#### 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.

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

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

\[\frac{dV}{dt} = v_0 – v\]

Steady-state CSTR energy balance

\[0=\sum F_{i0}C_{Pi}(T_0 – T_R) – \sum F_iC_{Pi} (T-T_R) + \Delta H_{rxn}(T_R)r_AV + \dot{Q} + \dot{W},\] 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.