Gas – Solid Reactions
Types of Models: Heterogeneous, shrinking core.
Homogeneous
Assumptions: Isothermal porous pellet, pseudo-steady state.
Heterogeneous Model
Simple shrinking-core model: An ash layer grows
without change in particle diameter
A( g )  bB( s )  R( g )  S ( s )
énole ùdV
dr
r s ê 3 ú c = r s 4p rc2 c = - kc (C Af - C As )4p R 2b
êëcm úû dt
dt
4
(Vcore  rc3 )
3
1.1. If external resistance is controlling CAs  C Ab
kc C Af 4p R 2b
dVc
=,
dt
rs
Particle burnout time: t * =
Vc = Vc 0 -
kc C Af 4p R 2b
rs
r sR
;
3bkcC Af
t
Vc
t
 1  fs  1  *
Vc 0
t
In Cartesian geometry: t * 
s R
bkcCAf
1.2 If diffusion through the ash layer is controlling:
De
d 2C A
 0,
dr 2
C A (r c )  0 ,
CA  r   
rc
J = De
R
dC A
dr
=
rc
r - rc
CA
=
CAf
R - rc
DeC Af
R - rc
,
C ( R ) = C Af
DeC Af
dVc
dr
= Ar s c = - A
b
dt
dt
R - rc
rs
R- rc
ò
( R - rc )d ( R - rc ) =
R- rc = 0
1


( R - rc )drc =
( R - rc )2 DeC Af b
=
t
2
rs
 2 DeC Af b 
t 

2
 s R 
r  t 2
1 c   *  ,
R t 
- DeC Af b
rs
dt

1
*
1.3 If an ash layer doesn’t grow, the particle diameter decreases.
rs
dVc
dr
= r s 4p rc2 c = - kc (C Af - C As )4p rc2b
dt
dt
To a first order approximate kc is assumed to be independent on rc, and C As  C Af :
kc C Af b
drc
,
=dt
rs
3
t 
 rc 

   1  f b  1  * 
R
 t 
rc
t
1 * ,
R
t
3
2. The general heterogeneous model: spherical pellet with both external and internal
resistances; fast reaction occurs at r=rc at a rate of k’CAC (k’=[cm/s])
Shell balance:
 2C A  0,
C A  C AC
De
dC A
dr
 kc  C Af  C A  ,
R
De
dC A
dr
 k ' C AC
rc

De  1 1
1 
 
k ' rc  rc r
 1 1

 B     C Af

De  1 
De  1
 rc r 
1 
  1 

 k ' rc  rc  kc R  R
To find rc(t) rewrite the solid phase balance (here Cs0=s denote molar solid
concentration):

dC A
d  4 rc 3
Cso   bDe

dt  3
dr

4 rc 2
rc
(*)
Which after substitution of Eqn. (*) yields:
r
 bC Af 
1  1
1  rc 2 
c
dt 

dr





 

c
 De k '  De k g R  R 
 Cso 
After integration from R to rc, the burning time is (c=rc/R):
bCAf
Cso
t
R 1 R 
R2
R
3

1



1  c2   1  c 



c 
3  kc De 
2 De
k'
Burnout time corresponds to c=0 and it varies like R for film-controlled or kineticscontrolled process and like R2 for pore-diffusion control
We can find a relation of time and conversion (x=1-c3).
When film-diffusion controls:
t
Cso R
x
3bkcCAf
while when pore-diffusion controls
t
Cso R 2 
23
1  3 1  x   2 1  x  

bDeC Af 
3. Nonisothermal pellet- Heterogeneous Model
The rate of heat generation and heat removal, assuming the particle to be isothermal, are
h TC  T f  R 2   H  k ' C AC rc2   H  C Af
De
rc 2

De
1 
 k ' rc
1 
De  1
  1 

 rc  kc R  R
rc2
After using Eqn. (*) for CAC. The heat generation curve is temperature dependent [k’(T)]
but also time-dependent (as rc varies with t)