Extending estimation of the critical deposition velocity in solid–liquid pipe flow to ideal and non-ideal particles at low and intermediate solid volume fractions​

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Chemical Engineering Science

journal homepage: www.elsevier.com/locate/ces

Chemical Engineering Science 211 (2020) 115308

Extending estimation of the critical deposition velocity in solid–liquid pipe flow to ideal and non-ideal particles at low and intermediate solid volume fractions

Hugh P. Rice a,1,, Jeffrey Peakall b, Michael Fairweather a, Timothy N. Hunter a

a School of Chemical and Process Engineering, University of Leeds, Leeds LS2 9JT, United Kingdom‌‌

b School of Earth and Environment, University of Leeds, Leeds LS2 9JT, United Kingdom

h i g h l i g h t s

  • Critical deposition velocity separates suspended and bed-forming particle-laden flows.
  • Data from new experiments using acoustic method combined with data from literature.
  • Relationship found between critical deposition velocity and material/flow properties.
  • Volume factor accounts for non-ideal behaviour of solid particles such as aggregation.
  • Correlation found between volume factor and packing fraction of solid phase.

a r t i c l e i n f o

Article history:

Received 24 May 2019

Received in revised form 3 October 2019 Accepted 18 October 2019

Available online 21 October 2019


Particle-laden flow Acoustic measurements Ultrasonics

Sediment transport Radioactive waste

a b s t r a c t

The critical deposition velocity in horizontal pipe flow of liquid-solid slurries separates bed-forming and fully suspended flows. A compilation of critical deposition velocity data is presented using new experi- mental data (for particles ranging from 9 to 690 mm in diameter) along with data from the literature, and a close correlation between the particle Reynolds number and the Archimedes number (which describe the properties of the flow and the liquid and solid phases) is found. The role of solid particle packing is discussed and suggestions are made for the incorporation of solid-phase material properties – specifically particle shape and angularity, and surface forces – into an empirical parameter, the volume

factor, a, to account for the deviation of particle behaviour from ideal, non-interacting, hard-sphere


© 2019 Elsevier Ltd. All rights reserved.

  1. IntroductionSolid-liquid slurry flows can be categorized into fully suspended and homogeneous (or pseudo-homogeneous); heterogeneous, in which a solids concentration gradient exists; flow with moving bed (or saltation flow or ‘‘two-layer” flow); flow with a stationary bed (sometimes ‘‘three-layer” flow) (Peysson et al., 2009; Rice et al., 2017); and plug flow, in which the solid span the conduit and move en masse (Crowe, 2006; Doron and Barnea, 1995; Wasp et al., 1977). Each of these flow regimes, which are illustrated in idealised form elsewhere (Doron and Barnea, 1996; Gillies and Shook, 1991; Rice et al., 2015), can be distinguished by a critical* Corresponding author.E-mail address: h.p.rice@leeds.ac.uk (H.P. Rice).1 Current address: School of Mechanical Engineering, University of Leeds, Leeds LS2 9JT, United Kingdom.velocity or flow rate, which are essential flow parameters for oper- ators working with high-value or hazardous substances in the food, nuclear and minerals processing industries, for example (Bux et al., 2017; Poloski et al., 2010; Thomas, 1961,1962) because of the asso- ciated pumping and energy requirements, the possibility of block- ages and, in the case of stationary deposits of chemically or radiologically active materials, the increased risk of corrosion, heat deposition and elevated radiation dose to operators.These critical velocities, such as the critical deposition velocity (CDV), which delineates bed-forming and fully suspended flows of liquid–solid mixtures, slurries and sludges, have received exten- sive theoretical and practical attention. They are defined in a vari- ety of ways, such as the pick-up velocity, saltation velocity, critical velocity and suspending velocity (Bain and Bonnington, 1970; Crowe, 2006; Peker and Helvacı, 2007; Rabinovich and Kalman, 2011; Spells, 1955) and some ambiguity over the relationship between the various definitions persists.https://doi.org/10.1016/j.ces.2019.1153080009-2509/© 2019 Elsevier Ltd. All rights reserved.A similarly large range of experimental and numerical methods, theoretical models and empirical correlations exist for measuring and predicting such transitional velocities (Gillies et al., 1991; Oroskar and Turian, 1980; Soepyan et al., 2014; Turian et al., 1987), e.g., minimum pressure drop with respect to flow rate (Goedde, 1978), acoustic bed depth measurement (Rice et al.,‌
  2. Theoretical background0.41In a review of a large number of data in the dilute limit (specif- ically 0 < / < 10-4, where / is the solids volume fraction), Soepyan et al. (2014) found that a large number of models, when optimized, could be reduced to the following simple form:2015), visual assessment of deposition (Thomas, 1961), and the onset of motion of individual particles (Clark et al., 2015). How-Repc0= 7.90Ar , (1)ever, the predictions of such models often disagree with each other and with experimental results (Al-lababidi et al., 2012; Miedema, 2016), and this lack of clarity gives operators less predictive cer- tainty in terms of flow assurance and safety: using too low a pump rate encourages solids deposition and increases the risk of plug- ging, whereas too high a flow rate increases energy costs and the risk of wear in conduits and pumping equipment.The UK and USA have large and complex nuclear legacy waste inventories that are physically, chemically and radiologically diverse. In the UK, the majority is stored at the Sellafield site, whereas in the USA the waste is more widely distributed; however, the Hanford Site, Washington, is the most contaminated. On both sites, an ongoing challenge exists to transport, store, process and dispose of the waste inventories safely and economically. A num- ber of reviews of models and correlations for predicting critical deposition velocities – as they pertain to nuclear waste slurries at the Hanford Site specifically – are available. Welch (2001) reviewed the most significant engineering and scientific obstacles present at Hanford and investigated the suitability of the Oroskar and Turian (1980) CDV correlation. Liddell and Burnett (2000) reviewed several correlations and recommended those of Gillies‌‌where Repc is the particle Reynolds number, with the subscript ‘‘0” indicating the low-/ limit, and Ar is the Archimedes number. The particle Reynolds can be given by:mRepc = Ucd , (2)where Uc is the CDV, d is the particle diameter and m is the kine- matic viscosity of the fluid. The Archimedes number, Ar, is given by:( — )gd3 s 1Ar = m2 , (3)where g is the acceleration due to gravity, s is the ratio of solid to liquid densities, i.e., s = qs/ql. Rice et al. (2015) presented results from an experimental study and a compilation of data from the lit- erature – selected according to strict criteria, which are discussed and revised later – and derived a relationship between Repc and Ar that extended beyond the dilute limit, of the following form:Repc = aArb 1 + a/0.5 (4)where a is referred to hereafter as the volume factor. In the dilute limit, the value of Re corresponding to pick-up, Re , is obtainedand Shook (1991) and Oroskar and Turian (1980) for slurry flow pcpc0prediction at Hanford. More recently, Poloski et al. (2010) described the difficulty in representing slurries with complex com- positions and presented a correlation for the CDV with two regimes separated by particle material properties. However, it is noted that both the Gillies and Shook (1991) and Oroskar and Turian (1980) correlations do not capture the correct behaviour at low solids loadings: both predict a CDV of zero in this limit, whereas in reality a non-zero value is obtained, i.e., for individual solid particles (Rice et al., 2015), referred to variously as the equilibrium velocity, pick- up velocity and saltation velocity, depending on the exact mecha- nism, by Soepyan et al. (2014), for example.Rice et al. (2015) presented an unambiguous acoustic bed- depth measurement method for determining the CDV in horizon- tal, solid-liquid, cylindrical-pipe flow, and synthesised new experimental data with others from the literature, using strict data selection criteria described later, to establish a correlation between flow parameters, material properties and the CDV. It was shown that the correlation extended beyond the very dilute limit (Soepyan et al., 2014) and that no existing mathematical form of the CDV could account for the experimental data in the literature. Key questions remain as to the universality of CDV correlations for different systems, which is the focus of the present study. In partic- ular, the CDV is measured for a wide range of particle sizes and densities, with three different species (glass, plastic and dense bar- ium sulphate) that cover a large range of particle sizes and densi- ties, using a high-resolution acoustic backscatter technique to characterise the consolidated sediment beds in a horizontal pipe loop. Importantly, the concentration ranges over which the CDV is measured is significantly increased over previous studies to clar- ify the range of validity of the empirical relationship previously determined by the current authors (Rice et al., 2015). In the follow- ing section (Sec. 2), the relationship between the CDV and material and flow parameters is reviewed and the objectives of the study are described, in terms of the range of solids loadings investigated, the data selection criteria employed and the effect of particle prop- erties not accounted for through the mean particle diameter.such that Repc0 = aArb in general. Rice et al. (2015) found the follow- ing correlation best fitted the available datasets up to volume frac- tions of several per cent:Repc = 12.4Ar0.493 1 + 8.91/0.5 (5)The aims of this study are: (a) to investigate whether the valid- ity of the /0.5 term in Eq. (4) extends beyond volume fractions of a few per cent; (b) to present new experimental data at higher vol- ume fractions; and (c) to supplement with additional datasets the 11 identified by Rice et al. (2015). The strict data selection cri- teria used by Rice et al. (2015) are also reviewed and relaxed. Lastly, the relationship between an empirical parameter that incor- porates the deviation of solid materials’ behaviour from ideal- sphere, the volume factor, a, and the maximum packing fraction of ideal and non-ideal particulate species is discussed in terms its influence on the CDV correlation.
  3. Experimental methodology and data selction
    1. Materials characterisationFive solid particle species were used in the experiments: two glass species, two plastic species and barium sulphate, commonly known as barytes (sources: all Guyson International, Ltd., UK, except for barytes: RBH Ltd., UK). The glass particles are spherical in shape, while the plastic and barytes have a roughened morphol- ogy with non-spherical aspect ratios (see micrograph images of barytes in Bux et al., 2017; glass and plastic in supplementary material and Rice, 2013). The glass and plastic are low-cost, engi- neered blast media, while the barytes is a milled mineral and a common simulant for fine nuclear fission wastes (Bux et al., 2017). Particle size distributions were measured with Malvern Instruments Mastersizer 2000E and Mastersizer 3000 laser diffrac- tion sizers and the densities with a Micromeritics AccuPyc 1300 pycnometer. In addition, the Corey shape factor, Fs (Corey, 1949; Dietrich, 1982) and Powers roundness factor, P (Powers, 1953;‌Syvitski, 2007) have been measured previously for the large plastic species (Rice et al., 2017) with a Retsch Camsizer XT optical sizing instrument and were found to be Fs = 0.842 (where 0 corresponds to a rod and 1 to a sphere) and P = 2.49 (i.e., sub-angular, where 0 corresponds to very angular and 6 to well rounded). Particle size distributions for the five species used in this study are shown in Fig. 1(a), along with log-normal fits to the measured data in Fig. 1(b); it is also noted that triangles, squares and circles are the symbols used to denote barytes, glass and plastic particles, respectively, in figures throughout this paper.
      The packing fraction, /m, of the particle species were deter- mined in two ways. The first, used for the glass and plastic species, was with dry particles in volumetric flasks of at least three sizes (50, 100, 250 and 500 ml) for each species, and no trend with flask size was found, confirming wall effects were not significant, as described by Rice et al. (2015). The second method, used for bary- tes, was to measure the settled bed depth in wet samples taken from the mixing tank of the pipe flow loop (see Section 3), and no variation in packing fraction was found with changes to nomi- nal volume fraction or sample mass. For non-interacting species like glass and plastic, the first and second method should yield the same result; for surface-charged species like barytes, the result from each method may differ due to the effects of the fluid phase on interparticle interactions and so the second method was used for barytes, as it better represents the conditions in the flow experiments.
      Fig. 1. (a) Particle size distributions by volume of species used in this study, measured with the Mastersizer instrument; dashed lines for legibility. (b) Cumu- lative distribution function (CDF) of particle size, with log-normal fits shown as dashed lines. (1) Barytes, (2) small glass, (3) large glass, (4) small plastic and (5) large plastic.
  4. Literature data selection and analysisA total of 14 datasets were compiled: five from the present experimental study and nine from the literature, for which strict selection criteria were applied (see below). For the current study, additional data for the glass and plastic systems from a previous study by Rice et al. (2015) at low volume fractions (/ < 2%) are included in the analysis that follows. As discussed, barytes was chosen for its utility as a nuclear waste simulant (Hunter et al., 2013; Paul et al., 2013) and is an ultrafine, micron-sized material with a surface charge (see Bux et al., 2017), a property that strongly influences sedimentation and resuspension behaviour.Data from the literature were selected according to strict crite- ria, which were relaxed somewhat compared to those used by Rice et al. (2015). First, at least three data points at several volume frac- tions had to be available to ensure a reliable fit and enable accurate extrapolation to / ? 0. Second, the data had to be selected over a range of / such that Uc shows a /0.5 dependence. Third, at least two particle size data had to be available, so that the distribution is known or can be inferred (e.g., modelled as log-normal). Fourth, the data must be specifically for critical deposition velocity, defined as that at which the solid phase is fully suspended, rather than any other critical velocity. These criteria were formulated in order that all flow and material properties are accounted for.The literature on experimental determination of critical veloci- ties is large – comprising thousands of data (Oroskar and Turian, 1980; Soepyan et al., 2014; Turian et al., 1987) – and mature, with few recent studies, but the majority of datasets did not satisfy the selection criteria given above and the scarcity of high-quality data is one motivation for this study. Of the datasets assessed (and some of the studies cited had both suitable and unsuitable datasets within them), many were rejected based principally on: the first or second criteria, which are similar (Goedde, 1978; Sinclair, 1962; Wasp et al., 1977; Worster and Denny, 1955); the third (Hayden et al., 1971; Parzonka et al., 1981; Spells, 1955); or the fourth, or were in a format that meant they were difficult to inter- pret or could not be converted into SI units (Babcock, 1970; Cairns et al., 1960; Durand and Condolios, 1952; Hughmark, 1961;Murphy et al., 1954; Newitt, 1955; Smith, 1955; Thomas, 1979;Thomas, 1961, 1962; Wilson, 1965).All measured and derived properties of the solids species used in this study are given in Table and Table and Tables S1 and S2 in the supplementary material, where the datasets from the lit- erature are also summarized (Al-lababidi et al., 2012; Graf et al., 1970; Parzonka et al., 1981; Sinclair, 1962), including several that are additional to those used by Rice et al. (2015). Particle size dis- tributions for the five species used in this study are shown in Fig. 1.
  5. Pipe flow loop and bed depth measurement methodThe bed depth measurement method, which includes a correc- tion for ambient suspended solids, is as described in detail by Rice et al. (2015) but is summarized here. First, a solid-liquid suspen- sion with a known volume fraction of solids was circulated in a pipe flow loop (internal diameter D = 42.6 mm) at a high flow rate to minimise deposition throughout the apparatus, then reduced to a low flow rate for several minutes in order that a flat, stationary bed of settled solid particles is established along a transparent, horizontal test section. The flow rate was then increased incremen- tally and the bed depth allowed to equilibrate over several minutes at each flow rate. The pump was then turned off and the solids allowed to settle for seconds or minutes, depending on the mate- rial. The bed depth could then be measured using an acoustic backscatter system (as a more accurate, distinct bed depth measurement is possible with a settled bed than with a movingTable 1Physical and derived properties of particle species used in this study. D = 42.6 mm in all cases.Substance Particle size data (lm) sa Ar




    Small glass, ‘‘Honite 22”






    Large glass, ‘‘Honite 16”






    Small plastic. ‘‘Guyblast 40/60”






    Large plastic, ‘‘Guyblast 30/40”






    Barium sulphate






    a s = qs/ql, where qs and ql are densities of the solid and liquid phases, respectively.b Updated using data at higher / or corrected following review of Rice et al. (2015).Table 2Log-normal, packing fraction and derived properties of species used in present study. M and S are log-normal parameters.








    Small glass







    Large glass







    Small plastic







    Large plastic







    Barium sulphate







    a M and S calculated using particle size data (in mm units) via linearization of Eq. (11).b Updated using data at higher / or corrected following review of Rice et al. (2015).one), and the flow rate was increased until no settled bed was visible.A geometric correction, dh, to the bed depth for ambient solids was applied; this correction accounts for solids that would remain suspended during flow, were the pump to be running, and is as follows:/mcdh = /Aflow (6)where /m is the maximum packing fraction of the solid species, c is the chord length of the bed surface (determined geometrically from the measured bed depth), / is the (known) volume fraction of solids overall and Aflow is the flow cross-sectional (i.e., that above the bed, also determined geometrically from the bed depth).An off-the-shelf ultrasonic system was used to measure bed depth, consisting of a UVP-DUO signal processor (Met-Flow, Switzerland) and a monostatic (i.e., emitter-receiver) transducer operating at 4 MHz (Imasonic, France), giving a spatial resolution of 0.37 mm. The transducer was mounted, perpendicular to the mean flow direction, on the test section. A variable centrifugal pump was used to control the flow rate, an impeller mixer to main- tain a suspension in the mixing tank (nominal capacity 100 L, i.e., 0.1 m3) and an ultrasonic flow meter (Omega Engineering, UK) to measure the flow rate, Q, from which the mean axial flow velocity,U, is calculated such that Q = pUD2/4.
  6. Results and discussionBed depth results for three of the particle species (large glass, large plastic and barytes) are given at three volume fractions in Figs. 2and 4, respectively, where the x-intercept of a linear fit to the corrected bed-depth data gives the CDV, Uc, which is observed to increase with /, for all solid species. CDV data for all particle species are listed in Table S1 in the supplementary mate- rial. It is noted that the volume fraction range accessible to mea- surement with barytes was lower than for either the glass or plastic particles. Indeed, at concentrations above / = 2%, determi- nation of the CDV from bed depth measurements became difficult as settling times became prohibitively long as a result of hindered settling (Richardson and Zaki, 1954; Vesilind, 1968). Very low, concentration-dependent settling velocities have been observedin suspensions of several nuclear-analogues (Paul et al., 2017) and magnesium hydroxide (Johnson et al., 2016).Fig. 5 shows the variation of the CDV, Uc, with solids volume fraction for some examples of particle species: large glass, large plastic and barytes (present study) in Fig. 5(a) and two datasets from the literature (Parzonka et al., 1981; Sinclair, 1962) in Fig. 5(b). The square-root dependence of CDV on solids concentra- tion was suggested by Thomas (1962) and Rice et al. (2015) and remains tentative. However, it is clear from Fig. 5 that Uc varies with /0.5 over at least the range 0 < /0.5 < 0.4 (0 < / < 0.16) or so for most particle species, i.e., far beyond the dilute limit. How- ever, caution must be taken regarding the upper concentration limit for fine particles with significant surface forces, such as the barytes, as discussed earlier. Nevertheless, for most systems studied (including the datasets taken from the literature) the square-root correlation appears valid up to a significant part of the solids volume fraction corresponding to the maximum CDV (as illustrated by Parzonka et al., 1981), which allows for the data selection criterion relating to volume fraction given by Rice et al. (2015) to be relaxed.As described by Rice et al. (2015), the pick-up velocity, Repc0,can be found from the y-intercept of the plot of Uc vs. /0.5, as in Fig. 5. Additionally, the volume factor, a (see the functional rela- tionship of Repc in Eq. (4)) is found from the ratio of the gradient and y-intercept in the limit of / ? 0. The relationship between Repc0 and Ar is then determined by fitting, in order to determine the parameters a and b (such that Repc0 = aArb), and a mean value of a for all particle species is taken to yield the following result for the five species used in this study, given in Fig. 6, where Repc0 vs. Ar is plotted for all datasets; Repc0 and a for all particle species used here are given in Table 2.The final, explicit relationship for Repc (and thus Uc) with expo- nents and coefficients derived from Fig. and Fig. 6 is given in Eq.(7) for the five species investigated in this study, while the corre- sponding relationship for all 14 datasets (i.e., those from both this study and the literature) is given in Eq. (8).Repc = 16.3Ar0.414 1 + 6.73/0.5 (7)Repc = 15.3Ar0.457 1 + 9.04/0.5 (8)
    Fig. 2. Measured bed depth, h, vs. mean axial flow velocity, U, for large glass particles at (a) / = 5%, (b) / = 10% and (c) / = 15%. Open symbols: uncorrected data; filled symbols: corrected data. Correction procedure given by Rice et al. (2015) and summarised in text. Dashed line: linear fit to corrected data; intercept with x-axis gives critical deposition velocity, Uc.
    Fig. 3. Measured bed depth, h, vs. mean axial flow velocity, U, for large plastic particles at (a) / = 5%, (b) / = 10% and (c) / = 15%. Symbols, etc., as Fig. 2.
    Fig. 4. Measured bed depth, h, vs. mean axial flow velocity, U, for barytes at (a) / = 0.5%, (b) / = 1% and (c) / = 2%. Symbols, etc., as Fig. 2.It is noted that the agreement between the limiting cases of Eqs.(7) and (8) (i.e., for / ? 0) with the pick-up velocity relationship given by Soepyan et al. (2014) in terms of the exponent b (0.414, present data; 0.457, all data) is very good (0.41 in Soepyan et al., 2014), but the present results (a = 16.2, present data; 15.3, all data) suggest the Soepyan et al. (2014) relationship (a = 7.90) underesti- mates Repc0 in general. It is important to note the subtle but impor- tant differences between the definitions of pick-up and incipient motion velocities used by Soepyan et al. (2014); however, in the case of zero bed depth, the mechanisms of pick-up and incipient motion are identical.Measured values of Repc and those predicted by Eqs. (7) (present data) and (8) (all data) are shown in frames (a) and (b) of Fig. 7, respectively. Bounds corresponding to ± 30% are indicated in frame(a) and ± 100% in frame (b), demonstrating very good agreement between experiment and prediction. The degree of scatter in the data compares very favourably to that shown in other studiesand reviews (Oroskar and Turian, 1980; Poloski et al., 2010; Turian et al., 1987).As was the case in the study of Rice et al. (2015), the volume factor a, as defined in Eq. (4) and evaluated in Eq. (8), was calcu- lated as a simple mean of the values for all particle species. This method serves not to give bias to any dataset with more points, but may not be ideal because the values of a (see Table 2 for par- ticle species used in the present study and Table S2 in the supple- mentary material for those from the literature) span more than an order of magnitude, and the material properties that are presum- ably responsible for the variation in a – most likely shape, roughness, tendency to aggregate and surface forces, as discussed later – are not accounted for. To address this issue, a correlation was sought between the derived values of a and the measured values of the packing fraction, /m, for the five species used here only (see Table 2; packing fraction information was not available for the datasets taken from the literature). Broadly, as is clear from
    Fig. 6. Critical particle Reynolds number in dilute limit (or pick-up Reynolds number), Repc0, vs. Archimedes number, Ar. Closed diamonds: species used in this study (five); open diamonds: all datasets (14). Solid line: fit to present data. Dashed line: fit to all data. Dashed-dotted (lower) line: pick-up (Repc0) correlation of Soepyan et al. (2014).‌‌‌
    cSquares: large glass, circles: large plastic, triangles: barytes; all present study, offsets for better visualisation indicated in figure. (b) Crosses: unknown materialFig. 5. Critical deposition velocity, U , vs. /0.5, where / is solids volume fraction. (a)by Farr (2013) is used here as it is simpler to implement, taking only a measure of the width of the particle size distribution as an argument, and is as given in Eq. (10):/Farr = 1 — 0.57exp(—S) + 0.2135exp(—0.57S/0.2135)+ 0.0019hcos 2ph1 — exp(—0.75S0.7 — 0.025S4)i — 1i(10)where S, along with M, are the logarithmic standard deviation and mean of a log-normal particle size distribution, respectively, such that the log-normal cumulative distribution function for a particle size d is as given in Eq. (11):2S,2C(d) = 1 1 + erf ln d —ffiffiffiM , (11)(d50 = 90 lm, s = 3.00), series 7, Fig. of Parzonka et al. (1981); pluses: iron in kerosene, from Sinclair (1962). All material data given in Tables 12 and S2 in supplementary material.Eq. (4), a material with a higher value of a will tend to have a higher critical deposition velocity at a given solids concentration, as a modifies the /0.5 term. The results are shown in Fig. and the relationship given in Eq. (9), where the goodness of fit was R2 = 0.843: maximum packing fraction correlates well with volume factor. Conversely, Eq. (9) allows maximum packing fraction, an important parameter for rheological modelling, to be estimated from CDV data alone.a = 0.160exp(6.68/m ). (9)Overall, the volume factor a might be viewed as a measure of the deviation from ideal hard-sphere behaviour, implicitly incor- porating factors such as surface interactions and cohesion/aggrega- tion, particle shape and roughness, for example, none of which are accounted for in the Archimedes number but which strongly influ- ence packing fraction and the tendency to form a space-filling net- work. In order to quantify this deviation from ideal behaviour, it is noted that both Brouwers (2014) and Farr (2013) give expressions for the maximum packing fraction, /m, for non-interacting, hard- sphere particles with a log-normal size distribution. That givenwhere erf is the error function. M and S for the particle species used here are given in Table 2 and were found via Eq. (11) using the siz- ing data (in mm units) given in Table 1.The Farr (2013) model applies to ideal, non-interacting spheres and so represents a theoretical maximum in the packing fraction. Very high packing fractions are possible for species with wide size distributions, ultimately because smaller particles can fit into the interstices between larger ones. In the simple case of a bidisperse species – i.e., two distinct sizes, and a large size ratio of the order of ten so that the particles of the smaller fraction fits fully between the larger (McGeary, 1961) – and assuming each size fraction reaches the random close packing (RCP) limit of /m = 0.64 (Gondret and Petit, 1997; Torquato et al., 2000), the ‘‘ultimate packing fraction” (Sudduth, 1993) in the bidisperse case is/m = 0.870. For a five-fraction species with similarly high size ratios, the figure is 99.4%. McGeary (1961) obtained real packings of up to 95.1% experimentally with metal spheres with quaternary size distributions; very high packings of spheres, cubes and cylin- ders were also modelled and compared with experimental data from the literature by Liu and Ha (2002). In numerical experi- ments, Desmond and Weeks (2014) found that both the skewness and relative width of spheres with linear, Gaussian and log-normal (i.e., continuous) size distributions strongly influenced the packing density.However, packings of real particles will exhibit some degree of deviation from this ideal behaviour and therefore a lower packing
    fraction, whether as a result of (a) physical properties like rough- ness and non-sphericity, for example, or (b) surface forces and interparticle interactions that may cause aggregation and/or a more loosely packed settled bed. In fact, packing fractions as low as 1% have been found with nanoparticle species (Mizuno et al., 1991), in which interparticle forces are very high (Dong et al., 2006).
    It is important also to highlight the influence of particle charac- teristics on measured maximum packing fractions. It is clear from Table 2 that the maximum packing fraction, /m, of the glass species is closest to the ideal hard-sphere value, /Farr, as would be expected as they are near-spherical and smooth (Rice et al., 2014) and therefore close to ideal. The plastic species are, however, rough and non-spherical as described in Section and (Rice et al., 2017) and /m//Farr is lower in those cases. The ratio /m//Farr is lower still for barium sulphate: its broad size distribution (as char- acterised by a high value of S) would yield a high packing fraction if the particles were ideal hard spheres. However, it is known that the barytes has a surface charge (Balastre et al., 1999; Bux et al., 2017) which reduces the potential for particle rearrangement dur- ing gravitational consolidation in a settled bed, resulting in a lower packing fraction in practice (Balastre et al., 2002; Mizuno et al., 1991).So, in suspensions of non-spherical, slightly aggregating or
    surface-charged minerals, the real packing fraction can differ vastly from the ideal hard-sphere packing fraction – calculated, for exam- ple, using the Farr (2013) model – because of the difficulty of surface-charged particle or aggregate network rearrangement dur- ing settling (Balastre et al., 2002; Balastre et al., 1999; Bergström, 1992; Bergström et al., 1992; Michaels and Bolger, 1962). A high value of the volume factor will tend to enhance the CDV for a given solids volume fraction, and ideal-sphere packing fraction estima- tions are not good for interacting or non-spherical particles. For highly aggregated, flocculated or surface-charged systems, the physical significance of a is to increase the effective volume of solids, which incorporates fluid entrained in the particle network structure (Michaels and Bolger, 1962). As such, the rightmost term
    effin Eq. (4), a/0.5 can be rewritten in a more intuitive form as /0.5,
    Fig. 7. Comparison of measured and predicted critical particle Reynolds number, Repc. (a) Data from present study and predicted values from Eq. (7), 26 data, grey dotted lines: ±30%; (b) all data and predicted values from Eq. (8), 64 data; grey dotted lines: ±100%.
    Fig. 8. Volume factor, a, vs. maximum packing fraction, /m, of solid phase for five particle species used in present study (two glass: squares, two plastic: circles, barytes: triangle) with exponential fit (dashed line). Values of a and /m given in Table 2.where /eff = a2/./It is noted that, perhaps counterintuitively, pipe diameter does not appear to be as significant an influence as other factors on CDV. The dependence of CDV on conduit diameter found by other reviews, e.g., Oroskar and Turian (1980) and Turian et al. (1987), is generally weak; Cabrejos and Klinzing (1994) for example, found a Uc D0.25 dependence. In addition, four datasets taken from Graf et al. (1970), as summarized in Tables S2 and S2 in the supplemen- tary material, show only a small variation in Repc between experi- ments in pipes with two diameters. However, the difference in D is small (D = 101.6 and 152.4 mm). The range of pipe diameters in the datasets used here cover only about an order of magnitude; assum- ing a D0.25 dependence, this corresponds to a ± 78% variation in Uc, i.e., within the bounds (±100%) indicated in Fig. 7(b).
  7. Conclusions

New experiments using five types of particles (two glass, two plastic and barytes) and solid concentrations up to 15% by volume have been undertaken, thereby extending a recent review (Soepyan et al., 2014) in the dilute limit and a previous study (Rice et al., 2015) at concentrations up to 5%. The results were inte- grated with data from the literature in order to establish a correla- tion between material and flow properties for prediction of the critical deposition velocity (CDV) in a range of multiphase flows consisting of both ideal and non-ideal solid particles, and a rela- tionship between particle packing fraction and CDV is presented.

With respect to the issue of packing fraction and its influence on CDV, in general, any correlation that is able to predict CDV in a range of flows of industrial interest must be able to account, implicitly or explicitly, for the highly non-ideal behaviour of the majority of particle species for which data exist in the literature. Not only is this information (most importantly, particle size distri- bution, maximum packing fraction particle shape/roundness, ten- dency to aggregate, and more generally, surface properties such as the zeta potential, for example) often not reported by research- ers, it is also difficult to account for theoretically in a model. The suggestion made here is that the maximum packing fraction and the discrepancy between it and the hard-sphere value for a given particle size distribution be used as a measure of, and proxy for, the deviation of a particle species from ideal behaviour. This dis- crepancy is quantified by the volume factor, which provides a valu- able simplification. The empirical volume factor, a, was found to correlate well with measured maximum packing fraction. Although other proxies may be more suitable, none are forthcom- ing in the literature, and the discussion presented here of the influ- ence of particle properties on the CDV is unlikely to be resolved without new high-quality experimental data that (a) incorporate full material characterisation, as outlined above and (b) cover the range of material properties likely to be encountered by operators in a range of industries in which hydraulic conveying of solids takes place. Consequently, the empirical volume factor provides an approach to extending the estimation of CDV to a much broader range of particles, and across a greater range of concentrations than has hitherto been demonstrated.‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌

As to the future of research in this area, the method of zero bed- depth identification used here breaks down for sediments of inter- acting particles or any solids close to the gel point or other limiting packing fraction (and it is noted that there is a range of overlapping definitions), as the onset of hindered or very slow settling means a bed will not form within the measurement timescale, so the con- cept of bed depth becomes nebulous. More ambiguous but well established experimental methods, such as minimum pressure- drop identification, must then be used, in which case care must be taken when comparing results derived using different methods (Doron and Barnea, 1995; Thomas, 1961): Bain and Bonnington (1970) found that the critical velocity corresponding to minimum pressure drop generally exceeds the critical deposition velocity, and it may be possible to develop a consistent method to estimate one from the other.‌‌‌‌

Declaration of Competing Interest

The authors declared that there is no conflict of interest.


The authors thank Peter Dawson, Bob Harris, Tony Windross, Gareth Keevil and Robert Thomas for their technical assistance, and Olivier Mariette at Met-Flow, Switzerland, for his support and advice. The authors also thank the U.K. Engineering and Phys- ical Sciences Research Council (EPSRC) for financial support of this study through grant EP/L014041/1: Decommissioning, Immobilisa- tion and Storage Solutions for Nuclear Waste Inventories (DISTINC- TIVE). We also warmly thank the editor and three anonymous reviewers for their comments, which improved the paper.

Appendix A. Supplementary material

Supplementary data to this article can be found online at https://doi.org/10.1016/j.ces.2019.115308.


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