5.3.1 A Single CSTR

5.3.1.1 First-Order Reaction

Let’s consider a first-order irreversible reaction for which the rate law is

–r_A = kC_A

For liquid-phase reactions, there is no volume change during the course of the reaction, so we can use Equation (4-12) to relate concentration and conversion,

4-12

We can combine the mole balance Equation (5-7), the rate law, and concentration Equation (4-12) to obtain

Rearranging

5-8

A plot of conversion as a function of τk using Equation (5-8) is shown in Figure 5-3.

Figure 5-3. First-order reaction in a CSTR.

We see that when we increase the reactor volume by a factor of two as we go from τk = 4 to τk = 8, the conversion only increases from 0.8 to 0.89.

We could also combine Equations (4-12) and (5-8) to find the exit reactor concentration of A, C_A,

5-9

5.3.1.2 A Second-Order Reaction in a CSTR

For a second-order liquid-phase reaction being carried out in a CSTR, the combination of the rate law and the design equation yields

5-10

Using our stoichiometric table for constant density, υ = υ₀, C_A = C_A0(1 – X), and F_A0 X = υ₀ C_A0X, then

Dividing by υ₀,

5-11

We solve Equation (5-11) for the conversion X:

5-12

The minus sign must be chosen in the quadratic equation because X cannot be greater than 1. Conversion is plotted as a function of the Damköhler parameter for a second-order reaction, Da = τkC_A0, in Figure 5-4. Observe from this figure that at high conversions (say 67%), a 10-fold increase in the reactor volume (or increase in the specific reaction rate by raising the temperature) will only increase the conversion to 88%. This observation is a consequence of the fact that the CSTR operates under the condition of the lowest reactant concentration (i.e., the exit concentration), and consequently the smallest value of the rate of reaction.

Figure 5-4. Conversion as a function of the Damköhler number (τkC_A0) for a second-order reaction in a CSTR.

5.3.1.3 The Damköhler Number

For a first-order reaction, the product τk is often referred to as the reaction Damköhler number, Da, which is a dimensionless number that can give us a quick estimate of the degree of conversion that can be achieved in continuous-flow reactors. The Damköhler number is the ratio of the rate of reaction of A to the rate of convective transport of A evaluated at the entrance to the reactor.

The Damköhler number for a first-order irreversible reaction is

For a second-order irreversible reaction, the Damköhler number is

It is important to know what values of the Damköhler number, Da, give high and low conversion in continuous-flow reactors. For irreversible reactions, a value of Da = 0.1 or less will usually give less than 10% conversion, and a value of Da = 10.0 or greater will usually give greater than 90% conversion; that is, the rule of thumb is

if Da < 0.1, then X < 0.1

if Da > 10, then X > 0.9

Equation (5-8) for a first-order liquid-phase reaction in a CSTR can also be written in terms of the Damköhler number