## Constant Pressure Mole Reactor¶

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

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

## Constant Pressure Mole Reactor¶

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

A Constant Pressure Mole Reactor is defined by the two state variables:

• $$H$$, the total enthalpy of the reactor's contents (in J)

• $$n_k$$, the number of moles for each species (in kmol)

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{n}_{in} \bar{h}_{in} - \bar{h} \sum_{out} \dot{n}_{out} \tag{1} \end{equation*}

Where the total specific enthalpy $$h$$ is defined as $$h = \sum_k{\bar{h}_k n_k}$$. 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 moles of each species in the reactor's contents changes as a result of flow through the reactor's inlets and outlets, and production of homogeneous gas phase species and reactions on the reactor Wall(). The rate of moles of species $$k$$ generated through homogeneous phase reactions is $$V \dot{\omega}_k$$, and the total rate at which moles of species $$k$$ changes is:

\begin{equation*} \frac{dn_k}{dt} = V \dot{\omega}_k + \sum_{in} \dot{n}_{k, in} - \sum_{out} \dot{n}_{k, out} + \dot{n}_{k, wall} \tag{2} \end{equation*}

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. Reactor Walls are defined here.

Equations 1-2 are the governing equations for a Constant Pressure Mole Reactor.