Chapter 12: SteadyState Nonisothermal Reactor Design: Flow Reactors with Heat Exchange
Topics
 Overview of User Friendly Energy Balance Equations
 Evaluating the Heat Exchanger Term
 Multiple Steady States
 Multiple Reactions with Heat Effects
 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 F_{i} and T 1. Adiabatic CSTR, PFR, Batch, PBR achieve this:
2. CSTR with heat exchanger, UA(T_{a}T) and large coolant flow rate.
3A. In terms of conversion, X
3B. In terms of molar flow rates, F_{i}
4. For Multiple Reactions
5. Coolant Balance


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 
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, T_{a1}T_{a2}T_{a}: 

Multiple Steady States (MSS)  top 
From page 593 we can obtain  


where 

Now we need to find X. We do this by combining the mole balance, rate law, Arrhenius Equation, and stoichiometry.
For the firstorder, irreversible reaction A > B, we have:


where 

At steady state: 

Substituting for k... 

Multiple Reactions with Heat Effects  top 
To account for heat effects in multiple reactions, we simply replace the term (delta H_{RX}) (r_{A}) in equations (1235) PFR/PBR and (1240) CSTR by:
PFR/PBR
CSTR
These equations are coupled with the mole balances and rate law equations discussed in Chapter 6.
Complex Reactions
Example: Consider the following gas phase reactions

We now substitute the various parameter values (e.g. delta H_{RX}, E, U) into equations (1)(13) and solve simultaneously using Polymath.
Applications of the PFR/PBR User Friendly Energy Balance Equations  top 
NOTE: The PFR and PBR formulas are very similar.
If we include pressure drop: 

C. $$ \frac{dp}{dW} = \frac{\alpha}{2p}\frac{F_{T}}{F_{T0}}\frac{T}{T_{0}} = \frac{\alpha(1+\epsilon X)}{2p}(\frac{T}{T_{0}}) = h(X,T) $$ 

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: 

CoCurrent Flow  
CounterCurrent Flow 
with T_{a} = T_{ao} 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.
 Consider an exothermic reaction where the coolant stream enters at the end of the reactor at a temperature T_{a0}, say 300 K.
 Assume a coolant temperature at the entrance (X = 0, V = 0) to the reactor T_{a2} =340 K.
 Calculate X, T, and T_{a} as a function of V. We can see that our guess of 340 K for T_{a2} 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.
 Now guess a coolant temperature at V = 0 and X = 0 of 330 K. We see that the exit coolant temperature of T_{a2} = 330 K will give a coolant temperature at V = V_{1} of 300 K.