Warning

This documentation is for an old version of Cantera. You can find docs for newer versions here.

Reactor Networks

Cantera’s Reactor Network module is designed to simulate networks of interconnected reactors. The contents of each reactor in the network are assumed to be homogeneous, a model variously referred to as the Continuously Stirred Tank Reactor (CSTR), Well-Stirred Reactor (WSR), or Perfectly Stirred Reactor (PSR) model. Cantera solves the time-dependent governing equations that describe the evolution of the chemical and thermodynamic state of the reactors.

The contents of each reactor can undergo chemical reactions according to a specified kinetic mechanism, and surface reactions may occur on the reactor walls. Each reactor in a network may be connected so that the contents of one reactor flow into another. Reactors may be also be in contact with one another or the environment via walls which move or conduct heat.

The purpose of this document is to describe the governing equations of reactor models as implemented in Cantera.

Wall Interactions

At each wall where there are surface reactions, there is a net generation (or destruction) of homogeneous phase species. The molar rate of production for each species \(k\) on wall \(w\) is \(\dot{s}_{k,w}\). The total (mass) production rate for species \(k\) on all walls is:

\[\dot{m}_{k,wall} = W_k \sum_w A_w \dot{s}_{k,w}\]

where \(W_k\) is the molecular weight of species \(k\) and \(A_w\) is the area of each wall. The net mass flux from all walls is then:

\[\dot{m}_{wall} = \sum_k \dot{m}_{k,wall}\]

The total rate of heat transfer through all walls is:

\[\dot{Q} = \sum_w f_w \dot{Q}_w\]

General Reactor

The state variables for Cantera’s general reactor model are

• \(m\), the mass of the reactor’s contents
• \(V\), the reactor volume
• \(U\), the total internal energy of the reactors contents
• \(Y_k\), the mass fractions for each species

Reactor Volume

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

\[\frac{dV}{dt} = \sum_w f_w A_w v_w(t)\]

where \(f_w = \pm 1\) indicates the facing of the wall, \(A_w\) is the surface area of the wall, and \(v_w(t)\) is the velocity of the wall as a function of time.

Mass Conservation

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 walls:

\[\frac{dm}{dt} = \sum_{in} \dot{m}_{in} - \sum_{out} \dot{m}_{out} + \dot{m}_{wall}\]

Species Conservation

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:

\[\dot{m}_{k,gen} = V \dot{\omega}_k W_k + \dot{m}_{k,wall}\]

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

\[\frac{d(mY_k)}{dt} = \sum_{in} \dot{m}_{in} Y_{k,in} - \sum_{out} \dot{m}_{out} Y_k + \dot{m}_{k,gen}\]

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

\[m \frac{dY}{dt} = \sum_{in} \dot{m}_{in} (Y_{k,in} - Y_k)+ \dot{m}_{k,gen} - Y_k \dot{m}_{wall}\]

Energy Conservation

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

\[\frac{dU}{dt} = - p \frac{dV}{dt} - \dot{Q} + \sum_{in} \dot{m}_{in} h_{in} - h \sum_{out} \dot{m}_{out}\]

Ideal Gas Reactor

The Ideal Gas Reactor model is similar to the General Reactor model, with the reactor temperature \(T\) replacing the total internal energy \(U\) as a state variable. For an ideal gas, we can rewrite the total internal energy in terms of the mass fractions and temperature:

\[U = m \sum_k Y_k u_k(T)\]\[\frac{dU}{dt} = u \frac{dm}{dt} + m c_v \frac{dT}{dt} + m \sum_k u_k \frac{dY_k}{dt}\]

Substituting the corresponding derivatives yields an equation for the temperature:

\[m c_v \frac{dT}{dt} = - p \frac{dV}{dt} - \dot{Q} + \sum_{in} \dot{m}_{in} \left( h_{in} - \sum_k u_k Y_{k,in} \right) - \frac{p V}{m} \sum_{out} \dot{m}_{out} - \sum_k \dot{m}_{k,gen} u_k\]

While this form of the energy equation is somewhat more complicated, it significantly reduces the cost of evaluating the system Jacobian, since the derivatives of the species equations are taken at constant temperature instead of constant internal energy.

Constant Pressure Reactor

For this reactor model, the pressure is held constant. The volume is not a state variable, but instead takes on whatever value is consistent with holding the pressure constant. The total enthalpy replaces the total internal energy as a state variable. Using the definition of the total enthalpy:

\[H = U + pV\]\[\frac{dH}{dt} = p \frac{dV}{dt} + V \frac{dp}{dt}\]

Noting that \(dp/dt = 0\) and substituting into the energy equation yields:

\[\frac{dH}{dt} = - \dot{Q} + \sum_{in} \dot{m}_{in} h_{in} - h \sum_{out} \dot{m}_{out}\]

The species and continuity equations are the same as for the general reactor model.

Ideal Gas Constant Pressure Reactor

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:

\[H = m \sum_k Y_k h_k(T)\]\[\frac{dH}{dt} = h \frac{dm}{dt} + m c_p \frac{dT}{dt} + m \sum_k h_k \frac{dY_k}{dt}\]

Substituting the corresponding derivatives yields an equation for the temperature:

\[m c_p \frac{dT}{dt} = - \dot{Q} - \sum_k h_k \dot{m}_{k,gen} + \sum_{in} \dot{m}_{in} \left(h_{in} - \sum_k h_k Y_{k,in} \right)\]