**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 [11–14]. 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*<sup>m</sup> 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 narrowgraded 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 Vm* ≈ *1.84 m/s and Cvd* ≈ *0.17 inclined at +25° (left) and −25° (right). Legend: Points = measurement, line = prediction by a layered model [12].*

#### *Settling Slurry Transport: Effects of Solids Grading and Pipe Inclination DOI: http://dx.doi.org/10.5772/intechopen.108436*

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** compares different dimensionless pressure gradients (*i* = Δ*p*/ρw/*g*/*L*, where Δ*p* = measured pressure differential, ρw = 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

#### **Figure 3.**

*Dimensionless pressure gradients at various angles of inclination for slurry flow of SP0612 sand at Vm* ≈ *1.84 m/s and Cvd* ≈ *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 Cvd; blue line = manometric gradient by Worster and Denny based on Cvi; black line = frictional gradient by Worster and Denny.*

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 partiallystratified 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 *d*50.

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 (*d*50 = 0.55 mm).

#### *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.

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 broadgraded 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.

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


#### **Table 2.**

*Proportions of fractions of broadly graded sand mixture.*

*Settling Slurry Transport: Effects of Solids Grading and Pipe Inclination DOI: http://dx.doi.org/10.5772/intechopen.108436*

#### **Figure 4.**

*Dimensionless pressure gradients at various angles of inclination for slurry flow of broad-graded sand at Vm* ≈ *2.5 m/s and Cvd* ≈ *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.*

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.

**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 (Cvi, im); magenta diamond = broad-graded slurry (Cvi, im); x and + = Cvd.*

#### *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.

The differences in the distribution of solids are quite small between the narrowgraded slurry and broad-graded slurry in ascending flows (**Figure 8**) where the

**Figure 7.**

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

*Settling Slurry Transport: Effects of Solids Grading and Pipe Inclination DOI: http://dx.doi.org/10.5772/intechopen.108436*

**Figure 8.**

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

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**).

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 *d*50 = 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


#### **Table 3.**

*Proportions of fractions of broadly graded sand mixtures.*

composed of three or four fractions of sand. Results for slurry flows of individual narrow-graded fractions are also added for comparison.
