15.4.2 Droplet Vaporization (Law 2)

Law 2 is applied to predict the vaporization from a discrete phase droplet. Law 2 is initiated when the temperature of the droplet reaches the vaporization temperature, $T_{\rm vap}$, and continues until the droplet reaches the boiling point, $T_{\rm bp}$, or until the droplet's volatile fraction is completely consumed:

The onset of the vaporization law is determined by the setting of $T_{\rm vap}$, a modeling parameter that has no physical significance. Note that once vaporization is initiated (by the droplet reaching this threshold temperature), it will continue to vaporize even if the droplet temperature falls below $T_{\rm vap}$. Vaporization will be halted only if the droplet temperature falls below the dew point. In such cases, the droplet will remain in Law 2 but no evaporation will be predicted. When the boiling point is reached, the droplet vaporization is predicted by a boiling rate, Law 3, as described in a section that follows.

Mass Transfer During Law 2

During Law 2, the rate of vaporization is governed by gradient diffusion, with the flux of droplet vapor into the gas phase related to the difference in vapor concentration at the droplet surface and the bulk gas:

where

	$N_i$	=	molar flux of vapor (kgmol/m $^2$-s)
	$k_c$	=	mass transfer coefficient (m/s)
	$C_{i,s}$	=	vapor concentration at the droplet surface (kgmol/m $^3$)
	$C_{i,\infty}$	=	vapor concentration in the bulk gas (kgmol/m $^3$)

Note that ANSYS FLUENT's vaporization law assumes that $N_i$ is positive (evaporation). If conditions exist in which $N_i$ is negative (i.e., the droplet temperature falls below the dew point and condensation conditions exist), ANSYS FLUENT treats the droplet as inert ( $N_i = 0.0$).

The concentration of vapor at the droplet surface is evaluated by assuming that the partial pressure of vapor at the interface is equal to the saturated vapor pressure, $p_{\rm sat}$, at the particle droplet temperature, $T_p$:

where $R$ is the universal gas constant.

The concentration of vapor in the bulk gas is known from solution of the transport equation for species $i$ for nonpremixed or partially premixed combustion calculations:

where $X_i$ is the local bulk mole fraction of species $i$, $p$ is the local absolute pressure, and $T_{\infty}$ is the local bulk temperature in the gas. The mass transfer coefficient in Equation  15.4-12 is calculated from the Sherwood number correlation [ 284, 285]:

where	$D_{i,m}$	=	diffusion coefficient of vapor in the bulk (m $^2$/s)
	Sc	=	the Schmidt number, $\frac{\mu}{\rho D_{i,m}}$
	$d_p$	=	particle (droplet) diameter (m)

The vapor flux given by Equation  15.4-12 becomes a source of species  $i$ in the gas phase species transport equation, (see this section in the separate User's Guide) or in the mixture fraction equation for nonpremixed combustion calculations.

The mass of the droplet is reduced according to

where	$M_{w,i}$	=	molecular weight of species $i$ (kg/kgmol)
	$m_p$	=	mass of the droplet (kg)
	$A_p$	=	surface area of the droplet (m $^2$)

ANSYS FLUENT can also solve Equation  15.4-16 in conjunction with the equivalent heat transfer equation using a stiff coupled solver. See this section in the separate User's Guide for details.

Defining the Vapor Pressure and Diffusion Coefficient

You must define the vapor pressure as a polynomial or piecewise linear function of temperature ( $p_{\rm sat}(T)$) during the problem definition. Note that the vapor pressure definition is critical, as $p_{\rm sat}$ is used to obtain the driving force for the evaporation process (Equations  15.4-12 and  15.4-13). You should provide accurate vapor pressure values for temperatures over the entire range of possible droplet temperatures in your problem. Vapor pressure data can be obtained from a physics or engineering handbook (e.g., [ 266]).

You must also input the diffusion coefficient, $D_{i,m}$, during the setup of the discrete phase material properties. Note that the diffusion coefficient inputs that you supply for the continuous phase are not used in the discrete phase model.

Heat Transfer to the Droplet

Finally, the droplet temperature is updated according to a heat balance that relates the sensible heat change in the droplet to the convective and latent heat transfer between the droplet and the continuous phase:

where	$c_p$	=	droplet heat capacity (J/kg-K)
	$T_p$	=	droplet temperature (K)
	$h$	=	convective heat transfer coefficient (W/m $^2$-K)
	$T_{\infty}$	=	temperature of continuous phase (K)
	$\frac{d m_{p}}{dt}$	=	rate of evaporation (kg/s)
	$h_{\rm fg}$	=	latent heat (J/kg)
	$\epsilon_p$	=	particle emissivity (dimensionless)
	$\sigma$	=	Stefan-Boltzmann constant (5.67 x 10 $^{-8}$ W/m $^2$-K $^4$)
	$\theta_R$	=	radiation temperature, $(\frac{I}{4\sigma})^{1/4}$, where $I$ is the radiation intensity

Radiation heat transfer to the particle is included only if you have enabled the P-1 or discrete ordinates radiation model and you have activated radiation heat transfer to particles using the Particle Radiation Interaction option in the Discrete Phase Model dialog box.

The heat transferred to or from the gas phase becomes a source/sink of energy during subsequent calculations of the continuous phase energy equation.