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 an ideal gas control volume mole reactor.

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 the case of the ideal gas control volume mole reactor model, the reactor temperature \(T\) is used instead of the total internal energy \(U\) as a state variable. For an ideal gas, we can rewrite the total internal energy in terms of the species moles and temperature:

\[ U = \sum_k \hat{u}_k(T) n_k \]

and differentiate it with respect to time to obtain:

\[ \frac{dU}{dt} = \frac{dT}{dt}\sum_k n_k \hat{c}_{v,k} + \sum \hat{u}_k \dot{n}_k \]

Substituting this into the energy equation for the control volume mole reactor (3) yields an equation for the temperature:

(3)#\[ \sum_k n_k \hat{c}_{v,k} \frac{dT}{dt} = \dot{Q} - \sum \hat{u}_k \dot{n}_k \]