### 1.5.2. Scalar Transport Equation

For a multicomponent fluid, scalar transport equations are solved for velocity, pressure, temperature and other quantities of the fluid. For details, see Governing Equations. However, additional equations must be solved to determine how the components of the fluid are transported within the fluid.

The bulk motion of the fluid is modeled using single velocity, pressure, temperature and turbulence fields. The influence of the multiple components is felt only through property variation by virtue of differing properties for the various components. Each component has its' own equation for conservation of mass. After Reynolds-averaging (see Turbulence Models) this equation can be expressed in tensor notation as:

| (1–145)

where:

is the mass-average density of fluid component in the mixture, that is, the mass of the component per unit volume,

is the mass-average velocity field,

is the mass-average velocity of fluid component ,

is the relative mass flux,

is the source term for component , which includes the effects of chemical reactions.

Note that if all the terms in the expanded form of Equation 1–145 are summed over all components, the result is the standard continuity equation,

| (1–146)

because the reaction rates must sum to zero.

The relative mass flux term accounts for differential motion of the individual components. This term may be modeled in a number of ways to include effects of concentration gradients, a pressure gradient, external forces or a temperature gradient. Of these possible sources of relative motion among the mixture components, the primary effect is that of concentration gradient. The model for this effect gives rise to a diffusion-like term in Equation 1–145.

| (1–147)

The molecular diffusion coefficient, , is assumed to be equal to , where is the Kinematic Diffusivity set on the Fluid Models tab for a domain in CFX-Pre. For details, see Fluid Models Tab in the CFX-Pre User's Guide. A detailed description of the effects of the relative mass flux term and various models for it may be found in reference [29].

Now, define the mass fraction of component to be:

| (1–148)

Note that, by definition, the sum of component mass fractions over all components is 1. Substituting Equation 1–148 and Equation 1–147 into Equation 1–145, you have:

| (1–149)

The turbulent scalar fluxes are modeled using the eddy dissipation assumption as:

| (1–150)

where is the turbulent Schmidt number. Substituting Equation 1–150 into Equation 1–149 and assuming now that you have mass weighted averages of :

| (1–151)

where:

| (1–152)

Equation 1–151 is simply a general advection-diffusion equation of the form common to the equations solved for each of the other dependent variables in the fluid flow calculation. Thus, it is convenient to solve for the in order to establish the composition of the fluid mixture.

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