### 1.14.1. Wall Condensation Model

A laminar boundary layer model (Equation 1–308 in the CFX-Solver Theory Guide) is employed for laminar flow. A turbulent boundary layer model (Equation 1–314 in the CFX-Solver Theory Guide) is employed for turbulent flow. This makes use of the turbulent wall functions.

The following topics are discussed:

- Laminar Boundary Layer Model
- Turbulent Boundary Layer Model

#### 1.14.1.1. Laminar Boundary Layer Model

In a laminar boundary layer, the molar fluxes , of a non-condensable component and a condensable component of a binary gaseous mixture through a plane parallel to, and at a distance from, a wall, are given by:

| (1–301)

| (1–302)

is the binary diffusion coefficient, is the molar convective flux of the mixture, is the molar density of the mixture and , are the molar fractions of the non-condensable and condensable components, respectively.

Adding Equation 1–301 and Equation 1–302, using , gives . Hence, substituting into (Equation 1–302) and rearranging gives:

| (1–303)

Hence, integrating from the wall () to the edge of the boundary layer () gives:

Finally, assuming = constant in the boundary layer gives:

| (1–304)

It is convenient to express this result in terms of mass concentrations , , , and mass fluxes , , . Component mass concentrations are related to component molar concentrations and molecular weights , as follows:

| (1–305)

Also, mixture concentrations are related to component concentrations as follows:

| (1–306)

From Equation 1–305 and Equation 1–306, we define a mixture molecular weight relating mixture mass and molar concentrations as follows:

| (1–307)

Finally, we can substitute Equation 1–307 into Equation 1–304 to express the condensable component mass flux as follows:

| (1–308)

where the mass transfer coefficient for laminar flow, , is given by:

| (1–309)

The condensable component molar fraction at the interface is determined by assuming that the vapor is in thermal equilibrium with the liquid film at the interface, and hence its partial pressure is equal to its saturation pressure at the interface temperature. In reality, the vapor at the edge of the boundary layer may be a supersaturated wet vapor, or mist. Only the dry part of the vapor will form the concentration gradient which drives the condensation mass flux. Hence, only the molar fraction of dry vapor must be used to determine the mass flux. This is achieved by clipping the molar fraction at the edge of the boundary layer as follows:

| (1–310)

where denotes the saturated molar fraction.

It is assumed that the liquid droplets suspended in the vapor are not transported to the interface, but merely act as a reservoir for the production of dry vapor to counter the depletion of the boundary layer by condensation.

#### 1.14.1.2. Turbulent Boundary Layer Model

For turbulent flows, the concentration boundary layer is modeled using turbulent wall functions. Consequently, we use discrete versions of Equation 1–301, Equation 1–302 employing turbulent wall functions to model the mass fluxes in the turbulent boundary layer. Thus:

| (1–311)

| (1–312)

Here, w subscripts refer to wall quantities, P subscripts refer to near wall mesh points, and is the wall multiplier determined by the form of the turbulent wall function ( for laminar flow; for turbulent flow, see [227]). Performing the same manipulations as in the previous section, we obtain:

| (1–313)

As the multi-component flow model solves for mass fractions rather than molar fractions, it is more convenient to work with mass fluxes and mass fractions rather than molar fluxes and mole fractions. This gives the following formula for the condensation mass flux:

| (1–314)

Note that the turbulent wall function array is only defined at no-slip walls. It is not defined at free slip walls, specified stress walls, and non-overlap boundary conditions. Hence, wall condensation is disallowed on these types of walls in the case of turbulent flows.

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