There are several different, but related variables to quantify the content of a component “” in an -component flow:

The four quantities are related as follows:

	5.442
	5.443
	5.444

where is the molecular weight of component ;  is the mixture density; and is the sum of the molar concentrations of all the components in a system:

5.445

where is the mixture molecular weight:

5.446

equation 5.445 indicates that with mass-fraction weighted molecular weight for the mixture, equation 5.442 applies also for the mixture of the -components.

In addition, from the definitions in equation 5.443 and equation 5.444, the sum of both the molar and mass fractions must be unity:

5.447

In CFD solvers, in general, the mass fraction of the arbitrary component , , is obtained directly by solving partial differential transport equations, while the other variables, , and are auxiliary variables, primarily used for post-processing.

In a multicomponent flow, the bulk motion of the mixture is modelled using single velocity, pressure, temperature and turbulence fields. For the mixing and transport of the chemical species, each component has its own governing equation for the conservation of mass. The influence of multiple components on the bulk flow is felt through the variation of the mixture properties (density, viscosity, etc.) with the component properties and the local mass fractions. The set of governing equations is presented in this subsection.

Mass Fraction Equations

For an -component mixture flow, if there are no chemical reactions, the transport of an arbitrary component “” is governed by the following mass fraction equation:

5.448

where and are mixture density and velocity; is any user-defined source; and is mass diffusion term. Note that for laminar flows, the velocity vector and the mass fraction are instantaneous variables, while for turbulent flows they are Favre-averaged quantities since the multicomponent flows are generally considered as variable density/compressible flows.

In equation 5.448, the mixture quantities and the mass diffusion term are defined in the following:

Mixture Density

The mixture density is, by default, the mass-averaged value of all the component densities:

5.449

For a mixture of gaseous species, the mixture density is often computed using the ideal gas law based on the mixture molecular weight, , calculated by equation 5.446:

5.450

where is the universal gas constant; is the mixture temperature; and is the absolute pressure. If the operating pressure (constant) is used, then equation 5.450 is reduced to the so-called incompressible ideal gas law. It is an appropriate assumption for the mixing and transport of species, where the gauge pressure is often negligible compared to the operating pressure.

Mixture Velocity

In definition, the mixture velocity is the mass-averaged value of all the component velocities:

5.451

However, since only a single velocity is solved, the mixture velocity and all the component velocities are assumed to have the same values.

Mass Diffusion Flux

The mass diffusion flux of the component "" consists of two parts: the laminar and turbulent diffusion terms, which may be expressed as:

5.452

In equation 5.452, is the laminar diffusion flux of the component,"", which arises due to gradients of concentration and temperature. By default, Simerics-MP uses the dilute approximation, also referred to as the Fick's law, to model the mass diffusion due to concentration gradients. The laminar diffusion flux has the following formulation:

5.453

where is the mass diffusion coefficient for the component “” in the mixture; and is the thermal (Soret) diffusion coefficient.

For turbulent flows, the fluctuating term, derived from Favre-averaging the advection in equation 5.448, is modelled as turbulent diffusion:

5.454

where is the turbulent viscosity; and  is the turbulent Schmidt number. By default, .

It may be noted that the turbulent diffusion generally overwhelms the laminar diffusion, and the specification of detailed laminar diffusion properties in turbulent flows is generally less important than the turbulent counterpart.

To derive the mass continuity equation for the mixture flow, we add all the component mass fraction equations together and apply equation 5.447:

5.455

Clearly, to satisfy the total mass conservation of the mixture flow, the sum of the diffusion terms for all components must be zero,

5.456

From equation 5.447 and equation 5.454, it is easy to see that the turbulent diffusion term is always zero. Therefore, for fully turbulent flows, equation 5.456 is usually considered automatically satisfied. However, for laminar flows, or when the laminar mass diffusion cannot be ignored in some turbulent flows, equation 5.456 reduces to the following form:

5.457

Then to satisfy equation 5.457, we impose the two separate constraints:

	5.458
	5.459

The continuity equation of the multicomponent flows has then the final form:

5.460

Diffusion Coefficients

To solve the transport equation 5.448, in particular, for multicomponent laminar flows, the mass diffusion coefficient () and the thermal diffusion coefficient () for each component in a mixture are required. The methods to determine and are described in this subsection.

Mass Diffusion Coefficients

The formulation of the mass diffusion flux in laminar flows, equation 5.453, is strictly valid when the mixture composition is not changing, or when is independent of the composition. This is an acceptable approximation in dilute mixtures when  is very small for all the components except the carrier gas. For non-dilute mixtures in multicomponent laminar flows, can be computed from the following formulation:

5.461

where is the binary mass diffusion coefficient of component "" in component "", which needs to be specified or calculated.

Specified Value

The binary mass diffusion coefficient is a constant or function of temperature if heat transfer is accounted for. It can be a directly specified value, or obtained from the specified Schmidt number:

5.462

where is the Schmidt number, defined as the ratio of the viscous diffusion rate to the molecular (mass) diffusion rate.

If one value or one function of temperature applies for all the components, equation 5.461 is reduced to

5.463

equation 5.463 is an appropriate approximation for modelling a dilute mixture, with the species present at low mass fractions in a "carrier'' fluid that is present at high concentration. In such cases,  may also be defined directly as a constant or a function of temperature.

However, for non-dilute mixtures, with the specified , equation 5.461 is used to compute the individual mass diffusion coefficient in the mixture, .

Kinetic Theory

For an ideal gas, the binary mass diffusion coefficient can also be obtained using kinetic theory 1]. H. A. McGee, “Molecular Engineering”, McGraw-Hill, New York, 1991.

5.464

where is the absolute pressure, and is the diffusion collision integral, which is a measure of the interaction of the molecules in the system. is a function of the quantity , defined as:

5.465

is the Boltzmann constant, which is defined as the universal gas constant divided by the Avogadro number.  for the mixture is the geometric average:

5.466

For a binary mixture, is calculated as the arithmetic average of the individual and :

5.467

and are the so-called Lennard-Jones parameters for component in the mixture. Specifically,  is the collision cross-section of the sphere molecule with the diameter of  (note that a molecule sweeps out an area given by twice its diameter, as the molecules with which it collides also have diameter ); and =1.38064852(79) ×10^-23(J/K) is the Boltzmann constant.

In Simerics-MP, to determine the two Lennard-Jones parameters, the diameter and the energy are specified values by user inputs.

Thermal Diffusion Coefficients

The thermal diffusion coefficients () can be defined as constants, polynomial functions of temperature, user-defined functions, or using the following empirically-based composition-dependent expression derived from 2K. K. Y. Kuo, “Principles of Combustion”, John Wiley and Sons, New York, 1986.:

5.468

This form of the thermal diffusion coefficient will cause heavy molecules to diffuse less rapidly, and light molecules to diffuse more rapidly, towards heated surfaces.

Momentum Equations

With the mass-weighted properties and velocities, the momentum equations for the mixture of all the components have the same expression as those for single fluid flows:

5.469

where the mixture density and velocity are calculated using equation 5.449 to equation 5.451. For the turbulent viscosity, it is directly computed from the turbulence models based on the mixture flow so that its value is independent of the components. For the laminar viscosity, it is computed as follows:

Mass-Averaged Laminar Viscosity

For non-ideal gas mixtures, the mixture viscosity is computed based on a simple mass fraction average of the pure chemical species (components) viscosities:

5.470

In Simerics-MP, this is the default method.

Kinetic Theory

For ideal gas mixtures, the mixture viscosity may be computed based on kinetic theory. For each component, the dynamic viscosity is based on the Boltzmann Equation:

5.471

As for the mass diffusivity, the Lennard-Jones parameters, and are required as inputs to calculate the viscosities of the gas components in a mixture.

The viscosity for the ideal gas mixture is then calculated as:

 

5.472

where

5.473

Energy Equation

As described in the module of Heat, the energy equation for the mixture of all the components takes the following form:

5.474

where and are the total internal energy and total enthalpy of the -component mixture. Along with the mixture specific heat () and static enthalpy (), they are obtained by mass-averaging the corresponding values of each component:

Mass-Averaged Mixture Heat Capacity

5.475

Mass-Averaged Mixture Energy and Enthalpy

	5.476
	5.477
	5.478

As described in the Heat module, the static enthalpy of a component consists of two parts: the standard state reference enthalpy and sensible enthalpy. For multicomponent flows, both parts of the enthalpy should be included (the absolute/total value) when is calculated. The detailed formulations of the component enthalpies for both incompressible and compressible flows can be found in the Heat module.

In equation 5.464, the first term on the right-hand side represents the diffusion of the energy. It consists of three parts: heat conduction, energy transport due to the diffusion of the species and viscous heating. For the mixture heat conduction, it is modelled in the same way as in the single fluid flow. However, special attention needs to be paid for the computation of the mixture heat conductivity. In Simerics-MP, it is calculated as following (the same approach as for the mixture laminar viscosity):

Mass-Averaged Heat Conductivity

For non-ideal gas mixtures, the mixture heat conductivity is computed based on a simple mass fraction average of the pure species (components) heat conductivities:

5.479

In Simerics-MP, this is the default method.

Kinetic Theory

For ideal gas mixtures, the mixture heat conductivity may be computed based on kinetic theory. For each component, the heat conductivity has the form:

5.480

where is the universal gas constant; is the molecular weight, is the component's specified or computed viscosity, and  is the component's specified or computed specific heat capacity. Note that as the laminar viscosity, , the specific heat may also be obtained using kinetic theory:

5.481

where is the number of modes of energy storage (degrees of freedom) for the gas component,””.

The heat conductivity for the ideal gas mixture is then calculated as:

5.482

where is expressed in equation 5.463.

The second diffusion term

5.483

represents the transport of enthalpy due to the diffusion of the chemical species in an -component flow. This term can have a significant effect on the enthalpy field and should not be neglected. In particular, when the Lewis number (the ratio of thermal diffusivity, , to mass diffusivity, ):

5.484

for any species is far from unity, neglecting this term can lead to significant errors.

The third diffusion term is the viscous heating contribution, . Though it is treated in the same way as in the single fluid flow, the shear is calculated using the mixture laminar and turbulent viscosities. The general source term is the total external/user heat source on all the components.

Turbulence Models

With the mixture density , molecular viscosity , and velocity , the turbulence modelling equations in both the standard k-ε and RNG k-ε models have the same general forms as in the single fluid turbulence models, described in Turbulence module. The turbulent viscosity for the mixture , is computed directly from the expression:

5.485

And the production of turbulent kinetic energy is calculated based on mixture turbulent viscosity and velocity gradients:

In a multicomponent flow, the boundary conditions for the flow, energy and turbulence modelling equations are the same as those in the single phase flows, described in Flow, Heat and Turbulence modules respectively. For the mass fractions of a component, the boundary conditions consist of specified value, specified volumetric flux, and/or gradient.

n-Component Inlet Boundary

At an inlet boundary, the net transport of a component can consist of both convection and diffusion contributions. The convection is fixed by the specified inlet species mass fraction, whereas the diffusion depends on the gradient of the computed mass fraction field, which is not known a priori. At very small convective inlet velocities, substantial mass can be gained or lost through the inlet due to diffusion. For this reason, the inlet diffusion is not included by default, but can be enabled as an option.

Specified Value

For an -component flow, the inlet mass fractions are pre-determined for () components, while the mass fraction of the  component is obtained using the physical constraint, equation 5.447:

	5.486
	5.487

And the mass fraction for each component must be non-negative.

Specified Volumetric Flux

Assuming that is the pre-described inlet volumetric flux for the component “” we have the mass flux of each component and the total mass flux at the inlet as:

5.488

where is the inlet density of the component “”.

By definition, the mass fraction is computed as:

5.489

Outlet/Symmetry/Wall Boundary

For () components, zero-gradient conditions apply for all the outlet, symmetry, and wall boundaries, while the phase is obtained using the physical constraint:

	5.490
	5.491

where is the boundary value obtained from equation 5.475.

The above governing equations, turbulence models and boundary conditions form the foundation of the multicomponent mixing model. Without external/user source terms and chemical reactions, they are a closed system of equations and can be solved numerically using a pressure-based finite volume solver. Since the mixture of the components has the similar set of governing equations to that in variable density single-fluid flows, no special treatment is required for the numerical algorithms and face volume flux formulation (see Numerics).

The mass-fraction transport equations are solved for all the components. To satisfy the physical constraint, the actual mass fractions are scaled by the sum of the solved values for all the components:

5.492

where is the value obtained from solving equation 5.448. The actual mass fraction is:

5.493