(SSL Module Only) The Detailed Slurry Calculation Method was developed for two-layer heterogeneous slurry flow with a newtonian carrier fluid. This method is similar to the Simplified Slurry Calculation Method, but includes more sophisticated methods to calculate M and V₅₀, as is discussed below. The following data is required for the Detailed Calculation Method:

• Temperature

• Solids concentration

• Density

• d₅₀ particle size

• d₈₅ particle size if the M exponent it so be calculated

• M exponent value if no d₈₅ particle size is available

• Terminal velocity parameter, using either the Shape Factor (K) or V_t / V_ts

Horizontal Pipes

The correlation for additional pressure loss in horizontal pipes due to the slurry is given by 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 130

(1)

Where i_m is the dimensionless head gradient and V_m is the mean velocity based on the full diameter regardless of whether any solids collection in the bottom of the pipe has reduced the effective diameter. By adding in the base fluid pressure loss the full head gradient is given by Wilson, et al, 2006, on page 134 (note that Wilson, et al, assume the fluid is water where here the fluid is generalized)

(2)

Noting that:

We get the following solution for the total head gradient:

(3)

Calculating the V₅₀ Value

Equations 2 and 3 above need the velocity at which 50% of the solids are suspended by the fluid, V₅₀. This is discussed by Wilson, et al, 2006, with a functional relationship given on page 130.

(4)

Where:

• w₅₀ is the particle associated velocity

• d₅₀ is the particle diameter

• D is the pipe inner diameter

W₅₀ used in Equation 4 above, is the particle associated velocity for the median particle size, and is defined in equation 5 below. The 3rd edition of Wilson (Wilson, et al, 2006, from which the equations for the rest of this discussion are taken) defines w using Figure 6.5 on page 136 of the text. The plot of w vs. particle size is based on an empirically-derived equation in the 2nd edition of the text (Wilson, et al, 1996), equation 6.12 found on page 134. The AFT Fathom SSL module uses that equation (Equation 5 below) in its solution methodology.

(5)

Where:

• v_t is the particle settling velocity for the particle size in question

• ρ_s is the slurry density

• ρ_f is the fluid density

Calculating the M Exponent Value

If both the d₅₀ and d₈₅ values are entered, the M exponent value can be calculated using the method shown in Wilson, et al, 2006, on page 140.

(6)

Where:

(7)

When using these calculations M is limited to between 0.25 and 1.7.