# 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 Control #### Integral Controller #### 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 and signify steady state values. Then at steady state: 7. Adding equations (6) and (7): 8. Linearizing by 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 ,   then is real (i.e., non-oscillatory behavior)

if b2 < 4c ,   then is imaginary (i.e., oscillatory behavior) Critically damped: b is positive and is real Unstable growth: b is negative and is real Oscillatory and damped: b is positive and is imaginary Oscillatory: b is zero and is imaginary Unstable growth oscillation: b is negative and is 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 .