11), we obtain:

(13)

Substituting equation (12) in (13), we obtain (14):

(14)

(14) takes into account the variation of the relative velocity as a function of the drop and the gas phase.

By considering uniform in the drop and that the fluid is an ideal gas , integration of the third of the systems (10) and (11) gives (15):

(15)

with

After integration by parts and consequently by whole series to the first order, we obtain from the second equation of the system (15) the system of equation (16):

(16)

We integrate the third equations of the system (10) and (11) and we find (17):

(17)

Taking into account that the drop is at rest, we obtain and applying the jump relations (9) to equation (17) we obtain (18):

(18)

The relations (13) written on the boundary of the drop taking into account the relation (17) give (19):

(19) gives the differential equation which describes the drop’s radius evolution versus the time. The integration of (19) gives a recession law for the diameter of the drop with a nonlinear function of time ¹⁶. To determine the evolution of the temperature, we must know the functions with regard to equation (17).

The functions and are determined using the boundary conditions and they are:

(20)

(21)

(22)

The substitution of relation (20) and (22) in the equation (19) and after the integration, we obtain the relations (23) and (24) respectively.

(23)

(24)

4. Results and Discussion

Figure 1 and Figure 2 represent the spatiotemporal evolution of the flow velocity inside droplet at order 0 and 1, introduced into a hotter gaseous environment of the same liquid during evaporation. They show that the velocity both at order 0 and 1 is continuous throughout the flow but with a profile varying over time. The velocity is non-monotonic in the drop where it can have extrema. These extrema show the existence of disturbances or discontinuities in the drop propagating towards the boundary of the drop. The same results are found by research carried out by ^{11, 17}.

Figures 3 to 5 show the spatiotemporal evolution of the density of ethanol during evaporation.

We note the presence of the disturbances which develop from the center of the drop towards the border. Its thickness and lifetime depend on τ and the initial velocities of the drop and the gas.

These disturbances propagate in the whole inside of the drop and disappear at the boundary (see Figure 4 and Figure 6).

These results confirm the work of ¹⁷ both at order 1 of the mass flow.

The spatial evolution of the temperature is presented in Figure 7 and Figure 8 and shows the intensity of the thermal shock wave inside the isolated drop.

This temperature trend is due to the existence of a transient warming period which precedes the evaporation of the droplets in the steady state ^{1, 17, 18}. This shows that the effective heat flow and the vapor flow are continuous inside the drop. When approaching the boundary of the drop, we noted a discontinuity characterized by a thermal shock wave.

Figure 9 and Figure 10 present the temporal evolutions of the square of the diameter of the drop at order 0 and 1 respectively. Note that these graphs do not present the periods of heating of the drops. Figure 9 reveals that the law d² is preserved throughout the lifetime of the drops, but this is not the case in Figure 10. However, for the small values of time, Figure 11 shows that the drop begins to evaporate according to the law d².

5. Conclusion

We solved analytically the Navier-Stokes equations for a subcritical drop in evaporation in quasi-steady conditions. The solution obtained in first and second approximations, gives recession laws which give the square of the diameter of the drops as a nonlinear function of time if the ratio is independent of x in the gaseous phase. We obtain the classical law of d² ⁸ if this ratio is a constant.

The study of the evolution of the macroscopic parameters both at order 0 and 1 shows that:

- The flow velocity is continuous throughout the flow except for the center of the drop and is strictly monotonic. It cancels out at the interface and changes sign.

- The density presents discontinuities at the center and at the boundary of the drop. Indeed, a shock wave whose intensity decreases as one moves away from the center of the drop develops in the drop and disappears near the liquid-gas interface. They are non-monotonic and have extrema that remain attached to the drop boundary until peeling.

- The temperature presents from the center and towards the border of the drop a continuous but not uniform progression. This shows that the effective heat flow and the vapor flow are continuous inside the drop. When approaching the boundary of the drop, we noted a discontinuity characterized by a thermal shock wave.

It follows that one of the main sources of the discontinuities is the transient heating of the drops persisting for most of the lifetime of the droplet. The existence of thermal and mass discontinuities during the evaporation process is proved whether at zero order or at order 1 of the asymptotic expansion method applied to analytical resolution of the Navier–Stokes equations. The control of these discontinuities during evaporation would therefore contribute enormously to the optimization of combustion in the combustion chambers of engines through an energy gain.

References