ChE 528
Lecture 35A
Runaway Reactions
CSTR
We are going to consider CSTR has a runaway reaction when it goes to the
upper steady state which is at an unacceptably high temperature. This transition
to the upper steady state will occur when the point of tangency between R(T)
and G(T) disappears and only the upper steady state exists.
r V 
GT  H Rx  A 
(8-68)
 FA0 


R T   CP 1  T  TC 
~
~
~
~
(8-69)
~
~
R(T)
G(T)
TC
T*
At the point of tangency T = T*,

i.e.,

G(T )  R(T )
and


R G

dR 
dG

dT T  dT T 



 
T  TC
dR 
R
 CP 1    CP 1   
 
dT T
T  TC
T  TC
R(T )2
T  TC 
E

1
Runaway Reactions (10/99)

Derive
Derive
dG H RxV drA 

dT
FA0
dT

E RT
drA  d Ae

dT
dT
fC 
assume fCi   constant
i
 Ae E RT fC i 
rA
E
E

r


A
RT 2
RT 2
dG HRx rA V E
GT

E
2 
dT
FA0
RT
RT 2
At the point of tangency T 
dG
E
 G(T ) 2

dT T 
RT
E
R
T  To
G
, recall TC  a
 2  C P 1     
1 
R(T )
T  TC

But : R   G 
E
1
 
 2
R(T )
T  TC
T  TC 
R(T  )2
E
This equation is Eqn. (8-77) on p.498 of the text
T 
E 
4RTC 
1

1
2R 
E 


Derive
Put in dimensionless (TD) form by letting
T
E
TD 
and r 
TC
RTC
Dividing by TC
2
T
RTC T 
 
1 
TC
E TC 
1 2
TD  TD  1  0
r
2
Runaway Reactions (10/99)
Derive
  4 1 2 
r1  1   
  r  
TD 
2
T   TC TD 
E 
4RTC 
1 1 

2R 
E 

dR dG
the point of tangency will disappear and the system will be

dT dT
considered an unstable because it moves to the upper steady state. Combining the Eqn.
directly below (8-74) and Eqn. (8-75) in the text
If
dG  HRx rAV E

dT
FA 0
RT 2
 GT 
stable if
E
RT 2
 
C P0 1     G T
 
Let S  G T 
E
 
R T
2
E
 
R T
2
 f A,E,  HRx , ,TC 
We are now going to calculate S  as a function of one parameter, say TC. Therefore
Fix the other parameters E, A,  HRx , 
1. Specify TC

2. Calc T 
E 
4RTC 
1

1
2R 
E 


3. Calc k  A e E RT

4. Calc X 

k

1k 
 


5. Calc G T  X H Rx 

6. Calc S 
E
 
RT
 2
 
G T
7. Increment TC and carry out steps 2 through 6 to arrive at the following
figures of S* as a function of TC
3
Runaway Reactions (10/99)
If CP0 1    50
S*
50
Stable
Unstable
CP (1+)
TC
If CP0 1    10
S*
50
CP (1+)
10
Stable
Unstable
TC


Consequently the line S  divides the place into regions for values of C P0 1    will be
either stable and unstable.
4
Runaway Reactions (10/99)
RUNAWAY IN PLUG FLOW REACTORS
Phase Plane Plots
We transform the above temperature and concentration profiles into a phase
plane in the following manner.
Figure 2. CA and T profiles
Figure 3. CA – T phase plane
For example, at volume V2 the temperature is T2 and the concentration is CA2
Now increase the entering concentration CA0
Runaway
Tm3
CA0 3
Temperature Profiles
for
CA0 4 > CA0 3 > CA0 2 > CA0 1
CA0 4
T
Tm2
CA0 2
Tm1
CA0 1
V
Figure 4. Temperature profiles
5
Runaway Reactions (10/99)
Figure 5. Concentration profiles for different values of CA0.
CA0 3
CA0 2
CA
CA0 1
CAm 3
CAm 1
CA0 3 > CA0 2 > CA0 1
T m1
T m2
T m3
T
Figure 6. CA – T phase plane Temperature-concentration phase plane plot
for different entering concentrations.
Now let’s plot the maximum temperature for each entering concentration, CA0,
and the corresponding reactor concentration at his maximum temperature.
2
CAm
1
3
TM
Tm
Figure 7. Phase plot of value temperature at maximum and corresponding
concentration at the maximum temperature.
6
Runaway Reactions (10/99)
Figure 4 shows the temperature profile for three different entering concentrations
[CA03 > CA02 > CA01]. Figure 5 shows the concentration profiles for the same set of
entering concentrations. Figure 7 shows a phase plot of C Am as a function of Tm. This
plot (Figure 7) could also have been obtained directly form the equation
Qr
C Am 
UaTm  Ta 
HRx Ae E RTm 
Derive
Qg
Derive
The governing equation for a PFR/PBR
Mole Balance:
vo
dCA
 rA
dV
dT UaTa  T   rA H Rx 

dV
 FiCp i
Energy Balance:
Solutions to these equations give the following typical profiles in the absence of
runaway.
Temperature and concentration profiles.
At the maximum temperature in the reactor, Tm
dT
 0,
dV
then
Ua Tm  Ta  rAm  HRx 
For a first order rxn: rA  kCA , then rAm  kCAm and we can solve for CAm to obtain
CAm 
UaTm  Ta 
, with k  AeE RTm
H Rx k
where Tm is the maximum temperature in the reactor and CAm is the corresponding
concentration of A at that maximum temperature. For this reactor To  Ta .
7
Runaway Reactions (10/99)
If Tm is above the TM (the temperature at which CAm is at its maximum value) the
C Am 
Qr
Q 
g
the Qg increases more rapidly than the heat generated “prime” (It’s “prime” because it
does not include CA). In other words the rate of increase of temperature dependent part
of Qg is greater than the rate of increase of the temperature dependent part of the heat
of removal, Qr
The following figure shows a plot of CAm vs. Tm for different ambient temperatures
(Ta  Tw)
C AM
C Am
Ta1=T o1=500K
500 550 600 650 700
Tm
Ta1 Ta2
Figure 8. Maximum curves for a few wall temperatures. Simplified curve.
We note that the concentration CAm goes through a maximum as the temperature Tm is
varied. Different values of this maximum result for different values of Ta(Tw). The line
Ps is the locus through these maximum values.
We want to see how the concentration of A at the maximum temperature CAm
varies with the maximum temperature Tm. Specifically we want to find the maximum
value of CAm and the corresponding temperature Tm.
dk
Ua kH Rx   Ua Tm  Ta HRx
dCAm
dT
0
2
dTm
 H Rxk
dk
E

k
dT RT 2
TM  Tm Max 
CAM  CA m Max 
1 E
E E



4T
a

2 
R R
R

UaTM  Ta 
with kM  Ae E RTM
HRxk M 
8
Runaway Reactions (10/99)
Criteria 1. The trajectory going through the maximum of the “maxima curve” is
considered as critical and therefore is the locus of the critical inlet conditions
for CA and T corresponding to a given wall temperature.
Figure 7. Critical trajectory on the CAm – Tm phase plane plot.
The critical trajectory goes through Tm. The locus of the maximum CAM from
Figure 8 is also shown on this figure. What is the inlet concentration that is related to
this critical trajectory? The safe inlet concentration can be found by using adiabatic
conditions
FA0 X H Rx  FA0  i C p i T  To 
FA0  FA 
C po
FA 0 
1
FA
C p o T  To 
HRx 

 v 

 CAm


v o 
C

A
0

C p o TM  To  
1



 HRx


Safe operation will occur
below this entering
concentration of A, C A0
CRITERIA BASED ON INFLECTION POINTS
9
Runaway Reactions (10/99)
Figure 9. Temperature profile showing inflection points.
Let Fi Cpi  CA0voCp
CA0Cp
At the inflection point
d 2T
d
2
dT
 rA HRx   UaTa  T 
d
0
CA0 Cp
d 2T
d
2
 0  H Rx 
drA 
d
 Ua
dT
d
At
 HRx  rAi   UaTa  Ti 

E

H


r


Ua
 k 
Rx
Ai
i rAi HRx   0
2



C A0 C p
RTi



Derive
rA  k fA c i 
drA 
d

dk
df
fn c i   k
d
d
df dC A dC A df

,
d dC A
d dC A
and from the mole balance
dCA
 rA
d
10
Runaway Reactions (10/99)
Derive
k
kdf kdf

rA  k  rA
d
dC A
dk
E dT kE dT
 Ae E RT

d
RT 2 d RT 2 d
E dT
dT
0   HRx  rA
 k  rA  Ua
2
 RT d

d
0
E

dT
H Rx rA  2  Ua
 k rA HRx 


d
RT
If we have a first order reaction
rAi  kiCAi  Ae E RTi CAi
Then we can solve this equation to find CAi as a function of Ti. To find the locus
of the inflection points.
For a first order reaction
rAi  kiCAi
12
CAi
2
Ti  Ta  i    i  Ta  Ti  i    i   4 i 2Ta  Ti 

2 i ki
Derive
Dividing by  HRx  multiplying by CA0Cp and rearranging we obtain

UaTa  Ti  
E
Ua  kiC A 0 C p
rAi 
rAi


rAi  0

 HRx  
RTi2  HRx 

 

 HRx 
rAi  Ta  Ti  rAi i   i  rAi   0
where

i 
i 
Ua
H Rx 
k i C A0 C p
H Rx 
E
RTi2
 i rAi    rAi Ta  Ti  i    i   2i Ta  Ti   0
2
rAi 
T  T   
i
a
i

2
2 i
11
Runaway Reactions (10/99)

 i  Ti  Ta  i    i   4 i2i Ta  Ti 
12
Derive
Criteria 2.
Runaway will occur when the trajectory starting at CA0 and To intersects the
locus of the inflection points
A slight increase in conditions
above the inflection locus and we
have runaway
CA03
CA02
CA01
Runaway
Ti – CAi
T
Figure 10 T–CA phase plane with the locus of inflection CAi–Ti points.
So how far do we back off from the inflection point conditions 10°C, 0.1 mole/dm3?
These are arbitrarily set numbers (e.g., 10°C) and we need a criterion based on the
intrinsic properties of the system and not on arbitrarily limited temperature range.
Therefore use the locus of CAM as a function of TM.
Why use locus of maximum, CAM vs. TM?
Figure 11. CA – T Phase plane showing critical trajectory intersecting the locus of the
inflection points and locus of maximum TM–CAM from Figure 8.
You want to stay away from the intersection of the CA/T trajectory and the locus of the
inflection points.
12
Runaway Reactions (10/99)
C A0
CA
CAi – T i
C AM – T M
T
Figure 12
Criteria 2A. (Conservative Criteria) Runaway will occur if the trajectory starting at C A0
and To intersects the locus of the maxima of CAm and Tm for different values of To (i.e.
Tw because To = Ta = Tw) (See Figure 5, p. 3)
CA
C AM (Ps)
CAm 
TM
Ua Tm  To 
k m H Rx
T
Figure 13. CA – T Phase plane plot showing locus of maxima.
Criteria 3.
Runaway reaction figure from Froment-Bischoff (FB) “Chemical Reactor Analysis
and Design”
T  To
H RxX
  ad

To
To   i C p i
For X  1 ,  
Comparing the term multiplying
(Eqn 8-56) for CP = 0
dT
in Froment-Bischoff (Eqn 11.5.2-2) with Fogler
dX
UsgCpm  Uo CA0 i Cpi ,
13
Runaway Reactions (10/99)
 HRx
 i C p i To
therefore:
  iCp i 

 g C pm
CA 0
HRx CA 0 

E HRxC A 0 


S    
RTo2  g Cp m 


g C p To

E
RTo
N
4
U
d t M m C p m rA 0 
N 4U
1
RTo2 C T

S
d t rA0 HRx  E C A0
Derive
Derive
N
4U
RTo2

  g Cp m
S d tM m C p rA 0 
EH Rx C A0
m


g
4U 
RTo2
P
 g 

,
 CT 
d t rA 0  M m  EHRx C A0
Mm
RTo

4U
dt
1
RTo2 C T
rA0 HRx  E CA0
For no pressure drop CT = CTo
Note: The equation for (N/S) in Froment and Bischoff is not correct. If you check the
units you will find (N/S) is not dimensionless.
where
g = gas density (kg/m3)
Mm mean molecular weight (kg/kmol)
dt = Tube diameter, (m)
CT = total concentration (kmol/m3)
CA0 = Entering concentration of A (mol/m3)

J

U = overall heat transfer coefficient
2
m s K 
 J 

Cpm = heat capacity 
kg K 
R = Gas constant (J/mol•K)
E = Activation Energy (J/mol)
14
Runaway Reactions (10/99)
HRx = Heat of Reaction (J/mol)
–rA0 = rate of reaction (mol/s• m3) at To, k  Ae E RTo
To = temperature (K) (To = Tw = Ta)
n = reaction order
Figure 14. Runaway diagram ( Adapted from G. F. Froment and K. B. Bischoff, Chemical
Reactor Design and Analysis, 2nd ed. New York: John Wiley and Sons, 1990).
15
Runaway Reactions (10/99)
Figure 9. CA – T Phase plane showing critical trajectory intersecting the locus of the inflection
points.
16
Runaway Reactions (10/99)