Solid-Liquid Pipeflows – Holdup and the Two-Layer Model

Abstract

The relative delay of the flow of solids in a pipe, holdup, is shown to be an important factor in the evaluation of the flow of settling particle-bearing liquids. Holdup can be determined using a new, modified interpretation of the two-layer model, originally published in the 1970s. In the original model, particles in the upper layer of the flow are supported by hydrodynamic forces only, while those in the lower layer are also supported by hindered settling and wall reactions. This concept has been retained in the new model. An initial approximated value of holdup can be refined to allow an increasingly accurate value to be calculated. Innovative coding of the new model overcomes instability and interface positioning problems. Applications for determinations of holdup and the model are examined. Pressure loss, flow rate and the prediction of the pipe velocity to avoid a stationary bed are established. The model allows a locus of stationary bed conditions to be plotted for a family of holdup values. A method to obtain the centre of concentration in the cross section is demonstrated. The locus of centre of concentration (LCC) gives an indication of the position and size of the particle burden as conditions change.

1. Introduction

R.A. Bagnold [1] identified two distinct mechanisms of particle support: fluid suspension and intergranular contact, in his studies of desert sand in Egypt. It was Professor K.C. Wilson [2, 3] who proposed the application to particle-bearing liquids in pipes, i.e., an upper fluid suspension and a lower layer taking support from hindered settling and the pipe walls: the Two-Layer Model. This is a gross simplification in some eyes, but a great help in identifying the velocity and magnitude of the stratified burden. Figure 1 shows a typical concentration map for a liquid with settling particles obtained by electrical resistance tomography (ERT) for low-velocity pipe flow (0.75 m/s). Interestingly, the supernatant phase (in pale blue) has not stacked itself in height order but has spread around the boundaries of the other pipe contents—actually emphasising the interface between two layers. The two-layer model was later modified for computer modelling by Professor C.A. Shook [4]. Shook proposed a contact load within an all-pervading suspension fraction.

Figure 1.

Concentration map (∼5.5% v/v) for 2 mm beads (relative density 1.4) in water, obtained by ERT. High concentration red and low concentration blue.

2. Computer models of slurry pipe flow

The model of two parts, settling and suspended, is not the only classification, which has been applied to component parts of a settling slurry flow. Shook and Roco [4] postulated a notional third layer without lift forces underneath the settling layer, and a three-layer model was proposed by Doron and Barnea [5] to overcome perceived “limitations” of the two-layer representation. The new model provides a way to identify the relative delay of the solids fraction in a duct or holdup, which none of the other classifications achieve, at least not in a direct way. A more recent four component model pioneered by Wilson and Sellgren [6] and revised by Visintainer et al. [7] produces useful pump derating calculations but is based on uniform particle distribution.

Since its invention by K.C. Wilson [2, 3, 8, 9], there have been several references to the two-layer model. At the time of writing a new reference volume, Visintainer et al. [10] contain detailed discussion of it. The author’s current version (2LM) described in the following pages has completely new coding to overcome instability and interface positioning difficulties of the original in obtaining estimates of the downstream velocity of the settling layer and holdup.

The limit of stationary deposition velocity over ranges of particle size, pipe diameters and relative density of grains was published by Wilson [3] in the form of an elegant nomogram constructed using techniques originated by Wood [11] (see Annexe C). The original nomogram did not account for volumetric concentrations, but in later work, Wilson et al. [8] modified the nomogram for a range of concentrations. The results are comparable with those obtained later in this chapter (Figure 2).

Figure 2.

Stationary bed loci for horizontal flow of sand in water. The concentration is 0.2 v/v, particle size is 2 mm, pipe diameter is 250 mm. U is the axial pipe velocity, Us is the velocity of the solids and dP/dz is the downstream pressure gradient. Note the classification of each locus by the relative velocity of solids, n.b ${\mathcal{H}}_{\mathit{SBL}}=\mathit{1}-\frac{U_s}{U}.$

Among other advantages, the simple geometry of the model provides an accessible alternative to an experimental image of a particle distribution, initially at least. For example, the first moment yields the centre of concentration of the particle burden about the centre of the duct. This is an indicator of its position and possibly its wear propensity. The second moment can indicate the energy required to achieve optimal particle suspension by swirling action. If available, experimental images of particle distribution can be obtained using, for example, electrical resistance tomography (ERT) or particle image velocimetry (PIV).

3. Inputs and outputs

Inevitably, one must gather the parameters of the slurry flow problem ahead of any model calculation. For example, the pipe bore, the particle diameter, the v/v concentration, etc. are all highly influential in the results obtained. Table 1 gives a list of the initial requirements for the model and suggestions for starting values if they are not known.

Parameter	Symbol	Unit		Typical Value
Pipe Bore	D	m
Particle diameter	d	m
Mean pipe velocity	U	m/s
Concentration of solids by volume [0,1]	CV		The required or delivered concentration of particles (v/v)
Internal roughness height	ε	m	Commercial steel pipe [12]; Cast iron [12]; Concrete, upper value [12]; Concrete, lower value [12]; Drawn Tubing [12]	$4.5\times {10}^{-5}$; $2.6\times {10}^{-4}$; $3.0\times {10}^{-3}$; $3.0\times {10}^{-4}$; $1.5\times {10}^{-6}$
Coefficient of friction	η	—	For the friction of settled particles with the duct walls. Starting estimate	0.6
Viscosity of liquid vehicle	μ	Pas	At 293°K; Water [13]; Glycerol [13]; Olive oil [13]; Paraffin oil [13]	0.001; 0.0015; 0.08; ∼1.0
Density of solid particles	ρS	kg/m^3
Density of Liquid vehicle	ρL	kg/m^3
Limiting concentration of lower layer	CLIM	—	Loosely-packed concentration of particles v/v. Starting estimate; NB dense particle packing will require a higher value.	0.6
Angle of inclination	θ	°

Table 1.

Preparatory data gathering.

4. Flags

The solution can hit obstacles, but these are almost always the result of unachievable entry data (Table 1). Convergence of the model is assessed by the convergence of actual efflux concentration with that desired. If this cannot be achieved a flag is raised. The axial pipe velocity for a stationary bed is the solution of a quadratic equation (see Eqs. (37)–(40)). If an imaginary solution is obtained, a discriminant flag is set. The SB flag is not really an obstacle, but an indication that the value obtained for the velocity of the lower layer (U₂) is zero or less than zero. If U₂ is calculated to a value less than zero, this does not generally indicate a reverse flow direction, but an established stationary bed. The model sets it to zero, and the SB flag is raised.

The outputs from the model are shown in Table 2.

$\mathcal{H}$	Holdup ratio: the relative delay to the solids fraction
${\mathcal{H}}_{\mathit{SBL}}$	Holdup if pipe velocity, U, were to be reduced to form a stationary bed
Cr	In-situ concentration v/v
β	Subtending angle for the interface chord
U1	Pipe velocity, upper layer
U2	Pipe velocity, lower layer
USBL	Pipe velocity for stationary bed locus
$\mathit{dP}/\mathit{dz}\mathit{\text{or}}i$	Pressure gradient
${\rho }_1$	Density, upper layer
${\rho }_2$	Density, lower layer
${\rho }_M$	Density, mixture
y	Distance of the notional interface chord from the centre line
$\overline{y}$	Centre of Concentration, vertical distance from centre line
$\stackrel{=}{y}$	Radius of gyration of the cross section including the settling bed
Cv	Efflux concentration v/v
Cv convergence flag	YES if Cv has converged satisfactorily to the value specified (Table 1)
Discriminant flag	YES if discriminant is positive
SB flag	YES for stationary bed.; NB if $U_2\le U_{\mathit{SBL}}$ U2 is always set to zero and the flag is raised

Table 2.

2LM outputs.

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 trial-correction 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, d_25, 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, C_LIM 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 (C_c), i.e.

Figure 3.

The two-layer model. After shook [4].

$$C_c=C_2\times \frac{A_2}{A}$$E1

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

$$C_r=C_1+C_c$$E2

The in situ, concentration takes no account of the differing rates of flow of the upper and lower layers. The efflux or delivered concentration (C_v) will be lower than the in situ value (C_r) 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 flows, remembering that C₂ represents only the concentration in the lower part of the section.

$$C_v\mathit{AU}=C_1\mathit{AU}+C_2{A}_2{U}_2$$E3

substituting from Eqs. (1) and (2)

$$C_c=\frac{C_2{A}_2}{A}=\frac{U\left(C_v-C_1\right)}{U_2}$$E4

Holdup ($\mathcal{H}$) 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 ($\frac{C_r}{C_v}$). 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.

$$\mathcal{H}=\frac{U-U_s}{U}$$E5

where U_s 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 (Q_s) can be calculated using the mean mixture velocity in the duct or the in situ velocity of the solids (U_s) 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

$$C_v=\frac{U_s}{U}{C}_r$$E6

Using Eq. (6), we obtain the influential relationship, which relates holdup ($\mathcal{H}$) to in situ concentration (C_r).

$$\mathcal{H}=1-\frac{C_v}{C_r}$$E7

In Eq. (4)

$$C_c=\frac{U{C}_v}{\left(U-U_2\right)}\frac{\mathcal{H}}{\left(1-\mathcal{H}\right)}$$E8

Eq. (8) yields the contact load C_c after the results of the first pass of the calculation have provided U₂ and $\mathcal{H}$. But one-off estimates of C_c and $\mathcal{H}$ 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 trial-correction 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 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.

Figure 4.

Holdup vs. W/U ratio from data collected by Lahiri and Ghanta [16]. Linear regression gives $\mathcal{H}=\mathit{3.1179}\times \left(\frac{W}{U}\right)$ with r² = 0.89.

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

$${\mathcal{H}}^{\left(0\right)}\sim 3.1179\left(\frac{W}{U}\right)$$E9

Richardson and Zaki [8] provide

$$W=v_0{\left(1-C_v\right)}^Z$$E10

in which v₀ is the unhindered settling velocity. The defining equations are

$$v_0=\left(\frac{g}{18\mu }\right)\left({\rho }_s-{\rho }_L\right)d^2\mathit{Re}<1\left(\text{Stokes}\mathrm{Law}\right)$$E11

$$v_0=\frac{0.2d^{1.18}{\left(\frac{{\rho }_s-{\rho }_L}{{\rho }_L}\right)}^{0.72}}{{\left(\mu /{\rho }_L\right)}^{0.45}}1<\mathit{Re}<800\left(\text{Alle}{\mathrm{n}}^{\prime }\mathrm{s}\mathrm{Law}\right)$$E12

$$v_0=1.74\sqrt{\mathit{gd}\left(\frac{{\rho }_s-{\rho }_L}{{\rho }_L}\right)}\mathit{Re}>1000\left(\text{Newto}{\mathrm{n}}^{\prime }\mathrm{s}\mathrm{Law}\right)$$E13

$$Z=4.65+1.95\left(\frac{d}{D}\right)0.002<\mathit{Re}\le 0.2$$E14

$$Z=\left(4.35+17.5\left(\frac{d}{D}\right)\right){\mathit{Re}}^{-0.03}0.2<\mathit{Re}\le 1.0$$E15

$$Z=\left(4.45+18\left(\frac{d}{D}\right)\right){\mathit{Re}}^{-0.1}1<\mathit{Re}$$E16

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 C_c, the first of which is

$$C_c^{\left(0\right)}\sim C_r^{\left(0\right)}\mathit{exp}\left[-0.124\times {\mathit{Ar}}^{-0.061}{\left(\frac{U^2}{\mathit{gd}}\right)}^{0.028}{\left(\frac{d}{D}\right)}^{-0.431}{\left(S_s-1\right)}^{-0.272}\right]$$E17

where Ar = Archimedes Number defined $\frac{4g{d}^3\left(S_s-1\right){\rho }_L^2}{3{\mu }_L^2}$

S_s = Relative density of solids

and, from Eq. (7) the in situ concentration $C_r^{\left(0\right)}=\frac{C_v}{1-{\mathcal{H}}^{\left(0\right)}}$

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.,

$$-{\left.\frac{\mathit{dP}}{\mathit{dz}}\right|}_{\mathit{\text{Layer}}2}=-{\left.\frac{\mathit{dP}}{\mathit{dz}}\right|}_{\mathit{\text{Layer}}1}$$E18

$$\mathit{\text{i.e.,}}\frac{{\tau }_1{S}_1+{\tau }_{12}{S}_{12}}{A_1}+{\rho }_1\mathit{\text{gsin}}\mathrm{\theta }=\frac{{\tau }_2{S}_2-{\tau }_{12}{S}_{12}}{A_2}+{\rho }_2\mathit{\text{gsin}}\mathrm{\theta }$$E19

where τ₁₂ 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

$${\tau }_{12}=\frac{1}{2}{f}_{12}{\left(U_1-U_2\right)}^2{\rho }_1$$E20

where f₁₂ is the friction factor at the notional interface. Nikuradse’s sand-roughened pipe tests [21] can be characterised as follows:

$$\frac{1}{\sqrt{f}}=1.14-0.86\mathit{ln}\left(\frac{\epsilon }{D}\right)$$E21

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

$$f_{12}=\frac{1}{{\left(1.736-1.98{\mathit{log}}_{10}\left(\frac{d}{D}\right)\right)}^2}$$E22

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.

$$f=2{\left[{\left(\frac{8}{\mathit{Re}}\right)}^{12}+\frac{1}{{\left(A+B\right)}^{1.5}}\right]}^{\frac{1}{12}}$$E23

in which:

$$A={\left\{-2.457\mathit{ln}\left[{\left(\frac{7}{\mathit{Re}}\right)}^{0.9}+\frac{0.27\epsilon }{D}\right]\right\}}^{16}$$E24

$$B={\left(\frac{37430}{\mathit{Re}}\right)}^{16}$$E25

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,

$${\tau }_1=\frac{1}{2}f{U}_1^2{\rho }_1$$E26

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

$${\tau }_2{S}_2={\tau }_{2m}{S}_2+{\tau }_{2s}{S}_2\mathit{\text{cosθ}}$$E27

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

$${\tau }_{2m}=\frac{1}{2}f{U}_2^2{\rho }_1$$E28

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].

$${\tau }_{2s}{S}_2={\eta }_s{C}_2\frac{D^2}{2}\frac{\left(1-C_{\mathit{LIM}}\right)}{\left(1-C_2\right)}\left({\rho }_s-{\rho }_L\right)g\left(\mathit{\text{sinβ}}-\mathit{\text{βcosβ}}\right)$$E29

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

8. Subtending angle 2β

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).

$$F\left(\beta \right)=\frac{4A_2}{D^2}=\beta -\frac{1}{2}\mathit{sin}2\beta$$E30

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

$$\mathit{sin}2\beta =2\beta -\frac{{\left(2\beta \right)}^3}{3!}+O\left({\beta }^5\right)$$E31

In Eq. (30), when β is small, $F\left(\beta \right)\approx \frac{2{\beta }^3}{3}$

i.e., for small values of segmental area A₂

$$\beta \left(F\right)\approx \sqrt[3]{\frac{3F}{2}}$$E32

For $F\left(\beta \right)\le 0.00032$ (approximately $\beta \le 4.5°$) Eq. (32) applies. Above that, given a value of F(β), the Taylor series gives us a simple interpolation.

$${\beta }_{i+1}\left(F\right)={\beta }_i+\beta {\prime }_i\times \left(F_{i+1}-F_i\right)+\beta {"}_i\times \frac{{\left(F_{i+1}-F_i\right)}^2}{2!}+\dots$$E33

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 ${\beta }_{i+1}\left(F\right)$ into Eq. (30) until $F\left({\beta }_i\right)\to \frac{4A_2}{D^2}$ to any prescribed accuracy.

9. Contact lengths

With a value of the subtending angle $\beta$, 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

$$S_2=R\times 2\beta$$E34

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

$$S_1=2\mathit{\pi R}-S_2$$E35

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

$$S_{12}=2\mathit{\text{Rsinβ}}$$E36

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.

$$a{U}_2^2+b{U}_2+c=0$$E37

where

$$a=0.5A_2{S}_{1}f{\rho }_1{\left(\frac{A_2}{A_1}\right)}^2+0.5A{S}_{12}{f}_{12}{\rho }_1{\left(\frac{A}{A_1}\right)}^2-0.5A_1{S}_{2}f{\rho }_1$$E38

$$b=-A_2{S}_{1}f{\rho }_1\left(\frac{{\mathit{AA}}_2}{A_1^2}\right)-A{S}_{12}{f}_{12}{\rho }_1{\left(\frac{A}{A_1}\right)}^{2}U$$E39

$$c=0.5A_2{S}_{1}f{\rho }_1{\left(\frac{\mathit{AU}}{A_1}\right)}^2-A_1{\tau }_{2s}{S}_2\mathit{\text{cosθ}}+0.5A{S}_{12}{f}_{12}{\rho }_1{\left(\frac{\mathit{AU}}{A_1}\right)}^2+A_1{A}_2\left({\rho }_1-{\rho }_2\right)\mathit{\text{gsinθ}}$$E40

From the final value of the bed velocity U_2, the velocity of the upper layer can be obtained from the principle of conservation of volume.

$$A_2{U}_2+A_1{U}_1=\mathit{AU}$$E41

11. The stationary bed

Conditions for a stationary bed can be obtained from Eq. (37), i.e.

if $U_2\to 0\text{then}c\to 0$

i.e.

$$U_{\mathit{\text{SBLθ}}}=\frac{A-A_2}{A}\sqrt{\frac{\left(A-A_2\right){\tau }_{2s}{S}_2\mathit{\text{cosθ}}+\left(A-A_2\right)A_2\left({\rho }_1-{\rho }_2\right)\mathit{\text{gsinθ}}}{0.5{\rho }_1\left(A_2{S}_{1}f+A{S}_{12}{f}_{12}\right)}}$$E42

For a horizontal duct, this is mercifully simplified

$$U_{\mathit{SBL}}=\frac{A-A_2}{A}\sqrt{\frac{\left(A-A_2\right){\tau }_{2s}{S}_2}{0.5{\rho }_1\left(A_2{S}_{1}f+A{S}_{12}{f}_{12}\right)}}$$E43

Note that the velocity for a stationary bed, $U_{\mathit{\text{SBLθ}}}$ or $U_{\mathit{SBL}}$, 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.

$$i=-{\left.\frac{\mathit{dP}}{\mathit{dz}}\right|}_{\mathit{\text{Layer}}1}=\frac{\frac{1}{2}{f}_1{U}_1^2{\rho }_1{S}_1+\frac{1}{2}{f}_{12}{\left(U_1-U_2\right)}^2{\rho }_1{S}_{12}}{A_1}+{\rho }_1\mathit{\text{gsinθ}}$$

For a stationary bed in horizontal flow, $U_2\to 0$, $U_1\to U_{\mathit{SBL}}\times \frac{A}{A_1}$ and $\theta \to 0$ so the loci of pressure gradient with the velocity for a stationary bed are described by Eq. (44).

$$i_{\mathit{SBL}}=\frac{{\rho }_1}{2A_1}{\left(U_{\mathit{SBL}}\frac{A}{A_1}\right)}^2\left(f_1{S}_1+f_{12}{S}_{12}\right)$$E44

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 (${\mathcal{H}}_{\mathit{SBL}}$) can be obtained. This value of ${\mathcal{H}}_{\mathit{SBL}}$ 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): $\mathcal{H}=\frac{U-U_s}{U}$. When ${\mathcal{H}}_{\mathit{SBL}}=1.0,\left(\frac{U_s}{U}=0\right)$, the solid particles have zero velocity, and a high pipe velocity is needed to prevent a stationary bed. For ${\mathcal{H}}_{\mathit{SBL}}=0.1$, $\left(\frac{U_s}{U}=0.9\right)$ 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 2LM 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.

Figure 5.

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

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.

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, C_v, and achieves this task after the second set of corrections arriving at a value within 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.

Figure 6.

2LM example — Fine particles in a viscous medium.

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.

15. Centre of concentration

The centre of concentration (G(r,θ) or G(x,y)) of particle suspensions can be as important as the centre of mass in the mechanics of solids. Uniform suspension is indicated if G is near the geometrical centre of the duct. If it is near the periphery of the duct, sluggish or static settlement can be inferred.

Figure 7 shows the locus of centre of concentration (LCC) for a series of concentrations The origin (0,0) is the centre of the pipe. The position of G for a given pipe velocity changes as concentration increases. This downward trend is a locus of centre of concentration (LCC), which can be used to study the state of suspension of the medium.

Figure 7.

Locus of Centre of concentration (LCC) from ERT measurements. (2 mm beads of RD 1.4 and concentrations 7.31%, 5.48%, 3.66% and 1.79% v/v in water).

The current version of the 2LM model computes the first moment of the two layers and combines these results to obtain the centre of concentration of the whole. All three results are useful in different ways. The position of the centre of concentration for the upper layer for example may be important when a stationary bed is indicated. The estimates are surprisingly close to those obtained from tomography measurements (see Figure 8). The 2LM estimates are above the parity line because the actual interface between layers is a diffuse boundary rather than a precise chord.

Figure 8.

Centre of concentration calculations using the two-layer model as an approximation to the actual concentration distribution in the cross section.

16. Conclusions

In my conclusions for this chapter, I must first pay tribute to the pioneers of the Two-Layer Model: Professors K.C. Wilson and C.A. Shook. They motivated the lengthy process of precise definition of the interface layer, calculation of holdup ratio, accurate estimation of the contact load, estimation of the velocity for a stationary bed, the locus of points for a stationary bed and the calculation of the centre of concentration. The algorithm has a long and useful history, which has been punctuated by a series of incremental modifications. A guiding principle in this model has been to keep trial-correction iterations to a minimum and the two-stage iteration process works extremely well in this endeavour. The first stage — six calculations using an empirical value for the holdup — invites the second stage in which all the variables (including the holdup and contact load) are refined. Figures 6 and 9 show the progress of the calculations for two widely different examples. They also show how the bed depth reduces as pipe velocity is increased.

Figure 9.

2LM example — Coarse particles in water.

Much of the original terminology (from reference (4)) has endured, and the author has been careful to keep to the original symbols where possible. This said much of the 2LM model described in the foregoing pages is truly new. From a rather unstable computation, a robust and widely applicable system has emerged. In its present form, it has already been used by the author in many calculations for particle-bearing liquids.

β	β	F(β)=	dβ/dF	d^2β/dF^2	d^3β/dF^3
degrees	radians	4*area/D^2
0	0	0	1	0
4	0.06981317	0.00022662	102.7545277	−2938.916414	8053120.117
8	0.13962634	0.001807662	25.81424314	−367.3557681	497829.108
12	0.20943951	0.006071189	11.56677223	−108.8347698	96554.43462
16	0.27925268	0.014293048	6.581029752	−45.90155642	29777.73395
20	0.34906585	0.027672046	4.274316085	−23.48717386	11801.60686
24	0.41887902	0.047306608	3.022340595	−13.57657624	5466.671326
28	0.488692191	0.074173404	2.268566019	−8.5331043	2813.653051
32	0.558505361	0.109108337	1.780535302	−5.698904249	1561.315876
36	0.628318531	0.152790273	1.447213595	−3.983837256	916.1578674
40	0.698131701	0.205727824	1.210138313	−2.884373363	561.0662394
44	0.767944871	0.268249457	1.036161515	−2.145953318	355.3309265
48	0.837758041	0.340497093	0.905363721	−1.630386313	231.1981656
52	0.907571211	0.422423348	0.805203615	−1.258188022	153.8334161
56	0.977384381	0.513792454	0.72748087	−0.981384085	104.3400823
60	1.047197551	0.614184849	0.666666667	−0.769800359	72
64	1.117010721	0.723005344	0.618941539	−0.603755918	50.50151797
68	1.186823891	0.839494706	0.581618596	−0.469978332	36.00772855
72	1.256637061	0.962744435	0.552786405	−0.359222381	26.12198067
76	1.326450232	1.09171445	0.531082227	−0.264827342	19.31211832
80	1.396263402	1.22525333	0.515545602	−0.181809199	14.57906665
84	1.466076572	1.362120726	0.50552345	−0.106265311	11.26033979
88	1.535889742	1.501011505	0.50060973	−0.034963354	8.91045152
92	1.605702912	1.640581149	0.50060973	0.034963354	7.226413536
96	1.675516082	1.779471927	0.50552345	0.106265311	6.00034357
100	1.745329252	1.916339324	0.515545602	0.181809199	5.088747506
104	1.815142422	2.049878203	0.531082227	0.264827342	4.392268907
108	1.884955592	2.178848218	0.552786405	0.359222381	3.84213259
112	1.954768762	2.302097947	0.581618596	0.469978332	3.39094193
116	2.024581932	2.418587309	0.618941539	0.603755918	3.006351647
120	2.094395102	2.527407804	0.666666667	0.769800359	2.666666667
124	2.164208272	2.6278002	0.72748087	0.981384085	2.35774786
128	2.234021443	2.719169306	0.805203615	1.258188022	2.070815141
132	2.303834613	2.80109556	0.905363721	1.630386313	1.800873504
136	2.373647783	2.873343196	1.036161515	2.145953318	1.545575938
140	2.443460953	2.935864829	1.210138313	2.884373363	1.304395663
144	2.513274123	2.988802381	1.447213595	3.983837256	1.078019326
148	2.583087293	3.032484316	1.780535302	5.698904249	0.867899432
152	2.652900463	3.067419249	2.268566019	8.5331043	0.675922485
156	2.722713633	3.094286046	3.022340595	13.57657624	0.504161945
160	2.792526803	3.113920608	4.274316085	23.48717386	0.354693941
164	2.862339973	3.127299605	6.581029752	45.90155642	0.229459909
168	2.932153143	3.135521465	11.56677223	108.8347698	0.130164791
172	3.001966313	3.139784991	25.81424314	367.3557681	0.058202636
176	3.071779484	3.141366034	102.7545277	2938.916414	0.014603838

From Wilson [3], with permission

1. Select the pipe diameter/m on the left hand axis

2. Place a straight edge to cut the particle diameter/mm curve to the middle axis (stationary deposit velocity/ms⁻¹ for grains with RD = 2.65)

3. Place the straight edge from the value on the middle axis to cut the right-hand function at the correct relative density.