15	Fluidization and Sedimentation

“I never teach my pupils. I only attempt to provide the conditions in which they can learn.”

—Albert Einstein, 1879–1955, Physicist

I.    FLUIDIZATION

When a fluid is passed upward through a bed of particles, as illustrated in Figure 15.1, the pressure drop increases as the fluid velocity increases. The product of the pressure drop and the bed crosssectional area represents the net upward force acting on the bed, and when this force becomes equal to the weight of the bed (solids plus fluid), the bed becomes suspended by the fluid. In this state, the particles can move freely within the “bed,” which thus behaves much like a boiling liquid. Under these conditions, the bed is said to be “fluidized.” This “freely flowing” or bubbling behavior results in a high degree of mixing in the bed, which provides a great advantage for heat or mass transfer efficiency as compared to that of a fixed bed. Fluid bed operations are found in refineries (i.e., fluid catalytic crackers), polymerization reactors, fluid bed combustors, drying of cohesive solids, etc. If the fluid velocity within the bed is greater than the terminal velocity of the particles, however, the fluid will tend to entrain the particles and carry them out of the bed. However, if the superficial velocity above the bed, which is less than the interstitial velocity within the bed, is less than the terminal velocity of the particles, they will fall back and remain in the bed. Thus, there is a specific range of velocity over which the bed remains in a fluidized state without particles being entrained by the gas fluid (gas or liquid).

The scope of this chapter concerns mainly the sedimentation and fluidization behavior of non-cohesive granular particles. More detailed discussion of these topics as well as the fluidization and sedimentation of fibrous systems (encountered in paper-pulp suspensions) and/or fine cohesive powders are available in other references, such as Yang (2003) and Millan (2013).

A.    GOVERNING EQUATIONS

The Bernoulli equation relates the pressure drop across the bed to the fluid flow rate and the bed properties:

$\frac{-\text{Δ}P}{{\text{ρ}}_f}-gh=e_f=\frac{f_{PM}h{V}_S^2}{d}\left(\frac{1-\text{ε}}{{\text{ε}}^3}\right)$

(15.1)

where the porous medium friction factor is given by the Ergun equation:

$f_{PM}=1.75+\frac{180}{N_{RePM}}$

(15.2)

and the porous medium Reynolds number is defined as:

$N_{RePM}=\frac{d{V}_S\text{ρ}}{\left(1-\text{ε}\right)\text{μ}}$

(15.3)

FIGURE 15.1  Schematic representation of a fluidized bed.

Now the criterion for incipient fluidization is that the force due to the pressure drop must balance the buoyant weight of the bed, that is,

$-\text{Δ}P=\frac{BedWt.}{A}={\text{ρ}}_s\left(1-\text{ε}\right)gh+\text{ρε}gh$

(15.4)

where the first term on the right-hand side is the pressure due to the weight of the solids and the second is that due to the weight of the fluid in the bed. When the pressure drop is eliminated from Equations 15.1 and 15.4, an equation for the “minimum fluidization velocity” (V_mf) results:

$\left({\text{ρ}}_s-\text{ρ}\right)\left(1-\text{ε}\right)g=\frac{\text{ρ}{e}_f}{h}=1.75\frac{\text{ρ}{V}_{mf}^2\left(1-\text{ε}\right)}{d{\text{ε}}^3}+180\frac{V_{mf}\text{μ}{\left(1-\text{ε}\right)}^2}{d^2{\text{ε}}^3}$

(15.5)

which can be written in dimensionless form:

$N_{Ar}=1.75\frac{{\widehat{N}}_{Re}^2}{{\text{ε}}^3}+180\left(\frac{1-\text{ε}}{{\text{ε}}^3}\right){\widehat{N}}_{Re}$

(15.6)

where

$N_{Ar}=\frac{\text{ρ}g\text{Δ}\text{ρ}{d}^3}{{\text{μ}}^2},{\widehat{N}}_{Re}=\frac{d{V}_S\text{ρ}}{\text{μ}}$

(15.7)

Equation 15.6 can be solved for the Reynolds number to give

${\widehat{N}}_{Re}={\left(C_1^2+C_2{N}_{Ar}\right)}^{1/2}-C_1$

(15.8)

where

$C_1=\frac{180\left(1-\text{ε}\right)}{3.5},C_2=\frac{{\text{ε}}^3}{1.75}$

(15.9)

Equation 15.8 gives the dimensionless superficial velocity (V_S) for incipient fluidization in terms of the Archimedes number and the bed porosity, ε.

B.    MINIMUM BED VOIDAGE

Before the bed can become fluidized, however, the particles must dislodge from their “packed” state, after which the bed can expand. Thus, the porosity (ε) in Equations 15.5 and 15.9 is not the initial “packed bed” porosity but is the “expanded bed” porosity at the point of incipient fluidization (ε_mf), that is, the “minimum bed voidage” in the bed just prior to fluidization. Actually, the values of C₁ and C₂ in Equation 15.8 that give the best results for fluidized beds of uniform spherical particles have been found from empirical observations to be

$C_1=27.2,C_2=\text{0}\text{.0408}$

(15.10)

By comparing these empirical values of C₁ and C₂ with Equation 15.9, the C₁ value of 27.2 is seen to be equivalent to ε_mf = 0.471, while the C₂ value of 0.0408 is equivalent to ε_mf = 0.415. In practice, the value of ε_mf may vary considerably with the nature of the solid particles, as shown in Figure 15.2.

C.    NONSPHERICAL PARTICLES

Many particles are not spherical and so will not have the same drag properties as spherical particles. The effective diameter for such particles is often characterized by the equivalent Stokes diameter, which is the diameter of the sphere that has the same terminal velocity as the nonspherical particle. This can be determined from a direct measurement of the settling rate of the particles and provides the best value of equivalent diameter for use in applications involving fluid drag on the particles.

An alternative description of nonspherical particles is often represented by the “sphericity factor” (ψ), which is the number that, when multiplied by the diameter of a sphere with the same volume as the particle (d_s), gives the particle effective diameter (d_p):

$d_p=\text{ψ}{d}_s$

(15.11)

The sphericity factor is defined as

$\text{ψ = }\frac{\text{Surface area of the sphere with the same volume as the particle}}{\text{Surface area of the particle}}$

(15.12)

FIGURE 15.2 Values of ε_mf for various solids. (From Azbel, D.S. and Cheremisinoff, N.P., Fluid Mechanics and Unit Operations, Ann Arbor Science, Ann Arbor, MI, 1983.)

Thus

$\text{ψ = }\frac{A_s}{A_p}=\frac{A_s/V_s}{A_p/V_p}=\frac{6/d_s}{a_s}$

(15.13)

Equations 15.11 and 15.13 show that d_p = 6/a_s, where a_s is the surface-to-volume ratio for the particle (A_p/V_p), as deduced in Chapter 14. Since V_p = V_S (by definition), an equivalent definition of ψ is

$\text{ψ = }\frac{6}{d_s}\left(\frac{V_p}{A_p}\right)={\left(6^2\text{π}\right)}^{1/3}\frac{V_p^{2/3}}{A_p}=\frac{4.8V_p^{2/3}}{A_p}$

(15.14)

The minimum bed porosity at incipient fluidization for nonspherical particles can be estimated from

${\text{ε}}_{mf}\cong {\left(14\text{ψ}\right)}^{-1/3}$

(15.15)

For spherical particles (ψ = 1), Equation 15.15 reduces to ε_mf = 0.415.

II.    SEDIMENTATION

Sedimentation, or thickening, involves increasing the solids content of a slurry or a suspension by gravity settling in order to effect separation (or partial separation) of the solids and the fluid. It differs from the gravity settling process that was previously considered in that the solids fraction is relatively high in these systems, so that particle settling rates are strongly influenced by the presence of the surrounding particles. This is referred to as hindered settling. Fine particles (10 μm or less) tend to behave differently than larger or coarse particles (100 μm or more), because fine particles may exhibit a high degree of flocculation due to the presence of surface forces and high specific surface area. Figure 13.1 shows a rough illustration of the effect of solids concentration and particle/fluid density ratio on the free and hindered settling regimes.

A.    HINDERED SETTLING

A mixture of particles of different sizes can settle in different ways, according to Coulson et al. (2002), as illustrated in Figure 15.3. Case (a) corresponds to a suspension with a range of particle sizes less than about 6:1. In this case, all of the particles settle at about the same velocity in the “constant composition zone” (B), leaving a layer of clear liquid above. As the sediment (D) builds up, however, the liquid that is “squeezed out” of this layer serves to further retard the particles just above it, resulting in a zone of variable composition (C). Case (b) in Figure 15.3 is less common and corresponds to a broad particle size range, in which the larger particles settle at a rate significantly greater than that of the smaller ones, and consequently there is no constant composition zone.

FIGURE 15.3 Modes of settling. (a) Narrow particle size range; (b) broad particle size range.

The sedimentation characteristics of hindered settling systems differ significantly from those of freely settling particles in several ways:

1.  The large particles are hindered by the small particles, which increase the effective resistance of the suspending medium for the large particles. At the same time, however, the small particles tend to be “dragged down” by the large particles, so that all particles tend to fall at about the same rate (unless the size range is very large, i.e., greater than 6:1 or so).

2.  The upward velocity of the displaced fluid flowing in the interstices between the particles is significant, so that the apparent settling velocity (relative to a fixed point) is significantly lower than the particle velocity relative to the fluid.

3.  The velocity gradient in the suspending fluid flowing upward between the particles is increased, resulting in greater shear forces.

4.  Because of the high surface area-to-volume ratio for small particles, surface forces are important resulting in flocculation or “clumping” of the smaller particles into larger effective particle groups. This effect is more pronounced in a highly ionic (conducting) fluid, because the electrostatic surface forces that would cause the particles to be repelled are “shorted out” by the conductivity of the surrounding fluid.

There are essentially three different approaches to describing hindered settling. One approach is to define a “correction factor” to the Stokes free settling velocity in an infinite Newtonian fluid (which we will designate V_o), as a function of the solids loading. A second approach is to consider the suspending fluid properties (e.g., viscosity and density) to be modified by the presence of fine particles. A third approach is to consider the collection or “swarm” of particles equivalent to a moving porous bed, the resistance to flow through the bed being determined by an equivalent of the Kozeny equation. There is insufficient evidence to say that any one of these approaches is any better or worse than the others. For many systems, they may all give comparable results, whereas for others one of these methods may be better or worse than the others.

If all of the solids are relatively fine and/or the slurry is sufficiently concentrated so that settling is extremely slow, the slurry can usually be approximated as a uniform continuous “pseudo-homogeneous” medium with properties (viscosity and density) that depend upon the solids loading, particle size and density, and interparticle forces (surface charges, conductivity, etc.). Such systems are generally quite non-Newtonian, with properties that can be described by the Bingham plastic or power law models. If the particle size distribution is broad and a significant fraction of the particles are fines (e.g., less than about 30 μm or so), the suspending fluid plus fines can be considered to be a continuous medium with a characteristic viscosity and density through which the larger particles must move. Such systems may or may not be non-Newtonian, depending on solids loading, particle size distribution, etc., but are most commonly non-Newtonian. If the solids loading is relatively low (e.g., below about 10% solids by volume) and/or the particle size and/or density are relatively large, the system will be “heterogeneous” and the larger particles will settle readily. Such systems are usually Newtonian. A summary of the flow behavior of these various systems has been presented by Darby (1986), Shook and Roco (1991), and Wilson et al. (2008).

B.    FINE PARTICLES

For suspensions of fine particles, or systems containing a significant amount of fines, the suspending fluid can be considered to be homogeneous with the density and viscosity modified by the presence of the fines. These properties depend primarily upon the solids loading of the suspension, which may be described in terms of either the porosity or void fraction (ε) or, more commonly, the volume fraction of solids, φ (φ = 1 – ε). The buoyant force on the particles is due to the difference in density between the solid (ρ_s) and the surrounding suspension (ρ_φ), which is

${\text{ρ}}_s-{\text{ρ}}_{\text{φ}}={\text{ρ}}_s-\left[{\text{ρ}}_s\left(1-\text{ε}\right)+\text{ρε}\right]=\text{ε}\left({\text{ρ}}_s-\text{ρ}\right)=\left(1-\text{φ}\right)\left({\text{ρ}}_s-\text{ρ}\right)$

(15.16)

where ρ is the density of the clear fluid phase.

The viscosity of the suspension (μ_φ) is also modified by the presence of the solids. For uniform spheres at a volumetric fraction of 2% or less, Einstein (1906) showed that

${\text{μ}}_{\text{φ}}=\text{μ}\left(1+2.5\text{φ}\right)$

(15.17)

where μ is the viscosity of the suspending fluid phase. For more concentrated suspensions, a wide variety of expressions have been proposed in the literature (see, e.g., Darby [1986], Mewis and Wagner [2012], and Chhabra [2016]). For example, Vand (1948) proposed the expression

${\text{μ}}_{\text{φ}}=\text{μ exp}\left(\frac{2.5\text{φ}}{1-0.609\text{φ}}\right)$

(15.18)

However, Mooney (1951) concluded that the numerical constant 0.609 in Equation 15.18 varies from 0.75 to 1.5, depending upon the system. Equation 15.17 or 15.18 (or equivalent) may be used to modify the viscosity and density in Stokes’ law, that is,

$V_o=\frac{\left({\text{ρ}}_s-\text{ρ}\right)g{d}^2}{18\text{μ}}$

(15.19)

In this equation, V_o is the relative velocity between the unhindered particle and the fluid. However, in a hindered suspension, this velocity is increased by the velocity of the displaced fluid, which flows back up through the suspension in the void space between the particles. Thus, if V_S is the (superficial) settling velocity of the suspension (e.g., “swarm”) and V_L is the velocity of the fluid, the total flux of solids and liquid is [φV_S + (1 – φ)V_L]. The relative velocity between the fluid and solids in the swarm is V_r = V_S – V_L. If the total net flux is zero (e.g., “batch” settling in a closed-bottom container with no outflow), elimination of V_L gives

$V_r=\frac{V_S}{1-\text{φ}}$

(15.20)

This also shows that V_L = −φV_S/(1 – φ), that is, V_L is negative relative to V_S in batch settling.

From Equations 15.16, 15.18, and 15.20, the ratio of the settling velocity of the suspension (V_S) to the terminal velocity of a single freely settling sphere (V_o) can be determined to be

$\frac{V_S}{V_o}=\frac{{\left(1-\text{φ}\right)}^2}{\exp \left(\frac{2.5\text{φ}}{1-k_2\text{φ}}\right)}$

(15.21)

where the value of the constant k₂ can vary from 0.61 to 1.5, depending upon the system. However, Coulson et al. (2002) remark that the use of a modified viscosity for the suspending fluid is more appropriate for the settling of large particles through a suspension of fines than for the uniform settling of a “swarm” of uniform particles with a narrow size distribution. They state that in the latter case, the increased resistance is due to the higher-velocity gradients in the interstices rather than to an increased viscosity. However, the net effect is essentially the same for either mechanism. This approach and the other two mentioned earlier all result in expressions of the general form:

$\frac{V_S}{V_o}={\text{ε}}^{2}fn\left(\text{ε}\right)\text{where}\text{ε}=\left(1-\text{φ}\right)$

(15.22)

which is consistent with Equation 15.21.

A widely quoted empirical expression for the function in Equation 15.22 is that of Richardson and Zaki (1954):

$fn\left(\text{ε}\right)={\text{ε}}^n$

(15.23)

where

$n=\{\begin{array}{ccc}4.65& \text{for}& N_{Rep}<0.2\\ 4.35N_{Rep}^{-0.03}& \text{for}& 0.2<N_{Rep}<1\\ 4.45N_{Rep}^{0.1}& \text{for}& 1<N_{Rep}<500\\ 2.39& \text{for}& N_{Rep}>500\end{array}$

and N_{Re p} is the single-particle Reynolds number in an “infinite” fluid. An alternative expression due to Davies et al. (1977) for the ratio of the two velocities is

$\frac{V_S}{V_o}=\exp \left(-k_1\text{φ}\right)$

(15.24)

which agrees well with Equation 15.23 for k₁ = 5.5. Yet another expression for fn(ε), deduced by Steinour (1944) from settling data on tapioca in oil, is

$fn\left(\text{ε}\right)={10}^{-1.82\left(1-\text{ε}\right)}$

(15.25)

Barnea and Mizrahi (1973) considered the effects of the modified density and viscosity of the suspending fluid, as represented by Equation 15.21, as well as a “crowding” or hindrance effect that decreases the effective space around the particles and increases the drag. This additional “crowding factor” is given by (1 + k₂φ^1/3), which, when included in Equation 15.21, gives

$\frac{V_S}{V_o}=\frac{{\left(1-\text{φ}\right)}^2}{\left(1+{\text{φ}}^{1/3}\right)\exp \left[5\text{φ}/3\left(1-\text{φ}\right)\right]}$

(15.26)

for the modified Stokes velocity, where the constant 2.5 in Equation 15.21 has been replaced by 5/3, and the constant k₂ set equal to unity, based upon experimental observations in a range of systems.

C.    COARSE PARTICLES

Coarser particles (e.g., ~100 μm or larger) have a relatively small specific surface, so that flocculation is not common. Also, the suspending fluid surrounding the particles is the liquid phase rather than a “pseudo continuous” phase of fines in suspension which modify the fluid viscosity and density properties. Thus, the properties of the continuous phase can be taken to be those of the pure fluid unaltered by the presence of fine particles. In this case, it can be shown by dimensional analysis that the dimensionless settling velocity V_S/V_o must be a function of the particle drag coefficient, which in turn is a unique function of the particle Reynolds number, N_Rep, the void fraction (porosity), ε = (1 – φ), and the ratio of the particle diameter to container diameter, d/D. Since there is a unique relationship between the drag coefficient, the Reynolds number, and the Archimedes number for settling particles, the result can be expressed in functional form as

$\frac{V_S}{V_o}=fn\left(N_{Ar},d/D,\text{ε}\right)$

(15.27)

It has been found that this relationship can be represented by the following empirical expression (Coulson et al., 2002):

$\frac{V_S}{V_o}={\text{ε}}^n{\left(1+2.4\frac{d}{D}\right)}^{-1}$

(15.28)

where the exponent n is given by

$n=\frac{4.8+2.4X}{X+1}$

(15.29)

and

$X=0.043N_{Ar}^{0.57}\left[1-2.4{\left(\frac{d}{D}\right)}^{2.7}\right]$

(15.30)

D.    ALL FLOW REGIMES

The above expressions give the suspension velocity (V_S) relative to the single-particle free settling velocity, V_o, that is, the Stokes velocity. However, it is not necessary that the particle settling conditions correspond to the Stokes regime to use these equations. As shown in Chapter 12, the Dallavalle equation can be used to calculate the single-particle terminal velocity V_o under any flow conditions from a known value of the Archimedes number, as follows:

$V_o=\frac{\text{μ}}{\text{ρ}d}{\left[{\left(14.42+\text{1}\text{.827}\sqrt{N_{Ar}}\right)}^{1/2}-\text{3}\text{.798}\right]}^2$

(15.31)

where

$N_{Ar}=\frac{d^3\text{ρ}g\text{Δ}\text{ρ}}{{\text{μ}}^2}$

(15.32)

This result can also be applied directly to coarse particle “swarms.” For fine particle systems, the suspending fluid properties are assumed to be modified by the fines in suspension, which necessitates modifying the fluid properties in the definitions of the Reynolds and Archimedes numbers accordingly. Furthermore, since the particle drag is a direct function of the local relative velocity between the fluid and the solid (i.e., the interstitial relative velocity, V_r), it is this velocity that must be used in the drag equations (e.g., the modified Dallavalle equation). Knowing that V_r = V_S/(1 – φ) = V_S/ε, the appropriate definitions for the Reynolds number and drag coefficient for the suspension (e.g., the particle “swarm”) are (after Barnea and Mizrahi, 1973)

$N_{Re\text{φ}}=\frac{d{V}_r\text{ρ}}{{\text{μ}}_{\text{φ}}}=N_{R{e}_0}\left(\frac{V_S}{V_o}\right)\frac{1}{\left(1-\text{φ}\right)\exp \left[5\text{φ}/3\left(1-\text{φ}\right)\right]}$

(15.33)

and

$C_{D\text{φ}}=C_{Do}{\left(\frac{V_o}{V_S}\right)}^2\frac{{\left(1-\text{φ}\right)}^2}{\left(1+{\text{φ}}^{1/3}\right)}$

(15.34)

where N_Reo = dV_oρ/μ and $C_{Do}=4gd\left({\text{ρ}}_S-\text{ρ}\right)/\left(3\text{ρ}{V}_o^2\right)$ are the Reynolds number and drag coefficient, respectively, for a single particle in an infinite fluid. Data presented by Barnea and Mizrahi (1973) show that the “swarm” dimensionless groups N_Reφ and C_Dφ are related by the same expression as the corresponding groups for single particles, for example, by the Dallavalle equation:

$C_{D\text{φ}}={\left(0.6324+\frac{4.8}{N_{Re\text{φ}}^{1/2}}\right)}^2$

(15.35)

The settling velocity or the terminal velocity of the “swarm” may therefore be determined from

$N_{Re\text{φ}}={\left[{\left(14.42+1.827N_{Ar\text{φ}}^{1/2}\right)}^{1/2}-3.798\right]}^2$

(15.36)

where

$N_{Ar\text{φ}}=\frac{3}{4}{C}_{D\text{φ}}{N}_{Re\text{φ}}^2=\left(\frac{d^3\text{ρ}g\left({\text{ρ}}_s-\text{ρ}\right)}{{\text{μ}}^2\left(1+{\text{φ}}^{1/3}\right)}\right)\exp \left(\frac{-10\text{φ}}{3\left(1-\text{φ}\right)}\right)$

(15.37)

III.    GENERALIZED SEDIMENTATION/FLUIDIZATION

The above relations all apply to hindered settling of a suspension (or “swarm”) of particles (in most cases of uniform size) in a stagnant suspending medium. Barnea and Mizrahi (1973) showed that these generalized relations may be applied to fluidization as well, since a fluidized bed may be considered a particle “swarm” suspended by the fluid flowing upward at the terminal velocity of the swarm. In this case, the above equations apply with V_S replaced by the velocity V_f, that is, the superficial velocity of the fluidizing medium. Once N_Reφ is found from Equations 15.36 and 15.37, the settling velocity (V_S) is determined from Equation 15.33. Barnea and Mizrahi (1973) presented data for both settling and fluidization, which cover a very wide range of the dimensionless parameters, as shown in Figure 15.4. In this figure, the x-coordinate is independent of velocity and the y-coordinate is independent of particle size.

FIGURE 15.4 Generalized dimensionless correlation of settling and fluidizing velocities. (From Barnea, E. and Mizrahi, J., A generalized approach to the fluid dynamics of particulate systems, Part 1: General correlation for fluidization and sedimentation in solid multi-particle systems, Chem. Eng. J., 5, 171, 1973.)

IV.    THICKENING

The process of thickening involves the concentration of a slurry, suspension, or sludge usually by gravity settling. Since concentrated suspensions and/or fine particle dispersions are often involved, the result is usually not a complete separation of the solids from the liquid but is instead a separation into a more concentrated (underflow) stream and a diluted (overflow) stream. Thickeners and clarifiers are essentially identical. The only difference is that the clarifier is designed to produce a clean liquid overflow with a specified purity, whereas the thickener is designed to produce a concentrated underflow product with a specified concentration (McCabe et al., 1993; Christian, 1994; Tiller and Tang, 1995).

A schematic of a thickener/clarifier is shown in Figure 15.5. As indicated in Figure 15.3, several settling regions or zones can be identified, depending upon the solids concentration and interparticle interactions. For simplicity, we consider three primary zones, as indicated in Figure 15.5 (with the understanding that there are transition zones in between). The top, or clarifying, zone contains relatively clear liquid from which most of the particles have settled out. Any particles remaining in this zone will settle by free settling. The middle zone is a region of varying composition through which the particles move by hindered settling. The size of this region and the settling rate depend upon the local solids concentration. The bottom zone is a highly concentrated settled or compressed region containing the settled particles. The particle settling rate in this zone is very slow.

FIGURE 15.5 Schematic of a thickener.

In the top (clarifying) zone, the relatively clear liquid moves upward and overflows the top. In the middle zone, the solid particles settle as the displaced liquid moves upward, and both the local solids concentration and the settling velocity vary from point to point. In the bottom (compressed) zone, the solids and liquid both move downward at a rate that is determined mainly by the underflow draw-off rate from the thickener. For a given feed rate and solids loading, the objective is to determine the area of the thickener and the optimum underflow (draw-off) rate to achieve a specified underflow concentration (φ_u), or the underflow rate and underflow concentration for a stable steady-state operation.

The solids concentration can be expressed in terms of either the solids volume fraction (φ) or the mass ratio of solids to fluid (R). If φ_f is the volume fraction of solids in the feed stream (flow rate Q_f) and φ_u is the volume fraction of solids in the underflow (flow rate Q_u), then the solids ratio in the feed, R_f = [(mass of solids)/(mass of fluid)]_feed, and in the underflow, R_u = [(mass of solids)/(mass of liquid)]_u, are given by

$R_f=\frac{{\text{φ}}_f{\text{ρ}}_s}{\left(1-{\text{φ}}_f\right)\text{ρ}},R_u=\frac{{\text{φ}}_u{\text{ρ}}_s}{\left(1-{\text{φ}}_u\right)\text{ρ}}$

(15.38)

These relations can be rearranged to give the solids volume fractions in terms of the solids ratio:

${\text{φ}}_f=\frac{R_f}{R_f\text{+}\left({\text{ρ}}_s/\text{ρ}\right)},{\text{φ}}_u=\frac{R_u}{R_u\text{+}\left({\text{ρ}}_s/\text{ρ}\right)}$

(15.39)

Now the total (net) flux of the solids plus liquid moving through the thickener at any point is given by

$q=\frac{Q}{A}=q_S+q_L=\text{φ}{V}_S+\left(1-\text{φ}\right)V_L$

(15.40)

where

q_s = φV_S is the local solids flux, defined as the volumetric settling rate of the solids per unit crosssectional area of the settler

q_L = (1 – φ)V_L is the local liquid flux

The solids flux depends upon the local concentration of solids, the settling velocity of the solids at this concentration relative to the liquid, and the net velocity of the liquid. Thus, the local solids flux will vary within the thickener because the concentration of solids increases with depth. The amount of liquid that is displaced (upward) by the solids decreases as the solids concentration increases, thus affecting the “upward drag” on the particles. As these two effects act in opposite directions, there will be some point in the thickener at which the actual solids flux is a minimum. This point determines the conditions for stable steady-state operation, as explained in the following.

The settling behavior of a slurry is normally determined by measuring the velocity of the interface between the top (clear) and middle suspension zones in a batch settling test using a closed system (e.g., a graduated cylinder) as illustrated in Figure 15.3. A typical batch settling curve is shown in Figure 15.6 (e.g., Foust et al., 1980). The initial linear portion of this curve corresponds to free (unhindered) settling, and the slope of this region is the free settling velocity, V_o. The nonlinear region of the curve corresponds to hindered settling in which the solids flux depends upon the local solids concentration, which can be determined from the batch settling curve, as follows (Kynch, 1952).

If the initial height of the suspension with a solids fraction of φ_o is Z_o, at some later time the height of the interface between the clear layer and the hindered settling zone will be Z(t), where the average solids fraction in this zone is φ(t). Since the total amount of solids in the system is constant, and assuming the amount of solids in the clear layer to be negligible, it follows that

$Z\left(t\right)\text{φ}\left(t\right)=Z_o{\text{φ}}_o\text{or}\text{φ}\left(t\right)=\frac{{\text{φ}}_o{Z}_o}{Z\left(t\right)}$

(15.41)

Thus, given the initial height and concentration (Z_o, φ_o), the average solids concentration φ(t) corresponding to any point on the Z(t) curve can be determined using Equation 15.41. Furthermore, the hindered settling velocity and batch solids flux at this point can be determined from the slope of the curve at that point, that is, V_Sb = −(dZ/dt) and q_sb = φV_Sb. Thus, the batch settling curve can be converted to a batch flux curve, as shown in Figure 15.7. The batch flux curve exhibits a maximum and a minimum, since the settling velocity is nearly constant in the free settling region and the flux is directly proportional to the solids concentration, whereas the settling velocity and the flux drop rapidly with the increasing solids concentration in the hindered settling region, as explained above. However, the solids flux in the bottom (compressed) zone is much higher because of the high concentration of solids in this zone. The minimum in this curve represents a “pinch” or “critical” condition in the thickener, which limits the total solids flux that can be obtained under steady (stable) operation.

FIGURE 15.6 Typical batch settling curve for a limestone slurry.

FIGURE 15.7 Typical batch flux curve with operating lines.

Because the batch flux data are obtained in a closed system with no outflow, the net solids flux is zero in the batch system and Equation 15.40 reduces to V_L = −φV_S/(1 – φ). Note that V_L and V_S are of opposite sign, since the displaced liquid moves upward as the solids settle downward. The relative velocity between the solids and liquid is V_r = V_S – V_L, which, from Equation 15.20, is V_r = V_S/(1 – φ). It is this relative velocity that controls the dynamics in the thickener. If the underflow draw-off rate from the thickener is Q_u, the additional solids flux in the thickener due to superimposition of this underflow is q_u = Q_u/A = V_u. Thus, the total solids flux at any point in the thickener (q_s) is equal to the settling flux relative to the suspension (i.e., the batch flux q_sb) at that point, plus the bulk flux due to the underflow draw-off rate, φV_u, that is, q_S = q_sb + φq_u. Furthermore, at steady state, the net local solids flux in the settling zone (q_S) must be equal to that in the underflow, that is, q_S = q_uφ_u. Eliminating q_u and rearranging leads to

$q_{sb}=q_S\left(1-\frac{\text{φ}}{{\text{φ}}_u}\right)$

(15.42)

This equation represents a straight line on the batch flux curve (q_sb vs. φ) that passes through the points (q_S, 0) and (0, φ_u), that is, the line intersects the φ axis at φ_u and the q_sb axis at q_S, which is the net local solids flux in the thickener at the point where the solids fraction is φ. This line is called the “operating line” for the thickener, and its intersection with the batch flux curve determines the stable operating point for the thickener, as shown in Figure 15.7. The “properly loaded” operating line is tangent to the batch flux curve (point A). At the tangent point, called the critical (or “pinch”) point, the local solids flux corresponds to the steady-state value at which the net critical (minimum) settling rate in the thickener equals the total underflow solids rate. The “underloaded” line represents a condition for which the underflow draw-off rate is higher than the critical settling rate, so that no solids layer can build up and excess clear liquid will eventually be drawn out of the bottom (i.e., the draw-off rate is too high). The “overloaded” line represents the condition at which the underflow draw-off rate is lower than the critical settling rate, so that the bottom solids layer will build up and eventually rise to the overflow (i.e., the underflow rate is too low).

Once the operating line is set, the equations that govern the thickener operation are determined from a solids mass balance as follows. At steady-state (stable) operating conditions, the net solids flux is

$q_S=\frac{Q_s}{A}=\frac{Q_f{\text{φ}}_f}{A}=\frac{Q_u{\text{φ}}_u}{A}$

(15.43)

This equation relates the thickener area (A) and the feed rate and loading (Q_f, φ_f) to the solids underflow rate (Q_u) and the underflow loading (φ_u), assuming that there are no solids in the overflow.

The area of a thickener required for a specified underflow loading can be determined as follows. For a given underflow solids loading (φ_u), the operating line is drawn on the batch flux curve from φ_u on the φ axis tangent to the batch flux curve at the critical point, (q_c, φ_c). The intersection of this line with the vertical axis (φ = 0) gives the local solids flux (q_S) in the thickener, which results in stable or steady (properly loaded) conditions. This value is determined from the intersection of the operating line on the q_sb axis or from the equation of the operating line that is tangent to the critical point (q_c, φ_c):

$q_S=\frac{q_c}{1-{\text{φ}}_c/{\text{φ}}_u}$

(15.44)

If the feed rate (Q_f) and solids loading (φ_f) are specified, the thickener area A is determined from Equation 15.43. If it is assumed that none of the solids are carried over with the overflow, the overflow rate Q_o is given by

$Q_o=Q_f\left(1-{\text{φ}}_f\right)-Q_f{\text{φ}}_f\frac{\left(1-{\text{φ}}_u\right)}{{\text{φ}}_u}$

(15.45)

or

$\frac{Q_o}{Q_f}=1-\frac{{\text{φ}}_f}{{\text{φ}}_u}$

(15.46)

Likewise, the underflow rate Q_u is given by

$Q_u=Q_f-Q_o=Q_f-Q_f\left(1-\frac{{\text{φ}}_f}{{\text{φ}}_u}\right)$

(15.47)

or

$\frac{Q_u}{Q_f}=\frac{{\text{φ}}_f}{{\text{φ}}_u}$

(15.48)

SUMMARY

The following are some of the major points covered in this chapter:

•  The principles governing the fluidization of solid particles, the minimum fluidization velocity, and the minimum bed voidage for both spherical and nonspherical particles

•  The difference in the settling characteristics of suspensions of fine particles and coarse particles, and the method of prediction of the settling rate in both cases

•  The use of the Archimedes number in predicting the settling velocity of particle “swarms” and the similarity to the fluidization velocity of the “swarm”

•  The use of batch settling curves and batch flux curves to determine the properly loaded stable operating point for a thickener

PROBLEMS

1.    Calculate the flow rate of air (in scfm) required to fluidize a bed of sand (SG = 2.4), if the air exits the bed at 1 atm, 70°F. The sand grains have an equivalent diameter of 500 μm and the bed is 2 ft in diameter and 1 ft deep, with a porosity of 0.35. What flow rate of air would be required to blow the sand away?

2.    Calculate the flow rate of water (in gpm) required to fluidize a bed of 1/16 in. diameter lead shot (SG = 11.3). The bed is 1 ft in diameter, 1 ft deep, and has a porosity of 0.38. What water flow rate would be required to sweep the bed away?

3.  Calculate the range of water velocities that will fluidize a bed of glass spheres (SG = 2.1) if the sphere diameter is (a) 2 mm, (b) 1 mm, and (c) 0.1 mm.

4.    A coal gasification reactor operates with particles of 500 μm diameter and a density of 1.4 g/cm³. The gas may be assumed to have properties of air at 1000°F and 30 atm. Determine the range of superficial gas velocity over which the bed is in a fluidized state.

5.    A bed of coal particles, 2 ft in diameter and 6 ft deep, is fluidized using a hydrocarbon liquid with a viscosity 15 cP and a density of 0.9 g/cm³. The coal particles have a density of 1.4 g/cm³ and an equivalent spherical diameter of 1/8 in. If the bed porosity is 0.4,

(a)    Determine the range of liquid superficial velocities over which the bed is fluidized.

(b)  Repeat the problem using the “particle swarm” (Barnea and Mizrahi) “swarm terminal velocity” approach, assuming (1) φ = 1 – ε; (2) φ = 1 – ε_mf.

6.    A catalyst having spherical particles with d_p = 50 μm and ρ_s = 1.65 g/cm³ is used to contact a hydrocarbon vapor in a fluidized reactor at 900°F, 1 atm. At operating conditions, the fluid viscosity is 0.02 cP and its density is 0.21 lb_m/ft³. Determine the range of fluidized bed operation, that is, calculate

(a)  Minimum fluidization velocity for ε_mf = 0.42

(b)  The particle terminal velocity

7.    A fluid bed reactor contains catalyst particles with a mean diameter of 500 μm and a density of 2.5 g/cm³. The reactor feed has properties equivalent to 35° API distillate at 400°F. Determine the range of superficial velocities over which the bed will be in a fluidized state.

8.    Water is pumped upward through a bed of 1 mm diameter iron oxide particles (SG = 5.3). If the bed porosity is 0.45, over what range of superficial water velocity will the bed be fluidized?

9.    A fluidized bed combustor is 2 m in diameter and is fed with air at 250°F, 10 psig, at a rate of 2000 scfm. The coal has a density of 1.6 g/cm³, and a shape factor of 0.85. The flue gas from the combustor has an average MW of 35 and leaves the combustor at a rate of 2100 scfm at 2500°F and 1 atm. What is the size range of the coal particles that can be fluidized in this system.

10.  A fluid bed incinerator, 3 m in diameter and 0.56 m high, operates at 850°C using a sand bed. The sand density is 2.5 g/cm³, and the average sand grain has a mass of 0.16 mg and a sphericity of 0.85. In the stationary (packed) state, the bed porosity is 35%. Find

(a)  The range of air velocities that will fluidize the bed.

(b)  The compressor power required, if the bed is operated at 10 times the minimum fluidizing velocity and the compressor efficiency is 70%. The compressor takes air in from the atmosphere at 20°C, and the gases leave the bed at 1 atm.

11.  Determine the range of flow rates (in gpm) that will fluidize a bed of 1 mm cubic silica particles (SG = 2.5) with water. The bed is 10 in. in diameter, 15 in. deep.

12.  Determine the range of velocities over which a bed of granite particles (SG = 3.5, a_s = 0.012 μm⁻¹, ψ = 0.8, d =0.6/a_s would be fluidized using the following fluids:

(a)  Water at 70°F

(b)  Air at 70°F and 20 psig

13.  Calculate the velocity of water that would be required to fluidize spherical particles with SG = 1.6 and a diameter of 1.5 mm, in a tube with a diameter of 10 mm. Also, determine the water velocity that would sweep the particles out of the tube. Use each of the two following methods, and compare the results:

(a)  The bed starts as a packed bed and is fluidized when the pressure drop due to friction through the bed balances the weight of the bed.

(b)  The bed is considered to be a “swarm” of particles, falling at the terminal velocity of the “swarm.” (Assume ε = 0.45.)

Comment on any uncertainties or limitations in your results.

14.  You want to fluidize a bed of solid particles using water. The particles are cubical, with a length on each side of 1/8 in., and a SG of 1.2.

(a)  What is the sphericity factor for these particles, and their equivalent diameter?

(b)  What is the approximate bed porosity at the point of fluidization of the bed?

(c)  What velocity of water would be required to fluidize the bed?

(d)  What velocity of water would sweep the particles out of the bed?

15.  Solid particles with a density of 1.4 g/cm³ and a diameter of 0.01 cm are fed from a hopper into a line where they are mixed with water, which is draining by gravity from an open tank, to form a slurry having 0.4 lb_m of solids/lb_m of water. The slurry is transported by a centrifugal pump, through a 6 in. sch 40 pipeline that is 0.5 mile long, at a rate of 1000 gpm. The slurry can be described as a Bingham plastic, with a yield stress of 120 dyn/cm² and a limiting viscosity of 50 cP.

(a)  If the pipeline is at 60°F, and the pump is 60% efficient with a required NPSH (net positive suction head) of 15 ft, what horsepower motor would be required to drive the pump?

(b)  If the pump is 6 ft below the bottom of the water storage tank, and the water in the line upstream of the pump is at 90°C (P_v = 526 mmHg), what depth of water in the tank would be required to prevent the pump from cavitating?

(c)  A venturi meter is installed in the line to measure the slurry flow rate. If the maximum pressure drop reading for the venturi is 29 in. of water, what diameter should the venturi throat be?

(d)  The slurry is discharged from the pipeline to a settling tank, where it is desired to concentrate the slurry to 1 lb_m of solids/lb_m of water (in the underflow). Determine the required diameter of the settling tank, and the volumetric flow rates of the overflow (Q_o) and underflow (Q_u), in gpm.

(e)  If the slurry were to be sent to a rotary drum filter instead, to remove all of the solids, determine the required size of the drum (assuming the drum length and diameter are equal). The drum rotates at 3 rpm, with 25% of its surface submerged in the slurry, and operates at a vacuum of 20 in. of mercury. Lab test data taken on the slurry with 0.5 ft² of the filter medium, at a constant flow rate of 3 gpm, indicated a pressure drop of 1.5 psi after 1 min of filtration and 2.3 psi after 2 min of operation.

16.  A sludge is clarified in a thickener, which is 50 ft in diameter. The sludge contains 35% solids by volume (SG = 1.8) in water, with an average particle size of 25 μm. The sludge is pumped into the center of the tank, where the solids are allowed to settle and the clarified liquid overflows the top. Estimate the maximum flow rate of the sludge (in gpm) that this thickener can handle. Assume that the solids are uniformly distributed across the tank and that all particle motion is vertical.

17.  In a batch thickener, an aqueous sludge containing 35% by volume of solids (SG = 1.6) with an average particle size of 50 μm is allowed to settle. The sludge is fed to the settler at a rate of 1000 gpm, and the clear liquid overflows the top. Estimate the minimum tank diameter required for this separation.

18.  Ground coal is slurried with water in a pit, and the slurry is pumped out of the pit at a rate of 500 gpm with a centrifugal pump and into a classifier. The classifier inlet is 50 ft above the slurry level in the pit. The piping system consists of an equivalent length of 350 ft of 5 in. sch 40 pipe and discharges into the classifier at 2 psig. The slurry may be assumed to be a Newtonian fluid, with a viscosity of 30 cP, a density of 75 lb_m/ft³, and a vapor pressure of 30 mmHg. The solid coal has a SG = 1.5.

(a)  How much power would be required to pump the slurry?

(b)  Using the pump characteristic charts in Appendix H, select the best one of these for this job. Specify the pump size, motor speed (rpm), and impeller diameter that you would use. Also determine the pump efficiency and NPSH requirement.

(c)  What is the maximum height above the level of the slurry in the pit that the pump could be located without cavitating?

(d)  A venturi meter is located in a vertical section of the line to monitor the slurry flow rate. The meter has a 4 in. diameter throat, and the pressure taps are 1 ft apart. If a DP cell (transducer) is used to measure the pressure difference between the taps, what would it read (in inches of water)?

(e)  A 90° flanged elbow is located in the line at a point where the pressure (upstream of the elbow) is 10 psig. What are the forces transmitted to the pipe by the elbow from the fluid inside the elbow (neglect the weight of the fluid)?

(f)  The classifier consists of three collection tanks in series that are full of water. The slurry enters at the top on the side of the first tank and leaves at the top on the opposite side, which is 5 ft from the entrance. The solids settle into the tank as the slurry flows into it and then overflows into the next tank. The space through which the slurry flows above the tank is 2 ft wide and 3 ft high. All particles for which the settling time in the space above the collection tank is less than the residence time of the fluid flowing in the space over the collection tank will be trapped in that tank. Determine the diameter of the largest particle that will not settle into each of the three collection tanks. Assume that the particles are equivalent spheres and that they fall at their terminal velocity.

(g)  The suspension leaving the classifier is transferred to a rotary drum filter to remove the remaining solids. The drum operates at a constant pressure difference of 5 psi and rotates at a rate of 2 rpm with 20% of the surface submerged. Lab tests on a sample of the suspension through the same filter medium were conducted at a constant flow rate of 1 gpm through 0.25 ft² of the medium. It was found that the pressure drop increased to 2.5 psi after 10 min, and the resistance of the medium was negligible. How much filter area would be required to filter the liquid?

19.  You want to concentrate a slurry from 5% (by vol.) solids to 30% (by vol.) in a thickener. The solids density is 200 lb_m/ft³ and that of the liquid is 62.4 lb_m/ft³. A batch settling test was run on the slurry, and the analysis of the tests yielded the following information:

φ (Vol. Fraction Solids)	Settling Rate (lbm/h ft2)
0.05	73.6
0.075	82.6
0.1	79.8
0.125	70.7
0.15	66
0.2	78
0.25	120
0.3	200

If the feed flow rate of the slurry is 500 gpm, what should the cross-sectional area of the thickener tank be? What are the overflow and underflow rates?

20.  You must determine the maximum feed rate that a thickener can handle to concentrate a waste suspension from 5% solids by volume to 40% solids by volume. The thickener has a diameter of 40 ft. A batch flux test in the laboratory for the settled height versus time was analyzed to give the following data for the solids flux versus solids volume fraction. Determine (a) the proper feed rate of liquid in gpm, (b) the overflow liquid rate in gpm, and (c) the underflow liquid rate in gpm.

φ (Solids Volume Fraction)	Solids Flux (ft3/h ft2)
0.03	0.15
0.05	0.38
0.075	0.46
0.10	0.40
0.13	0.33
0.15	0.31
0.20	0.38
0.25	0.60
0.30	0.80

NOTATION

A	Area, [L2]
Ap	Surface area of particle, [L2]
As	Surface area of equal volume sphere, [L2]
as	Particle surface area/volume, [1/L]
C1, C2	Constants, Equation 15.9, [—]
CD	Drag coefficient, [—]
CDO	Single-particle drag coefficient, [—]
CDφ	Swarm drag coefficient, [—]
D	Container diameter, [L]
d	Particle diameter, [L]
dp	Particle effective diameter, [L]
ds	Equal volume sphere diameter, [L]
ef	Energy dissipated per unit mass of fluid, [F L/M = L2/t2]
fPM	Porous media friction factor, [—]
g	Acceleration due to gravity, [L/t2]
h	Height of bed, [L]
NAr	Archimedes number, Equation 15.32, [—]
NArφ	Swarm Archimedes number, Equation 15.37, [—]
NRep	Unconfined single-particle Reynolds number, [—]
${\widehat{N}}_{Re}$	Reynolds number defined by Equation 15.7, [—]
NReφ	Swarm Reynolds number, Equation 15.33, [—]
NRe,PM	Porous media Reynolds number, Equation 15.3, [—]
n	Richardson–Zaki index, Equation 15.23, [—]
P	Pressure, [F/L2 = M/(L t2)]
Qf	Slurry feed rate, [L3/t]
Qo	Overflow rate, [L3/t]
Qu	Solids underflow rate (thickener), [L3/t]
qs	Solids flux, [L/t]
t	Time, [t]
V	Velocity, [L/t]
Vo	Stokes velocity, [L/t]
Vp	Particle volume, [L3]
Vr	Relative velocity between solid and fluid, Equation 15.20, [L/t]
Z(t)	Instantaneous height of liquid/suspension interface, [L]

GREEK

Δ()	()2 – ()1
ε	Porosity or void fraction, [—]
μ	Viscosity, [M/(L t)]
φ	Volume fraction of solids, [—]
ρ	Density, [M/L3]
ψ	Sphericity factor, [—]

SUBSCRIPTS

c	Critical point
i	Inlet
s	Superficial
f	Fluid, feed
L	Liquid
mf	Minimum fluidization condition
o	Infinitely dilute condition, overflow
p	Particle
s	Solid
u	Underflow
φ	Solid suspension of volume fraction φ

REFERENCES

Azbel, D.S. and N.P. Cheremisinoff, Fluid Mechanics and Unit Operations, Ann Arbor Science, Ann Arbor, MI, 1983.

Barnea, E. and J. Mizrahi, A generalized approach to the fluid dynamics of particulate systems, Part I, General correlation for fluidization and sedimentation in solid multi-particle systems, Chem. Eng. J., 5, 171–189, 1973.

Chhabra, R.P., Rheology: From simple fluids to complex suspensions, in Lignocellulosic Fibers and Wood Handbook, N. Belgacem and A. Pizzi, (Eds.), pp. 407–438, Scrivener, New York, 2016.

Christian, J.B., Improve clarifier and thickener design and operation, Chem. Eng. Prog., 90(7), 50–56, 1994.

Coulson, J.M., J.F. Richardson, J.R. Blackhurst, and J.H. Harker, Chemical Engineering, Vol. 2, 5th edn., Butterworth-Heinemann, Oxford, U.K., 2002.

Darby, R., Hydrodynamics of slurries and suspensions, Chapter 2, in Encyclopedia of Fluid Mechanics, Vol. 5, N.P. Cheremisinoff, (Ed.), Gulf, Houston, TX, 1986, pp. 49–92.

Davies, L., D. Dollimore, and G.B. McBride, Sedimentation of suspensions: simple methods of calculating sedimentation parameters, Powder Technol., 16, 45–49, 1977.

Einstein, A., Eine neue Bestimmung der Molekul-dimensionen, Ann. Phys., 19, 289–306, 1906.

Foust, A.S., L.A. Wenzel, C.W. Clump, L. Maus, and L.B. Anderson, Principles of Unit Operations, 2nd edn., Wiley, New York, 1980.

Kynch, G.J., A theory of sedimentation, Trans. Faraday Soc., 48, 166, 1952.

McCabe, W.L., J.C. Smith, and P. Harriott, Unit Operations of Chemical Engineering, 5th edn., McGraw-Hill, New York, 1993.

Mewis, J. and N.J. Wagner, Colloid Suspension Rheology, Cambridge University Press, New York, 2012.

Millan, J.M.V., Fluidization of Fine Powders: Cohesive versus Dynamical Aggregation, Springer, New York, 2013.

Mooney, M., The viscosity of a concentrated suspension of spherical particles, J. Colloid Sci., 6, 162–170, 1951.

Richardson, J.F. and W.N. Zaki, Sedimentation and fluidization, Trans. Inst. Chem. Engrs., 32, 35, 1954.

Shook, C.A. and M.C. Roco, Slurry Flow: Principles and Applications, Butterworth-Heinemann, Oxford, U.K., 1991.

Steinour, H.H., Rate of sedimentation, Ind. Eng. Chem., 36, 618, 840, 901, 1944.

Tiller, F.M. and D. Tarng, Try deep thickeners and clarifiers, Chem. Eng. Prog., 91, 75–80, March 1995.

Vand, V., Viscosity of solutions and suspensions, J. Phys. Colloid Chem., 52, 277–299, 1948.

Wilson, K.C., G.R. Addie, A. Sellgren, and R. Clift, Slurry Transport Using Centrifugal Pumps, 3rd edn., Springer, New York, 2008.

Yang, W.-C., Handbook of Fluidization and Fluid-Particle Systems, Taylor & Francis, New York, 2003.