Ideal Gas Constant Pressure Reactor

A constant pressure reactor using temperature as a state variable, as implemented by the C++ class IdealGasConstPressureReactor and available in Python as the IdealGasConstPressureReactor class. It is defined by the state variables:

• \(m\), the mass of the reactor’s contents (in kg)

• \(T\), the temperature (in K)

• \(Y_k\), the mass fractions for each species (dimensionless)

Equations 1-3 below are the governing equations for this reactor model. While the class name is historical, this formulation is applicable to non-ideal equations of state as well.

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 surfaces:

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

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.

Species Equations

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:

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

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

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

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

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

Energy Equation

In this reactor model, the reactor temperature \(T\) is used as a state variable instead of the total enthalpy \(H\). For the mass-based form, write:

\[ H = H(T, P, m_1, \ldots, m_K), \qquad m_k = mY_k \]

At constant pressure, applying the chain rule gives:

\[ \frac{dH}{dt} = m c_p \frac{dT}{dt} + \sum_k \frac{\bar{h}_k}{W_k} \frac{dm_k}{dt} \]

where \(\bar{h}_k\) are the partial molar enthalpies and \(W_k\) are the molecular weights. Substituting the species and mass equations into the constant pressure reactor energy equation (3) yields:

(3)\[ m c_p \frac{dT}{dt} = \dot{Q} - \sum_k \frac{\bar{h}_k}{W_k} \dot{m}_{k,\t{gen}} + \sum_\t{in} \left(\dot{m}_\t{in} h_\t{in} - \sum_k \frac{\bar{h}_k}{W_k} \dot{m}_{k,\t{in}} \right) \]