Constant Pressure Reactor#

For this reactor model, the pressure is held constant and the energy equation is defined in terms of the total enthalpy. This model is implemented by the C++ class ConstPressureReactor and available in Python as the ConstPressureReactor class. A constant pressure reactor is defined by the state variables:

\(m\), the mass of the reactor’s contents (in kg)
\(H\), the total enthalpy of the reactor’s contents (in J)
\(Y_k\), the mass fractions for each species (dimensionless)

Equations 1-3 below are the governing equations for a constant pressure reactor.

Mass Conservation#

The total mass of the reactor’s contents changes as a result of flow through the reactor’s inlets and outlets, and production of homogeneous phase species on surfaces:

(1)#\[ \frac{dm}{dt} = \sum_\t{in} \dot{m}_\t{in} - \sum_\t{out} \dot{m}_\t{out} + \dot{m}_\t{wall} \]

Where the subscripts in and out refer to the sum of the superscripted property over all inlets and outlets respectively. A dot above a variable signifies a time derivative.

Species Equations#

The rate at which species \(k\) is generated through homogeneous phase reactions is \(V \dot{\omega}_k W_k\), and the total rate at which species \(k\) is generated is:

\[ \dot{m}_{k,\t{gen}} = V \dot{\omega}_k W_k + \dot{m}_{k,\t{wall}} \]

The rate of change in the mass of each species is:

\[ \frac{d(mY_k)}{dt} = \sum_\t{in} \dot{m}_\t{in} Y_{k,\t{in}} - \sum_\t{out} \dot{m}_\t{out} Y_k + \dot{m}_{k,\t{gen}} \]

Expanding the derivative on the left hand side and substituting the equation for \(dm/dt\), the equation for each homogeneous phase species is:

(2)#\[ m \frac{dY_k}{dt} = \sum_\t{in} \dot{m}_\t{in} (Y_{k,\t{in}} - Y_k) + \dot{m}_{k,\t{gen}} - Y_k \dot{m}_\t{wall} \]

Energy Equation#

Writing the first law for an open system gives:

\[ \frac{dU}{dt} = - p \frac{dV}{dt} + \dot{Q} + \sum_\t{in} \dot{m}_\t{in} h_\t{in} - h \sum_\t{out} \dot{m}_\t{out} \]

where positive \(\dot{Q}\) represents heat addition to the system and \(h\) is the specific enthalpy of the reactor’s contents.

Differentiating the definition of the total enthalpy, \(H = U + pV\), with respect to time gives:

\[ \frac{dH}{dt} = \frac{dU}{dt} + p \frac{dV}{dt} + V \frac{dp}{dt} \]

Noting that \(dp/dt = 0\) and substituting into the energy equation yields:

(3)#\[ \frac{dH}{dt} = \dot{Q} + \sum_\t{in} \dot{m}_\t{in} h_\t{in} - h \sum_\t{out} \dot{m}_\t{out} \]