12. Steady-State Non-Isothermal Reactor Design : The Steady State Energy Balance and Adiabatic PFR Applications *

Topics

  1. Overview of User Friendly Energy Balance Equations
  2. Evaluating the Heat Exchanger Term
  3. Multiple Steady States
  4. Multiple Reactions with Heat Effects
  5. Applications of the PFR/PBR User Friendly Energy Balance Equations


User Friendly Energy Balance Equations top

The user friendly forms of the energy balance we will focus on are outlined in the following table.


User friendly equations relating X and T, and Fi and T

1. Adiabatic CSTR, PFR, Batch, PBR achieve this:

(1.A)

 

(1.B)

2. CSTR with heat exchanger, UA(Ta-T) and large coolant flow rate.

(2)


3 . PFR/PBR with heat exchange

3A. In terms of conversion, X

(3.A)

3B. In terms of molar flow rates, Fi 

(3.B)

4. For Multiple Reactions

(4)

5. Coolant Balance

Co-Current Flow

(5)

 

These equations are derived in the text.  These are the equations that we will use to solve reaction engineering problems with heat effects.

Evaluating the Heat Exchanger Term top


Energy transferred between the reactor and the coolant:

Assuming the temperature inside the CSTR, T, is spatially uniform:
 

At high coolant flow rates the exponential term will be small, so we can expand the exponential term as a Taylor Series, where the terms of second order or greater are neglected, then:

Since the coolant flow rate is high, Ta1Ta2Ta:


Multiple Steady States (MSS) top

CSTR with Heat Effects
From pagem 593 we can obtain

           

where


Finding when R(T) = G(T)
Finding MSS for an Endothermic Reaction

Now we need to find X. We do this by combining the mole balance, rate law, Arrhenius Equation, and stoichiometry.

For the first-order, irreversible reaction A --> B, we have:

where

At steady state:

Substituting for k...

Generating G and R verse T: Single Reaction


Multiple Reactions with Heat Effects top

To account for heat effects in multiple reactions, we simply replace the term (-delta HRX) (-rA) in equations (12-35) PFR/PBR and (12-40) CSTR by:


PFR/PBR


CSTR


These equations are coupled with the mole balances and rate law equations discussed in Chapter 6.

Textbook Example (Alternative Solution)

Complex Reactions

Example: Consider the following gas phase reactions


Combined Mole Balance, Rate Law, Stoichiometry, and Energy Balance:



We now substitute the various parameter values (e.g. delta HRX, E, U) into equations (1)-(13) and solve simultaneously using Polymath.

Multiple Reactions in a PFR with Variable Coolant Temperature


Applications of the PFR/PBR User Friendly Energy Balance Equations top

NOTE: The PFR and PBR formulas are very similar.

Heat exchange for a PFR:
a = heat exchange area per unit volume of reactor; for a tubular reactor, a = 4/D
Catalyst weight is related to reactor volume by:
Heat exchange for a PBR:
Steady State Energy Balance (with no work):
Final Form of the Differential Equations in Terms of Conversion:

A.        

Final Form in terms of Molar Flow Rates

 

B.        

If we include pressure drop:

C.       

Note: the pressure drop will be greater for exothermic adiabatic reactions than it will be for isothermal reactions

Balance on Heat Exchanger Coolant

Solve simultaneously using an ODE solver (Polymath/MatLab). If Ta is not constant, then we must add an additional energy balance on the coolant fluid:

Co-Current Flow
Counter-Current Flow

with Ta = Tao at W = 0

For an exothermic reaction: with counter current heat exchange

A Trial and Error procedure for counter current flow problems is required to find exit conversion and temperature.

  1. Consider an exothermic reaction where the coolant stream enters at the end of the reactor at a temperature Ta0, say 300 K.
  2. Assume a coolant temperature at the entrance (X = 0, V = 0) to the reactor Ta2 =340 K.
  3. Calculate X, T, and Ta as a function of V. We can see that our guess of 340 K for Ta2 at the feed entrance (X = 0) gives a coolant temperature of 310 K, which does not match the actual entering coolant temperature of 300 K.
  4. Now guess a coolant temperature at V = 0 and X = 0 of 330 K. We see that the exit coolant temperature of Ta2 = 330 K will give a coolant temperature at V = V1 of 300 K.
A ↔ B Liquid Phase Adiabatic
A ↔ B Liquid Phase Constant Ta
A ↔ B Liquid Phase Variable Ta, Co-Current
A ↔ B Liquid Phase Variable Ta, Counter Current
Sketch the Ambient Temperature as a function of V.
Elementary Liquid Phase Reaction
Exothermic, Reversible Reaction
Adiabatic Reaction in a PBR.
PBR with heat exchange.
PBR with heat exchange and variable coolant flow rate.
Nonisothermal Reactions.
LEP 12-T12-3: Variable Coolant Temperature.

Objective Assessment of Chapter 12
 

* All chapter references are for the 1st Edition of the text Essentials of Chemical Reaction Engineering .

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