Ideal Gas Constant Pressure Mole Reactor

An ideal gas constant pressure mole reactor, 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 an ideal gas constant pressure mole reactor.

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:

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

where positive \(\dot{Q}\) represents heat addition to the system and \(h\) is the specific 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:

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

As for the ideal gas mole reactor, we replace the total enthalpy as a state variable with the temperature by writing the total enthalpy in terms of the species moles and temperature:

\[ H = \sum_k \hat{h}_k(T) n_k \]

and differentiating with respect to time:

\[ \frac{dH}{dt} = \frac{dT}{dt}\sum_k n_k \hat{c}_{p,k} + \sum \hat{h}_k \dot{n}_k \]

Making this substitution and rearranging yields an equation for the temperature:

(2)\[ \sum_k n_k \hat{c}_{p,k} \frac{dT}{dt} = \dot{Q} - \sum \hat{h}_k \dot{n}_k \]