## Control Volume Reactor¶

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

More information on the Control Volume Reactor class can be found here.

## Control Volume Reactor¶

A homogeneous zero-dimensional reactor. By default, they are closed (no inlets or outlets), have fixed volume, and have adiabatic, chemically-inert walls. These properties may all be changed by adding appropriate components such as Wall(), ReactorSurface(), MassFlowController(), and Valve().

A Control Volume Reactor is defined by the four state variables:

• $$m$$, the mass of the reactor's contents (in kg)

• $$V$$, the reactor volume (in m3)

• $$U$$, the total internal energy of the reactors 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 corresponding property over all inlets and outlets respectively. A dot above a variable signifies a time derivative.

The reactor volume changes as a function of time due to the motion of one or more walls:

\begin{equation*} \frac{dV}{dt} = \sum_w f_w A_w v_w(t) \tag{2} \end{equation*}

where $$f_w = \pm 1$$ indicates the facing of the wall (whether moving the wall increases or decreases the volume of the reactor), $$A_w$$ is the surface area of the wall, and $$v_w(t)$$ is the velocity of the wall as a function of time.

The equation for the total internal energy is found by writing the first law for an open system:

\begin{equation*} \frac{dU}{dt} = - p \frac{dV}{dt} + \dot{Q} + \sum_{in} \dot{m}_{in} h_{in} - h \sum_{out} \dot{m}_{out} \tag{3} \end{equation*}

Where $$\dot{Q}$$ is the net rate of 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{4} \end{equation*}

Equations 1-4 are the governing equations for a Control Volume Reactor.