**5. Solution of the equations: iteration and approximation strategy**

A guiding principle in the development of this version of the model has been to keep trial-correction iterations to a minimum. However, a small number of trialcorrection cycles are needed for the solution because many of the variables are only known in an experimental or empirical sense. The first phase, usually six iterations, is concerned with estimating the concentrations in the cross section. These values are established using the best estimate of the vital holdup ratio, the relative delay to the solids fraction. The holdup ratio can be estimated using an empirical correlation and is generally within �3% of its experimentally determined value. The second, and last, phase of the solution (usually a further six iterations) is concerned with refining the holdup ratio, the velocity in the lower layer, the pressure gradient, the velocity for a stationary bed and, importantly, the convergence of the solids concentration to the specified value.

*2LM* requires a representative particle diameter, so care should be taken when there is a broad distribution of particle sizes. Wilson [2] suggests that distributions be partitioned into a small series of percentiles, *for example, d02, d25, d50 d75*, and *d98,* so that the model can be run with the fines fraction, the coarse fraction and prominent intermediate size ranges. Individual researchers will approach the results in different ways, perhaps simply taking a weighted mean of a key outcome and applying a tolerance band. Inevitably, such a procedure ignores the packing propensity of small particles to occupy voids between larger particles and the segregation of larger particles when pipe velocities are low. In **Table 1**, a value of limiting concentration, *CLIM* of *0.6,* is advised as a starting estimate*.* Published datasets [14] give values between *0.58* and *0.78*, depending on the packing propensity of slurries with broad-size distributions.

A flowchart summarising the stages in the calculation is presented in Annexe A.

### **6. Concentrations**

**Figure 3** shows the basis of the idealised concentration distribution. The barchart illustrates the principle used by Shook and Roco [4] to express the concentration of the settling layer, *C2*, as an addition to a pervading medium composed of the suspended fraction (a "carrier" medium [6, 7]). This is particularly relevant when the suspended fraction is composed of finely divided solids or particles of a completely different character to the settling medium. When referred to the whole cross section, *C2* is termed the *Contact Load* (*Cc*), i.e.

$$\mathbf{C}\_{\mathbf{c}} = \mathbf{C}\_{2} \times \frac{A\_{2}}{A} \tag{1}$$

In this form, it can be added to the concentration in layer 1 to form the *in situ* concentration.

$$\mathbf{C}\_{\mathbf{r}} = \mathbf{C}\_{1} + \mathbf{C}\_{\mathbf{c}} \tag{2}$$

The *in situ,* concentration takes no account of the differing rates of flow of the upper and lower layers. The *efflux* or *delivered* concentration (*Cv*) will be lower than the *in situ* value (*Cr*) because of the delay or holdup of the lower layer. It can be obtained by expressing the total flow rate as the combination of the two constituent

**Figure 3.** *The two-layer model. After shook [4].*

flows, remembering that *C2* represents only the concentration in the lower part of the section.

$$\mathbf{C}\_{v}AU = \mathbf{C}\_{1}AU + \mathbf{C}\_{2}\mathbf{A}\_{2}U\_{2} \tag{3}$$

substituting from Eqs. (1) and (2)

$$C\_c = \frac{C\_2 A\_2}{A} = \frac{U(C\_v - C\_1)}{U\_2} \tag{4}$$

Holdup (H) is the key variable in this version of the model. It is quite difficult to determine with accuracy from experiments, and one must expect some scatter in experimental results. It can be defined in one of two possible ways. Some authors define it as the ratio of the *in situ* concentration to the delivered concentration (*Cr Cv* ). A better definition is given by the ratio of the velocity reduction of the solids to the free stream velocity, and this is the form used here.

$$\mathcal{H} = \frac{\mathbf{U} - \mathbf{U}\_s}{\mathbf{U}} \tag{5}$$

where *Us* is the axial velocity of the solids at any section. Fortunately, the two definitions of holdup are algebraically linked as shown below.

The delivery flow rate of solids (*Qs*) can be calculated using the mean mixture velocity in the duct or the *in situ* velocity of the solids (*Us*) at the delivery section. These two expressions must be equal, i.e.

$$Q\_s = C\_v \left( U \frac{\pi D^2}{4} \right) = U\_s \left( C\_r \frac{\pi D^2}{4} \right).$$

From which

$$\mathbf{C}\_{v} = \frac{U\_{s}}{U} \mathbf{C}\_{r} \tag{6}$$

Using Eq. (6), we obtain the influential relationship, which relates holdup (H) to *in situ concentration* (*Cr*).

$$\mathcal{H} = \mathbf{1} - \frac{\mathbf{C}\_v}{\mathbf{C}\_r} \tag{7}$$

In Eq. (4)

$$\mathbf{C}\_{t} = \frac{\mathbf{U}\mathbf{C}\_{v}}{(U - U\_{2})} \frac{\mathcal{H}}{(\mathbf{1} - \mathcal{H})} \tag{8}$$

Eq. (8) yields the contact load *Cc* after the results of the first pass of the calculation have provided *U2* and H. But one-off estimates of *Cc* and H are needed at the start of the calculation to make a first estimate of *U2*. They are strongly related for a given slurry, but no direct analytical relationship solely between the two is available. A trialcorrection strategy for two variables simultaneously is fraught with difficulties, so a good solution procedure is to fix one of them as accurately as possible in the initial

#### *Advances in Slurry Technology*

stages. Unsurprisingly, the holdup ratio is highly influential in the two-layer model (Jones [14, 15]), so this would be a good choice for a variable to be determined as accurately as possible at outset. Later, the value of the holdup ratio can be refined by a second set of iterations. So, the first task was to search for a strong correlation for holdup in terms of the input variables.

Lahiri and Ghanta [16] have a neural network design, which they claim predicts holdup ratio with an absolute average accuracy of 2.5%. Seshadri et al. [17] demonstrate a strong relationship between holdup and hindered settling velocity using equations from Richardson and Zaki [18]. They suggest a relationship with a dimensionless parameter *W/U\*,* where *W* is the hindered settling velocity and *U\** is the shear velocity. This cannot be used directly because the shear velocity *U\** is not available at this stage of the calculation. However, a simpler direct correlation with holdup emerges when the hindered settling velocity is expressed as a non-dimensional ratio with pipe velocity (*W/U*). **Figure 4** shows a good general relationship when applied to data collected by Lahiri and Ghanta [16] (predominantly from publications by Hsu [19]), but scatter and uncertainty, particularly at low values of holdup ratio, must be considered. A second set of iterations to refine the estimate of holdup is a necessary precaution with this important variable.

Hence, an initial estimate of the holdup ratio can be obtained from

$$\mathcal{H}^{(0)} \sim \mathbf{3.1179} \left( \frac{W}{U} \right) \tag{9}$$

Richardson and Zaki [8] provide

$$\mathcal{W} = \nu\_0 (\mathbf{1} - \mathbf{C}\_v)^Z \tag{10}$$

in which *v0* is the *un*hindered settling velocity. The defining equations are

$$w\_0 = \left(\frac{\text{g}}{18\mu}\right) (\rho\_s - \rho\_L) d^2 \text{ } \text{Re} < \mathbf{1} \text{ (Stokes Law)}\tag{11}$$

**Figure 4.**

*Holdup vs. W/U ratio from data collected by Lahiri and Ghanta [16]. Linear regression gives* <sup>H</sup> <sup>¼</sup> *<sup>3</sup>:<sup>1179</sup>* � *<sup>W</sup> U with r2 = 0.89.*

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

$$w\_0 = \frac{0.2d^{1.18} \left(\frac{\rho\_r - \rho\_l}{\rho\_L}\right)^{0.72}}{\left(\mu/\rho\_L\right)^{0.45}} \text{ 1} < Re < 800 \text{ (Allen's Law)}\tag{12}$$

$$w\_0 = 1.74 \sqrt{\text{gd} \left(\frac{\rho\_s - \rho\_L}{\rho\_L}\right)} \text{ } Re > 1000 \text{ (Newton's Law)}\tag{13}$$

$$Z = 4.65 + 1.95 \left( \frac{d}{D} \right) \ 0.002 < Re \le 0.2 \tag{14}$$

$$Z = \left( 4.35 + 17.5 \left( \frac{d}{D} \right) \right) Re^{-0.03} \ 0.2 < Re \le 1.0 \tag{15}$$

$$Z = \left( 4.45 + 18\left(\frac{d}{D}\right) \right) Re^{-0.1} \text{ 1} < Re \tag{16}$$

where *Re =* Reynolds number.

A later correlation for the exponent *Z* for power-law liquids is given by Coulson and Richardson [20].

Shook and Roco [4] suggest two empirical correlations for *Cc*, the first of which is

$$C\_c^{(0)} \sim C\_r^{(0)} \exp\left[ -0.124 \times Ar^{-0.061} \left(\frac{U^2}{gd}\right)^{0.028} \left(\frac{d}{D}\right)^{-0.431} (\text{S}\_i - 1)^{-0.272} \right] \tag{17}$$

where *Ar =* Archimedes Number defined <sup>4</sup>*gd*<sup>3</sup> ð Þ *Ss*�<sup>1</sup> *<sup>ρ</sup>*<sup>2</sup> *L* 3*μ*<sup>2</sup> *L*

*Ss* = Relative density of solids

and, from Eq. (7) the *in situ* concentration *C*ð Þ <sup>0</sup> *<sup>r</sup>* <sup>¼</sup> *Cv* <sup>1</sup>�Hð Þ <sup>0</sup>

Note that Eq. (17) is simply a starting estimate from which a short- trial correction sequence can be initiated. Initial estimates of concentrations in hand, the solution of the pressure balance equations can proceed.

### **7. Pressure balance**

In a steady-state situation, the pressure per unit length in the upper and lower component must be equal, i.e.,

$$-\frac{dP}{dz}\bigg|\_{\text{Layer 2}} = -\frac{dP}{dz}\bigg|\_{\text{Larger 1}}\tag{18}$$

$$i.e., \frac{\tau\_1 \mathbf{S}\_1 + \tau\_{12} \mathbf{S}\_{12}}{A\_1} + \rho\_1 \mathbf{g} \sin \theta = \frac{\tau\_2 \mathbf{S}\_2 - \tau\_{12} \mathbf{S}\_{12}}{A\_2} + \rho\_2 \mathbf{g} \sin \theta \tag{19}$$

where *τ<sup>12</sup>* and *S12* are shear stress and surface area at the interface between layers. The essential strategy now is to populate each of the terms in Eq. (19) as a means to provide a solution.

In most cases, the interfacial shear stress is an artefact of the model since there is generally no sharp discontinuity here unless *U2* approaches zero. Shook and Roco [4] put

$$
\pi\_{12} = \frac{1}{2} f\_{12} (U\_1 - U\_2)^2 \rho\_1 \tag{20}
$$

where *f12* is the friction factor at the notional interface. Nikuradse's sandroughened pipe tests [21] can be characterised as follows:

$$\frac{1}{\sqrt{f}} = \mathbf{1.14} - \mathbf{0.86} \ln\left(\frac{\varepsilon}{D}\right) \tag{21}$$

Putting roughness height, *ε*, to half the screen size of the sediment and changing the base of the logarithm yields.

$$f\_{12} = \frac{1}{\left(1.736 - 1.98 \log\_{10} \left(\frac{d}{D}\right)\right)^2} \tag{22}$$

Colebrook [22] used Nikuradse's results to develop an empirical transition function for the region between laminar flow and fully developed turbulence, which was the basis of the well known Moody diagram [12, 23].

Eq. (21) can be used again to determine pipe friction factor (*f*), but it has been claimed that a later correlation from a short paper by Churchill [24] delivers better accuracy over a wide range of fluid regimes.

$$f = 2\left[\left(\frac{8}{Re}\right)^{12} + \frac{1}{\left(\mathcal{A} + \mathcal{B}\right)^{1.5}}\right]^{\frac{1}{12}}\tag{23}$$

in which:

$$A = \left\{-2.457 \ln \left[ \left( \frac{7}{Re} \right)^{0.9} + \frac{0.27e}{D} \right] \right\}^{16} \tag{24}$$

$$B = \left(\frac{37430}{Re}\right)^{16} \tag{25}$$

There is a trap for the unwary here. Churchill's correlation is for the *Fanning* friction factor, one quarter of the value usually found on Moody charts. Churchill's formula for *f* is implied in subsequent calculations.

In the upper layer,

$$
\pi\_1 = \frac{1}{2} f U\_1^2 \rho\_1 \tag{26}
$$

In the lower layer, we must take account of the Coulombic friction of the particle burden and the effect of pipe inclination, *i.e.,*

$$
\tau\_2 \mathbf{S}\_2 = \tau\_{2m} \mathbf{S}\_2 + \tau\_{2s} \mathbf{S}\_2 \cos \theta \tag{27}
$$

The first part of the Eq. (27) uses the density and friction factor of the invested medium.

$$
\pi\_{2m} = \frac{1}{2} f U\_2^2 \rho\_1 \tag{28}
$$

The second part of the Eq. (27) requires analysis of the weight of the particle burden on the wall of the duct and the application of the coefficient of friction [15].

$$
\pi\_2 \mathbf{S}\_2 = \eta\_s \mathbf{C}\_2 \frac{D^2}{2} \frac{(\mathbf{1} - \mathbf{C}\_{\rm LIM})}{(\mathbf{1} - \mathbf{C}\_2)} (\rho\_s - \rho\_L) \mathbf{g} (\sin \beta - \beta \cos \beta) \tag{29}
$$

Surface peripheral areas *S1, S2 and S12* (per downstream length) are functions of the subtending angle *β* (**Figure 3**) and will be derived below.
