Chapter 13: Unsteady State Nonisothermal Reactor Design
Balance on a system volume that is well-mixed:
\( \dot{Q} - \dot{W}_S + \sum_i F_{i.o.} H_{i.o.} = \frac{d E_{sys}}{dt} = \frac{d \sum_i E_i}{dt} = \frac{d \sum_i H_i}{dt} = \sum_i H_i \frac{d N_i}{dt} + \sum_i \frac{d H_i}{dt} \)
\( F_{i.o.} - F_i + r_i V = \frac{dN_i}{dt} \)
\( \frac{dH_i}{dt} = C_{P,i} \frac{dT}{dt} \)
\( \dot{Q} - \dot{W}_S + \sum F_{i.o.} H_{i.o.} - \sum F_i H_i = \sum H_i \left( F_{i.o.} - F_i + r_i V \right) + \sum N_i C_{P,i} \frac{dT}{dt} = \sum H_i F_{i.o.} - \sum H_i F_i + \sum r_i V H_i + \sum N_i C_{P,i} \frac{dT}{dt} \)
\( r_i = \nu_i \left( -r_A \right) \)
\( \dot{Q} - \dot{W}_S + \sum F_{i.o.} H_{i.o.} - H_o = -r_A V \sum \nu_i H_i + \sum N_i C_{P,i} \frac{dT}{dt} \)
\( \dot{Q} - \dot{W}_S + \dot{i_A} V \left| \Delta H_R \right| (T) = \sum F_{i.o.} \left( H_i - H_{i.o.} \right) + \sum N_i C_{P,i} \frac{dT}{dt} \)