Chapter 13: Unsteady State Nonisothermal Reactor Design

Balance on a system volume that is well-mixed:

\( \dot{Q} - W_S + \dot{i}_A V | \Delta H_R | T_i | = \sum_{i,o} i H_i - H_{i,o} + \sum_{N_i} C_{P_i} \frac{dT}{dt} \)

\( \dot{Q} - W_S - F_{A0} \sum \theta_i C_{P_i} | T - T_o | + \dot{i}_A V | \Delta H_R | T_i | = \sum_{N_i} C_{P_i} \frac{dT}{dt} \)

Adiabatic batch reactor with no work: \( T = T_o + \frac{X | \Delta H_R | T_o |}{\sum \theta_i C_{P_i} + X \Delta C_P} \)

Polymath

The following reaction occurs in a batch reactor:

\( A + B \rightleftharpoons 2C \)

1) \( \frac{dT}{dt} = \frac{(r_A V)(\Delta H_E(T)) + U A (T_1 - T)}{N_{Ao} \left( \sum \theta_i C_i^n + \Delta C_P X \right)} \)

2) \( \frac{dX}{dt} = \frac{-r_A V}{N_{Ao}} \)

3) \( -r_A = -k \left[ C_A C_B - \frac{C_C^2}{K_C} \right] \)

4) \( k = k_1 \exp \left( \frac{E}{R} \left( \frac{1}{T_1} - \frac{1}{T} \right) \right) \)

5) \( K_C = K_{C2} \exp \left( \frac{\Delta H_E}{R} \left( \frac{1}{T_2} - \frac{1}{T} \right) \right) \)

6) \( C_A = C_{A0} (1 - X) \)

\( C_B = C_{A0} (1 - X) \)

\( C_C = 2 C_{A0} X \)

7) Parameter Values \( \Delta H_Rx, E, C_{A0}, E, \text{etc.} \)

Adiabatic Reaction \( X = \frac{(C_{P_A} + C_{P_B}) (T - T_0)}{-\Delta H_{RX}} \) \( -r_A = k(t) F_{A0} \left[ (1 - x)^2 - \frac{2 x^2}{K_c(T)} \right] \)

or use one of the integration formulas, e.g.: \( \int_{X_0}^{X_2} f(X) \, dX = \frac{h}{3} \left[ f(X_0) + 4 f(X_1) + f(X_2) \right] \) , to find the reaction time, t. Even better, use Polymath.

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Cooling:

\( \frac{dT}{dt} = \frac{r_a V (\Delta H_{RX}) - UA (T - T_a)}{\sum N_i C_{pi}} \)

\( \frac{dT}{dt} = \frac{Q_g - Q_R}{\sum N_i C_{pi}} \)

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

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