### 1.5.7. Multicomponent Energy Diffusion

The energy equation can be simplified in the special case that all species diffusivities are the same and equal to thermal conductivity divided by specific heat capacity,

| (1–160)

Equation 1–160 holds when the Lewis number is unity for all components: . For turbulent flow, assuming for all components is usually just as good as the common practice of using the fluid viscosity for the default component diffusivity (unity Schmidt number, ). For , the energy equation (Equation 1–159) simplifies exactly to the following:

| (1–161)

Equation 1–161 has the advantage that only a single diffusion term must be assembled, rather than one for each component plus one for heat conduction. This can significantly reduce numerical cost, in particular when the fluid consists of a large number of components.

When component-dependent turbulent Schmidt numbers are specified, the turbulent energy flux must be generalized. This is achieved by splitting the turbulent fluctuation of enthalpy into the contributions from temperature fluctuation, pressure fluctuation and fluctuations of component mass fractions:

| (1–162)

Using this transformation, the turbulent energy flux can be modeled by applying the eddy diffusivity model with turbulent Prandtl number to the temperature fluctuations plus the sum of the secondary enthalpy transport terms derived from the component mass fluxes:

| (1–163)

| (1–164)

| (1–165)

Inserting the eddy diffusivity model for the turbulent component mass fluxes yields the following model for the turbulent enthalpy flux:

| (1–166)

The previous relation is equivalent to the eddy diffusivity model applied to enthalpy and generalized for different and , as can be derived by expanding the enthalpy gradient term according to the chain rule :

| (1–167)

| (1–168)

| (1–169)

Using the above relations, the energy equation for generalized turbulent component transport becomes:

| (1–170)

And for the special case that the eddy diffusivity assumption is used for modeling the turbulent mass flux for each component:

| (1–171)

For the Thermal Energy heat transfer model the pressure gradients are assumed to be small compared to the other terms, and the corresponding terms are neglected from the equation.

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