### 5.9.4. Derivation of the Algebraic Slip Equation

The phasic and bulk momentum equations are first transformed to nonconservative form by combining with the phasic and bulk continuity equations. The phasic momentum equation then becomes:

| (5–185)

and the bulk momentum equation becomes:

| (5–186)

Equation 5–185 and Equation 5–186 are combined to eliminate the pressure gradient term, yielding:

| (5–187)

Several assumptions are now made:

1. The dispersed phase is assumed to instantaneously reach its terminal velocity, so the transient term on the drift velocity is neglected.

2. The approximation is made that:

   | (5–188)

3. The viscous stresses and apparent diffusion stresses are neglected.

With these approximations, Equation 5–187 simplifies to:

| (5–189)

In addition, it is assumed that the interphase momentum transfer is due only to drag and that the particles are spherical:

| (5–190)

which leads to the following closed relationship for the slip velocity:

| (5–191)

Note that, for rotating reference frames, the apparent accelerations are automatically included by taking the derivative of the absolute frame velocity rather than relative frame velocity on the right-hand-side.

The effect of in the bulk momentum equation is neglected in the current implementation.

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