Page 269 - Biomass Gasification, Pyrolysis And Torrefaction Practical Design and Theory

Chapter | 7 Gasification Theory 245

where r k (T s ,L r ) is the surface reaction rate of the kth reaction (one of
R1 R5) at the reactor’s surface temperature, T s , and length, L r . A is its
cross-section area.
Gaseous reactants diffuse to the char surface to participate in k reactions.
Thus, if a jk is the mass of the jth gas, required for the kth reaction, the over-
all diffusion rate of this gas from free stream concentration, y j , to the char
surface, y js , may be related to the total of all reactions consuming the jth gas
as follows:
5
X 2 D gj P
a jk r k ðT s ; L r Þ 5 4πr ðy j 2 y js Þ (7.86)
c
RT g r c
k51
where y js and y j are mole fractions of gas on the char surface and in the bulk
gas, respectively; P is the reactor pressure; and D gj is the diffusion coeffi-
cient of the jth gas in the mixture of gases.
The surface reaction rate, r k (T s ,L r ), may be written in nth-order form as:
2
n
r k ðT s ; L r Þ 5 4πr K sk ðT s ÞðPy js Þ mol=s (7.87)
c
where n is the order of reaction, and K sk (T s ) is the surface reaction rate con-
stant at temperature T s . For conversion of gaseous species, we can write:
5
dF gj X
56 N c A a j;k r k ðT s ; L r Þ (7.88)
dZ
k51
where a j,k is the stoichiometric coefficient for the jth gas in the kth reaction.
The total molar flow-rate of the jth gas is found by adding the contribu-
tion of each of nine gas solid and gas gas reactions:
X
F gj 5 F gj0 1 a jk ξ (7.89)
k
where F gj0 is the initial flow-rate of the gas.

7.6.3.1 Energy Balance
Some of the five equations (reactions R1 R5) are endothermic while some
are exothermic. The overall heat balance of reacting char particles is known
from a balance of a particle’s heat generation and heat loss to the gas by
conduction and radiation.

" #
5
dðW c C pc T s Þ X
52 N c A r k ðT s ; L r Þ ΔH k ðT s Þ
dz
k51 (7.90)

4
4
1 4πr c 2 λ g ðT s 2 T g Þ 1 e p σðT 2 T Þ
g
c
r c
where C pc is the specific heat of the char, ΔH k is the heat of reaction of the
kth reaction at the char surface at temperature T s , e p is the emissivity of the