A significant variation in particle size distribution (PSD) is generally encountered in slurry transportation. Clift et al. (1982) found that the value of exponent M is nearly equal to 1.7 for narrow size grading and the same value was observed for broader grading. Wilson et al. (1990) performed tests on a broader range of particle size and found out that the slope is now much flatter, with M ≈ 0.9. The influence of particle grading on distribution parameter M is evident here. So, it has been observed that the value of M decreases as the grading becomes broader. A plausible reason for this behavior can be found by considering how M is related to standard deviation of the integrated log-normal distribution representing the ogee curve. The magnitude of M is the same as the slope of a log-log plot of the integrated probability function. If this slope is evaluated at some fixed value of V_{m}/V_{50}, it will be found that M is inversely proportional to the standard deviation of log(V_{m}/V_{50}), where log indicates the logarithm to the base 10. This standard deviation is denoted here by σ_{10}.

For the case of uniform particles, a condition approximated by narrow particle grading, V_{50} is constant, so that the variation which accounts for the observed value of σ_{10} must be associated with the inherently stochastic nature of the turbulent flow field. As the value of σ_{10} for this uniform-particle condition depends only on fluid turbulence it is denoted by the symbol σ_{f}. For fully-developed turbulent flow it may be assumed that σ_{f} will be sensibly constant.

When the particles are not uniform, the effect of the solids grading can be expressed by the standard deviation of logV_{50}, denoted by σ_{s}. It can be shown that the standard deviations σ_{f} and σ_{s} combine on a summed-squares basis, i.e.

With M inversely proportional to σ_{10}, and σ_{f} effectively constant, it follows the expression for M is of the form

This equation includes the proviso that a calculated M larger than 1.7 must be reduced to that value.

__Calculation of σ _{s}__

Let us understand the calculation with an example using d_{85} and d_{50}. The value of cosh(60d_{85} /D) is readily calculated, and the associated value of w(i.e. w_{85}) can be obtained from graph below. The value of σ_{s} is then given by

Here, log specifies the logarithm to the base 10.

* Figure:* Particle-associated velocity w for sand-water slurries

__Symbols__

M → Exponent in stratification-ratio equation

D → internal diameter of a pipe

d_{50} → Mass-median particle diameter

d_{85} → diameter for which 85 % (by mass) of the particles are finer

w → Particle-associated velocity

V_{m} → Mean velocity of mixture

V_{50} → Value of V_{m} at which 50% of solids is suspended by fluid

σ_{10} → Standard deviation of log (V_{m}/V_{50})

σ_{f} → Component of σ_{10} due to fluid turbulence

σ_{s} → Component of σ_{10} associated with particle grading

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