Ideal Gas Constant Pressure Reactor#

An ideal gas constant pressure reactor, as implemented by the C++ class IdealGasConstPressureReactor and available in Python as the IdealGasConstPressureReactor class. It is defined by the state variables:

\(m\), the mass of the reactor’s contents (in kg)
\(T\), the temperature (in K)
\(Y_k\), the mass fractions for each species (dimensionless)

Equations 1-3 below are the governing equations for an ideal gas 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 corresponding 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:

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

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#

As for the ideal gas reactor, we replace the total enthalpy as a state variable with the temperature by writing the total enthalpy in terms of the mass fractions and temperature and differentiating with respect to time:

\[ \begin{align}\begin{aligned} H &= m \sum_k Y_k h_k(T)\\\frac{dH}{dt} &= h \frac{dm}{dt} + m c_p \frac{dT}{dt} + m \sum_k h_k \frac{dY_k}{dt} \end{aligned}\end{align} \]

Substituting the corresponding derivatives into the constant pressure reactor energy equation (3) yields an equation for the temperature:

(3)#\[ m c_p \frac{dT}{dt} = \dot{Q} - \sum_k h_k \dot{m}_{k,\t{gen}} + \sum_\t{in} \dot{m}_\t{in} \left(h_\t{in} - \sum_k h_k Y_{k,\t{in}} \right) \]