**8. Subtending angle 2***β*

The area of the segment at the base of the circular cross section can be expressed as a non-dimensional function of the subtending angle *β* (**Figure 3** shows this angle).

$$F(\beta) = \frac{4A\_2}{D^2} = \beta - \frac{1}{2}\sin 2\beta \tag{30}$$

The inverse function *β(F)* is difficult to obtain analytically, especially for small values (*F(β)* has an infinite gradient as *β* ! 0). However, the sine expansion for the second part of the Eq. (30) gives

$$\sin 2\beta = 2\beta - \frac{\left(2\beta\right)^3}{3!} + O\left(\beta^5\right) \tag{31}$$

In Eq. (30), when *<sup>β</sup>* is small, *<sup>F</sup>*ð Þ *<sup>β</sup>* <sup>≈</sup> <sup>2</sup>*β*<sup>3</sup> 3 i.e., for small values of segmental area *A2*

$$
\beta(F) \approx \sqrt[3]{\frac{3F}{2}} \tag{32}
$$

For *F*ð Þ *β* ≤0*:*00032 (approximately *β* ≤ 4*:*5°) Eq. (32) applies. Above that, given a value of *F(β)*, the Taylor series gives us a simple interpolation.

$$\beta\_{i+1}(F) = \beta\_i + \boldsymbol{\beta}\_i^\prime \times (F\_{i+1} - F\_i) + \boldsymbol{\beta}\_i^{\prime\prime} \times \frac{\left(F\_{i+1} - F\_i\right)^2}{2!} + \dots \tag{33}$$

A look-up table for *F(β)* and the first three derivatives of *β* is given at Annexe B. The first lookup and interpolation should be enough for most purposes, but an iterative procedure can be applied by substituting *<sup>β</sup><sup>i</sup>*þ<sup>1</sup>ð Þ *<sup>F</sup>* into Eq. (30) until *<sup>F</sup> <sup>β</sup><sup>i</sup>* ð Þ! <sup>4</sup>*A*<sup>2</sup> *<sup>D</sup>*<sup>2</sup> to any prescribed accuracy.

### **9. Contact lengths**

With a value of the subtending angle *β*, we are now in a position to define accurate lengths of contact between layers and the pipe wall. Referring to **Figure 3**, the length of the arc of contact with layer 2 is given by

$$\mathbf{S}\_2 = \mathbf{R} \times \mathbf{2}\boldsymbol{\beta} \tag{34}$$

The length of the arc of contact with layer 1 is given by

$$\mathbf{S}\_1 = \mathbf{2}\pi \mathbf{R} - \mathbf{S}\_2 \tag{35}$$

The length of the interface between the two layers is given by

$$\mathbf{S}\_{12} = \mathbf{2}\mathbf{R}\sin\beta\tag{36}$$

### **10. Gathering terms**

Eq. (19) can now be populated with finished values from the foregoing analyses. By good fortune, it resolves into a quadratic equation in *U2.*

$$aU\_2^2 + bU\_2 + c = 0\tag{37}$$

where

$$\mu = 0.5A\_2 \text{S} \circ \rho\_1 \left(\frac{A\_2}{A\_1}\right)^2 + 0.5A \text{S}\_{12} f\_{12} \rho\_1 \left(\frac{A}{A\_1}\right)^2 - 0.5A\_1 \text{S}\_2 f \rho\_1 \tag{38}$$

$$b = -A\_2 S\_4 f \rho\_1 \left(\frac{A A\_2}{A\_1^2}\right) - A S\_{12} f\_{12} \rho\_1 \left(\frac{A}{A\_1}\right)^2 U \tag{39}$$

$$c = 0.5A\_2S\_{\text{f}}f\rho\_1 \left(\frac{AU}{A\_1}\right)^2 - A\_1\tau\_2S\_2\cos\theta + 0.5AS\_{12}f\_{12}\rho\_1 \left(\frac{AU}{A\_1}\right)^2 + A\_1A\_2(\rho\_1 - \rho\_2)\text{g}\sin\theta \tag{40}$$

From the final value of the bed velocity *U2,* the velocity of the upper layer can be obtained from the principle of conservation of volume.

$$A\_2 U\_2 + A\_1 U\_1 = AU \tag{41}$$

### **11. The stationary bed**

Conditions for a stationary bed can be obtained from Eq. (37), i.e. if *U*<sup>2</sup> ! 0 then *c* ! 0 i.e.

$$U\_{\rm SBL\theta} = \frac{A - A\_2}{A} \sqrt{\frac{(A - A\_2)\tau\_2 S\_2 \cos \theta + (A - A\_2)A\_2(\rho\_1 - \rho\_2)\text{g}\sin \theta}{0.5\rho\_1(A\_2 S\_1 + A S\_{12} f\_{12})}}\tag{42}$$

For a horizontal duct, this is mercifully simplified

$$U\_{\rm SBL} = \frac{A - A\_2}{A} \sqrt{\frac{(A - A\_2)\tau\_{2\prime}\mathbf{S}\_2}{\mathbf{0}.5\rho\_1(A\_2\mathbf{S}\_2\mathbf{f} + A\mathbf{S}\_{12}\mathbf{f}\_{12})}}\tag{43}$$

*Solid-Liquid Pipeflows – Holdup and the Two-Layer Model DOI: http://dx.doi.org/10.5772/intechopen.112023*

Note that the velocity for a stationary bed, *USBL<sup>θ</sup>* or *USBL*, is not exactly equivalent to the *critical deposition velocity* to indicate the lowest velocity at which a slurry might be pumped to avoid settling. There is a slight difference between the definition of these two velocities. The *critical deposition velocity* can be defined as the velocity at which the first stationary particle layer will form at the bottom of a duct. Clearly, the stationary bed is not usually a single layer of particles, but in many ways, the velocity for a stationary bed is as useful a concept as the *critical deposition velocity*, if not more useful for the designer.

### **12. The stationary bed locus**

Practitioners often make use of a plot of the pressure gradient (*-dP/dz* or *i*) against pipe velocity (the "*i-v diagram"*). On this useful plot, the stationary bed envelope can be plotted to indicate safety from a settling bed and potential blockage. Pump characteristics can also be superimposed. The locus of points with *U2 = 0* is called the *Stationary Bed Locus* or *SBL*. Delineation between the two layers is most obvious when the bed is stationary. Hence, *2LM* is particularly adept at plotting this locus and identifying a predicted operating envelope to be avoided.

The pressure gradient can be computed from the final values of density, layer velocities, bed friction and surface areas. In Eq. (18), the pressure gradients in both layers must be equal so the layer 1 computation can be used.

$$\dot{a} = -\frac{dP}{dx}\Big|\_{\text{Layer 1}} = \frac{\frac{1}{2}f\_1 U\_1^2 \rho\_1 \mathbf{S}\_1 + \frac{1}{2}f\_{12} (U\_1 - U\_2)^2 \rho\_1 \mathbf{S}\_{12}}{A\_1} + \rho\_1 \mathbf{g} \sin \theta$$

For a stationary bed in horizontal flow, *<sup>U</sup>*<sup>2</sup> ! 0, *<sup>U</sup>*<sup>1</sup> ! *USBL* � *<sup>A</sup> <sup>A</sup>*<sup>1</sup> and *θ* ! 0 so the loci of pressure gradient with the velocity for a stationary bed are described by Eq. (44).

$$i\_{\rm SBL} = \frac{\rho\_1}{2A\_1} \left( U\_{\rm SBL} \frac{A}{A\_1} \right)^2 \left( f\_{\rm 1} \text{S}\_1 + f\_{\rm 12} \text{S}\_{12} \right) \tag{44}$$

**Figure 2** shows stationary bed loci of pressure gradient with pipe velocity, *U*. The advantage of this graph is that pump characteristics can be overlaid to investigate the possibility of a stationary bed. If a stationary bed is implicated, a specific holdup ratio (H*SBL*) can be obtained. This value of H*SBL* only applies when there is a stationary bed and should not be confused with the value obtained when both beds are flowing freely.

To interpret **Figure 2**, one must refer to the fundamental definition of holdup given by Eq. (5): <sup>H</sup> <sup>¼</sup> *<sup>U</sup>*�*Us <sup>U</sup>* . When <sup>H</sup>*SBL* <sup>¼</sup> <sup>1</sup>*:*0, *Us <sup>U</sup>* <sup>¼</sup> <sup>0</sup> , the solid particles have zero velocity, and a high pipe velocity is needed to prevent a stationary bed. For <sup>H</sup>*SBL* <sup>¼</sup> <sup>0</sup>*:*1, *Us <sup>U</sup>* <sup>¼</sup> <sup>0</sup>*:*<sup>9</sup> solid particles are relatively mobile, and only a low pipe velocity is required to keep the settling layer in suspension. Note the shallow slope of the loci at low pressures—a very small change in pressure gradient yields a large change in velocity requirement. The safest policy is, of course, to maintain a pipe velocity entirely to the right of the SBL envelope, 2.84 m/s in this case This is the "maximum velocity at the limit of stationary deposition" in Annexe C. The value from Wilson's original nomogram [8, 11] (Annexe C) is approximately 2.70 m/s. The slightly reduced value (approximately �5%) is attributed to the lack of compensation for volumetric concentration. A later version of the nomogram, including allowance for a range of concentrations is available [9].

### **13. Validating the 2***LM* **model**

*2LM* works on a significant simplification of particle concentrations in a duct. Shou [25] suggests a three-layer model for large pipes with a low-concentration supernatant layer at the top of the section. There have been other criticisms, and it is important to contrast predictions with experimental data. For example, the model proposes a sharply defined discontinuity or interface between upper and lower layers not strictly observed in practice. It is closest to reality when the lower layer is stationary. Even then, local effects have been observed at the interface such as bouncing, gusting (where groups of particles leave the interface altogether), surface rippling and cycles of deposition and re-entrainment. Lahiri and Ghanta [16] provide a dataset of 43 slurries with values of holdup for each one. These data can be compared with predictions from *2LM*. Pressure differences can be accurately measured in experiments and **Figure 5b** shows this. Note the greater scatter at higher pressures. **Figure 5a** shows a significant agreement between experiment and model determinations of holdup but considerably more scatter than those for pressure gradient data. At high values of holdup, the local effects of bouncing, gusting, *etc.* can be expected to have a strong influence on the experimental accuracy of holdup measurements.

In testing the model, it was important to explore as wide a selection of examples as possible. The work by Shook and Roco [4] was the motivation for this development, and their example was the first example to be tested [15]. Application and modification over several years have resulted in a utility with wide applicability and stability. The model has been tested over many examples of slurry flow, two of which will be shown in the next section.

**Figure 5.**

*2LM model validation against experimental results from Lahiri and Ghanta [16].*

### **14. Example calculations from the dataset**

Results are presented in two parts. The first part displays iteration progress for a specific example from the dataset. This has a fixed velocity. The second set shows the depth of the settling layer as the velocity is varied.

The corrections shown in **Figure 6** (fine particles in moderately viscous medium) are relatively minor. The algorithm strives to match the given volume concentration, *Cv*, and achieves this task after the second set of corrections arriving at a value within

#### *Solid-Liquid Pipeflows – Holdup and the Two-Layer Model DOI: http://dx.doi.org/10.5772/intechopen.112023*

**Figure 6.**

*2LM example — Fine particles in a viscous medium.*

approximately 3% on the original estimate. Only 12 corrections are applied in two batches of 6. The depth of the settling layer reduces systematically as the mean pipe velocity is increased.

Corrections required for the second example (very coarse particles in water **Figure 9**) are clearly much greater, but this is to be expected in view of the large particles specified in the case. The second case is the most interesting challenge of the two. Large variations in pressure gradient and volume concentration are becalmed by the second set of iterations. The depth of the settling bed starts from a low value before reducing systematically as in the first example.
