TRANSFER: BASICS OF MASS AND HEAT TRANSFER - Biomass as a Sustainable Energy Source for the Future - page 123

The thermal conductivity of a porous particle λ eff depends on the conductivity of

the solid and gaseous parts and the porosity ε :

" !

λ C

#

ρ g
ρ g , 0 λ B +1

− ρ g
ρ g , 0

λ eff = ε λ g +1 ð − ε Þ ð Eq : 4 : 66 Þ

In general, also radiation can contribute to transfer of heat through the particle, and
provided the temperature differences are small (less than a few 100 K), this effect
can be taken into account by an additive term in the thermal conductivity (see Kaviany
(1995), page 340, and Chapter 11, Appendix).
To obtain a complete description of the problem, initial conditions and boundary
conditions must be known. As initial conditions, we can assume an isothermal bio-
mass particle with only inert gas in the pores:

p t = 0, r ð Þ = p atm

Tt=0,r ð Þ =T 0

u t = 0, r ð Þ =0 ð Eq : 4 : 67 Þ

Y I t = 0, r ð Þ =1

Y T t = 0, r ð Þ =Y C t = 0, r ð Þ =Y G t = 0, r ð Þ =0

As boundary conditions, we have to impose that at the particle center of symmetry, all
fluxes are zero:

r=0 r=0 r=0

∂ p t, r ðÞ

∂

= ∂

Tt,r ðÞ

∂

= ∂

Y i t, r ðÞ

∂

=0

r r r

ð Eq : 4 : 68 Þ

u t, r = 0 ð Þ =0

As boundary conditions at the particle outer surface, we have to impose equality of the
pressure and equality of the fluxes at both sides of the particle boundary. The flux at
the inside is the effective diffusive flux; the flux at the outside is the sum of the con-
vective and radiative fluxes:

= p atm

p t, r = r p

r=r p

λ eff ∂

Tt,r ðÞ

∂

T w − T 4

= θ T hR SA T b − ð T Þ + R SA ε e σ

r

ð Eq : 4 : 69 Þ

r=r p

D eff ∂

Y i t, r ðÞ

∂

= θ m kR SA Y i , ∞ − Y i , s

r

Here, h and k are heat and mass transfer coefficients valid in the absence of outflow of
gas from the particle. θ m and θ T are factors representing the effect of outflow of gas