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2.4. Single Droplet Modeling 37
through solid layer due to pressure difference in pores is not considered, and it is the
scope of the future study. Moreover, the influence of internal circulation within the
droplet is neglected, which can be modeled by a correction for the diffusion coefficient
rather than adding a convection term [151].
The heat conduction equation, describing the conductive heat transfer within the
droplet, is written as
∂T
∂t = α [ ( ∂
r 2 ∂T )]
, (2.56)
r 2 ∂r ∂r
where T is the liquid temperature and α denotes the thermal diffusivity. The above
equation is solved with the following initial and boundary conditions: At t = 0 s, the
droplet is at uniform temperature, T = T 0 .
At the droplet center, r = 0 m, zero
gradient condition prevails at any time, ∂T /∂r = 0. The energy balance at the droplet
surface is given through the boundary condition,
∂T
k l
∂r = h(T ∂R
g − T s ) + L V (T s )ρ l
∂t
(2.57)
at r = R, where R is the droplet radius. In Eq. (2.57), T s denotes droplet surface
temperature, T g stands for gas temperature in the bulk, k l is the liquid thermal conductivity,
h is the convective heat transfer coefficient, and L V (T s ) is the latent heat of
vaporization at the surface temperature, T s .
In this work, first Eq. (2.56) is solved numerically with initial and boundary condition
as defined above using a finite difference method. It is observed that the gradient
in droplet temperature from the center to the droplet surface is very small as the computed
Biot number, which is a measure of heat transfer resistances within and outside
the droplet, (Bi = h/k s R = k g /(2k s )Nu), always remains below 0.5. Therefore, in the
remaining simulations, uniform temperature within the droplet is assumed, which is a
valid assumption as per the revelations made by Mezhericher et al. [173]. The droplet
temperature continuously changes due to heat transfer from ambient gas to the binary
liquid droplet, and it is computed using the energy balance across the droplet, which
gives the net heat transferred into the droplet [62], as
[ ]
dT s
mC pL
dt = Q CpLf (T g − T s )
L = ṁ
− L V (T s ) , (2.58)
B T
where m is the total droplet mass, m = Σ N i=1m i , C pL , C pLf are the specific heat capacity
of liquid and in the film, respectively and B T is the Spalding heat transfer number.
This equation can be used to calculate the time evolution of droplet temperature. Here,
the heat transfer number, B T , is calculated in terms of mass transfer number defined
by Eq. (2.47).
Equation (2.58) needs modification in order to account for the solid layer formation
at droplet surface, and this is achieved through the equation written in terms of the