The isomerization reaction
is carried out in a CSTR with a heat exchanger. We want to construct a stability diagram. First we will consider a zero order reaction with
Next, we will consider a first order reaction to be with
Construct the stability curves for a zero order reaction, (S_{0}), and a first order reaction, (S_{1}), as a function of T_{C}.
Additional Information
The ratio of inerts to A fed is 2 to 1.
Solution for Zero Order Reaction
(1) 
For X^{*}<1: 
(2) 
Otherwise X^{*} = 1 
Choose T_{C} and solve for T^{*} using the Polymath program in Table 1.

(3) 
We shall use the same Polymath program shown in Table 1 to solve for S_{0} and S_{1}. Note, in the Polymath program for this zero order reaction we let T_{1} = T^{*} and then solve Equation (4) for T_{1}.

(4) 
After solving for T_{1} (i.e., T^{*}), we substitute for T_{1} to find X^{*} in Equation (2) and then also into Equation (1) to find S_{0} at the chosen T_{C}

(5) 
We can calculate the conversion (X_{1}) in the Polymath program for this zero order reaction

(6) 
Provided X_{1} ≤ 1. Otherwise X_{1} = 1. Choose another T_{C} and repeat solution (Equations (3) through (6)) to construct S_{0} versus T_{C} in Table 2 which we will use to find regions of stability
Solution for First Order Reaction
Use Polymath to solve in Equation (7) for T^{*}

(7) 
Note: In the Polymath program we let T^{*} ≡ T for a first order reaction
Evaluate X^{*} at this T_{C}

(8) 
Calculate S_{1} at this T_{C}

(9) 
The Polymath solution for T_{C} = 475K is given below. We now choose another value of T_{C} and repeat to find T^{*} and then S_{1} to construct a table of S_{1} versus T_{C} as shown in Table 2.
Table 1
Table 2
When this program is repeated for different values of T_{C}, Figure 1 can be developed using Table 2 where both S_{0} and S_{1} are plotted as a function of T_{C} on Figure 1.
Figure 1. Stability plot.
We now can find the region of stability and of runaway

(10) 
Recall that

(11) 
then taking the derivative

(12) 

(13) 
also recall

(14) 
When

(15) 
runaway will NOT occur. However,
when (i.e., )
runaway WILL occur.
We see that for the first order reaction is globally stable for all T_{C}. However, we see the zero order reaction will run away when the entering temperature T_{C} is increased above 460K. We also note the runaway reaction becomes stable again as T_{C} is further increased to values above 685K.
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