This document discusses the mathematical modeling of a continuous stirred tank reactor (CSTR). It begins by describing a CSTR and its approximation as a continuously ideally stirred tank reactor. It then presents the mass and energy balances used to develop a model of a CSTR, including a list of variables and assumptions. The balances derived are for total mass, mass of component A, and total energy in the reactor. The document concludes by referencing additional sources on control systems modeling.

1. ICE401: PROCESS INSTRUMENTATION
AND CONTROL
Class 10:
Mathematical Modeling of Continuous
Stirred Tank Reactor Systems (CSTR)
Dr. S. Meenatchisundaram
Email: meenasundar@gmail.com
Process Instrumentation and Control (ICE 401)
Dr. S.Meenatchisundaram, MIT, Manipal, Jan – May 2015

2. Continuous Stirred Tank Reactor (CSTR):
Process Instrumentation and Control (ICE 401)
Dr. S.Meenatchisundaram, MIT, Manipal, Jan – May 2015
• 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 behavior 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):
Process Instrumentation and Control (ICE 401)
Dr. S.Meenatchisundaram, MIT, Manipal, Jan – May 2015
• 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):
Process Instrumentation and Control (ICE 401)
Dr. S.Meenatchisundaram, MIT, Manipal, Jan – May 2015
• A simple exothermic reaction A→B 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):
Process Instrumentation and Control (ICE 401)
Dr. S.Meenatchisundaram, MIT, Manipal, Jan – May 2015
• 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.
• Consider a temperature increase in the feed Ti, causes an increase
in the temperature of the reacting mixture T2'.

6. Continuous Stirred Tank Reactor (CSTR):
Process Instrumentation and Control (ICE 401)
Dr. S.Meenatchisundaram, MIT, Manipal, Jan – May 2015
• 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.
• 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.

7. Mathematical Model of CSTR:
Process Instrumentation and Control (ICE 401)
Dr. S.Meenatchisundaram, MIT, Manipal, Jan – May 2015
• Consider a CSTR shown in figure. A typical model with
associated variables is shown in the RHS.

8. Mathematical Model of CSTR:
Process Instrumentation and Control (ICE 401)
Dr. S.Meenatchisundaram, MIT, Manipal, Jan – May 2015
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

9. Mathematical Model of CSTR:
Process Instrumentation and Control (ICE 401)
Dr. S.Meenatchisundaram, MIT, Manipal, Jan – May 2015
Assumptions:
• Perfect mixing ― it indicates that everywhere in the tank
temperature and concentration are identical.
• Liquid density ρ 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.
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.

10. Mathematical Model of CSTR:
Process Instrumentation and Control (ICE 401)
Dr. S.Meenatchisundaram, MIT, Manipal, Jan – May 2015
Now let us apply the conservation principle on the three
fundamental quantities.
Total mass balance:
(8.1)
where, ρi and ρ are the densities of inlet and outlet stream. Since ρ is
constant, the above equation can be re written as,
(8.2)
Accumulation Input of Output of Total mass generated
of total mass total mass total mass or consumed
=
Time Time Time Time
− ±
( ) 0i i
d V
F F
dt
ρ
ρ ρ= − ±
i
dV
F F
dt
= −

11. Mathematical Model of CSTR:
Process Instrumentation and Control (ICE 401)
Dr. S.Meenatchisundaram, MIT, Manipal, Jan – May 2015
Mass balance on component A:
(8.3)
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,
(8.4)
k0=pre exponential kinetic constant; E=activation energy for the
reaction; R=ideal gas constant.
Accumulation Input of Output of Disappearanceof A
of A A A due to reaction
=
Time Time Time Time
− −
( ) ( )A A Ai i A
d d
n C V C F C F rV
dt dt
= = − −
/
0
E RT
Ar k e C−
=

12. Mathematical Model of CSTR:
Process Instrumentation and Control (ICE 401)
Dr. S.Meenatchisundaram, MIT, Manipal, Jan – May 2015
Substituting, eqn. 8.4 into eqn. 8.3,
(8.5)
(8.6)
(8.7)
Substituting Eqn. 8.2 into 8.7 and rearranging,
(8.8)
( ) ( )/
0
E RT
A Ai i A A
d
C V C F C F k e C V
dt
−
= − −
( ) ( )/
0
E RTiA
Ai A A
FdC
C C k e C
dt V
−
= − −
( ) ( )/
0
E RTA
A A Ai i A A
dCd dV
C V C V C F C F k e C V
dt dt dt
−
= + = − −
( )/
0
E RTA
Ai i A A A
dC dV
V C F C F k e C V C
dt dt
−
= − − −

13. References:
• Modern Control Engineering, 5th Edition, by Katsuhiko Ogata.
• Advanced Control Systems Engineering, Ronald Burns
• Control Systems, Nagoor Kani.
• A course in Electrical, Electronic Measurements and
Instrumentation, A.K. Sawhney.
Process Instrumentation and Control (ICE 401)
Dr. S.Meenatchisundaram, MIT, Manipal, Jan – May 2015