## Plug Flow Reactor¶

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

## Plug Flow Reactor¶

A Plug Flow Reactor (PFR) represents a steady-state flow in a channel. Perpendicular to the flow direction, the gas is considered to be completely homogeneous. In the axial direction $$z$$, the state of the gas is allowed to change. However, all diffusion processes are neglected.

In addition, the interior surface of the reactor may consist of one or more catalytically active surfaces where heterogenous reactions occur.

Plug-Flow Reactors are often used to simulate emission formation and catalytic processes.

A Plug Flow Reactor is defined by the state variables:

• $$\rho$$, the density of the fluid phase (in kg/m3)

• $$u$$, the velocity of the fluid phase (in m/s)

• $$p$$, the pressure (in Pa)

• $$T$$, the temperature (in K)

• $$Y_k$$, the mass fractions for each fluid phase species (dimensionless)

• $$\theta_{i,j}$$, the coverage of each surface species $$i$$ on each surface $$j$$ (dimensionless)

The reactor geometry is defined by the length $$L$$, total volume $$V$$, and the surface area $$A_{s,j}$$ for each surface, where the surface area per unit length is assumed to be a constant along the length of the reactor.

The governing equations for a PFR are a system of differential-algebraic equations, which depend on the spatial derivatives of some but not all of the state variables. The plug flow reactor model in Cantera is implemented by class FlowReactor.

Equation of state

The fluid satisfies the ideal gas law:

\begin{equation*} \rho = p \frac{R}{\overline{W}} T \end{equation*}

where $$R$$ is the universal gas constant and $$\overline{W}$$ is the mixture molecular weight.

Mass conservation:

The net rate at which fluid phase species are generated by reactions on the walls (per unit cross sectional area) can be defined as

\begin{equation*} \dot{m}_s = \sum_j \frac{A_{s,j}}{V} \left(\sum_k \dot{s}_{k,j} W_k\right) \end{equation*}

where $$\dot{s}_{k,j}$$ is the net rate of production of species $$k$$ on surface $$j$$ and $$W_k$$ is the molecular weight of species $$k$$. The overall mass conservation equation for the reactor can then be written as:

\begin{equation*} u \frac{d\rho}{dz} + \rho \frac{du}{dz} = \dot{m}_s \tag{1} \end{equation*}

Momentum conservation in the axial direction:

\begin{equation*} \rho u \frac{du}{dz} = - u \sum_j \frac{A_{s,j}}{V} \dot{m}_s - \frac{dp}{dz} \tag{2} \end{equation*}

Energy conservation:

\begin{equation*} \rho u c_p \frac{dT}{dz} = - \sum_k \hat{h}_k \dot{\omega}_k - \sum_j \frac{A_{s,j}}{V} \left(\sum_k \hat{h}_k \dot{s}_{k,j}\right) \tag{3} \end{equation*}

where $$c_p$$ is the specific heat capacity of the mixture and $$\hat{h}_k$$ is the molar enthalpy of species $$k$$. Kinetic and potential energies are neglected.

Conservation equation for fluid species $$k$$:

\begin{equation*} \rho u \frac{d Y_k}{dz} = - Y_k \dot{m}_s + \dot{\omega}_k W_k + \sum_j \frac{A_{s,j}}{V} \dot{s}_k W_k \tag{4} \end{equation*}

Conservation equation for surface species $$i,j$$:

Because the PFR is modeled as steady state, net rate of production for each surface species must be zero. To satisfy the constraint that the total surface coverage is 1, the conservation equation for the first surface species on each surface is replaced by this constraint:

\begin{equation*} \dot{s}_{i,j} = 0,\quad i \ne 0 \tag{5} \end{equation*}
\begin{equation*} \dot{s}_{0,j} = 1 - \sum_{i\ne 0} \theta_{i,j} \tag{6} \end{equation*}

### Integrating the PFR¶

Because diffusion is neglected, downstream parts of the reactor have no influence on upstream parts. Therefore, PFRs can be integrated as initial value problems, starting from the composition at the inlet. Some care is required to determine initial values for the algebraic variables (the surface species coverages) and the time derivatives for the differential variables (the other state variables) that are consistent with the governing equations.

To do this, we first solve the steady-state problem for each surface, holding the fluid phase composition, temperature, and pressure fixed at the inlet values to determine the values of $$\theta_{i,j}(z=0)$$. Then, we construct a linear system comprising the ideal gas law, differentiated with respect to $$z$$ and equations (1), (2), (3), and (4) evaluated at $$z=0$$. This system is then solved to obtain the values of $$d\rho/dz$$, $$du/dz$$, $$dp/dz$$, $$dT/dz$$, and $$dY_k/dz$$ at $$z=0$$.