Governing equations#

Multicomponent mixture#

Let \(\mathbf{v}\) be the mass-averaged velocity of the mixture, defined as:

\[\mathbf{v} := \sum_{i = 1}^{n} Y_i \mathbf{u}_i\]

where:

\(Y_i\) is the mass fraction of species \(i\)
\(\mathbf{u}_i\) is the velocity of species \(i\)
\(n\) is the number of species (components) in the mixture

In an analogous way, we can define the molar-averaged velocity of the mixture, \(\mathbf{u}\), as:

\[\mathbf{u} := \sum_{i = 1}^{n} X_i \mathbf{u}_i\]

where:

\(X_i\) is the mole fraction of species \(i\)
\(\mathbf{u}_i\) is the velocity of species \(i\)
\(n\) is the number of species (components) in the mixture

At a given point in space and time, transport of physical quantities in a multicomponent mixture can be described by the following set of governing equations written in the conservative (strong) form:

Continuity equation#

\[\frac{\partial \rho}{\partial t} = - \nabla \cdot \rho \mathbf{v}\]

where:

\(\rho\) is the mixture mass density
\(\mathbf{v}\) is the mass-averaged velocity of the mixture

Species mass conservation equation#

\[\frac{\partial \rho Y_i}{\partial t} = - \nabla \cdot \rho Y_i \mathbf{v} - \nabla \cdot \mathbf{j}_i + \omega_i\]

where:

\(\rho\) is the mixture mass density
\(Y_i\) is the mass fraction of species \(i\)
\(\mathbf{v}\) is the mass-averaged velocity of the mixture
\(\mathbf{j}_i\) is the mass diffusive flux of species \(i\) relative to a mass-averaged velocity
\(\omega_i\) is the net mass production rate of species \(i\)

Species moles conservation equation#

\[\frac{\partial c X_i}{\partial t} = - \nabla \cdot c X_i \mathbf{u} - \nabla \cdot \mathbf{J}_i + s_i\]

where:

\(c\) is the mixture molar density
\(X_i\) is the mole fraction of species \(i\)
\(\mathbf{u}\) is the molar-averaged velocity of the mixture
\(\mathbf{J}_i\) is the molar diffusive flux of species \(i\) relative to a molar-averaged velocity
\(s_i\) is the net molar production rate of species \(i\)

Momentum equation#

\[\frac{\partial \rho \mathbf{v}}{\partial t} = - \nabla \cdot \rho \mathbf{v} \mathbf{v} - \nabla \cdot \pmb{\tau} - \nabla \cdot p \mathbf{I} + \rho \sum_{i=1}^{n} Y_i \mathbf{f}_i\]

where:

\(\rho\) is the mixture mass density
\(\mathbf{v}\) is the mass-averaged velocity of the mixture
\(\pmb{\tau}\) is the viscous momentum flux tensor
\(p\) is the pressure
\(\mathbf{I}\) is the identity matrix
\(Y_i\) is the mass fraction of species \(i\)
\(\mathbf{f}_i\) is the net acceleration from body forces acting on species \(i\)
\(n\) is the number of species (components) in the mixture

Total internal energy equation#

\[\frac{\partial \rho e_0}{\partial t} = - \nabla \cdot \rho e_0 \mathbf{v} - \nabla \cdot \mathbf{q} - \nabla \cdot \pmb{\tau} \cdot \mathbf{v} - \nabla \cdot p \mathbf{v} + \sum_{i=1}^{n} \mathbf{f}_i \cdot \mathbf{n}_i\]

where:

\(\rho\) is the mixture mass density
\(e_0\) is the mixture total internal energy
\(\mathbf{v}\) is the mass-averaged velocity of the mixture
\(\mathbf{q}\) is the heat flux
\(\pmb{\tau}\) is the viscous momentum flux tensor
\(p\) is the pressure
\(\mathbf{f}_i\) is the net acceleration from body forces acting on species \(i\)
\(\mathbf{n}_i\) is the total mass flux of species \(i\)
\(n\) is the number of species (components) in the mixture

Internal energy equation#

\[\frac{\partial \rho e}{\partial t} = - \nabla \cdot \rho e \mathbf{v} - \nabla \cdot \mathbf{q} - \pmb{\tau} : \nabla \mathbf{v} - p \nabla \cdot \mathbf{v} + \sum_{i=1}^{n} \mathbf{f}_i \cdot \mathbf{j}_i\]

where:

\(\rho\) is the mixture mass density
\(e\) is the mixture internal energy
\(\mathbf{v}\) is the mass-averaged velocity of the mixture
\(\mathbf{q}\) is the heat flux
\(\pmb{\tau}\) is the viscous momentum flux tensor
\(p\) is the pressure
\(\mathbf{f}_i\) is the net acceleration from body forces acting on species \(i\)
\(\mathbf{j}_i\) is the mass diffusive flux of species \(i\) relative to a mass-averaged velocity
\(n\) is the number of species (components) in the mixture

Enthalpy equation#

\[\frac{\partial \rho h}{\partial t} = - \nabla \cdot \rho h \mathbf{v} - \nabla \cdot \mathbf{q} - \pmb{\tau} : \nabla \mathbf{v} + \frac{Dp}{Dt} + \sum_{i=1}^{n} \mathbf{f}_i \cdot \mathbf{j}_i\]

where:

\(\rho\) is the mixture mass density
\(h\) is the mixture enthalpy
\(\mathbf{v}\) is the mass-averaged velocity of the mixture
\(\mathbf{q}\) is the heat flux
\(\pmb{\tau}\) is the viscous momentum flux tensor
\(p\) is the pressure
\(\mathbf{f}_i\) is the net acceleration from body forces acting on species \(i\)
\(\mathbf{j}_i\) is the mass diffusive flux of species \(i\) relative to a mass-averaged velocity
\(n\) is the number of species (components) in the mixture

Temperature equation#

\[\rho c_p \frac{DT}{D t} = - \nabla \cdot \mathbf{q} + \alpha T \frac{Dp}{Dt} - \pmb{\tau} : \nabla \mathbf{v} + \sum_{i=1}^{n} \big( h_i (\nabla \cdot \mathbf{j}_i - \omega_i) + \mathbf{f}_i \cdot \mathbf{j}_i \big)\]

where:

\(\rho\) is the mixture mass density
\(c_p\) is the mixture isobaric specific heat capacity
\(T\) is the mixture temperature
\(\mathbf{q}\) is the heat flux
\(\alpha\) is the coefficient of thermal expansion
\(p\) is the pressure
\(\pmb{\tau}\) is the viscous momentum flux tensor
\(\mathbf{v}\) is the mass-averaged velocity of the mixture
\(h_i\) is the enthalpy of species \(i\)
\(\mathbf{j}_i\) is the mass diffusive flux of species \(i\) relative to a mass-averaged velocity
\(\omega_i\) is the net mass production rate of species \(i\)
\(\mathbf{f}_i\) is the net acceleration from body forces acting on species \(i\)
\(n\) is the number of species (components) in the mixture

Entropy equation#

\[\frac{\partial \rho s}{\partial t} = - \nabla \cdot \rho s \mathbf{v} - \nabla \Big( \frac{1}{T} \big( \mathbf{q} - \sum_{i=1}^{n} \tilde{\mu}_i \mathbf{j}_i \big) \Big) + \mathbf{q} \cdot \nabla \Big( \frac{1}{T} \Big) - \sum_{i=1}^{n} \mathbf{j}_i \cdot \nabla \Big( \frac{\tilde{\mu}_i}{T} \Big) - \frac{1}{T} \pmb{\tau} : \nabla \mathbf{v} + \frac{1}{T} \sum_{i=1}^{n} \mathbf{f}_i \cdot \mathbf{j}_i - \frac{1}{T} \sum_{i=1}^{n} \tilde{\mu}_i \omega_i\]

where:

\(\rho\) is the mixture mass density
\(s\) is the mixture entropy
\(\mathbf{v}\) is the mass-averaged velocity of the mixture
\(T\) is the mixture temperature
\(\mathbf{q}\) is the heat flux
\(\tilde{\mu}_i\) is the chemical potential of species \(i\)
\(\mathbf{j}_i\) is the mass diffusive flux of species \(i\) relative to a mass-averaged velocity
\(\pmb{\tau}\) is the viscous momentum flux tensor
\(n\) is the number of species (components) in the mixture