(SSL Module Only) Flow in vertical pipes offer a special-case flow regime. The basic equation for flow rising in a vertical pipe is given in Wilson, et al, 2006Wilson, K.C., G.R. Addie, A. Sellgren and R. Clift, Slurry Transport Using Centrifugal Pumps, 3rd Ed., published by Springer, 2006., on page 178.

(1)

It is important to note that this is the total pressure drop per unit length whereas the horizontal pipe equations have components for the pure fluid and the solids separately.

Noting that for water, for which Equation 1 is based, we can substitute:

and obtain the following basic relationship:

(2)

Wilson, et al, states that for particles smaller than 150 microns, the density in the final term should be the mixture density:

Substituting this in to Equation 2 we obtain:

(3)

The friction factor f could be found with homogenous methods, but it is often reasonable to use the friction factor for a water flow f_w with the same mean velocity. For larger particles Wilson, et al, suggests that the density term should be that of water which, when substituted in to Equation 2 yields:

(4)