Binary Gas Mixture Flow Model

This model builds on the compressible flow model using a binary (two-component) gas mixture. In this section we do not repeat the aforementioned theory but just highlight the differences introduced by the additional gas component. The mass, momentum, and energy equations remain largely unchanged, except that in most cases, the properties correspond to the mixture properties. Note the following notation: the subscript corresponds to the primary gas component, the subscript corresponds to the secondary gas component, and no subscript is used for mixture quantities. An additional equation is added to the system for the continuity equation for the species , as derived in (Hansel et al., 2018):

(1)

where is the mass fraction for component , is the diffusive flux of component , approximated using Fick's first law, and is the binary diffusion coefficient.

Additionally, an energy term is added to the energy equation:

(2)

Spatial Discretization

Here we describe how the energy diffusion term in Eq. (2) is computed in our spatial discretization. The mass diffusion term in Eq. (1) is computed similarly.

After integrating over an element for the finite volume discretization,

Considering a single edge ,

where the density and the diffusion coefficient are computed using linear interpolation between the and cell-center values:

and the mass fraction gradient is computed using the slope between the adjacent cell-center values:

The specific total enthalpy is computed as follows, where edge quantities are computed as linear interpolations as shown above:

Binary Diffusion Coefficient

The binary diffusion coefficient depends upon the following:

  • mixture composition

  • temperature,

  • pressure,

For a binary mixture of ideal gases, kinetic theory shows the following (Incropera et al., 2002) relationship for temperature and pressure:

This is useful for estimating the diffusion coefficient when the value is known at some other and conditions. For reference, at one atmosphere, the binary diffusion coefficient between water and air at 298 K is m/s (Incropera et al., 2002), which is a typical order of magnitude for this coefficient. The diffusion coefficient can be computed from first principles, which generally is within an error of 10%, but in general this is not practical for non-specialists. Instead it is recommended to search literature for appropriate values, which may or may not be available, depending on the how common the gas pairing is.

Initial Conditions

Initial conditions must now specify the initial, secondary gas mass fraction , in addition to the initial pressure , initial temperature , and initial velocity :

where the initial density and initial specific total energy are computed as follows:

References

  1. Joshua E. Hansel, Matthias S. Kunick, Ray A. Berry, and David Andrs. Non-condensable gases in RELAP-7. Technical Report INL/EXT-18-51163, Idaho National Laboratory, 2018.[BibTeX]
  2. Frank P. Incropera, David P. DeWitt, Theodore L. Bergman, Adrienne S. Lavine, and others. Fundamentals of Heat and Mass Transfer. Wiley New York, sixth edition, 2002.[BibTeX]