THERMOCHEMICAL CONVERSION: DIRECT COMBUSTION - Biomass as a Sustainable Energy Source for the Future - page 281

Using this temperature profile, we know the (conductive) heat transfer to the surface
of the droplet, which should balance the heat of vaporization:

r=r s

r s ∂

T
∂

4 πλ = mh fg _ ð Eq : 9 : 22 Þ

r

in which h fg is the heat of vaporization. Taking the derivative of the temperature at the
particle surface,

r s

0 1

r=r s Z m

ð T ∞ − T boil Þ exp −

∂ T

∂

= Z m
r s

@ A

r s

ð Eq : 9 : 23 Þ

r Z m

1 − exp −

the mass flow rate becomes

ln B q +1

4 πλ r s

_ m=

ð Eq : 9 : 24 Þ

c p

with B q =c p (T ∞ − T boil )/h fg being the Spalding number B for heat transfer (q) by evap-

oration. If, e.g., combustion is considered, we also have to take into account the
release of heat by the conversion. This can be easily taken care of by involving this
quantity in the definition of the Spalding number (this is shown in Problem 9.4).
Now, the lifetime of a droplet is determined from the mass flow rate according to

∂ m d

∂

= − m ð Eq : 9 : 25 Þ

t

d d = with the mass of a droplet as a function of the diameter being m d = ρ d V d = ρ d π 6.

d d
∂

For the change of the diameter as a function of time, we can write ∂

t =3d d ∂

d d
∂

t

and thus

∂ d d

∂

λ
ρ d c p d d ln B q +1

4

= − ð Eq : 9 : 26 Þ

t

In the expression for the lifetime of a droplet, the diameter squared is generally used,
so we write Equation (9.26) as

d d
∂

=

∂

= −

λ
ρ d c p ln B q +1

8

: :

− K ð Eq 9 27 Þ

t

which shows that the diameter squared decreases linearly with time. This equation is
important since it is the definition of the evaporation constant K. Integration yields

d d

ð ð

t d

d d d = − K d t ð Eq : 9 : 28 Þ

0

d d , 0