### 5.3.1. Inhomogeneous Hydrodynamic Equations

The inhomogeneous hydrodynamic equations are as follows:

- Momentum Equations

- Continuity Equations

- Volume Conservation Equation

- Pressure Constraint

#### 5.3.1.1. Momentum Equations

| (5–21)

- describes momentum sources due to external body forces, and user-defined momentum sources. For details, see Sources.

- describes the interfacial forces acting on phase due to the presence of other phases. Additional information for the models available for interfacial forces is available in Interphase Momentum Transfer Models.

- The term:

  | (5–22)

  represents momentum transfer induced by interphase mass transfer. For details, see Interphase Mass Transfer.

- The above momentum equations are valid for fluid phases only. For dispersed solid phases, additional terms are present representing additional stresses due to particle collisions.

Note: The viscous stress term contains the product of and . Consequently, as the volume fraction approaches zero, so does the dissipation. However for a dilute phase, the magnitude of the dissipation is still significant because the mass of the dilute phase also goes to zero as approaches zero. This is true not only for momentum but also for any other transported quantity that includes a diffusion term (such as energy). In particular, if the volume fraction gradient is very large, the diffusion term can cause severe convergence problems because the cell with the smaller volume fraction will “see” a very large relative flux. This problem can be significantly alleviated by using harmonic averaging for the diffusion coefficients since the magnitude of the flux will be more sensitive to the volume fraction of the cell with the smaller mass. To change to harmonic averages, use the expert parameter settings `diffusion coef averaging type` = `3` for scalar equations and `stress coef averaging type` = `3` for momentum equations. For simulations that are laminar, or that have regions of low turbulent viscosity so that molecular diffusion is locally significant, a simple workaround is to modify the viscosity and thermal conductivity using CEL so that, below a volume fraction value of interest that is physically suitable, viscosity and thermal conductivity approach zero as the volume fraction approaches zero.

#### 5.3.1.2. Continuity Equations

| (5–23)

- describes user specified mass sources. For details, see Sources.

- is the mass flow rate per unit volume from phase to phase . This term only occurs if interphase mass transfer takes place. For details, see Interphase Mass Transfer.

#### 5.3.1.3. Volume Conservation Equation

This is simply the constraint that the volume fractions sum to unity:

| (5–24)

This equation may also be combined with the phasic continuity equations to obtain a transported volume conservation equation. Take Equation 5–23, divide by phasic density, and sum over all phases. This yields:

| (5–25)

Interpreting this equation is simpler if you consider the special case of incompressible phases with no sources, in which it simplifies to:

| (5–26)

which requires the volume flows to have zero divergence. Equation 5–25 is the volume continuity equation solved by the CFX-Solver.

#### 5.3.1.4. Pressure Constraint

The complete set of hydrodynamic equations represent equations in the unknowns , , , , . You need more equations to close the system. These are given by constraints on the pressure, namely that all phases share the same pressure field:

| (5–27)

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