THERMOCHEMICAL CONVERSION: DIRECT COMBUSTION - Biomass as a Sustainable Energy Source for the Future - page 275

has decreased to zero some distance away from the carbon surface. However, CO 2 can
then act as an oxidizing agent and react with the surface carbon to form CO. This CO
will be transported away from the surface by convection, and at some distance, it will
be converted with O 2 to form CO 2 . Thus, high-temperature combustion of carbon
particles can be modeled using a two-step process:

C+CO 2 ! 2CO

ð RX : 9 : 7 Þ

2CO+O 2 ! 2CO 2 ð RX : 9 : 8 Þ

with the first reaction occurring at the surface and the second in the gas phase.

9.3.1 Shrinking Density Model

Here, the shrinking density model of Figure 9.2, in which reaction takes place inside
the particle, resulting in a density decrease, is considered. If the particle is assumed
to be isothermal, the mass conservation equation for oxygen in spherical coordinates
can be written as

+ s

+ 1 = 1

∂
∂ r 2 ∂ r 2 ∂

ρ g D eff ∂

Y O 2
∂

r 2 r 2

ε ρ g Y O 2 ε u ρ g Y O 2 ð Eq : 9 : 1 Þ

t ∂ r ∂ r r

where ρ g is the gas density, u is the gas velocity, and D eff is

the effective diffusion coefficient. The first term on the left hand side is the temporal
change, and the second term represents convection of oxygen. On the right hand side,
the first term represents diffusion of oxygen, and the second term, s, is the source term
that describes the chemical consumption of oxygen due to conversion. The relation
between the effective diffusion coefficient inside a char particle and the molecular
diffusion coefficient D can be estimated from the porosity of the particle:

ε is the particle porosity,

D eff = ε
τ

2 D

D ε ð Eq : 9 : 2 Þ

This is an approximate equation where the inverse of the tortuosity is assumed to be

almost equal to the porosity. So the transport equation for oxygen Equation (9.1) can
be used for the situation inside and outside the particlewhere the effective diffusion coef-
ficient is the molecular diffusion coefficient outside the particle and the porosity modu-
lateddiffusion coefficient inside theparticle. Furthermore, weassume that the systemis in
quasi-steady state and that the carbon present in the char is converted to CO 2 . The latter
assumption (which is a simplification since carbon also can be converted to CO) makes
the reaction equimolar, and no gas flowwill therefore be present inside or outside the char
particle. The equation tobe solved is then reduced to a balance containing only a diffusion
term and a source term. Assuming first-order kinetics, the source term can be written as

τ

s= − k O 2 ρ g Y O 2 ð Eq : 9 : 3 Þ