## M 2 1 V

Vr K

For example, if the entering solids fraction ' is 0.4, the corresponding values of the local solids fraction'm for relative slip velocities (Vr) of 0.01, 0.1, and 0.5 are 0.403, 0.424, and 0.525, respectively. There are many "theoretical" expressions for slip, but practical applications depend on experimental observations and correlations (which will be presented later). In gas-liquid or gas-solid flows, 'm will vary along the pipe, because the gas expands as the pressure drops and speeds up as it expands, which tends to increase the slip, which in turn increases the holdup of the denser phase.

The mass fraction (x) of the less dense phase (which, for gas-liquid flows, is called the quality) is x = mL/(mS + mL), so the mass flow ratio can be written mL x

This can be rearranged to give the less dense phase volume fraction in terms of the mass fraction and slip ratio:

The local density of the mixture is given by

which depends on the slip ratio S through Eq. (15-11). The corresponding expression for the local in situ holdup of the more dense phase is

Note that both the local mixture density and the holdup increase as the slip ratio (S) increases. The "no slip'' (S = 1) density or volume fraction is identical to the equilibrium value entering (or leaving) the pipe.

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