## Homogeneous Slurries

Equivalent-Fluid Calculations As mentioned in the previous section, pseudo-homogeneous flow (for example aqueous slurries of silt or fine sand) shares with truly homogeneous flow the property that the pressure gradient increases with througput velocity in a fluid-like fashion. An increase of this type can be expressed, at least to a reasonable approximation, by the statement that the pressure drop for turbulent flow of a homogeneous or pseudo-homogeneous mixture is proportional to that obtained for an equal discharge of carrier fluid alone.

Before this statement can be put into mathematical form, it is necessary to consider various ways of expressing the pressure drop due to friction. The change of pressure per unit length of pipe, Ap!Ax, is not commonly employed in practice, in part because it includes the effect of differences in elevation as well as friction losses. For liquids, especially water, the hydraulic gradient, i, is normally used. This quantity is the slope of the hydraulic grade line, which is based on the levels to which the fluid would rise in a series of imaginary tubes tapped into the pipeline. As the slope of the hydraulic grade line, the hydraulic gradient represents the drop in level per unit length of pipe. For water (density pw) flowing in a horizontal pipe, i is related to the pressure gradient as follows

where g is gravitational acceleration.

As water forms the carrier liquid in the majority of slurry flows, it is convenient to use it as a standard for comparison of frictional pressure losses. Thus, throughout this chapter the usual expression for frictional losses is in terms of i (height of water per unit length of pipe). The value for i for clear-water flow is written iw, and that for a mixture (i.e. frictional loss in height of water per unit length) is written im. The statement made previously for homogeneous and pseudo-homogeneous flows amounts to a direct proportionality between im and iw.

Calculation of iw uses the Stanton-Moody friction factor, f, in the equation f

V2 2gD

For turbulent flow, the friction factor depends on both the Reynolds number (Re = pVmD!m) and the relative roughness e!D. These two quantities can be entered on Figure 2, which is

FIGURE 2 Friction factor in normal operating range

used to obtain f For example, with Re = 1.5 X 106 and e/D = 0.0001, f = 0.013 (a typical value). As f is approximately constant for turbulent flow in a given pipe, Eq. 2 shows that the hydraulic gradient varies roughly as V2 (or as Q2).

In non-slurry applications, if a liquid other than water is conveyed, the hydraulic gradient concept is applied by substituting the density of this liquid for that of water in Eq. 1, so that the hydraulic gradient is expressed in height of flowing liquid per unit length. In considering, say, a pseudo-homogeneous slurry of fine sand, it may be of interest to compare its behavior to that of a liquid with density equal to that of the flowing mixture pm, which is given by

Here Ss is the relative density of the solids (compared to water) and Cv is the volumetric concentration of solids.

Substituting pm for pw in Eq. 1 gives the mixture-height gradient (i.e. measured in height of mixture rather than of water). For clarity, a different symbol must be used, and j is employed for this purpose. For a mixture flowing in a horizontal pipe:

Although j is expressed in height of mixture per unit length of pipe, actual measurements based on columns of mixture in vertical tubes would not be feasible, and j would have to be obtained indirectly from pressure measurements. It can be seen from Eqs. 1 and 4 that the ratio of i to j equals the relative density of the mixture Sm (i.e. pm/pw).

The equivalent-fluid model of slurry flow assumes that the solids have little effect on friction factor, and that the mixture acts as a liquid as far as the relative-density effect is concerned. The resulting hydraulic gradient for homogeneous mixture flow, imh, is equivalent to the product of Sm and iw. Although the equivalent-fluid model has been widely employed in the past, it is not generally supported by the experimental evidence. For example, sand-water experiments by Carstens & Addie (1981) show that for some pseudo-homogeneous flows imh does not exceed iw at all. An appropriate equation for the hydraulic gradient is:

Setting the coefficient A equal to unity gives the relative-density effect of an equivalent-fluid model, whereas A = 0 gives the behavior observed by Carstens & Addie (1981). Intermediate types can be represented by values of A between zero and unity. Equation 5 will be referred to as the "homogeneous flow" equation, with the specific case of A = 1.0 called the "equivalent fluid" model.

Modeling Non-Newtonian Flows In pipeline transport of non-Newtonian materials, the variation of pressure drop with velocity is typically rather flat for laminar flow, while it is much steeper for turbulent flow. Figure 3 shows these features. This figure is a plot, on logarithmic axes, of hydraulic gradient jm (i.e. frictional losses in height of equivalent fluid per unit length of pipe) versus mean velocity Vm (volumetric discharge/cross-sectional area). The data shown on this plot refer to various concentrations of a red mud tested at the GIW Hydraulic Laboratory in a pipeline with internal diameter of 3.19 in. (81 mm). The plotted data are for mixtures with relative densities from 1.14 to 1.28. Note that where laminar flow occurs the slope of the lines (referred to the logarithmic coordinates) is rather small and does not vary significantly with particle concentration. However, the vertical position of the laminar lines increases strongly with increasing concentration. Conversely, for turbulent flow, concentration has very little effect; all the points fall close to a single line with slope (for logarithmic coordinates) close to 2.0. This behavior implies that the friction factor f has a near-constant value for the observed turbulent flows. Note that the transitions between laminar and turbulent flow are rather abrupt.

Rheograms, i.e. curves of shear stress versus strain rate, are obtained from tests under laminar conditions, using either rotary viscometers or tube viscometers. The latter are

FIGURE 3 Friction gradients of a red-mud slurry

FIGURE 3 Friction gradients of a red-mud slurry preferable, as they are geometrically similar to the pipeline configuration. For a Newtonian fluid of viscosity m, shear stress t is related to strain rate du/dy by the classic expression t = m( du/dy) (6)

Rheograms for non-Newtonian materials are not so simple. For example, Figure 4 shows a rheogram for a fine-particle slurry. This rheogram does not pass through the origin (the strain rate remains zero until a certain yield stress ty is exceeded) and is not straight (although in this case the behavior is approximately linear at large values of strain rate). The definition of viscosity given by Eq. 6 can be retained, although this viscosity no longer represents the slope of the rheogram itself, but rather that of the secant line shown on the figure. Hence m as defined by Eq. 6 can also be called the "secant" viscosity. The tangent to the rheogram, though less meaningful physically, is often referred to. This "tangent" viscosity will be denoted rt. Both m and r vary with position, and hence they both depend on du/dy (or, alternatively, on t). The stippling of Figure 4 indicates the area beneath the leftward portion of the rheogram. The area ratio a (which will be used below) is the ratio between the stippled area and the traingular area below the secant line. As with m and rt, both the area beneath the rheogram and the area ratio a depend on du/dy (or on t).

It is common practice to represent rheograms by simple mathematical functions (i.e. functions having only two or three parameters). For materials exhibiting a yield stress, the most common engineering choice is the Bingham model, a straight line given by t = tB + hB( du/dy) (7)

Here the Bingham viscosity rB is the tangent viscosity given by the slope of the fit line, and the Bingham yield stress tB is the intercept of the fit line with the shear-stress axis. For the data of Figure 4, tB is larger than the observed stress at zero strain rate, a situation that is commonly encountered.

FIGURE 4 Rheogram for phosphate slimes tested at GIW Hydraulic Laboratory (after Wilson, 1986).

FIGURE 4 Rheogram for phosphate slimes tested at GIW Hydraulic Laboratory (after Wilson, 1986).

The dimensionless ratio t/tb (denoted d and called the stress ratio) is a useful parameter when dealing with Bingham physics. For example, in laminar flow, the secant viscosity ("apparent viscosity") m is related to the stress ratio by the equation m/hB = e/(e - 1) (8)

This equation indicates a gradual decrease of m with increasing stress ratio. Another parameter appropriate for Bingham plastics is the Hedstrom number, He

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