Settling Slurry Transport: Effects of Solids Grading and Pipe Inclination

Abstract

In many industrial applications, settling slurries composed of coarse solid particles (typically sand or gravel) and Newtonian-carrying fluid (typically water) are transported in pipelines. Turbulent flow of such slurries consumes significantly more energy than flow of the carrying fluid alone. A contribution of transported solids to the energy loss is sensitive to solids grading and to the related distribution of solids in a pipe. Also related to the solid’s distribution are changes in energy losses caused by an inclination of a pipe transporting settling slurry. We report on recent advances in the description and modeling of pipe flows of settling slurries with a special focus on the effects that the solids grading and the flow inclination have on flow friction. The description includes results of laboratory experiments and model predictions.

1. Introduction

The size of a solid particle is one of the key parameters affecting the behavior of solid–liquid slurry flow in a pipe. The size determines the type of slurry and is responsible for prevailing mechanisms governing particle support and friction in slurry flow. A majority of predictive models for slurry flows consider just one characteristic size of transported particles and hence cannot take into account the profound effect a broad grading of solids can have on the slurry flow structure and energy loss due to friction.

For flows of settling slurries, it is well known that a considerable reduction of friction loss can be achieved if broadly graded solids are transported instead of uniformly graded solids of the same mass-median grain size [1, 2, 3]. Laboratory experiments on the effect of solids grading have been carried out primarily for bimodal slurries composed of two narrow-graded fractions of solids, each of different particle size [4, 5, 6, 7].

Recent extensive experiments with various combinations of up to four different fractions of solids of the same density (each representing one type of settling slurry: fine, pseudo-homogeneous, heterogeneous, and stratified, respectively) [8] demonstrated a considerable effect that finer fractions added to coarse stratified flow have on the friction loss and enabled its quantification. An analysis of bimodal slurries tested in these experiments and composed of the coarse stratified fraction and one additional finer fraction (either the fine or pseudo-homogeneous or heterogeneous) showed that the effect of each of the added fractions on the friction loss is different and thus caused by a different mechanism [9]. Other laboratory experiments showed that the broad grading also affects the solids distribution in settling slurry flows [6, 7].

Our experimental investigation, combining observations of flow friction and solids distribution in settling slurry flows of various solids compositions, has aimed to identify mechanisms responsible for the effects that the particle size and the particle size distribution have on friction losses in horizontal and inclined flows. The first results have been published for a limited number of tested solids and flow conditions recently [10, 11, 12, 13, 14, 15]. New results are discussed below.

2. Experimental work

The experiments were carried out in a laboratory loop at the Institute of Hydrodynamics in Prague.

2.1 Experimental setup and measuring techniques

The loop is composed of horizontal and vertical pipe sections, each with an internal diameter of 100 mm (Figure 1). The geometry of the loop, its measuring equipment, measuring techniques, and experimental procedures for slurry flow tests are described in detail in [15]. During tests, experimental data are collected from both the pipeline and the pump of the loop. Pipe measurements include the mean velocity of slurry V_m (defined as the ratio of the total volumetric discharge of slurry divided by the cross-sectional area of the pipe) by a magnetic flowmeter, the differential pressure over several measuring sections of a pipeline measured by a differential pressure transducer (DPT), and the chord-averaged vertical distributions of local volumetric concentration c by gamma-ray radiometric profilers mounted to some of the pressure-drop measuring sections. The delivered concentration C_vd (defined as the ratio of the solids discharge and the total discharge) is obtained from the differential pressures measured either in the invert U-tube if it is set to the vertical position, or in the vertical pipe section below the flowmeter (Figure 1).

Figure 1.

Layout of experimental pipe loop at Institute of Hydrodynamics.

For horizontal-flow tests, local velocities of individual particles are measured at the bottom of the transparent pipe section (No. 7 in Figure 1) in the horizontal pipe section. The instantaneous velocities are obtained from images collected by a high-speed camera. The particle image velocimetry (PIV) method is used to analyze the motion of coarse particles in one-species-slurry flow and the motion of fine particles in bimodal slurry flow. The particle tracking velocimetry (PTV) method is used to determine the motion of coarse particles in the flow of bimodal slurry. More details on the velocity measuring techniques used are given in [15].

2.2 Solids fractions used in experiments

Five narrowly graded fractions of sand from fine-to-medium to very coarse (Table 1) were used to produce one-species (narrowly graded), bimodal, or multispecies (broadly graded) slurries for laboratory experiments.

Fraction code	d50 [mm]	d85 [mm]	density [kg/m^3]
SS2030	2.19	2.49	2555
ST1040	1.56	2.41	2630
SP0612	0.87	1.20	2620
SP3031	0.55	0.78	2597
STJ25	0.22	0.35	2630

Table 1.

Properties of tested sand fractions.

3. Inclined flow of settling slurry

Our previous investigations focused on the effect of pipe inclination on partially stratified flow of narrow-graded medium (SP3031) sand [11, 12, 13, 14]. The work reported here extends the investigation to inclined flows of narrow-graded coarse-sand (SP0612) slurry and broad-graded medium-sand slurry to test the effects of particle size and its distribution.

3.1 Experimental procedure

Inclined flow tests were carried out for slurries of constant mean delivered concentration C_vd flowing at constant mean flow velocity V_m through the ascending limb and the descending limb of the inclinable, invert U-tube of the laboratory loop at various inclination angles ω up to ±45 degrees from the horizontal. In test runs, V_m and C_vd are the same in both measuring sections irrespective of the inclination of the U-tube. The other measured parameters, manometric pressure gradient obtained from the installed DPTs and the mean spatial volumetric concentration C_vi, obtained by an integration of the solid’s distribution over a cross-sectional area of the pipe, differ in the two measuring sections of the U-tube inclined to any angle different from zero. The manometric pressure gradient is composed of the static pressure gradient and the pressure gradient due to friction and both these gradients are affected by the flow inclination. The static gradient depends on the density of slurry in a measuring section and that density is determined from C_vi. If C_vi is not available, then C_vd is used as a less accurate alternative.

3.2 Effect of particle size on inclined flow

The previous experiments showed that the flow of aqueous slurry of narrow-graded medium SP3031-sand exhibited a very different degree of stratification in ascending and descending pipes inclined to the same slope between ±5 and ± 40 degrees. The partially stratified flow of the SP3031-sand produced higher friction than expected (and conventionally predicted) if inclined to slopes not steeper than approximately ±30 degrees [13, 14]. The frictional gradient reached the highest values at the mild negative slopes of say, −10 to −20 degrees. This anomalous trend was also seen in the course of the manometric gradient. It was demonstrated that the observed effect is associated with changes in solid’s distribution across the pipe cross-section caused by a variation in the angle of inclination.

The slurry flow of a coarser sand (SP0612) tested in the same laboratory loop using the same procedure exhibits the same effects. Plots in Figure 2 compare solids distributions (the local volumetric concentration, c, related to the vertical distance from the pipe bottom, y, relative to the internal diameter of a pipe, D) measured with the SP0612-sand slurry simultaneously in the ascending and descending limbs of the lab-loop U-tube set to 25 degrees. The flows of the same flow velocity V_m and delivered concentration C_vd differ significantly in the degree of stratification; the descending flow is fully stratified, while the ascending flow exhibits a gradual change in the local concentration across the pipe’s entire cross-section. This effect can be predicted by a layered model [11], which captures the variation of the solid’s distribution with the inclination angle as shown in Figure 2. The flow becomes fully stratified in the descending pipe because the bed slides faster (due to the submerged-weight component acting as an additional driving force in the flow direction) than in the ascending pipe. The top of the faster sliding bed is less eroded and hence the flow more stratified.

Figure 2.

Distribution of narrow-graded SP0612-sand in flow of V_m ≈ 1.84 m/s and C_vd ≈ 0.17 inclined at +25° (left) and −25° (right). Legend: Points = measurement, line = prediction by a layered model [12].

The variations in the thickness of the sliding bed cause considerable differences in the friction loss between the ascending limb and the descending limb of the U-tube when inclined up to ±45 degrees from the horizontal (Figure 3). Due to the significantly stronger stratification, the descending flow exhibits more resistance than the ascending flow at the same average velocity and delivered concentration if the flow is not very steep. This is particularly the case at mild slopes, where it may lead to a substantially larger frictional gradient in the descending flow than in the ascending flow.

Figure 3.

Dimensionless pressure gradients at various angles of inclination for slurry flow of SP0612 sand at V_m ≈ 1.84 m/s and C_vd ≈ 0.17. Comparison is made with predictions by Woster-Denny method. Legend: Square = measured manometric gradient; black + = measured frictional gradient; red line = manometric gradient by Worster and Denny based on C_vd; blue line = manometric gradient by Worster and Denny based on C_vi; black line = frictional gradient by Worster and Denny.

Figure 3 compares different dimensionless pressure gradients (i = Δp/ρ_w/g/L, where Δp = measured pressure differential, ρ_w = density of standard water of 1000 kg/m³, g = gravitational acceleration, L = length of measuring section) obtained from the U-tube measurements with the inclined flow of the SP0612-sand slurry. The manometric gradient exhibits both a general trend of an increase in the gradient with the increasing angle of inclination due to the increasing static part of the gradient and the local deviation from this trend for negatively sloped flow at the inclination angle ω of −25 degrees. The frictional gradient exhibits a peak at the same slope. Note that C_vi (obtained from measured solids distributions) in the ascending and descending pipes have been used to calculate the frictional gradient from the manometric gradient measurements in Figure 3.

The widely used Worster-Denny method [16] to predict the manometric and frictional gradients in inclined slurry flows is used to compare predictions with the experimental results (Figure 3). The method gives poor estimates of the inclination effect on the frictional pressure gradient as it does not account for the variable stratification that occurs as a result of variable bed shear in flow inclined to different angles. It gives also poor estimates of the manometric gradient if the measured C_vi is used to get the static term. If it uses C_vd instead of C_vi, the two contradictory trends affecting the manometric gradient compensate each other so that a successful prediction of the manometric gradient is reached for ascending partially stratified flow (but not for descending flow). This case, however, is not general and the compensation effect can be considerably less successful in other stratified flows.

3.3 Effect of broad grading on inclined flow

In order to include the effect of broad grading of solids on inclined partially-stratified flow, an experiment was carried out with inclined flow of broad-graded sand slurry comparable with the previously tested inclined flow of narrow-graded SP3031-sand slurry. Both slurries had same mass median particle size d₅₀.

For the SP3031-slurry, the inclined flows were observed for the conditions given by the delivered concentration C_vd = 0.24 and flow velocity V_m = 2.5 m/s in the 100 mm pipe inclined to ±45 degrees [13, 14]. The same conditions were maintained in slurry tests with the broad-graded sand. It was made up by blending three sand fractions as shown in Table 2. The proportion of the individual fractions produced a smooth particle size distribution curve of the broadly graded sand which was significantly less steep than the curve for the SP3031 sand and provided the same mass-median size as the SP3031 fraction (d₅₀ = 0.55 mm).

Fraction code	Ratio in mixture [%]
SP0612	38
SP3031	31
STJ25	31
Mixture	100

Table 2.

Proportions of fractions of broadly graded sand mixture.

3.3.1 Dimensionless pressure gradients

The measured gradients are shown in Figure 4. The anomalous values of the manometric gradient occur in the range of angles between −5 degrees and − 25 degrees with a local peak at −15 degrees, which is consistent with the behavior of the narrow-graded slurry (Figure 5). The comparison shows that the sensitivity of the manometric gradient to the pipe slope is higher in the flow of the narrow-graded slurry than in the flow of the broad-graded slurry.

Figure 4.

Dimensionless pressure gradients at various angles of inclination for slurry flow of broad-graded sand at V_m ≈ 2.5 m/s and C_vd ≈ 0.24. Comparison is made with predictions by Woster-Denny method. Legend: as in Figure 3.

Figure 5.

Measured manometric gradient for broadly graded slurry and corresponding narrowly graded slurry. Legend: Blue square = narrow-graded slurry; magenta diamond = broad-graded slurry.

The frictional gradients are compared for both types of slurries in Figure 6. The results for the broad-graded slurry confirm the trend in the development of the gradient observed previously for the narrow-graded slurry. Furthermore, they give a clearer picture of the trends because of the larger number of data points (measurements at a larger number of inclination angles) and because of the smaller scatter of the data. The results show that the broad-graded slurry exhibits less friction losses than the narrow-graded slurry. The variation in the frictional gradient with the inclination angle is significantly smoother and exhibits smaller peaks in flow of the broad-graded slurry than in flow of the narrow-graded slurry, suggesting that the presence of broader solids grading diminishes effects of a pipe incline on settling-slurry flow.

Figure 6.

Measured frictional gradient for broadly graded slurry and corresponding narrowly graded slurry (left). Measured mean concentrations for broadly graded slurry and corresponding narrowly graded slurry (right). Legend: Blue square = narrow-graded slurry (C_vi, i_m); magenta diamond = broad-graded slurry (C_vi, i_m); x and + = C_vd.

The frictional gradient is higher than predicted by the Worster-Denny method at slopes milder than ±30 degrees (Figure 4). A comparison of the measured gradients (both manometric and frictional) at all flow inclinations confirms that the Worster-Denny method is reliable in predicting the gradients only if the slurry flow is not stratified. For our tested conditions, it is the case only at steep ascending and descending flows.

After examining the effect of the pipe inclination on the dimensionless pressure gradients, it is interesting to evaluate relation between the friction loss and the mean concentration of solids in a pipe and a relation between the friction loss and the solids distribution in the inclined flows.

3.3.2 Mean concentrations and slip

Although C_vd was kept constant in the test runs for the broad-graded slurry at different inclination angles, corresponding values of C_vi varied with the inclination angle. Figure 6 shows that the course of C_vi correlates tightly with the course of the frictional gradient and it is consistent with the courses for the narrow-graded slurry. The C_vd values tend to exceed the C_vi values in the descending flows steeper than −30 degrees, indicating a negative slip at which the mean velocity of particles is higher than the mean velocity of water in the flowing slurry.

3.3.3 Solids distribution

The shapes of concentration distributions differ in ascending flows and descending flows of the broad-graded slurry (Figure 7), which is consistent with the distributions in the previously observed flows of the narrow-graded sand.

Figure 7.

Comparison of solids distribution of broadly graded sand (d₅₀ = 0.55 mm) slurry flow at V_m ≈ 2.5 m/s and C_vd ≈ 0.24 in a 100 mm pipe inclined to ±15° (left) and ± 25° (right). Legend: Black = up; red = down.

The differences in the distribution of solids are quite small between the narrow-graded slurry and broad-graded slurry in ascending flows (Figure 8) where the degree of stratification is weak and the sliding bed thin. Contrary to ascending flows, descending flows exhibit distributions which are quite different in broad-graded slurry than in the corresponding narrow-graded slurry (Figure 8).

Figure 8.

Measured solids distributions in sand-water flow at V_m ≈ 2.5 m/s and C_vd ≈ 0.24 in a 100 mm pipe inclined to +25° (left) and −25° (right). Legend: Blue = narrow-graded slurry; magenta = broad-graded slurry.

Comparisons of the developments in the solid’s distribution and frictional gradient with the flow inclination angle confirm that more stratified flows produce higher frictional gradients than less stratified flows. Note that the flow of the broadly graded slurry tends to be more energy efficient (produces lower friction losses) than the flow of the narrow-graded slurry at the same flow condition, even though the degree of stratification seems to be very similar. This effect will be discussed further in connection with horizontal bimodal flows.

To summarize, the experimental comparison of inclined flows of narrow-graded and broad-graded sand-water slurries of the same d₅₀ = 0.55 mm showed that the anomalous pressure gradient occurred at mild negative slopes in both slurries although it tended to be less pronounced in the broad-graded slurry. The reason for the anomalous gradient—a sharp stratification of descending flow—was the same too although the broad-graded slurry tended to produce a thinner and more concentrated sliding bed than the narrow-graded slurry at the same flow conditions. The friction loss tended to be smaller in the broad-graded flow than in the narrow-graded flow at various tested flow inclines between −45 to +45 degrees. The Worster-Denny method predicted friction loss successfully only if the flow was not stratified. For the stratified flows, the friction loss and other parameters should be predicted by a layered model.

4. Horizontal flow of settling slurry

Although the beneficial effect of a broad particle size distribution on friction loss has been experimentally observed in horizontal flows of settling slurries, for example [3, 6, 7], it has not been investigated systematically yet. Mechanisms through which an interaction of individual solids sub-fractions affects the overall flow resistance are not well understood. The simplest slurries with which to start an investigation on the interaction mechanisms are bimodal slurries. Our first results with horizontal flows of bimodal slurries were reported in [10, 15]. In this chapter, additional results for horizontal bimodal flows are discussed together with results for broad-graded slurries composed of three or four fractions of sand. Results for slurry flows of individual narrow-graded fractions are also added for comparison.

4.1 Experimental procedure

Tests were carried out for different sand slurries at flow conditions allowing easy mutual comparisons. During testing, sand mixtures of different degrees of grading were produced by combining the individual narrow-graded sands at various proportions of each sand in a resulting mixture (Table 3).

Fraction code	Ratio in mixture [%]
SS2030	70	22.5	18
ST1040	—	44.5	36
SP0612	—	33	27
STJ25	30	—	19
Mixture:	2S	3S	4S

Table 3.

Proportions of fractions of broadly graded sand mixtures.

In test runs, the differential pressures were measured over the measuring sections along the pipe loop and the mean delivered concentration C_vd was determined from the inverted U-tube or the other vertical-pipe section at various installed velocities V_m. The local quantities—the concentration of solids in different vertical positions in the pipe cross-section and the solid’s velocity at the bottom of the pipe—were measured as well.

4.2 Bimodal slurry flow

The recent extensive broad-graded-slurry tests revealed that the flow of bimodal slurry composed of gravel and fine sand exhibited a considerably reduced friction loss compared to flow of gravel-slurry without the fine-sand addition [9]. It was hypothesized [10] that the major reason for the observed loss reduction was the reduction of mechanical friction between the sliding bed and the pipe wall and that this reduction was caused by a presence of a thin layer of fine-sand particles at the bottom of the pipe which at least partially separated the coarse sliding bed from the pipe wall. This hypothesis was indirectly supported by other results from the same experimental campaign, namely by those for bimodal slurry composed of medium-to-coarse sand and the same fine sand as in the former bimodal slurry. Flow of this bimodal slurry showed a negligible loss reduction by the fine sand. Since this flow did not contain a sliding bed, the result suggested that the friction reduction was effective only if the sliding bed was present. The actual presence of the fine separation layer could not be detected during the experiment due to the lack of flow visualization options.

A more detailed experiment was required to provide information about the internal structure of bimodal slurry flows. Such experiments had to include measurements of local concentrations and velocities of solids in the flow domain under investigation. The first runs from a series of such experiments were carried out for bimodal flows of ST1040-sand and STJ25 in our laboratory in 2020 [15]. Additional experiments followed with a different coarse-sand fraction (SS2030) and their results are discussed in this chapter.

The bimodal 2S-slurry was composed of the coarse SS2030-sand and the fine-to-medium sand STJ25 (Table 3). Tests were carried out for flow of the 2S-slurry at C_vd ≈ 0.27 (0.19 of SS2030 and 0.08 of STJ25) and for flow of the corresponding one-species SS2030-slurry at C_vd ≈ 0.19. Both flows were strongly stratified. The stratification was detected visually in the transparent section of the laboratory pipe and its degree was measured by a radiometric concentration meter in the measuring section of the laboratory pipe.

4.2.1 Visual observation

The visualization confirmed the presence of the thin layer composed of the STJ25 sand at the bottom of the pipe. Furthermore, images of a camera directed to the bottom of the transparent pipe captured an interaction of coarse particles and fine particles at the pipe wall and clearly recognized that the finer STJ25 particles reduced the contact of the coarse SS2030 particles with the wall (Figure 9).

Figure 9.

Camera images (magnified) of slurry at the bottom of the transparent pipe section (flow at V_m = 3.0 m/s). Left: Coarse slurry (SS2030). Right: Corresponding bimodal 2S-slurry (SS2030 + STJ25).

4.2.2 Solids distribution and local velocity of solids at pipe bottom

Measured distributions of solids in the two comparable flows detected that the flow remained strongly stratified even when the finer sand fraction was added (Figure 10). Particles of the finer fraction increased the local concentration at all vertical positions in a pipe cross-section, including the positions in the sliding bed.

Figure 10.

Comparison of measured solids distributions in flow of coarse SS2030-slurry flow and of corresponding bimodal 2S-slurry at V_m ≈ 3.0 m/s. Legend: Blue = SS2030-slurry, magenta = 2S-slurry.

Developments in the longitudinal component of the instantaneous velocity of particles at the bottom of pipe were processed from high-speed-camera images over the period of a few seconds and then time-averaged. Figure 11 compares time series of velocities of particles of the two fractions in the bimodal slurry at V_m = 3.0 m/s. It shows that the finer STJ25-particles (red line) of the thin layer covering the pipe wall move slower than the coarser SS2030-particles at the bottom of the sliding bed. This suggests that the faster coarse particles slide over the slower finer particles of the thin separation layer.

Figure 11.

Measured time series of local velocity of particles of two fractions at pipe bottom for 2S-slurry flow at V_m = 3.0 m/s. Legend: Blue = coarse particles in 2S-slurry (horizontal line gives mean value); red = fine particles in 2S-slurry.

Figure 12 plots the development in the local velocity of coarse particles in the one-species SS2030-sand slurry corresponding with the bimodal slurry in Figure 11. Figure 12 shows that the coarse particles at the bottom of the sliding bed are slower in this flow without the STJ25-additive than the same coarse particles in the bimodal slurry flow at the same flow velocity V_m = 3.0 m/s and the same concentration of the coarse sand. This suggests that the presence of the STJ25-particles at the bottom of the pipe promotes the sliding of the coarse bed.

Figure 12.

Measured time series of local velocity of coarse particles at pipe bottom for SS2030-sand slurry flow at V_m = 3.0 m/s.

Time-averaged longitudinal velocities, u, plotted as horizontal lines in Figures 11 and 12 were collected at two more velocities additional to V_m = 3.0 m/s. The results are collected in Figure 13 and give a more complete picture of the behavior of particles at the bottom of the pipe. Figure 13 summarizes the velocities u_c of coarse particles (in the bimodal flow and corresponding one-species flow) and u_f of fine particles (in the bimodal flow). The plot shows that the coarse particles are faster than the fine particles at the bottom of the pipe (u_c > u_f) at all tested flow velocities V_m in the bimodal 2S-slurry flow. Moreover, the coarse particles in the 2S-slurry are faster than the same coarse particles in the corresponding one-species coarse slurry at the same flow velocity V_m.

Figure 13.

Measured time-averaged local velocity (normalized by mean flow velocity) at pipe bottom for the flow of bimodal 2S-slurry and for flow corresponding SS2030-sand slurry. Legend: Magenta square = coarse particles in bimodal slurry (u_c/V_m), magenta triangle = fine particles in bimodal slurry (u_f/V_m), blue square - coarse particles in coarse slurry (u_c/V_m).

4.2.3 Friction loss

The frictional gradients measured for flows of the bimodal 2S-slurry and of the corresponding coarse SS2030-slurry are compared in Figure 14. The bimodal slurry flow exhibits lower friction losses than the corresponding coarse-slurry flow and thus the addition of the STJ25-sand caused the loss reduction as in the ST1040-based bimodal slurry tested previously. Note that the STJ25-sand particles have little effect on the friction loss if they are transported in one-species STJ25-slurry at a low concentration similar to the concentration at which they were added to the ST2030-sand slurry (Figure 14).

Figure 14.

Measured dimensionless pressure gradient for bimodal 2S-slurry (SS2030 + STJ25 at C_vd ≈ 0.19 + 0.08) and corresponding one-species slurries of coarse sand (SS2030 at C_vd ≈ 0.19) and medium to fine sand (STJ25 at C_vd ≈ 0.12). Legend: Blue square = SS2030-sand slurry; red triangle = STJ25-sand slurry; magenta diamond = bimodal 2S-slurry; black + = water.

4.2.4 Identification of friction-reduction mechanism

The collected experimental information about the bimodal slurry flow and about flow of the corresponding coarse slurry without the finer-sand additive provides sufficient support for reasonable identification of the prevailing mechanism responsible for the friction reduction in the observed bimodal flow. As shown in Figure 10, both flows are strongly stratified. Therefore, the solid’s contribution to their friction loss must be due primarily to mechanical friction between the sliding bed and the pipe wall. Hence, the observed friction loss reduction (Figure 14) must result from the reduction of this mechanical friction. The detected presence of a thin layer of STJ25-particles (Figure 9) between the pipe wall and the bottom of the sliding bed composed preferably of coarse particles (Figure 10) suggests that this layer is responsible for the loss reduction.

The information on the local velocities of particles at the bottom of the pipe helps to clarify the role of the thin layer. The observations suggest that its role is as follows:

• the thin layer effectively reduces contacts of coarse particles of the sliding bed with the pipe wall,

• the coarse sliding bed slides over the moving thin layer (u_c > u_f), which itself slides over the pipe wall (u_f > 0),

• the thin layer makes sliding of the coarse bed easier (u_c in one-species slurry < u_c in the bimodal slurry) even though the pressure gradient, seen as the driving factor for bed sliding, is smaller in the bimodal slurry flow than in the coarse slurry flow (i_m in one-species slurry > i_m in bimodal slurry) at the same V_m.

The coarse bed sliding easier and faster in the bimodal flow than the coarse bed of the same thickness (Figure 10) in the one-species flow, which also has a higher pressure drop, is a very strong indicator that there is a considerably smaller resisting force acting on the sliding bed from the pipe wall in the bimodal flow than in the coarse flow. This reduction of the wall resisting force must be due to the presence of the thin layer at the wall and, therefore the layer is responsible for the friction reduction. In summary, the thin fine layer reduces contacts of coarse particles of the sliding bed with the pipe wall and so reduces resistance and energy consumption associated with mechanical friction between the sliding bed and the pipe wall.

4.3 Broad-graded slurry flow

Besides the bimodal 2S-slurry, two more broad graded slurries were tested. The 3S-slurry contains a mixture of three sands, SS2030, ST1040, and SP0612 (Table 3), and its d₅₀ = 1.48 mm is very similar to d₅₀ = 1.56 mm of the narrow-graded ST1040-sand. Measured friction losses are compared between the 3S-slurry flow and the ST1040-slurry flow of similar values of C_vd in the left plot of Figure 15. The broader graded slurry obeys considerably less friction in the range of measured flow velocities.

Figure 15.

Measured dimensionless pressure gradient for broad-graded slurry and comparable narrow-graded slurry (ST1040-sand slurry at C_vd ≈ 0.28). Left: 3S-slurry of d₅₀ = 1.48 mm at C_vd ≈ 0.25. Right: 4S-slurry of d₅₀ = 1.28 mm at C_vd ≈ 0.31. Legend: Blue square = narrow-graded slurry; magenta diamond = broad-graded slurry; black + = water.

The STJ25-sand is added to produce the 4S-slurry (Table 3) and to increase C_vd from 0.25 to 0.31. As shown in the right plot of Figure 15, the friction loss reduces further by the STJ25-addition, exhibiting the same effect that STJ25 exhibited in the previously discussed bimodal slurry flows.

4.4 Prediction of friction loss

There are several predictive models available for the frictional dimensionless pressure gradient in horizontal flow of settling slurry. As mentioned previously, settling slurry flows exhibit a wide spectrum of flow regimes from pseudo-homogeneous for fine slurries to fully stratified for very coarse slurries. Traditionally and pragmatically, a piecewise application of theories has been employed for each regime and different predictive models have been used for the different regimes. Examples of such models include the Vsm-model for the fully stratified flow, V50-model for the heterogeneous flow, and the equivalent liquid model (ELM) for the pseudo-homogeneous flow, for example, Ref. [17]. A 4-component model (4CM) was developed to predict friction loss over a range of settling-slurry compositions and flow regimes, for example, Ref. [8, 18]. In the 4-component methodology, friction losses are calculated by a weighted average approach of the standard Newtonian carrier-fluid flow model, combined with the ELM, V50-model, and Vsm-model. The broad-graded solids are partitioned into four volume fractions or “components” and each is assigned to one of the four sub-models. If the solids are narrowly graded and fall within one of the components, then the 4CM reduces to the sub-model for the specific flow regime.

4.4.1 Narrow-graded slurries

An example of the use of 4CM for a prediction of the friction loss in slurry flow of narrow-graded sand falling within one component of the 4-component pattern is shown in the left plot of Figure 16. The ST2030-sand matches the fully-stratified solids and its slurry flow can be predicted by both the 4CM and the Vsm-model. The plot shows an excellent agreement between the measured losses and the predictions for different concentrations of solids in the slurry.

Figure 16.

Measured and predicted dimensionless pressure gradient for slurry flows of individual sand fractions. Left: SS2030-sand slurry at C_vd ≈ 0.12, 0.19, and 0.28. Right: ST1040-sand slurry at C_vd ≈ 0.12, 0.17, and 0.28. Legend: Blue points = measurements (square for C_vd of 0.12, triangle for 0.17 and 0.19, circle for 0.28); red line = prediction by 4CM; black line with + = prediction by Vsm-model, black line with x = prediction by V50-model, plain black line = prediction for water.

The ST1040-sand is an interesting case of a narrowly graded sand, which does not fall within one component (in a 100 mm pipe), because its d₅₀ is very close to the threshold size for the two components (d_threshold = 0.015D = 1.5 mm; D is the internal diameter of a pipe). Therefore, the friction loss prediction is not successful by using any of the two individual-component models (either the Vsm-model or the V50-model), and it is successful only if the 4CM is used (Figure 16, right plot).

4.4.2 Bimodal and broad-graded slurries

The individual-component models are not suitable for friction loss predictions in broad-graded slurry flows because they use the particle size only to identify the flow. Instead, the entire particle size distribution must be considered, as in the 4CM. As shown in Figure 17, an individual-component model can seriously overestimate the friction loss in the flow of bimodal slurry composed of coarse sand and medium-to-fine sand. The two plots of Figure 17 compare the bimodal flows with flows of their coarse-only counterparts at two different solids concentrations and show that, contrary to the Vsm-model, the 4CM predicts the effect of adding the STJ25-sand to the coarse SS2030-sand slurry very well.

Figure 17.

Measured and predicted dimensionless pressure gradient for 2S-slurry (SS2030 + STJ25) and corresponding coarse slurry (SS2030). Left: as in Figure 14. Right: SS2030 + STJ25 at C_vd ≈ 0.12 + 0.06 and corresponding SS2030 at C_vd ≈ 0.12. Legend: Blue square = measurement for coarse slurry; magenta diamond = measurement for bimodal slurry; red line = prediction by 4CM; black line with + = prediction by Vsm-model; plain black line = prediction for water.

The 3S-slurry, and 4S-slurry, have d₅₀ < 1.5 mm (d₅₀ = 1.48 mm for the 3S-slurry, d₅₀ = 1.28 mm for the 4S-slurry) and thus their individual-component model is the V50-model. Figure 18 shows that the V50-model prediction of the friction loss in the two broadly graded slurries is less successful than the prediction by the 4CM.

Figure 18.

Measured and predicted dimensionless pressure gradient for broad-graded slurry and comparable narrow-graded slurry as in Figure 15. Legend: blue square = measurement for narrow-graded slurry; magenta diamond = measurement for broad-graded slurry; red line = prediction by 4CM; black line with + = prediction for narrow-graded slurry by Vsm-model; black line with x = prediction for broad-graded slurry by V50-model; plain black line = prediction for water.

To conclude, a comparison of the new experimental results for various sand slurries in a horizontal 100-mm pipe with predictions of the 4-component model confirms that the 4CM is capable of very reasonable predictions of friction losses in bimodal and broad-graded settling slurry flows in horizontal pipes.

5. Summary conclusions

Extensive resources of experimental work have been called on to elucidate the effects of solids grading and pipe inclination on settling slurry transport. If flowing in horizontal and inclined pipes, broadly graded slurries exhibit lower friction losses than narrowly graded slurries of the same mean particle size, mean flow velocity, and slurry density. The difference in friction loss is particularly important in slurries that are stratified. In broad-graded slurries, fine particles are able to reduce mechanical friction associated with sliding of coarse particles over a pipe wall by developing a thin layer effectively separating the coarse particles from the pipe wall. Also associated with flow stratification and with a presence of a sliding bed is the observed anomalously high friction loss in mildly negatively sloped flows. This anomalous friction loss reduces but does not disappear if the slurry is broadly graded. The reason for the loss reduction is the same as in horizontal flows.

The observed variations of friction loss with the degree of solids grading can be predicted with reasonable success by the 4-component model. A prediction of the observed variation of friction loss with solids distribution, as in the tested inclined flows, requires a layered model.

Acknowledgments

The research has been supported by the Czech Science Foundation through grant project no. 20-13142S.