#### Energy Balances in Stirred Tank Reactors: Screencasts

Explains terms in the unsteady-state energy balance for stirred tank reactors and shows the energy balances for batch, semibatch, and steady-state CSTRs.

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Setup of energy balances for stirred tank reactors. Comparisons are made with batch and semi-batch reactors.

We suggest you list the important points in this screencast as a way to increase retention.

Optional screencast: Energy Balance on a Semibatch Reactor

##### Important Equations:

Energy balance for unsteady-state tank reactor or semibatch reactor. This equation can also be used for start-up of a CSTR. For a semibatch reactor, the inlet flow rate or the outlet flow rate may be zero.

$\sum N_iC_{Pi} \frac{dT}{dt} = \sum F_{i0}C_{Pi}(T_0 – T_R) – \sum F_iC_{Pi} (T-T_R) + \Delta H_{rxn}(T_R)r_AV + \dot{Q} + \dot{W}$$\dot{Q} = -UA(T-T_C)$For a semibatch reactor

$\frac{dV}{dt} = v_0 – v$

$0=\sum F_{i0}C_{Pi}(T_0 – T_R) – \sum F_iC_{Pi} (T-T_R) + \Delta H_{rxn}(T_R)r_AV + \dot{Q} + \dot{W},$ which, if $$T_R = T$$ can be simplified to
$0=\sum F_{i0}C_{Pi}(T_0 – T_R) + \Delta H_{rxn}(T_R)r_AV + \dot{Q} + \dot{W},$ or, if $$T_R = T_0$$ $0=\sum F_{i0}C_{Pi}(T – T_R) + \Delta H_{rxn}(T_R)r_AV + \dot{Q} + \dot{W}$

Batch reactor energy balance

$\sum N_iC_{Pi} \frac{dT}{dt} = \Delta H_{rxn}(T_R)r_AV + \dot{Q} + \dot{W},$

Reactor residence time

$$\tau = \frac{V}{v}$$           Space Time = $$\frac{V}{v_0}$$

where $$N_i$$ is the number of moles of species i in the tank,
$$C_{Pi}$$ is the molar heat capacity of component i,
$$t$$ is time,
$$v_0$$ is the inlet volumetric flow rate,
$$v$$ is the outlet volumetric flow rate,
$$F_{i0}$$ is the inlet molar rate (mole per time) of species i to the reactor,
$$F_i$$ is the effluent molar flow of species i,
$$C_{io}$$ is the concentration of species i in the feed,
$$C_i$$ is the concentration of species i in the reactor (and also in thee effluent),
$$T_0$$ is the temperature of the feed (assuming one feed stream),
$$T$$ is the reactor temperature (and also the effluent temperature),
$$T_R$$ is a reference temperature, which is arbitrary,
$$T_C$$ is the coolant temperature, which is assumed constant,
$$\Delta H_{rxn}(T_R)$$ is the heat of reaction per mole of reactant A, and its evaluated at temperature $$T_R$$,
$$r_A$$ is the rate of formation of species A ($$r_A$$ is negative when A is a reactant, as assumed here),
$$\dot{Q}$$ is the rate heat is added to the reactor,
$$U$$ is a heat transfer area,
$$\dot{W}$$ is the rate that work is added, and
$$\tau$$ is the residence time.