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

Chapter 2 Literature Review Page 2.19
2.9.3 Continuum Approximation
From section 2.9.1 and section 2.9.2 it can be seen that wall roughness and <strong>the</strong> presence <strong>of</strong>
solid <strong>particle</strong>s will affect <strong>the</strong> velocity gradient. This fact should be stressed as virtually all
researchers in <strong>the</strong> field <strong>of</strong> hydrotransport describe homogeneous solid-liquid suspensions
using continuum models. The definition <strong>of</strong> continuum as given by Parker (1994) is:
"The study <strong>of</strong> <strong>distribution</strong>s <strong>of</strong> energy. matter and o<strong>the</strong>r physical quannnes under
.
circumstances where <strong>the</strong>ir discrete (composed <strong>of</strong> separate and distinct pans) nature is
unimportant and <strong>the</strong>y may be regarded as (in general. complex) continuous ftmctions <strong>of</strong>
position. "
The problem with using continuum models is that homogeneous solid-liquid suspensions can
never be truly homogeneous. The continuum nature <strong>of</strong><strong>the</strong>se slurries is an approximation and
is a state to which <strong>the</strong> slurries tend asymptotically (Shook & Roco, 1991). Although this
approximation is deemed to hold good, it only does so as long as <strong>the</strong> scale <strong>of</strong> fineness
required by <strong>the</strong> subsequent modelling is not surpassed by <strong>the</strong> <strong>particle</strong> <strong>size</strong> (Lumley, 1978).
Hence, when considering <strong>the</strong> viscous sub-layer, <strong>the</strong> continuum approximation MUST be
compromised (Slatter 1994, Slatter et al 1996). This can clearly be seen in Figure 2.7, a
magnified view <strong>of</strong> a rough pipe wall showing <strong>the</strong> viscous sub-layer thickness and <strong>the</strong> slurry
<strong>particle</strong>s, which indicates that <strong>the</strong> solid <strong>particle</strong>s are large compared with <strong>the</strong> scale <strong>of</strong>
modelling. It is <strong>the</strong>refore imperative that <strong>the</strong> <strong>the</strong>oretical model account for <strong>the</strong> <strong>effect</strong> <strong>of</strong> <strong>the</strong>
<strong>particle</strong>s in turbulent flow.
Maude & Whitmore (1956, 1958) and Slatter (1994) are at <strong>the</strong> time <strong>of</strong> writing <strong>the</strong> only
known researchers to have taken <strong>particle</strong> <strong>size</strong> into account in <strong>the</strong> turbulent flow analyses <strong>of</strong>
<strong>the</strong>ir <strong>the</strong>oretical models. Their analyses are presented in section 2.11.6 and section 2.11.5
respectively.
2.9.4 Turbulence
Blasius (1913) was <strong>the</strong> first to suggest a standard empirical relationship between <strong>the</strong> Reynolds
number and <strong>the</strong> friction factor for fully developed turbulenLNewtonian flow.