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## Q. 5.P.3

Explain the various mechanisms by which particles may be maintained in suspension during hydraulic transport in a horizontal pipeline and indicate when each is likely to be important.
A highly concentrated suspension of flocculated kaolin in water behaves as a pseudo-homogeneous fluid with shear-thinning characteristics which can be represented approximately by the Ostwald–de Waele power-law, with an index of 0.15. It is found that, if air is injected into the suspension when in laminar flow, the pressure gradient may be reduced, even though the flowrate of suspension is kept constant. Explain how this is possible in “slug” flow, and estimate the possible reduction in pressure gradient for equal volumetric, flowrates of suspension and air.

## Verified Solution

If u is the superficial velocity of slurry, then:
For slurry alone:
The pressure drop in a pipe of length l is: $K u^n l$.
If the air: slurry volumetric ratio is R, there is no slip between the slurry and the air and the system consists of alternate slugs of air and slurry, then:
The linear velocity of slurry is $(R+1) u$

Fraction of pipe occupied by slurry slugs is $\frac{1}{R+1}$

Assuming that the pressure drop is the sum of the pressure drops along the slugs, then:
the new pressure drop is: $K\{(R+1) u\}^n\left(\frac{l}{R+1}\right)=K u^n l(R+1)^{n-1}$

Then:  $r=\frac{\text { pressure gradient with air }}{\text { pressure gradient without air }}=\frac{K u^n l(R+1)^{n-1}}{K u^n l}=(R+1)^{n-1}$

For     n = 0.15 and: R = 1

$r=2^{-0.85}=\underline{\underline{0.55}}$