INAUGURAL–DISSERTATION zur Erlangung der Doktorwürde der ...

2.4. Single Droplet Modeling 31
the fluid flows in which there is simultaneous momentum and mass diffusion, and
it is defined as the ratio between momentum diffusion and mass diffusion, written
as Sc = µ f /(ρ g D f ).
In Eq. (2.43), the function f(Re d ) depends upon the droplet
Reynolds number, and in case of low Reynolds number, it may be calculated as defined
by Abramzon and Sirignano [62], with f(Re d ) = 1 for Re d ≤ 1 and f(Re d ) = Re d
0.077
for Re d ≤ 400.
In Eq. (2.41), B M is the Spalding mass transfer number, expressed in terms of the
mass fraction of vaporized liquid as,
B M = Y s − Y ∞
1 − Y s
. (2.44)
Here Y s and Y ∞ are mass fractions of the water at the droplet surface and in the bulk
of surrounding gas, respectively.
Y s is computed from the vapor-liquid equilibrium
through the vapor pressure of water, which is written as [148]
Y s =
M w
M w + ¯M(¯p/p w − 1) . (2.45)
The quantities M w and p w denote molar mass and vapor pressure of water while
and ¯p represent molar mass and mean pressure of the surrounding gas, respectively.
Although the initial temperatures of gas and the droplet are equal and are at
room temperature, the droplet temperature is subject to change due to evaporation.
Time evolution of droplet temperature for water spray is computed using the uniform
temperature model [62],
[ ]
dT s
mC pL
dt = Q CpLf (T ∞ − T s )
L = ṁ
− L V (T s ) , (2.46)
B T
where m is the droplet mass, Q L is the net heat transferred to the droplet per unit
time, C pL and C pLf are the specific heat capacity of the liquid and in film, respectively,
T s is the temperature at droplet surface, T ∞ is the temperature of the surrounding
gas, and L V (T s ) is the temperature dependent latent heat of vaporization at T s . B T is
the Spalding heat transfer number, which is calculated in terms of the mass transfer
number using the relation [62]
where the exponent φ is given by [62]
B T = (1 + B M ) φ − 1, (2.47)
φ = C pL
˜Sh 1
C pg Ñu Le . (2.48)
Here C pg is the specific heat capacity of the gas, Le is the Lewis number, and
¯M
Ñu is
the modified Nusselt number, which accounts for convective droplet heating, and it is