THERMOCHEMICAL CONVERSION: DIRECT COMBUSTION - Biomass as a Sustainable Energy Source for the Future - page 280

The basic derivation in case of vaporization proceeds as follows. We start with the
observation that in a steady conversion, the mass flow rate, r 2 m=4 _ π ρ g v r , remains the

same in the gas surrounding the particle (continuity):

r 2

∂ ρ g v r

=0 ð Eq : 9 : 14 Þ

∂ r

Now, the energy equation can be written as a balance equation for convection and
diffusion, which in terms of temperature becomes

=0

− ∂
∂

∂
∂

ρ g D ∂

T
∂

r 2 r 2 ρ g v r T ð Eq : 9 : 15 Þ

r r r

We can use continuity, r 2 ρ g v r = m = 4 π , to arrive at

=0

m
4

_ ∂ T

∂ r − ∂

ρ g D ∂

T
∂

r 2

ð Eq : 9 : 16 Þ

π ∂ r r

and assume for the Lewis number, being Le = is the ther-

mal conductivity and c p the specific heat capacity. Then Equation (9.16) becomes

λ /(c p ρ g D ), Le = 1. Herein, λ

=0

c p
λ

m
4

∂ T

∂ r − ∂

ρ g D ∂

T
∂

r 2

ð Eq : 9 : 17 Þ

π ∂ r r

Introducing Z =c p /(4 πλ ) and integrating give

r 2 ∂

T
∂

r = Z

mT + C 1 _ ð Eq : 9 : 18 Þ

Integrating once again results in a solution for the temperature

= −

C 1 Z _ m

r

Tr ðÞ

m +C 2 exp

− ð Eq : 9 : 19 Þ

Z _

Boundary conditions can be formulated at the droplet surface and far away from the
droplet, T(r = r s )=T boil and T(r ! ∞ )=T ∞ . Substituting these solutions, it is easy to

derive

T ∞ − T boil

r s

C 2 = ð Eq : 9 : 20 Þ

Z m

1 − exp −

and C 1 =C 2 − ð T ∞ Þ Z m, leading eventually to the temperature distribution

_

r s +T boil

−

Z m
r

Z m

ð T ∞ − T boil Þ exp − T ∞ exp −

r s

Tr ðÞ = ð Eq : 9 : 21 Þ

Z m

1 − exp −