# Interphase Coupling

## Deposition scheme

The following inputs must be preceded by “mfix.”

| Description | Type | Default
--- | --- | --- | ---
deposition_scheme | The algorithm that will be used to deposit particles quantities to the Eulerian grid. Available methods are: ‘centroid’ ‘trilinear’ ‘true-dpvm’ divided particle volume method ‘trilinear-dpvm-square’ dpvm with square filter | String | ‘trilinear’
deposition_scale_factor | The deposition scale factor. Only applies to ‘true-dpvm’ and ‘trilinear-dpvm-square’ methods. Its value must be in the interval [0, dx/2], where dx is the cell edge size. | Real | 1.0
deposition_diffusion_coeff | If a positive value is set, a diffusion equation with this diffusion coefficient is solved to smooth deposited quantities. | Real | -1.0

In the following subsections, the four possible deposition methods are briefly described and illustrated.

### Centroid

In the centroid deposition scheme, particles’ quantities are deposited only to the Eulerian grid cell to which the particle’s center belongs.

Fig. 1 Example of centroid deposition.

### Trilinear

In the trilinear deposition scheme, particles’ quantities are deposited linearly to the eight Eulerian grid cells that surround its center.

Fig. 2 Example of trilinear deposition.

### Divided Particle Volume Method (DPVM)

In the DPVM method, particles’ quantities are deposited to the Eulerian grid cells that they intersect, and the deposition weights are equal to the percentage of the particle’ volume that intersects the given cell.

Fig. 3 Example of dpvm deposition.

### Square DPVM

In the square DPVM method, particles’ quantities are deposited to the Eulerian grid similarly to the DPVM method, but with a filter applied to the deposition scheme.

Fig. 4 Example of square dpvm deposition.

## Drag

The following inputs must be preceded by “mfix.”

| Description | Type | Default
--- | --- | --- | ---
drag_type | Which drag model to use | String | None

The options currently supported in mfix are `WenYu`, `Gidaspow`, `BVK2`, or `UserDrag`.

If one of these is not specified, the code will abort with

    amrex::Abort::0::"Don't know this drag type!!!

The drag models are defined in src/src_des/des_drag_K.H

If the user wishes to use their own drag model, they must

> - specify `mfix.drag_type = UserDrag` in the inputs file

> - provide the code in the ComputeDragUser routine in a local usr_drag.cpp file. An example can be found in tests/DEM06-x.

With the variables defined as follows:

>     /*
>      *  \brief Returns: the calculated drag coefficient.
>      *
>      *  Inputs:
>      *    EPg     - gas volume fraction
>      *    Mug     - gas laminar viscosity
>      *    ROpg    - gas density * EP_g
>      *    vrel    - magnitude of gas-solids relative velocity
>      *    DPM     - particle diameter of solids phase M
>      *    DPA     - average particle diameter
>      *    PHIS    - solids volume fraction of solids phases
>      *    fvelx   - x component of the fluid velocity at the particle position
>      *    fvely   - y component of the fluid velocity at the particle position
>      *    fvelz   - z component of the fluid velocity at the particle position
>      *    i, j, k - particle cell indices
>      *    pid     - particle id number
>      */

The `WenYu` model is defined as

>      RE = (Mug > 0.0) ? DPM*vrel*ROPg/Mug : DEMParams::large_number;

>     if (RE <= 1000.0)
>     {
>         C_d = (24.0/(RE+DEMParams::small_number)) * (1.0 + 0.15*std::pow(RE, 0.687));
>     }
>     else
>     {
>         C_d = 0.44;
>     }

>     if (RE < DEMParams::eps) return 0.0;
>     return 0.75 * C_d * vrel * ROPg * std::pow(EPg, -2.65) / DPM;

The `Gidaspow` model is defined as

>     ROg = ROPg / EPg;

>     RE = (Mug > 0.0) ? DPM*vrel*ROPg/Mug : DEMParams::large_number;

>     // Dense phase - EPg <= 0.8
>     Ergun = 150.0*(1.0 - EPg)*Mug / (EPg*DPM*DPM) + 1.75*ROg*vrel/DPM;

>     // Dilute phase - EPg > 0.8
>     if (RE <= 1000.0)
>     {
>         C_d = (24.0/(RE+DEMParams::small_number)) * (1.0 + 0.15*std::pow(RE, 0.687));
>     }
>     else
>     {
>         C_d = 0.44;
>     }

>     WenYu = 0.75*C_d*vrel*ROPg*std::pow(EPg, -2.65) / DPM;

>     // switch function
>     PHI_gs = atan(150.0*1.75*(EPg - 0.8))/M_PI / DPM;

>     // blend the models
>     if (RE < DEMParams::eps) return 0.0;
>     return (1.0 - PHI_gs)*Ergun + PHI_gs*WenYu;

The `BVK2` model is defined as

>     amrex::Real RE = (Mug > 0.0) ? DPA*vrel*ROPg/Mug : DEMParams::large_number;

>     if (RE > DEMParams::eps)
>     {
>         oEPgfour = 1.0 / EPg / EPg / EPg / EPg;

>         // eq(9) BVK J. fluid. Mech. 528, 2005
>         // (this F_Stokes is /= of Koch_Hill by a factor of ep_g)
>         F_Stokes = 18.0*Mug*EPg/DPM/DPM;

>         F = 10.0*PHIS/EPg/EPg + EPg*EPg*(1.0 + 1.5*sqrt(PHIS));

>         F += RE*(0.11*PHIS*(1.0+PHIS) - 4.56e-3*oEPgfour +
>              std::pow(RE, -0.343)*(0.169*EPg + 6.44e-2*oEPgfour));

>         // F += 0.413*RE/(24.0*EPg*EPg) *
>         //     (1.0/EPg + 3.0*EPg*PHIS + 8.4/std::pow(RE, 0.343)) /
>         //     (1.0 + std::pow(10.0, 3.0*PHIS)/std::pow(RE, 0.5 + 2.0*PHIS));

>         return F*F_Stokes;
>     }
>     else
>     {
>         return 0.0;
>     }

## Heat Transfer Coefficient

The following inputs must be preceded by “mfix.”

| Description | Type | Default
--- | --- | --- | ---
convection_type | Which HTC model to use | String | RanzMarshall

The options currently supported in mfix are `RanzMarshall` (default) and `Gunn`. In both models the HTC is determined from a Nusslet number corelation.

The RanzMarshall Nusselt number correlation is defined as:

>     amrex::Real N_Nu = 2.0 + 0.6 * std::sqrt(N_Re) * std::pow(N_Pr, 0.333);

The Gunn Nusselt number correlation is defined as:

>     amrex::Real N_Nu =
>         (7 - 10*EPg + 5*EPg*EPg)*(1 + .7*std::pow(N_Re, 0.2)*std::cbrt(N_Pr))
>         + (1.33 - 2.4*EPg + 1.2*EPg*EPg)*std::pow(N_Re, 0.7)*std::cbrt(N_Pr);