## Ideal Gas Constant Pressure Reactor¶

This page shows the derivation of the governing equations used in Cantera's Ideal Gas Constant Pressure Reactor model.

More information on the Ideal Gas Constant Pressure Reactor class can be found here.

## Ideal Gas Constant Pressure Reactor¶

An Ideal Gas Constant Pressure Reactor is defined by the three 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)

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 ReactorSurface() objects:

\begin{equation*} \frac{dm}{dt} = \sum_{in} \dot{m}_{in} - \sum_{out} \dot{m}_{out} + \dot{m}_{wall} \tag{1} \end{equation*}

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

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:

\begin{equation*} H = m \sum_k Y_k h_k(T) \end{equation*}
\begin{equation*} \frac{dH}{dt} = h \frac{dm}{dt} + m c_p \frac{dT}{dt} + m \sum_k h_k \frac{dY_k}{dt} \end{equation*}

Substituting the corresponding derivatives yields an equation for the temperature:

\begin{equation*} m c_p \frac{dT}{dt} = \dot{Q} - \sum_k h_k \dot{m}_{k,gen} + \sum_{in} \dot{m}_{in} \left(h_{in} - \sum_k h_k Y_{k,in} \right) \tag{2} \end{equation*}

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:

\begin{equation*} \dot{m}_{k,gen} = V \dot{\omega}_k W_k + \dot{m}_{k,wall} \end{equation*}

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

\begin{equation*} \frac{d(mY_k)}{dt} = \sum_{in} \dot{m}_{in} Y_{k,in} - \sum_{out} \dot{m}_{out} Y_k + \dot{m}_{k,gen} \end{equation*}

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

\begin{equation*} m \frac{dY_k}{dt} = \sum_{in} \dot{m}_{in} (Y_{k,in} - Y_k)+ \dot{m}_{k,gen} - Y_k \dot{m}_{wall} \tag{3} \end{equation*}

Equations 1-3 are the governing equations for an Ideal Gas Constant Pressure Reactor.