# 9. Unsteady State Non-Isothermal Reactor Design*

## Topics

1. Batch Reactors with Heat Effects Example
2. Control of Chemical Reactors
3. Linearized Stability Theory
4. Predicting the Behavior of a CSTR using LST

 Batch Systems with Heat Effects Example top

Balance on a system volume that is well-mixed:

Adiabatic batch reactor with no work:

#### Polymath

The following reaction occurs in a batch reactor:

 1) 2) 3) 4) 5) 6) 7) Parameter Values

or use one of the integration formulas, e.g.: , to find the reaction time, t. Even better, use Polymath.

Cooling:

Batch Reactor with External Heating
Text Example 9-2
Questions related to Problem 9-2

 Control of Chemical Reactors top
 Unsteady State CSTR: For a batch reactor, FAO = 0

#### Integral Control

 For the reaction     in a CSTR:

#### Proportional-Integral Control

 Linearized Stability Theory top
 Energy Balance (Applied to a CSTR) 1. 2. 3. 4. CSTR Mole Balance 5. 6. Manipulating the Energy and Mole Balances Let andsignify steady state values. Then at steady state: 7. Adding equations (6) and (7): 8. Linearizingby expanding it in a Taylor Series: To obtain: 9. 10. Let 11. 12. 13. Using these substitutions, we can arrive at the following equations that describe the behavior of temperature and concentration, when the steady state conditions are perturbed in a CSTR: 14. 15. 16. 17. 18. 19. 20. 21. 22.

 Predicting the Behavior of a CSTR using LST top

At time t = 0, y1 = y10, where:

Making use of Equation 20

we notice that for the case of b2 = 4c:

if b < 0 ,   then the amplitude (i.e., T - TS) will increase

if b > 0 ,   then the amplitude (i.e., T - TS) will decrease

and that:

if b2 > 4c ,   thenis real (i.e., non-oscillatory behavior)

if b2 < 4c ,   thenis imaginary (i.e., oscillatory behavior)

 Critically damped: b is positive andis real Unstable growth: b is negative andis real Oscillatory and damped: b is positive andis imaginary Oscillatory: b is zero andis imaginary Unstable growth oscillation: b is negative andis imaginary Reference: R. Aris, Elementary Chemical Reactor Analysis, Prentice Hall, New Jersey, (1969).

Object Assessment of Chapter 9

* All chapter references are for the 4th Edition of the text Elements of Chemical Reaction Engineering .