Control Volume Reactor#

This model represents a homogeneous zero-dimensional reactor, as implemented by the C++ class Reactor and available in Python as the Reactor class. A control volume reactor is defined by the state variables:

  • \(m\), the mass of the reactor’s contents (in kg)

  • \(V\), the reactor volume (in m3)

  • \(U\), the total internal energy of the reactors contents (in J)

  • \(Y_k\), the mass fractions for each species (dimensionless)

Equations 1-4 below are the governing equations for a control volume reactor.

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 homogeneous 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 changes 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#

The equation for the total internal energy is found by writing the first law for an open system:

(4)#\[ \frac{dU}{dt} = - p \frac{dV}{dt} + \dot{Q} + \sum_\t{in} \dot{m}_\t{in} h_\t{in} - h \sum_\t{out} \dot{m}_\t{out} \]

Where \(\dot{Q}\) is the net rate of heat addition to the system.

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