Ideal Gas Control Volume Reactor#

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

\(m\), the mass of the reactor’s contents (in kg)
\(V\), the reactor volume (in m³)
\(T\), the temperature (in K)
\(Y_k\), the mass fractions for each species (dimensionless)

Equations 1-4 below are the governing equations for this reactor model. While the class name is historical, this formulation is now 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 gas 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.

Volume Equation#

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

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

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 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:

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

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

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

Energy Equation#

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

\[ U = U(T, V, m_1, \ldots, m_K), \qquad m_k = mY_k \]

and apply the chain rule:

\[ \frac{dU}{dt} = \frac{\partial U}{\partial T}\Bigg|_{V,m_k}\frac{dT}{dt} + \frac{\partial U}{\partial V}\Bigg|_{T,m_k}\frac{dV}{dt} + \sum_k \frac{\partial U}{\partial m_k}\Bigg|_{T,V,m_{j\ne k}} \frac{dm_k}{dt} \]

Here we can make the substitutions \((\partial U / \partial T)_{V,m_k} = m c_v\); \(\pi_T \equiv (\partial U / \partial V)_{T,m_k}\) is the internal pressure and \((\partial U / \partial m_k)_{T,V,m_{j\ne k}} = \tilde{u}_k /W_k \) where \(\tilde{u}_k\) are the partial molar internal energies at constant temperature and volume. This gives:

\[ \frac{dU}{dt} = m c_v \frac{dT}{dt} + \pi_T \frac{dV}{dt} + \sum_k \frac{\tilde{u}_k}{W_k}\frac{dm_k}{dt} \]

Substituting into the control-volume energy equation (4) gives:

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

While this form of the energy equation is somewhat more complicated, it significantly reduces the cost of evaluating the system Jacobian, since the derivatives of the species equations are taken at constant temperature instead of constant internal energy. It also eliminates the need to iteratively solve the equation of state to find \(T\) given \(u\) and \(v\).

In the case of an ideal gas, \(\pi_T = 0\) and \(\tilde{u}_k\) are equal to the usual partial molar internal energies \(\hat{u}_k\).