TRANSFER: BASICS OF MASS AND HEAT TRANSFER - Biomass as a Sustainable Energy Source for the Future - page 112

steady-state fuel vapor conservation equation, but taking into account a fixed
droplet radius as boundary condition. Following a similar reasoning, provided
the evaporation process is fast compared to the transport of heat and mass to
the surface, the fuel mass fraction at the surface Y s can be obtained from equilib-
rium thermodynamics. The equilibrium vapor mole fraction at the surface, X s ,at
saturation temperature T sat is obtained from the saturation pressure p sat determined
by the Clausius - Clapeyron equation, relating saturation temperature and saturation

pressure:

X s = p sat

h fg
R u =

1
T boil −

1
T sat p atm = exp

ð Eq : 4 : 34 Þ

MW F

where h fg is the latent heat of vaporization of the fuel, R u is the universal gas constant,
and MW F is the MW of the fuel.
The vapor mass fraction is obtained using the relation between mole fraction and
mass fraction in a mixture of vapor and air:

MW F
X s MW F +1

: :

Y s =X s ð Eq 4 35 Þ

ð − X s Þ MW air

Deviations from the phase equilibrium at the surface can occur when the evaporation
is very fast, occurring for small droplets and for droplets with a temperature very close
to the boiling point. Bellan and Summerfield (1978) have presented expressions for
evaporation rate taking into account nonequilibrium effects.
The solution of Equation (4.32) is (if

ρ D is assumed constant)

exp ½ − m d = ð 4 π r d ρ D Þ ð 1 − ð r d = r Þ Þ − 1

Yr ðÞ =Y s +Y b − ð Y s Þ for r ≥ r d ð Eq : 4 : 36 Þ

exp ½ − _ ð m d = 4 π r d ρ D Þ − 1

From Equation (4.31) and Equation (4.36), it follows that

D ∂

Y
∂

1

r 2

m d = m d Y s +4 π ρ = m d Y s + m d Y s − ð Y b Þ ð Eq : 4 : 37 Þ

r m d _

r d

exp − − 1

4 π r d ρ D

This equation can be solved for m d with the following result: _

m d = _ − 4 π r d ρ D ln B M +1 ð Þ ð Eq : 4 : 38 Þ

Here, B M is the Spalding mass coefficient:

B M = Y s −

Y b

ð Eq : 4 : 39 Þ

1 − Y s