### 5.6.3. Solids Shear Viscosity

#### 5.6.3.1. Constitutive Equation Models

The simplest constitutive equation model for solids shear viscosity was presented by Miller and Gidaspow (1992) [106]. They successfully modeled gas-solid flow in a riser using a solids shear viscosity linearly proportional to the solids phase volume fraction.

| (5–97)

Note that their constant of proportionality is dimensional, and is likely to require modification for different fluid-solid material properties.

More complex models for solids shear stress enable the shear stress to become very large in the limit of maximum packing. A wide range of such models is discussed in the review article by Enwald et al [97].

It is possible for you to implement any of these models, using a CEL expression for the solids shear viscosity. It is important to note that any solids volume fraction dependence must be included in the CEL expression. This is because the CFX-Solver treats material shear viscosities and solids collisional shear viscosities differently. In Eulerian multiphase flow applications, the effective material shear viscosity, , is assumed to be of the form:

| (5–98)

where is the single-phase material shear viscosity and is the volume fraction of that phase. is defined in the material properties library and the multiplication of Equation 5–98 is performed automatically by the CFX-Solver. However, if using a custom CEL expression for solids shear viscosity, the full dependence on volume fraction must be included in the CEL expression, that is, no internal multiplication by volume fraction will occur automatically.

#### 5.6.3.2. Kinetic Theory Models for Solids Shear Viscosity

Typically, the shear viscosity is expressed as a sum of at least two contributions: the kinetic and collisional contributions:

| (5–99)

There is wide agreement on the correct form of the collisional contribution. As in the kinetic theory of gases, it is proportional to the square root of the granular temperature:

| (5–100)

However, there are many proposals in the literature for the correct form of the kinetic contribution. For example:

Gidaspow (1994) [18]:

| (5–101)

Lun and Savage (1986) [100]:

| (5–102)

Here, .

Kinetic contributions are omitted from ANSYS CFX.

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