**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.

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 *d*50 is very close to the threshold size for the two components (*d*threshold = 0.015*D* = 1.5 mm; *D* is the internal

#### **Figure 15.**

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

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 mediumto-fine sand. The two plots of **Figure 17** compare the bimodal flows with flows of

#### **Figure 16.**

*Measured and predicted dimensionless pressure gradient for slurry flows of individual sand fractions. Left: SS2030-sand slurry at Cvd* ≈ *0.12, 0.19, and 0.28. Right: ST1040-sand slurry at Cvd* ≈ *0.12, 0.17, and 0.28. Legend: Blue points = measurements (square for Cvd 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.*

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

#### **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 Cvd* ≈ *0.12 + 0.06 and corresponding SS2030 at Cvd* ≈ *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.*

#### **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 narrowgraded slurry by Vsm-model; black line with x = prediction for broad-graded slurry by V50-model; plain black line = prediction for water.*

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.

The 3S-slurry, and 4S-slurry, have *d*50 < 1.5 mm (*d*50 = 1.48 mm for the 3S-slurry, *d*50 = 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.

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.
