the effect of the particle size distribution on non-newtonian turbulent ...

Appendix B Conference Paper B.7
and can never be truly homogeneous. Treating <strong>the</strong>m as a continuum is <strong>the</strong>refore an
approximation, which works well in <strong>the</strong> laminar regime. The term homogeneous, when
applied to slurries is taken as meaning uniform, stable, spatial <strong>distribution</strong> <strong>of</strong><strong>particle</strong>s.
2.4.6.1 The Laminar Sub-layer
Velocity components normal to <strong>the</strong> pipe axis cannot exist at <strong>the</strong> pipe wall, and
turbulence is suppressed in this region. Viscous forces are dominant and a laminar sublayer
exists for some finite thickness o. It has been shown (Slatter, 1994) that <strong>the</strong>
thickness <strong>of</strong> <strong>the</strong> laminar sub-layer is <strong>of</strong> <strong>the</strong> same order <strong>of</strong> magnitude as <strong>the</strong> larger
<strong>particle</strong>s in <strong>the</strong> sluny over <strong>the</strong> range <strong>of</strong>wall shear stresses found in turbulent flow. Since
<strong>the</strong> larger <strong>particle</strong>s will be larger than <strong>the</strong> thickness <strong>of</strong> <strong>the</strong> laminar sub-layer at <strong>the</strong> higher
wall shear stress values, <strong>the</strong> <strong>particle</strong>s must <strong>the</strong>refore have an obstructing <strong>effect</strong> on <strong>the</strong>
shear in <strong>the</strong> laminar sub-layer. The continuum approximation must <strong>the</strong>refore be
compromised in this region.
2.4.6.2 Wall Roughness and <strong>the</strong> Effect <strong>of</strong> Solid Particles
Under microscopic investigation it can be seen that <strong>the</strong> inner wall <strong>of</strong> <strong>the</strong> pipe is not
smooth but many protrusions exist. The term wall, pipe or surface roughness is used to
describe <strong>the</strong> complex <strong>size</strong>, shape and spacing <strong>of</strong><strong>the</strong>se protrusions. For a given fluid and
velocity, roughness in a pipe increases <strong>the</strong> pressure drop compared with conditions
existing in a "smooth" pipe (Bowen 1961, Cheng 1975). At low Reynolds numbers when
<strong>the</strong> laminar sub-layer is sufficiently thick to cover <strong>the</strong> protrusions on <strong>the</strong> pipe wall, <strong>the</strong>
roughness will not be <strong>effect</strong>ive. At high Reynolds numbers when <strong>the</strong> laminar sub-layer
is not sufficiently thick to cover <strong>the</strong> protrusions, roughness will be <strong>effect</strong>ive. The
progressive penetration <strong>of</strong> <strong>the</strong> laminar sub-layer by <strong>the</strong> wall roughness will generate a
wake <strong>of</strong> eddies, stimulating turbulence. This physical phenomenon is <strong>of</strong> fundamental
importance to <strong>the</strong> understanding <strong>of</strong> rough wall turbulent flow.
Particles will also have a similar <strong>effect</strong> to that <strong>of</strong> <strong>the</strong> wall roughness, in causing a
decrease in <strong>the</strong> velocity gradient and should be taken into account in turbulent flow
analysis.
The change in velocity as <strong>the</strong> pipe wall is approached is very rapid, <strong>the</strong> velocity gradient
being in <strong>the</strong> order <strong>of</strong> lmls over <strong>the</strong> diameter <strong>of</strong> a typical <strong>particle</strong> in <strong>the</strong> region <strong>of</strong> <strong>the</strong>
pipe wall. Ifsolid <strong>particle</strong>s are present in <strong>the</strong> fluid <strong>the</strong>y will resist shear and impede such
rapid changes in velocity.
A roughness Reynolds number can be used to determine various regions <strong>of</strong> turbulent
flow in a pipe (Schlichting, 1960). The inadequate formulation <strong>of</strong> <strong>the</strong> roughness
Reynolds number for non-Newtonian slurries is a problem with previous models eg a
formulation excluding <strong>the</strong> yield stress (Torrance, 1963 and Hanks & Dadia, 1971).