Ideal Gas Control Volume Mole Reactor#

An ideal gas control volume mole reactor, as implemented by the C++ class IdealGasMoleReactor and available in Python as the IdealGasMoleReactor class. It is defined by the state variables:

\(T\), the temperature (in K)
\(V\), the reactor volume (in m³)
\(n_k\), the number of moles for each species (in kmol)

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

Volume Equation#

The reactor volume can change as a function of time due to the motion of one or more walls:

(1)#\[ \frac{dV}{dt} = \sum_w f_w A_w v_w(t) \]

Where \(f_w = \pm 1\) indicates the facing of the wall (whether moving the wall increases or decreases the volume of the reactor), \(A_w\) is the surface area of the wall, and \(v_w(t)\) is the velocity of the wall as a function of time.

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:

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

Energy Equation#

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

\[ U = U(T, V, n_1, \ldots, n_K) \]

and differentiate it with respect to time to obtain:

\[ \frac{dU}{dt} = N \hat{c}_v \frac{dT}{dt} + \pi_T \frac{dV}{dt} + \sum_k \tilde{u}_k \frac{dn_k}{dt} \]

where

\[ \pi_T \equiv \left.\frac{\partial U}{\partial V}\right|_{T, n} ,\qquad \tilde{u}_k \equiv \left.\frac{\partial U}{\partial n_k}\right|_{T, V, n_{j\ne k}} \]

Here, \(\pi_T\) is the internal pressure and \(\tilde{u}_k\) are the partial molar internal energies at constant temperature and volume.

Combining this expression for \(dU/dt\) with the total energy equation for the general control volume mole reactor (3) yields an equation for the temperature:

\[ N \hat{c}_v \frac{dT}{dt} = -(p + \pi_T) \frac{dV}{dt} + \dot{Q} + \sum_\t{in} \dot{n}_\t{in} \hat{h}_\t{in} - \hat{h} \sum_\t{out} \dot{n}_\t{out} - \sum_k \tilde{u}_k \frac{dn_k}{dt} \]

Substituting the species equation (2) for \(dn_k/dt\) and making some conversions between mass and moles gives the final form of the energy equation:

(3)#\[\begin{split} m c_v \frac{dT}{dt} = & -(p + \pi_T) \frac{dV}{dt} + \dot{Q} - \sum_k \tilde{u}_k \left(\dot{\omega}_k V + \dot{n}_{k, \t{wall}} \right) \\ & - \frac{pV}{m} \sum_\t{out} \dot{m}_\t{out} + \sum_\t{in} \left(h_\t{in} \dot{m}_\t{in} - \sum_k \tilde{u}_k \dot{n}_{k,\t{in}} \right) \end{split}\]