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

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Chapter 2 Literature Review Viscous Sub-layer Thickness ---------------------._----------._-.-._----- Page 2.18 Figure 2.6: Magnified view <strong>of</strong> a rough wall pipe showing regions <strong>of</strong> turbulent flow 2.9.2 The Effect <strong>of</strong> Solid Particles Although a <strong>particle</strong> <strong>size</strong> <strong>effect</strong> on energy gradients in turbulent flow has been reported by Maude & Whitmore (1956, 1958), Mun (1988) and Slatter (1994), it is still customary in homogenous non-Newtonian slurries to ignore <strong>the</strong> fact that solid <strong>particle</strong>s are present. As previously mentioned in Chapter 1 it has been found that <strong>particle</strong> <strong>size</strong> and <strong>distribution</strong> do influence flow behaviour (philipp<strong>of</strong>f 1944, Hedstrom 1952, Orr & Blocker 1955, Zettlemoyer & Lower 1955, Maude & Whitrnore 1956, Thomas 1983, Mun 1988, Slatter 1994) and yet it is <strong>the</strong> most overlooked piece <strong>of</strong> data in turbulent flow analysis (Mun, 1988). Particles will cause a decrease in <strong>the</strong> velocity gradient similar to <strong>the</strong> <strong>effect</strong> <strong>of</strong> <strong>the</strong> wall roughness (Slatter, 1994) and should <strong>the</strong>refore be taken into account in turbulent flow analyses. This can be understood if <strong>the</strong> following is considered. The change in velocity as <strong>the</strong> pipe wall is approached is very rapid. The magnitude <strong>of</strong> change in <strong>the</strong> region <strong>of</strong> <strong>the</strong> pipe wall is in <strong>the</strong> order <strong>of</strong> hn/s (Slatter, 1994) over <strong>the</strong> diameter <strong>of</strong>a typical <strong>particle</strong>. Ifsolid <strong>particle</strong>s are present in <strong>the</strong> fluid <strong>the</strong>y will resist-shear and hence impede <strong>the</strong> rapid changes in velocity.
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.
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