12.5.2 Heat-Generated Term, G(T)

The heat-generated term, Equation (12-29), can be written in terms of conversion. [Recall: X = –r_AV/F_A0.]

12-31

To obtain a plot of heat generated, G(T), as a function of temperature, we must solve for X as a function of T using the CSTR mole balance, the rate law, and stoichiometry. For example, for a first-order liquid-phase reaction, the CSTR mole balance becomes

Solving for X yields

5-8

Substituting for X in Equation (12-31), we obtain

12-32

Finally, substituting for k in terms of the Arrhenius equation, we obtain

12-33

Note that equations analogous to Equation (12-33) for G(T) can be derived for other reaction orders and for reversible reactions simply by solving the CSTR mole balance for X. For example, for the second-order liquid-phase reaction

the corresponding heat generated term is

12-34

Let’s now examine the behavior of the G(T) curve. At very low temperatures, the second term in the denominator of Equation (12-33) for the first-order reaction can be neglected, so that G(T) varies as

[Recall that means that the standard heat of reaction is evaluated at T_R.]

At very high temperatures, the second term in the denominator dominates, and G(T) is reduced to

G(T) is shown as a function of T for two different activation energies, E, in Figure 12-9. If the flow rate is decreased or the reactor volume increased so as to increase τ, the heat generated term, G(T), changes, as shown in Figure 12-10.

Figure 12-9. Variation of G(T) curve with activation energy

Figure 12-10. Variation of G(T) curve with space-time

Can you combine Figures 12-10 and 12-8 to explain why a Bunsen burner goes out when you turn up the gas flow rate to a very high rate?