Reactors and Reactor Networks#

In Cantera, a reactor network represents a set of one or more homogeneous reactors and reacting surfaces that may be connected to each other and to the environment through devices representing mass flow, heat transfer, and moving walls. The system is generally unsteady – that is, all states are functions of time. In particular, transient state changes due to chemical reactions are possible.

Homogeneous Reactor Types and Governing Equations#

A Cantera reactor represents the simplest form of a chemically reacting system. It corresponds to an extensive thermodynamic control volume \(V\), in which all state variables are homogeneously distributed. By default, these reactors are closed (no inlets or outlets) and have adiabatic, chemically-inert walls. These properties may all be changed by adding components such as walls, surfaces, mass flow controllers, and valves, as described below.

The specific governing equations defining Cantera’s reactor models are derived and described below. These models represent different combinations of whether pressure or volume are held constant, whether they support any equation of state or are specialized for ideal gas mixtures, and whether mass fractions or moles of each species are used as the state variables representing the composition.

Control Volume Reactor: A reactor where the volume is prescribed by the motion of the reactor’s walls.
Constant Pressure Reactor: A reactor where the pressure is held constant.
Ideal Gas Control Volume Reactor: A reactor where the volume is prescribed by the motion of the reactor’s walls, specialized for ideal gas mixtures.
Ideal Gas Constant Pressure Reactor: A reactor where the pressure is held constant, specialized for ideal gas mixtures.
Control Volume Mole Reactor: A reactor where the volume is prescribed by the motion of the reactor’s walls, with the composition stored in moles.
Constant Pressure Mole Reactor: A reactor where the pressure is held constant and the composition is stored in moles.
Ideal Gas Control Volume Mole Reactor: A reactor where the volume is prescribed by the motion of the reactor’s walls, specialized for ideal gas mixtures and with the composition stored in moles.
Ideal Gas Constant Pressure Mole Reactor: A reactor where the pressure is held constant, specialized for ideal gas mixtures and with the composition stored in moles.

Plug Flow Reactors#

A plug flow reactor (PFR) represents a steady-state flow in a channel. The fluid is considered to be homogeneous perpendicular to the flow direction, while the state of the gas is allowed to change in the axial direction. However, all diffusion processes are neglected.

These assumptions result in a system of equations that is similar to those used to model homogeneous reactors, with the time variable replaced by the axial coordinate. Because of this mathematical similarity, PFRs are also solved by Cantera’s reactor network model. However, they can only be simulated alone, and not part of a network containing time-dependent reactors.

Plug Flow Reactor: A reactor modeling one-dimensional steady-state flow in a channel that may contain catalytically active surfaces where heterogeneous reactions occur.

Reactor Interactions#

Reactors can interact with each other and the surrounding environment in multiple ways. Mass can flow from one reactor into another can be incorporated, heat can be transferred, and the walls between reactors can move. In addition, reactions can occur on surfaces within a reactor. The models used to establish these interconnections are described in the following sections:

All of these interactions can vary as a function of time or system state. For example, heat transfer can be described as a function of the temperature difference between the reactor and the environment, or wall movement can be modeled as depending on the pressure difference. Interactions of the reactor with the environment are defined using the following models:

Reservoirs: Reservoirs are used to represent constant conditions defining the inlets, outlets, and surroundings of a reactor network.
Flow Devices: Flow devices are used to define mass transfer between two reactors, or between reactors and the surrounding environment as defined by a reservoir.
Walls: Walls between reactors are used to allow heat transfer between reactors. By moving the walls of the reactor, its volume can be changed and expansion or compression work can be done by or on the reactor.
Reacting Surfaces: Reactions may occur on the surfaces of a reactor. These reactions may include both catalytic reactions and reactions resulting in net mass transfer between the surface and the fluid.

Reactor Networks#

While reactors by themselves define the governing equations, the time integration is performed by assembling reactors into a reactor network. A reactor network is therefore necessary even if only a single reactor is considered.

Cantera uses the CVODES and IDAS solvers from the SUNDIALS package to integrate the governing equations for the reactor network, which are a system of stiff ODEs or DAEs.

Preconditioning#

Some of Cantera’s reactor formulations (specifically, the Ideal Gas Control Volume Mole Reactor and the Ideal Gas Constant Pressure Mole Reactor) provide implementations of a sparse, approximate Jacobian matrix, which can be used by ReactorNet to generate a preconditioner and use a sparse, iterative linear solver within CVODES. This sparse, preconditioned method can significantly accelerate integration for reactors containing many species. A derivation of the derivative terms and benchmarks demonstrating the achievable performance gains can be found in Walker et al. [2023]. Examples demonstrating the use of this feature can be found in preconditioned_integration.py (large mechanism, single reactor) and preconditioned_network.py (network with several coupled reactors).

For reactor networks containing multiple reactors, each reactor contributes the Jacobian rows corresponding to its own governing equations. Terms involving flow devices, walls, and reacting surfaces may therefore appear in columns for other reactors in the same network.

The connector implementation is split between connector-specific scalar derivatives and reactor-specific state derivatives. Connectors such as valves, pressure controllers, and walls know simple derivatives such as \(\partial \dot m / \partial (P_1 - P_2)\), \(\partial \dot V / \partial (P_\t{left} - P_\t{right})\), or \(\partial \dot Q / \partial T_\t{left}\). Reactor objects know how pressure, temperature, and flow-carried composition depend on their own state variables. Helper methods such as addPressureJacobian, addTemperatureJacobian, addSpeciesMassFractionJacobian, and addEnthalpyJacobian combine these pieces.

For an IdealGasMoleReactor, the pressure derivatives used in connector terms are

\[ \frac{\partial P}{\partial T}\bigg|_V = \frac{\pi_T + P}{T}, \qquad \frac{\partial P}{\partial V}\bigg|_T = -\frac{1}{V \kappa_T}, \qquad \frac{\partial P}{\partial n_j} \approx \frac{RT}{V}, \]

where \(\pi_T = T(\partial P/\partial T)_V - P\) is the internal pressure and \(\kappa_T = -(1/V)(\partial V/\partial P)_T\) is the isothermal compressibility. The first two derivatives are evaluated exactly using the equation of state via ThermoPhase::internalPressure() and ThermoPhase::isothermalCompressibility(), so they are correct for both ideal and non-ideal phases. The species-mole pressure derivative uses an ideal-gas approximation \(\partial P/\partial n_j = RT/V\), which is exact for ideal gases. This approximation avoids EOS-specific partial-molar pressure evaluation and these terms are skipped by default (see skip-connector-pressure-composition-dependence).

For an IdealGasConstPressureMoleReactor, the reactor pressure is treated as fixed for these connector derivatives, so pressure-coupling terms from this reactor type are zero. Composition carried by a flow device is based on mass fractions because flow-device species fluxes are defined as \(\dot m Y_k\). Mole reactors therefore convert those composition derivatives back to their native species-mole state variables using

\[ Y_k = \frac{W_k n_k}{m}, \qquad \frac{\partial Y_k}{\partial n_j} = \frac{W_k \delta_{kj}}{m} - \frac{Y_k W_j}{m} \]

where \(\delta_{kj}\) is the Kronecker delta function.

For inlets carrying enthalpy into a reactor, the connector terms also include derivatives of the upstream specific enthalpy with respect to the upstream reactor’s state. The temperature contribution is

\[ \frac{\partial h_\t{in}}{\partial T_\t{in}} = c_{p,\t{in}}, \]

and the optional composition contribution is

\[ \frac{\partial h_\t{in}}{\partial n_{k,\t{in}}} = \frac{\bar{h}_k - h_\t{in} W_k}{m_\t{in}}, \]

where \(\bar{h}_k\) is the partial molar enthalpy of species \(k\) in the upstream reactor. The composition contribution adds one entry per species to the preconditioner and is controlled by the skip-connector-composition-dependence setting.

The wall heat-transfer terms illustrate the difference between the full Jacobian and the sparse approximation used for preconditioning. For a wall contribution to reactor \(i\) written as

\[ \dot{T}_i = \frac{f_i \dot{Q}_w}{C_i}, \]

where \(C_i = n_i \bar{c}_{v,i}\) for an IdealGasMoleReactor and \(C_i = n_i \bar{c}_{p,i}\) for an IdealGasConstPressureMoleReactor, the full derivative is

\[ \frac{\partial \dot{T}_i}{\partial y_j} = \frac{f_i}{C_i}\frac{\partial \dot{Q}_w}{\partial y_j} - \frac{f_i \dot{Q}_w}{C_i^2}\frac{\partial C_i}{\partial y_j}. \]

The second term includes temperature and composition derivatives of the reactor heat capacity. These terms are omitted from connector preconditioner entries because they add cost and, for composition derivatives, can add substantial fill-in. The implemented wall connector terms retain the numerator derivatives, for example \((f_i / C_i)\partial\dot{Q}_w/\partial T_\t{neighbor}\), which are the sparse cross-reactor couplings that most directly affect the iterative linear solve. Similar denominator-derivative terms are neglected for connector contributions involving enthalpy, internal energy, and pressure-expansion work.