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

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

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Chapter 2 Literature Review Page 2.38 (2.77) Tests conducted by Slatter (1994) confirmed ong>theong> model to be more accurate than ong>theong> Torrance (1963) model and Wilson & Thomas (1985, 1987) model against which experimental data was compared. This model was also supported by ong>theong> experimental data ong>ofong>Park et al (1989) and Xu et al (1993). 2.11.6 Maude & Whitmore Correlation Maude & Whitmore (1956, 1958) are two ong>ofong> ong>theong> few researchers, if not ong>theong> only two before Slatter (1994), to take any account ong>ofong> ong>theong> ong>effectong> ong>particleong>s play in turbulent flow. Tests ·were conducted in thin vertical 2,2mm, 3,5mm and 5mm diameter tubes using emery slurries at six different concentrations. The ong>particleong> ong>sizeong> ong>distributionong> ong>ofong> ong>theong> emery slurries fell within ong>theong> range 20 to 40JLm. Data was presented in ong>theong> form ong>ofong>curves ong>ofong> friction factors against Reynolds numbers. The main observation ong>ofong> Maude & Whitmore (1958) was that ong>theong> turbulent friction factor (based on ong>theong> wall viscosity) was initially higher than ong>theong> Newtonian friction factor but fell below at increased Reynolds numbers. This observation is contradictory to ong>theong> more widely publicised finding that ong>theong> non-Newtonian friction factor is always less than ong>theong> Newtonian friction factor (eg Wilson, 1986). The phenomena encountered by Maude & Whitmore (1958) were explained in ong>theong> following ong>theong>oretical terms. The mixing length, as developed by Prandtl, is defined as ong>theong> mean distance which fluid elements move at right angles to ong>theong> direction ong>ofong> flow before ong>theong>y acquire ong>theong> velocity ong>ofong> ong>theong> layer ong>theong>y have entered. Hence, ong>theong> fluid element would pursue an oscillating passage down ong>theong> tube. However, inertial forces would prevent any high denSity ong>particleong>s from following exactly ong>theong> velocity changes ong>ofong> ong>theong> fluid. The ong>particleong>s Would instead oscillate with a smaller amplitude and ong>theong> mean mixing length ong>ofong> ong>theong> suspension would decrease with an increase in solids concentration as compared to that ong>ofong> ong>theong> raw fluid. They postulated that ong>theong> ong>effectong>ive mixing length is reduced by a ong>theong>oretical

Chapter 2 Literature Review Page 2.39 mixing length factor p p = P, (I - Cv> + q Pp Cv , PI (1 - Cv> + Pp Cv where Cv = volume ong>ofong> ong>particleong>s per unit volume ong>ofong> suspension (2.78) q = ong>theong> ratio ong>ofong> ong>theong> amplitude ong>ofong> ong>theong> ong>particleong>s to ong>theong> amplitude ong>ofong> fluid, given by; where I = vot, = average amplitude ong>ofong> ong>theong> liquid oscillation and K is given by: where a = shape factor. (2.79) (2.80) Maude & Whitmore (1958) hence stated that ong>theong> pressure loss and friction factor should be reduced in turbulent flow. It must be noted however, that while this accounts for flow behaviour at high ReynoIds numbers, ong>theong> approach does not take into account low Reynolds number behaviour. In order to account for low Reynolds number behaviour and to explain why higher-than­ Newtonian equivalent friction factors were being observed, Maude & Whitmore (1958) considered ong>theong> ong>effectong> ong>ofong> ong>theong> suspended ong>particleong> on ong>theong> viscosity ong>ofong> ong>theong> medium forming ong>theong> viscous sub-layer. It is known that as ong>theong> Reynolds number increases ong>theong> viscous sub-layer becomes narrower to ong>theong> extent that ong>theong> mean ong>particleong> diameter is greater. Thus ong>theong> viscous SUb-layer is said to consist only ong>ofong> ong>theong> suspending medium. At lower Reynolds numbers, ong>theong> viscous sub-layer thickens and ong>theong> whole suspension is sheared by ong>theong> viscous flow in ong>theong> SUb-layer. Hence, ong>theong> ong>effectong>ive viscosity becomes that ong>ofong> ong>theong> suspension and explains ong>theong> higher friction factors.

Chapter 2 Literature Review Page 2.39<br />

mixing length factor p<br />

p = P, (I - Cv> + q Pp Cv ,<br />

PI (1 - Cv> + Pp Cv<br />

where Cv = volume <str<strong>on</strong>g>of</str<strong>on</strong>g> <str<strong>on</strong>g>particle</str<strong>on</strong>g>s per unit volume <str<strong>on</strong>g>of</str<strong>on</strong>g> suspensi<strong>on</strong><br />

(2.78)<br />

q = <str<strong>on</strong>g>the</str<strong>on</strong>g> ratio <str<strong>on</strong>g>of</str<strong>on</strong>g> <str<strong>on</strong>g>the</str<strong>on</strong>g> amplitude <str<strong>on</strong>g>of</str<strong>on</strong>g> <str<strong>on</strong>g>the</str<strong>on</strong>g> <str<strong>on</strong>g>particle</str<strong>on</strong>g>s to <str<strong>on</strong>g>the</str<strong>on</strong>g> amplitude <str<strong>on</strong>g>of</str<strong>on</strong>g> fluid, given by;<br />

where I = vot, = average amplitude <str<strong>on</strong>g>of</str<strong>on</strong>g> <str<strong>on</strong>g>the</str<strong>on</strong>g> liquid oscillati<strong>on</strong> and K is given by:<br />

where a = shape factor.<br />

(2.79)<br />

(2.80)<br />

Maude & Whitmore (1958) hence stated that <str<strong>on</strong>g>the</str<strong>on</strong>g> pressure loss and fricti<strong>on</strong> factor should be<br />

reduced in <strong>turbulent</strong> flow. It must be noted however, that while this accounts for flow<br />

behaviour at high ReynoIds numbers, <str<strong>on</strong>g>the</str<strong>on</strong>g> approach does not take into account low Reynolds<br />

number behaviour.<br />

In order to account for low Reynolds number behaviour and to explain why higher-than­<br />

Newt<strong>on</strong>ian equivalent fricti<strong>on</strong> factors were being observed, Maude & Whitmore (1958)<br />

c<strong>on</strong>sidered <str<strong>on</strong>g>the</str<strong>on</strong>g> <str<strong>on</strong>g>effect</str<strong>on</strong>g> <str<strong>on</strong>g>of</str<strong>on</strong>g> <str<strong>on</strong>g>the</str<strong>on</strong>g> suspended <str<strong>on</strong>g>particle</str<strong>on</strong>g> <strong>on</strong> <str<strong>on</strong>g>the</str<strong>on</strong>g> viscosity <str<strong>on</strong>g>of</str<strong>on</strong>g> <str<strong>on</strong>g>the</str<strong>on</strong>g> medium forming <str<strong>on</strong>g>the</str<strong>on</strong>g><br />

viscous sub-layer. It is known that as <str<strong>on</strong>g>the</str<strong>on</strong>g> Reynolds number increases <str<strong>on</strong>g>the</str<strong>on</strong>g> viscous sub-layer<br />

becomes narrower to <str<strong>on</strong>g>the</str<strong>on</strong>g> extent that <str<strong>on</strong>g>the</str<strong>on</strong>g> mean <str<strong>on</strong>g>particle</str<strong>on</strong>g> diameter is greater. Thus <str<strong>on</strong>g>the</str<strong>on</strong>g> viscous<br />

SUb-layer is said to c<strong>on</strong>sist <strong>on</strong>ly <str<strong>on</strong>g>of</str<strong>on</strong>g> <str<strong>on</strong>g>the</str<strong>on</strong>g> suspending medium. At lower Reynolds numbers, <str<strong>on</strong>g>the</str<strong>on</strong>g><br />

viscous sub-layer thickens and <str<strong>on</strong>g>the</str<strong>on</strong>g> whole suspensi<strong>on</strong> is sheared by <str<strong>on</strong>g>the</str<strong>on</strong>g> viscous flow in <str<strong>on</strong>g>the</str<strong>on</strong>g><br />

SUb-layer. Hence, <str<strong>on</strong>g>the</str<strong>on</strong>g> <str<strong>on</strong>g>effect</str<strong>on</strong>g>ive viscosity becomes that <str<strong>on</strong>g>of</str<strong>on</strong>g> <str<strong>on</strong>g>the</str<strong>on</strong>g> suspensi<strong>on</strong> and explains <str<strong>on</strong>g>the</str<strong>on</strong>g><br />

higher fricti<strong>on</strong> factors.

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