The
vaporizing
Vaporization (or vaporisation) of an element or compound is a phase transition from the liquid phase to vapor. There are two types of vaporization: evaporation and boiling. Evaporation is a surface phenomenon, whereas boiling is a bulk phenomeno ...
droplet
A drop or droplet is a small column of liquid, bounded completely or almost completely by free surfaces. A drop may form when liquid accumulates at the lower end of a tube or other surface boundary, producing a hanging drop called a pendant ...
(droplet vaporization) problem is a challenging issue in
fluid dynamics
In physics and engineering, fluid dynamics is a subdiscipline of fluid mechanics that describes the flow of fluids—liquids and gases. It has several subdisciplines, including ''aerodynamics'' (the study of air and other gases in motion) and ...
. It is part of many engineering situations involving the transport and computation of sprays:
fuel injection
Fuel injection is the introduction of fuel in an internal combustion engine, most commonly automotive engines, by the means of an injector. This article focuses on fuel injection in reciprocating piston and Wankel rotary engines.
All comp ...
,
spray painting
Spray painting is a painting technique in which a device sprays coating material (paint, ink, varnish, etc.) through the air onto a surface. The most common types employ compressed gas—usually air—to atomize and direct the paint particles.
...
,
aerosol spray
Aerosol spray is a type of dispensing system which creates an aerosol mist of liquid particles. It comprises a can or bottle that contains a payload, and a propellant under pressure. When the container's valve is opened, the payload is forced out ...
, flashing releases… In most of these engineering situations there is a relative motion between the droplet and the surrounding gas. The gas flow over the droplet has many features of the gas flow over a rigid sphere:
pressure gradient
In atmospheric science, the pressure gradient (typically of air but more generally of any fluid) is a physical quantity that describes in which direction and at what rate the pressure increases the most rapidly around a particular location. The pr ...
, viscous
boundary layer
In physics and fluid mechanics, a boundary layer is the thin layer of fluid in the immediate vicinity of a bounding surface formed by the fluid flowing along the surface. The fluid's interaction with the wall induces a no-slip boundary condi ...
,
wake. In addition to these common flow features one can also mention the internal liquid circulation phenomenon driven by surface-
shear forces and the
boundary layer
In physics and fluid mechanics, a boundary layer is the thin layer of fluid in the immediate vicinity of a bounding surface formed by the fluid flowing along the surface. The fluid's interaction with the wall induces a no-slip boundary condi ...
blowing effect.
One of the key parameter which characterizes the gas flow over the droplet is the droplet
Reynolds number
In fluid mechanics, the Reynolds number () is a dimensionless quantity that helps predict fluid flow patterns in different situations by measuring the ratio between inertial and viscous forces. At low Reynolds numbers, flows tend to be domin ...
based on the relative velocity, droplet diameter and gas phase properties. The features of the gas flow have a critical impact on the exchanges of mass, momentum and energy between the gas and the liquid phases and thus, they have to be properly accounted for in any vaporizing droplet model.
As a first step it is worth investigating the simple case where there is no relative motion between the droplet and the surrounding gas. It will provide some useful insights on the physics involved in the vaporizing droplet problem. In a second step models used in engineering situations where a relative motion between the droplet and the surrounding exists are presented.
Single spherically symmetric droplet
In this section we assume that there is no relative motion between the droplet and the gas,
, and that the temperature inside the droplet is uniform (models that account for the non-uniformity of the droplet temperature are presented in the next section). The time evolution of the droplet radius,
, and droplet temperature,
, can be computed by solving the following set of ordinary differential equations:
[Crowe, C., Sommerfeld, M., Tsuji, Y. (1998). ''Multiphase flows with droplets and particles'', CRC Press LLC, .]
:
:
where:
*
is the liquid density (kg.m
−3)
*
is the vaporization rate of the droplet (kg.s
−1)
*
is the liquid specific heat at constant pressure (J.kg
−1.K
−1)
*
is the heat flux entering the droplet (J.s
−1)
The heat flux entering the droplet can be expressed as:
:
where:
*
is the heat flux from the gas to the droplet surface (J.s
−1)
*
is the latent heat of evaporation of the species considered (J.kg
−1)
Analytical expressions for the droplet vaporization rate,
, and for the heat flux
are now derived. A single, pure, component droplet is considered and the gas phase is assumed to behave as an ideal gas. A spherically symmetric field exists for the gas field surrounding the droplet. Analytical expressions for
and
are found by considering heat and mass transfer processes in the gas film surrounding the droplet.
[Abramzon, B., Sirignano, W. A. (1989). Droplet vaporization model for spray combustion calculations, ''Int. J. Heat Mass Transfer'', Vol. 32, No. 9, pp. 1605-1618.] The droplet vaporizes and creates a radial flow field in the gas film. The vapor from the droplet convects and diffuses away from the droplet surface. Heat conducts radially against the convection toward the droplet interface. This process is called Stefan convection or
Stefan flow.
[Sirignano, W. A. (2010). ''Fluid dynamics and transport of droplets and sprays - Second edition'', Cambridge University Press, .]
The gas phase conservation equations for mass, fuel-vapor mass fraction and energy are written in a spherical coordinate system:
:
:
:
where:
*
density of the gas phase (kg.m
−3)
*
radial position (m)
*
Stefan velocity (m.s
−1)
*
Fuel mass fraction in the gas film (-)
*
Mass diffusivity (m
2.s
−1)
*
Enthalpy of the gas (J.kg
−1)
*
Gas film temperature (K)
*
Thermal conductivity of the gas (W.m
−1.K
−1)
*
Number of species inside the gas phase, i.e. air + fuel (-)
It is assumed that the gas phase heat and mass transfer processes are quasi-steady and that the thermo-physical properties might be considered as constant. The assumption of quasi-steadiness of the gas phase finds its limitation in situations in which the gas film surrounding the droplet is in a near-critical state or in a situation in which the gas field is submitted to an acoustic field. The assumption of constant thermo-physical properties is found to be satisfying provided that the properties are evaluated at some reference conditions
[Hubbard, G. L., Denny, V. E., Mills, A. F. (1975). Droplet vaporization: effects of transients and variable properties, ''Int. J. Heat Mass Transfer'', Vol. 18, pp. 1003-1008.]
:
:
where:
*
is the reference temperature (K)
*
is the temperature at the droplet surface (K)
*
is the temperature of the gas far away from the droplet surface (K)
*
is the reference fuel mass fraction (-)
*
is the fuel mass fraction at the droplet surface (-)
*
is the fuel mass fraction far away from the droplet surface (-)
The ''1/3'' averaging rule,
, is often recommended in the literature
[Yuen, M. C., Chen, L. W. (1976). On drag of evaporating liquid droplets, ''Combust. Sci. Technol.'', Vol. 14, pp. 147-154.]
The conservation equation of mass simplifies to:
:
Combining the conservation equations for mass and fuel vapor mass fraction the following differential equation for the fuel vapor mass fraction
is obtained:
:
Integrating this equation between
and the ambient gas phase region
and applying the boundary condition at
gives the expression for the droplet vaporization rate:
:
and
:
where:
*
is the Spalding mass transfer number
Phase equilibrium is assumed at the droplet surface and the mole fraction of fuel vapor at the droplet surface is obtained via the use of the
Clapeyron's equation.
An analytical expression for the heat flux
is now derived. After some manipulations the conservation equation of energy writes:
:
where:
*
is the enthalpy of the fuel vapor (J.kg
−1)
Applying the boundary condition at the droplet surface and using the relation
we have:
:
where:
*
is the specific heat at constant pressure of the fuel vapor (J.Kg
−1.K
−1)
Integrating this equation from
to the ambient gas phase conditions (
) gives the variation of the gas film temperature (
) as a function of the radial distance:
:
The above equation provides a second expression for the droplet vaporization rate:
:
and
:
where:
*
is the Spalding heat transfer number
Finally combining the new expression for the droplet vaporization rate and the expression for the variation of the gas film temperature the following equation is obtained for
:
:
Two different expressions for the droplet vaporization rate
have been derived. Hence, a relation exists between the Spalding mass transfer number and the Spalding heat transfer number and writes:
:
where:
*
is the gas film
Lewis number The Lewis number (Le) is a dimensionless number defined as the ratio of thermal diffusivity to mass diffusivity. It is used to characterize fluid flows where there is simultaneous heat and mass transfer. The Lewis number puts the thickness of the t ...
(-)
*
is the gas film specific heat at constant pressure (J.Kg
−1.K
−1)
The droplet vaporization rate can be expressed as a function of the Sherwood number. The Sherwood number describes the non-dimensional mass transfer rate to the droplet and is defined as:
:
Thus, the expression for the droplet vaporization rate can be re-written as:
:
Similarly, the convective heat transfer from the gas to the droplet can be expressed as a function of the Nusselt number. The Nusselt number describes a non-dimensional heat transfer rate to the droplet and is defined as:
:
and then:
:
In the limit where
we have
which corresponds to the classical heated sphere result.
Single convective droplet
The relative motion between a droplet and the gas results in an increase of the heat and mass transfer rates in the gas film surrounding the droplet. A convective boundary layer and a wake can surround the droplet. Furthermore, the shear force on the liquid surface causes an internal circulation that enhances the heating of the liquid. As a consequence, the vaporization rate increases with the droplet Reynolds number. Many different models exist for the single convective droplet vaporization case. Vaporizing droplet models can be seen to belong to six different classes:
# Constant droplet temperature model (d
2-law)
# Infinite liquid conductivity model
# Spherically symmetric transient droplet heating model
# Effective conductivity model
# Vortex model of droplet heating
# Navier-Stokes solution
The main difference between all these models is the treatment of the heating of the liquid phase which is usually the rate controlling phenomenon in droplet vaporization.
The first three models do not consider internal liquid circulation. The effective conductivity model (4) and the vortex model of droplet heating (5) account for internal circulation and internal convective heating. The direct resolution of the Navier-Stokes equations provide, in principle, exact solutions both for the gas phase and the liquid phase.
Model (1) is a simplification of model (2) which is in turn a simplification of model (3). The spherically symmetric transient droplet heating model (3) solves the equation for heat diffusion through the liquid phase. A droplet heating time τ
h can be defined as the time required for a thermal diffusion wave to penetrate from the droplet surface to its center. The droplet heating time is compared to the droplet lifetime, τ
l. If the droplet heating time is short compared to the droplet lifetime we can assume that the temperature field inside the droplet is uniform and model (2) is obtained. In the infinite liquid conductivity model (2) the temperature of the droplet is uniform but varies with time. It is possible to go one step further and find the conditions for which we can neglect the temporal variation of the droplet temperature. The liquid temperature varies in time until the
wet-bulb temperature
The wet-bulb temperature (WBT) is the temperature read by a thermometer covered in water-soaked (water at ambient temperature) cloth (a wet-bulb thermometer) over which air is passed. At 100% relative humidity, the wet-bulb temperature is equal ...
is reached. If the
wet-bulb temperature
The wet-bulb temperature (WBT) is the temperature read by a thermometer covered in water-soaked (water at ambient temperature) cloth (a wet-bulb thermometer) over which air is passed. At 100% relative humidity, the wet-bulb temperature is equal ...
is reached in a time of the same order of magnitude as the droplet heating time, then the liquid temperature can be considered to be constant with regard to time; model (1), the d
2-law, is obtained.
The infinite liquid conductivity model is widely used in industrial spray calculations:
[Aggarwal, S. K., Peng, F. (1995). A review of droplet dynamics and vaporization modeling for engineering calculations, ''Journal of Engineering for gas turbines and power'', Vol. 117, p. 453.][Aggarwal, S. K., Tong, A. Y., Sirignano, W. A. (1984). A comparison of vaporization models in spray calculations, ''AIAA Journal'', Vol. 22, No 10, p. 1448.] for its balance between computational costs and accuracy. To account for the convective effects which enhanced the heat and mass transfer rates around the droplet, a correction is applied to the spherically symmetric expressions of the Sherwood and Nusselt numbers
:
:
Abramzon and Sirignano
suggest the following formulation for the modified Sherwood and Nusselt numbers:
:
:
where
and
account for surface blowing which results in a thickening of the boundary layer surrounding the droplet.
and
can be found from the well-known Frössling, or Ranz-Marshall, correlation:
:
:
where
*
is the
Schmidt number
Schmidt number (Sc) is a dimensionless number defined as the ratio of momentum diffusivity (kinematic viscosity) and mass diffusivity, and it is used to characterize fluid flows in which there are simultaneous momentum and mass diffusion convecti ...
,
*
is the
Prandtl number
The Prandtl number (Pr) or Prandtl group is a dimensionless number, named after the German physicist Ludwig Prandtl, defined as the ratio of momentum diffusivity to thermal diffusivity. The Prandtl number is given as:
: \mathrm = \frac = \frac ...
,
*
is the
Reynolds number
In fluid mechanics, the Reynolds number () is a dimensionless quantity that helps predict fluid flow patterns in different situations by measuring the ratio between inertial and viscous forces. At low Reynolds numbers, flows tend to be domin ...
.
The expressions above show that the heat and mass transfer rates increase with increasing Reynolds number.
References
{{Reflist
Fluid dynamics