Ideal Gas Constant Pressure Mole Reactor

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

• \(T\), the temperature (in K)

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

Equations 1 and 2 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.

Species Equations

The moles of each species in the reactor 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 surfaces. The rate at which species \(k\) is generated through homogeneous phase reactions is \(V \dot{\omega}_k\), and the total rate at which moles of species \(k\) changes is:

(1)\[ \frac{dn_k}{dt} = V \dot{\omega}_k + \sum_\t{in} \dot{n}_{k, \t{in}} - \sum_\t{out} \dot{n}_{k, \t{out}} + \dot{n}_{k, \t{wall}} \]

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.

Energy Equation

Writing the first law for an open system gives:

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

where positive \(\dot{Q}\) represents heat addition to the system and \(\hat{h}\) is the molar enthalpy of the reactor’s contents.

Differentiating the definition of the total enthalpy, \(H = U + pV\), with respect to time gives:

\[ \frac{dH}{dt} = \frac{dU}{dt} + p \frac{dV}{dt} + V \frac{dp}{dt} \]

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

\[ \frac{dH}{dt} = \dot{Q} + \sum_\t{in} \hat{h}_\t{in} \dot{n}_\t{in} - \hat{h} \sum_\t{out} \dot{n}_\t{out} \]

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

\[ H = H(T, P, n_1, \ldots, n_K) \]

At constant pressure, applying the chain rule gives:

\[ \frac{dH}{dt} = N \hat{c}_p \frac{dT}{dt} + \sum_k \bar{h}_k \frac{dn_k}{dt} \]

where \(N\) is the total number of moles, \(\hat{c}_p\) is the mixture molar heat capacity, and \(\bar{h}_k\) are the partial molar enthalpies. Making this substitution yields:

\[\begin{split} N \hat{c}_p \frac{dT}{dt} = & \dot{Q} + \sum_\t{in} \dot{n}_\t{in} \hat{h}_\t{in} - \hat{h} \sum_\t{out} \dot{n}_\t{out} \\ & - \sum_k \bar{h}_k \left(V \dot{\omega}_k + \dot{n}_{k,\t{wall}} + \sum_\t{in} \dot{n}_{k,\t{in}} - \sum_\t{out} \dot{n}_{k,\t{out}} \right) \end{split}\]

Rearranging and simplifying gives the final energy equation:

(2)\[ N \hat{c}_p \frac{dT}{dt} = \dot{Q} - \sum_k \bar{h}_k \left(V \dot{\omega}_k + \dot{n}_{k,\t{wall}} \right) + \sum_\t{in} \left(\dot{n}_\t{in} \hat{h}_\t{in} - \sum_k \bar{h}_k \dot{n}_{k,\t{in}} \right) \]