## Constant Pressure Reactor¶

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

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

## Constant Pressure Reactor¶

For this reactor model, the pressure is held constant. The energy equation is defined by the total enthalpy.

A Constant Pressure Reactor is defined by the three 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)

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 the reactor Wall():

\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. A Reactor wall is defined here.

Using the definition of the total enthalpy:

\begin{equation*} H = U + pV \end{equation*}
\begin{equation*} \frac{d H}{d t} = \frac{d U}{d t} + p \frac{dV}{dt} + V \frac{dp}{dt} \end{equation*}

Noting that $$dp/dt = 0$$ and substituting into the energy equation yields:

\begin{equation*} \frac{dH}{dt} = \dot{Q} + \sum_{in} \dot{m}_{in} h_{in} - h \sum_{out} \dot{m}_{out} \tag{2} \end{equation*}

Where the total specific enthalpy $$h$$ is defined as $$h = \sum_k{h_k Y_k}$$. The enthalpy terms in equation 2 appear due to enthalpy flowing in and out of the reactor. The rate of heat transfer $$\dot{Q}$$ can replace $$\frac{d U}{d t} + p \frac{dV}{dt}$$ in the above equation due to the first law of thermodynamics, which states $$\dot{Q} = \dot{H}$$ in a closed system where no work is done. Positive $$\dot{Q}$$ represents heat addition to the system.

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 a Constant Pressure Reactor.