Mathematical modeling of continuous stirred tank reactor systems (cstr) dependent quantities for CSTR:

1. Mathematical Modelling of
Continuous Stirred Tank
Reactor Systems (CSTR)

2. Continuous Stirred Tank Reactor (CSTR):
 The continuous flow stirred-tank reactor (CSTR), also known as vat-
or backmix reactor, is a common ideal reactor type in chemical
engineering.
 A CSTR often refers to a model used to estimate the key unit operation
variables when using a continuous agitated-tank reactor to reach a
specified output.
 The mathematical model works for all fluids: liquids, gases, and
slurries.
 The behaviour of a CSTR is often approximated or modeled by that of
a Continuous Ideally Stirred-Tank Reactor (CISTR). All calculations
performed with CISTRs assume perfect mixing.

3. Continuous Stirred Tank Reactor (CSTR):
 In a perfectly mixed reactor, the output composition is identical to
composition of the material inside the reactor, which is a function of
residence time and rate of reaction.
 If the residence time is 5-10 times the mixing time, this approximation
is valid for engineering purposes.
 The CISTR model is often used to simplify engineering calculations
and can be used to describe research reactors.
 In practice it can only be approached, in particular in industrial size
reactors.

4. Continuous Stirred Tank Reactor (CSTR):
 A simple exothermic reaction AB takes place in the reactor, which is in
turn cooled by the coolant that flows through a jacket around the
reactor.
 The curve that describes the amount of heat released by the
exothermic reaction is a sigmoidal function of the temperature T in the
reactor as shown in the figure (Curve A).

5. Continuous Stirred Tank Reactor (CSTR):
 On the other hand, the heat removed by the coolant is a linear function
of the Temperature T (Curve B).
 When CSTR is at steady state, the heat produced by the reaction
should be equal to the heat removed by the coolant.
 This yields the steady states P1, P2 and P3 at the interaction of the
curves A and B.
 Steady states P1 and P3 are called stable, whereas P2 is unstable.
The stability can be explained as:
 Assume that the reactor is started with a temperature T2 and the
concentration CA2.

6. Continuous Stirred Tank Reactor (CSTR):
 Consider a temperature increase in the feed Ti, causes an increase in
the temperature of the reacting mixture T2'.
 At T2', the heat released by the reaction (Q2') is more than the heat
removed by the coolant, (Q2''), thus leading to higher temperatures in
the reactor and consequently to increased rates of reaction.
 Increased rates of reaction produce larger amounts of heat released by
the exothermic reaction, which in turn lead to higher temperatures and
so on.

7. Continuous Stirred Tank Reactor (CSTR):
 An increase in temperature will eventually reach the value of steady
state T3 as well a decrease in temperature will reach T1 shown in
figure.

8. Mathematical Model of CSTR:
 Consider a CSTR shown in figure. A typical model with associated
variables is shown in the RHS.

9. Mathematical Model of CSTR:
 List of Variables:
Fi = input flow rate
 F0 = output flow rate
 Fc = coolant flow rate
 cAi, cA = input and output concentration of A (moles/volume)
 Ti = input temperature of feed
 T = output temperature
 Tci = input temperature of coolant
 Tco = output temperature of coolant
 V = volume of the reacting mixture in the tank

10. Mathematical Model of CSTR:
Assumptions:
 Perfect mixing it indicates that everywhere in the tank temperature
and concentration are identical.
 Liquid density and heat capacity Cp are constant.
 No heat loss to the surrounding from the reactor.
 Coolant is perfectly mixed and no energy balance for coolant.
 The momentum of the CSTR does not change under any operating
conditions and will be neglected.

11. Mathematical Model of CSTR:
Fundamental dependent quantities for CSTR:
 Total mass of the reacting mixture in the tank.
 Mass of chemical A in the reacting mixture.
 Total energy of the reacting mixture in the tank.
Now let us apply the conservation principle on the three fundamental
quantities.
Total mass balance:

12. Mathematical Model of CSTR:
 where, i and are the densities of inlet and outlet stream. Since is
constant, the above equation can be re written as,
 Mass balance on component A:

13. Mathematical Model of CSTR:
 where, r is the rate of reaction per unit volume; CAi, CA is the molar
concentrations (moles/volume) of A in the inlet and outlet streams and
nA is the number of moles of A in reacting mixture,
 k0=pre exponential kinetic constant; E=activation energy for the
reaction; R=ideal gas constant.

14. Mathematical Model of CSTR:
 Substituting, eqn. 8.4 into eqn. 8.3,
 Substituting Eqn. 8.2 into 8.7 and rearranging,