may be specified directly by the user, or from the assumption of local equilibrium in a transport equation model. The latter is based on:

(5–104)

where denotes the solids shear stress tensor Equation 5–88, and:

(5–105)

Expand the production term:

(5–106)

(5–107)

where:

(5–108)

In order to determine from Equation 5–104, it is useful to take into account the dependence of solids pressure and shear bulk viscosities on . You have:

(5–109)

so you may write:

(5–110)

(5–111)

Hence, substituting into Equation 5–107, you may express the production term in terms of granular temperature as follows:

(5–112)

where:

(5–113)

Similarly, the dissipation term Equation 5–105 may be simplified as follows:

(5–114)

where:

(5–115)

Equating Equation 5–112 and Equation 5–114, and dividing by gives a quadratic equation for :

(5–116)

Note that is strictly positive if, and only if, the coefficient of restitution is strictly less than unity. In this case, in view of the fact that , Equation 5–116 has a unique positive solution:

(5–117)

The Algebraic Equilibrium model has the flaw that unphysically large granular temperatures can be generated in regions of low solid particle volume fraction. To circumvent this, specify an upper bound for the granular temperature. The square of the mean velocity scale is a reasonable estimate for this.

The Zero Equation Model implements the simpler algebraic model of Ding and Gidaspow [98].

(5–118)