Gas-Liquid Reaction-Two-Film Model
Assumptions: (1) The two phases are well mixed
(2) At the interphase Henry Law is obeyed PAi = HCAi
(3) CAb, CBb are determined by the overall reactor balance. Rection occurs in liquid
GAS
phase
Liquid
CBb
(4) NA = kg (PAb - PAi)
Case I: Slow reaction: reaction in boundary layer is negligible
PAb
N A  kL (CAi  CAb )
i.e
PAb  k g C Ab k L
k
PAb  HC Ai  L  C Ai  C Ab   C Ai 
kg
kL kg  H
or :
 1 H
NA   PAb  HC Ab    
k

 g kL 
1
(1)
Case II: Fast reaction: reaction occurs mostly in b.l.
DA2C A  rA
C A (0)  C Ai , C A ( yL )  C Ab ,
DB2CB   b a  rA
CB (0)  CBi , CB ( yL )  CBb ,

 or :


dCB
 0
dy 0 
First order reaction, rA = kCACB0
C A sinh  1  y yL    C Ab C Ai   sinh   y yL  

C Ai
sinh 
use
kL 
N A   DA
DA
yL
to
estimate
  y L k DA
yL
    C Ab 1 
dC A
 kLC Ai 

 1 
dy 0
 tgh   C Ai cosh  
Liquid utilization factor =  L 
N AaV 1  C Ab  C Ai cosh  

kCAi
tgh
NAav - actual rate/vol., kCAi - rate withtout resistance.
  1  L ,
1
DA
 av
sh
k
sh  k L / a vD A
PAi
CAi
y
CAb
Eliminating CAi from NA and NA = kg (PAb - PAi) yields

HC Ab   1 H tgh 
N A   PAb 

 
cosh    k g k L  

1
(2)
which reduces to theslow reaction solution for   1while for   1 , NA  RAb / k g
Enhancement factor Fe=
NA
rate

k LC Ai simple transport
Fe 


 1
C Ab
1
1


 
tgh 
C Ai cosh  
Case III: Immediate reaction (any order): the reaction occurs at a narrow zone at y = y1
GAS
y  y1 ,
NA
 CA  0
2
y  y1 ,
 CB  0
DB CBb
bN A
a

  y L  y1  N A  DB C Bb
yL  y1
a
b
add , N A
Fe 

a DB CBb 
 k L C Ai  1 
 , where
b
D
C
A
Ai 

D
yL  A
kL

NA
a DB CBb 
 1 

k L C Ai
b DAC Ai 

Fe increase with CBb up to CBb* where y1=CAi=CBi=0
i.e,
For CBb > CBb
*
yL 
a DB CBb*
b DA k g PA
 CBb* 
b NA
a DB k L
the rate is independent of CBb.


aDB
Equating N A  kL  CAi 
CB   k g  PA  HCAi 
bD A


Yields

aDB CB 
N A  K  PA 
,
bD A H 

 1 H
K   
k

 g kL 
CBb
PAb
2
D  C Ai  C A 
D C
 A
 A Ai  y1 N A  DAC Ai
y
y1
NB 
Liquid
1
(3)
CAi
PAi
y
y1
Design of Gas-Liquid Reactor
F [gmol/h]
PAin
C*-C* model:
(a)
F
(b)
F
(c ) N A
yL
PA in  PA
PT
PA in  PA
PT
 NA

L [L/h]
aV V (1   G )
y 0
CB
CBin
int erphasearea

PA

a
L CBin  CB  LC A
b
(4)
aV V (1   G )  rA 1  aV yL V (1   G )  LC A
transport to bulk  reaction in bulk  flow out
Eqn. (c) is of importance only for fast reactions.
Case I: slow reaction
F
PA in  PA
PT
 K  PA  HC A  aV V (1   G )  rAV (1   G )  LC A
with K from Eqn.(1). e.g., rA  kC A ,  l 
CA 
F
V (1   G )
L
KaV PA
KaV H  k 
PA in  PA
PT
1
l
 KaV V (1   G ) PA
Case II: Fast reaction
k l  1
 find PA
KaV H l  k l  1

HC A 
N A 0  K  PA 
,
cosh  

(5)
 1
H 
Very fast (>>1, CA0) NAKPA, use K from eqn.(2) K   

k
 g k L 
F
PA in  PA
PT
 KPAVaV (1   G ) 

a
L CBin  CB
b
1

CAout
CBout
PAout
PFR-PFR co-current
(a)
F
 PA V  PA V dV   N A
PT
PA in  PA
(b)
F
(c )

 NA

PT
yL

y 0
aV dV (1   G )  

F dPA
 NA
PT dV
y 0
aV (1   G )

a
L CBin  CB  LC A
b
PA
CA
CB
dV
(6)

aV  rA  dV (1   G )   L  C A V  C A V  dV   LdC A

If the reaction is very fast or instantaneous, CA=0, and Eqn. (c) is
irrelevant; e.g for instantaneous reaction:
N A  k LC Ai Fe
PAin
CBin
PAout
CBin
aDB CB 

Fe  1 
 enhancement factor 
bD A C A 


aDB CB 
F dPA
  K  PA 
 aV (1   G ),
PT dV
bD A H 

Substitute Eqn. (4b) CB  CBin 

 1 H
K   
k

 g kL 
1

F
b
PA in  PA
PT
aL
To find
 D
dPA
F  DB  a CBin FPA in 

 PA 1  B




aV (1   G ) KPT dV
 DA PT LH  DA  b H PT LH 
F
And solve by simple integration.
PFR-PFR counter-current
Eqns. (6a & c) still apply. With change of sign to -LdCA.
PA
CA
CB
dV
Eqn. (6b) should be modified.
F
PA in  PA
PT



a
L CB  CBout  L(C Aout  C A )
b
PAin
CBout
CAout