## Mole Reactor¶

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

## Mole 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 Mole Reactor is defined by the three state variables:

• $$U$$, the total internal energy of the reactor's contents (in J)

• $$V$$, the reactor volume (in m3)

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

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

Where $$\dot{Q}$$ is the net rate of heat addition to the system.

The reactor volume changes as a function of time due to the motion of one or more Wall()s:

\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 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{3} \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.

Equations 1-3 are the governing equations for a Mole Reactor.