Extending acoustic in-line pipe rheometry and friction factor modeling to low-Reynolds-number, non-Newtonian slurries

The rheology of non-Newtonian slurries is measured in a recirculating pipe loop using an acoustic velocimetry-pressure drop technique at very low flow rates and variable solids loadings. The technique avoids (a) settling at low solids concentration, a short-coming of bench rheometry, by using a vertical test section, and (b) physical sampling, providing greater safety. Speed of sound in the suspensions is also modeled. In-line and off-line data are used to assess the suitability of several non-Newtonian models to describe observed flow behavior. Measured and predicted values of the friction factor are compared with the Madlener et al Herschel – Bulkley extended model found to be superior. The dependence of yield stress and viscosity on solids loading and particle size is investigated, showing complexities from aggregation on the particle size distribution require more interpretation than the choice of rheological or friction-factor model.


| INTRODUCTION
The need for sophisticated, data-rich but rapid and affordable in-line measurement for complex, multiphase flows exists across a range of industries, including high-value chemicals manufacturing, food and minerals processing, and waste/wastewater processing. This requirement is exacerbated when additional hazards are present, such as extreme temperature and pressure, chemical or corrosion hazards and radioactivity. The UK civil nuclear industry in particular has a mature and diverse inventory of active waste in storage, subject to processing and disposal, accumulated over the last six decades since the inception of the industry in the 1950s. 1,2 Much of this waste is chemically, physically, and rheologically complex, [3][4][5] and difficulties exist in understanding its composition and flow behavior due to hazards and limited accessibility. In the nuclear industry and beyond, there remains an urgent, ongoing need for robust, easily deployed techniques with which operators can quickly characterize complex flows in a range of storage and transfer geometries. In a series of recent studies by the authors, a suite of acoustic techniques has been developed, specifically for: tracking of solids settling 6,7 ; monitoring of settled bed resuspension by jet impingement 8 ; solids volume fraction determination [9][10][11] ; critical deposition velocity in pipe flow 12,76 and monitoring and categorization of moving beds of settled solids-that is, bedforms-in pipe flow. 13 Here the suite of methods is extended to in-line rheometry of complex slurries at low flow rates, 3 ≲ Re ≲ 150, where the Reynolds number, Re, for non-Newtonian flows is defined later.
The rheological properties of complex, multiphase, and non-Newtonian flows are of central importance to operators for flow prediction and assurance, and in-line rheometry offers the potential for greater safety as the method precludes the need for sampling. Traditional tube rheometry 14,15 is well established and yields high-quality results 16,17 but requires precise characterization of fluid properties and entry conditions. 18,19 The method requires that the shear stress and shear rate be calculated from the measured pressure drop and volumetric flow rate, with the rheological parameters being derived by fitting to a suitable constitutive model. 20,21 A more recent variant of tube rheometry that employs velocimetry for determination of the shear rate, either via an acoustic method, [22][23][24] or via electrical resistance tomography and particle-image velocimetry, 25 has not found widespread industrial use.
However, it offers the significant advantage over the traditional method that a single run yields a full rheological curve, that is, a set of viscosity data over a range of shear rates, as the shear rate varies across the pipe cross-section, rather than a single viscosity at a single shear rate determined from the mean flow velocity. 26 In-line rheometry satisfies the drive for minimally intrusive, remote, in-line measurement. It also addresses the trend for manufacturers to move toward in-line measurement, as high data acquisition rates are now possible for real-time product/formulation information, thereby avoiding physical sampling and, therefore, the possibility of altering the shear history of a suspension by sampling. Furthermore, in-line rheometry has the potential to be used to characterize suspensions at very low solids loadings that would otherwise settle in bench rheometers.
The main objective of this study is to assess the suitability of inline rheometry via acoustic velocimetry for a range of complex, non-Newtonian slurry flows at low flow rates and low and intermediate solids fractions. Through this we extend the lower limit of the method in terms of Reynolds number (3 ≲ Re ≲ 150). Suspensions of barium sulfate (barytes), calcium carbonate (calcite), and magnesium hydroxide were used. Rheological characterization is by application of several constitutive rheological models to these complex, non-Newtonian slurry flows.
We demonstrate a novel recirculating in-line pipe-loop method with a vertical test section that minimizes solids settling. This versatile apparatus can be used for any flowable suspension and avoids the shortcomings of other methods. We also compare the efficacy of two expressions from the literature for predicting the Fanning friction factor and therefore pressure drop, as the latter is required by operators for flow assurance and determining minimum pumping power requirements. The utility of the in-line rheometry technique is illustrated by assessing the influence of several measures of particle size on yield stress and effective mixture viscosity. To do this, the in-line data are compared to off-line bench rheometer data and modeled rheological parameters. With the friction factor modeling, the study provides a full methodology for monitoring slurries noninvasively, a key requirement for hazardous flows.

| Solid-phase characterization
Three solid particle species were used to form the suspensions-barytes, calcite, and magnesium hydroxide-and were chosen because they are common nuclear waste analogues. 4,11,27 They have the additional benefits of being nonhazardous and widely available, and cover a range of material properties. The physical properties of the particle species (sources: barytes: RBH Ltd, Meriden, UK; calcite: Omya, Aberdeen, UK; magnesium hydroxide: Rohm and Haas, Surrey, UK) are summarized in Tables S1 and S2, including the bulk modulus and com-pressibility, which are used in the speed of sound correction referred to later. Samples were taken from the mixing tank before and after each run and allowed to settle for 7 days, at which point the bed packing fraction, ϕ m , in each sample was calculated from the height of the settled solids. Samples were not dried so that the settled solids retained any aggregates that may have formed. Mean values of ϕ m are also given in Table S1. Although not explicitly accounted for here, the shape of dry samples of the particle species has been investigated elsewhere. Barytes and calcite are both irregular-shaped milled minerals powders, 28 and calcite has been shown to aggregate to a small degree at neutral pH conditions. 29 Magnesium hydroxide consists of plate-like nanocrystalline agglomerated particles and exhibits complex behavior, forming fractal aggregates in suspension. 4 Particle size distributions were measured with a Mastersizer 2000 laser diffraction instrument (Malvern Instruments, Malvern, UK). Samples of the solids species were mixed with a small amount of mains water to form a paste, several drops of which were added into a flow cell of the Mastersizer once it was filled with water and set to (a) agitate the mixing cell at 1800 rpm and (b) circulate through the measurement cell. The Mastersizer yields the volume-or massweighted mean particle diameter, d [3,4], but also the Sauter or areaweighted mean, d [2,3], and the 50th-percentile mean, d 50 , for example, and a binned size distribution. The notation d[a,b] is a shorthand for the following: d[a,b] = m a /m b , where m n is the nth moment of the particle size distribution such that, for a sampled size distribution with N size bins: where w i is the weight/frequency of the ith bin. Expressions for determination of the mean, μ, and standard deviation, σ, of a size distribution when modeled as a log-normal distribution are given in Data S1.
Particle size data are shown in Figure 1 and summarized in Table S2, including d [2,3] and d [3,4] for completeness as these are commonly used measures for d p. 30,31 Log-normal fits to the size data are also shown in Figure S1. The distributions given in Figure 1 represent the particle sizes in the dilute, well mixed limit. However, all three particle species, and calcite and magnesium hydroxide especially, may aggregate depending on the shear regime and pH level. 4,29,32 Calcite has been found to aggregate weakly at moderate pH, with a relatively low proportion of larger aggregates, 29 whereas aggregates larger than 100 μm have been measured with similar magnesium hydroxide sludges under low-shear, settling conditions. 4 The aggregation, settling behavior and sediment compactness of barium sulfate in suspension has been found to be strongly influenced by pH 32 ; it also has a surface charge that influences solid packing. 11,33 The experimental apparatus is shown in Figure S3. A variable centrifugal pump (Ebara Pumps, Didcot, UK) was used to control the flow rate and a mixer with impeller to maintain a suspension in the mixing tank (nominal capacity 40 liters).

| Pressure drop measurement
The vertical configuration of the test section of the flow loop dictated that a correction for hydrostatic pressure be applied to the recorded pressure drop, such that ΔP = ΔP m -ρgΔh, where ΔP m is the measured pressure drop between two pressure transducers and the second term is the hydrostatic correction, where ρ is the suspension density, g is the acceleration due to gravity, and Δh is the vertical distance between measurement points (i.e., Δh = L = 0.5 m). The suspension density was calculated using the data sources given in Table S1.
The temperature, T, was measured from the mixing tank during each run and was accounted for in the water density, as given in Table S1, when evaluating ΔP.

| Velocity and flow rate measurement
Compressibility data given in Table S1, along with density data, were used to correct (a) distance and (b) velocity measurements taken with the acoustic instrument (since both require the speed of sound to be specified); the process is described in the "Speed of sound variation" section of Data S1, specifically Equation (S10) onwards. As with density, temperature variation was accounted for in the compressibility of water as given in Table S1 Table S2 in Data S1; see Figure S1 in Data S1 for goodness of fit where U max is the velocity in the central plug region that is a result of the yield stress and r p is the plug radius, within which the mixture flow velocity is constant. The shear stress, τ, varies linearly with radial position, according to r/R = τ(r)/τ w , where τ w is the wall shear stress. As in the flow of any yield-stress substance, the plug radius corresponds to that at which the shear stress equals the yield stress, such that r p / R = τ 0 /τ w . The wall shear stress is inferred from the measured pressure drop according to τ w = DΔP/4L, where D is the inner pipe diameter (D = 42.6 mm). The Fanning friction factor, f, is calculated  20 The ability of two models 40,41 to predict frictional losses is described in the friction factor modeling section; both are given in Data S1.
The shear stress distribution varies linearly with radial position according to r p /R = τ 0 /τ w and was determined from the measured pressure drop, ΔP, via the relationship τ w = DΔP/4L and the shear rate from the velocity field measured with the ultrasonic system as follows 42 :˙γ The suspension viscosity, η(r), was then calculated according to τ = η˙γ . This is illustrated in the results section in Figure 4 for velocimetric data, which is transformed into rheometric data via the expressions above, as illustrated in Figure 5. The volumetric flow rate, Q, was measured by numerical integration of the velocity field as measured with the acoustic system (see Figure S4 and associated text in Data S1). The flow loop was filled with suspensions at several nominal (weighed) concentrations, ϕ n , to verify mixing efficiency. Samples were taken from the mixing tank after the suspension had been circulated for 1 min at a low flow rate. A comparison of nominal and physically sampled volume fractions, ϕ s , is shown in Figure S5 of Data S1, from which it is clear that calcite and magnesium hydroxide remained almost fully suspended at all volume fractions, whereas barytes, having a significantly higher material density, were depleted relative to the nominal volume fraction (by 50,56,64,26, and 12% at ϕ = 0.5, 1, 2, 5, and 10%, respectively). However, measured-rather than nominal-values of the volume fraction were used in all calculations.

| Determination of rheological parameters
The constitutive equation in terms of shear stress, applied shear rate, γ, where˙γ = dγ/dt, and apparent viscosity, η, for the HBE model are, respectively 41 : where τ 0 is the yield stress, n is the flow index or viscosity index, K is the consistency index, η ∞ is a limiting viscosity at high shear rates sometimes referred to as the Bingham or infinite-shear viscosity. The HBE model, based on previous models of non-Newtonian flow, 43,44 differs from the HB model, 38 in that it includes η ∞ , potentially allowing a greater range of materials to be characterized than with the HB model. Expressions that allow yield stress and effective suspension viscosity to be predicted based on particle size and concentration are presented and discussed in the results section.
The velocimetry system used in this study allowed for rheological parameters to be calculated in two ways: first, directly by fitting the recorded mean velocity profile (see Equation (7)); and second, by a fit to a rheological curve via Equation (4)  In the velocimetric case (i.e., Method 1, Figure 2), by using Equations (2) and (3), the following expression is obtained for the mean axial velocity profile in the HB case: A value for the plug radius, r p , was determined by visual inspection of the velocity profile; τ 0 could then be found, as r p /R = τ 0 /τ w , and U max was calculated as the mean value of U(r) for r < r p . The parameter n was then found by nonlinear fit of the data to Equation (7) for r > r p ; K was then calculated from Equation (2)  3 | RESULTS AND DISCUSSION

| Off-line rheometry
Off-line rheometric data are shown in Figure S6 of Data S1 for the three particle species. We used a nonlinear least-squares fit to the HB model to estimate the yield stress and the other rheological parameters, and we note that some data at (a) very low and (b) very high shear rates were excluded from the fits (data selections are given in Table S3) because (a) some were deemed as falling below the sensitiv-

| In-line rheometry
Velocity profiles at ϕ = 10% are presented in Figure 4, in order to illustrate the goodness of fit with Equation (7) and as examples of Method 1 (see Figure 2). The goodness of fit compares very well with similar studies of velocimetry for in-line rheometry. 23,50,51 Error bars are shown for every fifth datum and represent the standard deviation over 3,000 samples (119 ms sample period, i.e., 6 min total). Figure 4 also serves to demonstrate the advantage that velocimetric in-line rheometry has over traditional tube rheometry in that a full rheological curve is produced from a run at a single flow rate. Included in the plots are velocity profiles of equivalent Newtonian flows with the F I G U R E 2 Flowchart for determination of rheological parameters same values of the flow rate, Q (and therefore bulk flow velocity), and pressure drop (and therefore wall shear stress); U e is parabolic such that: and U e,max = U e (r = 0) = τ w R/2η e , where η e is the viscosity of an equivalent Newtonian fluid. The flow rate is as follows:  dictates that the higher the yield stress, the narrower the region (specifically the region r p > r > R) in which Equation (9) is satisfied and the fewer data that can therefore be used in a fit. The criterion is stricter still for the HBE case (i.e., Equation (8)).
Lastly, the runs shown in Figure 5b,c were chosen because in these cases both in-line and off-line data were available. The agreement between the off-line (open symbols) and in-line data (closed symbols) is excellent for calcium carbonate (Figure 5b, ϕ = 10%) and good for magnesium hydroxide (Figure 5c, ϕ = 10%). Similarly good agreement has been found by several other researchers using, for example, xanthan gum suspensions and a polynomial fit to the velocity profile 53 ; several flowable food suspensions 42 and wastewater sludge 50 and a HB fit (as used here, see Equations (2) and (3)); polymer gels 51 ; and polymer melts 54 with an Eyring-type velocity fit. 55 In most cases, discrepancies between data obtained using in-line and other methods are of the order of tens of percent, as here, which is encouraging, since an objective of this study was to operate at lower flow rates (3 ≲ Re ≲ 150) than had been achieved in other studies-for example, Re = 320, Kotzé et al 50 -in order to test the limit of the method.

| Friction factor modeling
For the expression f = 16/Re to hold in non-Newtonian suspensions, a variety of expressions exists for substances obeying, for example, the power-law 56 and HB 57 relationships. A commonly used model suitable for the HB and lesser cases is that of Chilton and Stainsby. 40 A recent analytical solution for the general HBE case was presented by Madlener et al. 41 In order to examine the suitability of these models friction factor data are given in Table S4 of Data S1. For fit (1) in  Thomas 16,17 applied the Bingham plastic model to capillary-tube rheometry data using a range of particle species and found that the following expressions for the yield stress and (Bingham) viscosity, respectively, provided good fits to the data:

| Yield stress and relative viscosity modeling
where ϕ is the dry-weight volume fraction of suspended solids, d p is a representative particle diameter, and η f is the viscosity of the fluid phase. The general forms of the expressions, which can be found by allowing the numerical terms to vary, are given in Equations (14) and (15). They allow rheological properties to be predicted based on particle size and concentration, but particle shape and aggregation state are not explicitly accounted for: η ∞,r = exp 2:5 + a 2 d b2 Measured values of the yield stress and relative viscosity from this study are presented in Figure 7. Both in-line and off-line data were available for yield stress (Figure 7a), whereas for relative  41 Dashed line: one-to-one relation. All friction factor data given in Table S4 of Data S1. HB, Herschel-Bulkley; HBE, Herschel-Bulkley extended viscosity (Figure 7b) all data are from off-line measurements, as the flow rates in the in-line experiments were too low to determine η ∞ at ϕ = 10%. However, in-line and off-line viscosity at a shear rate of˙γ = 10 s −1 are given in Table S5 in Data S1. As described in Equation (14) for yield stress, the fit lines in Figure 7a vary as ϕ 3 , and as in Equation (15) for relative viscosity, the fit lines in Figure 7b vary according to exp(k 2 ϕ).
It is clear from Figure 7a that for the magnesium hydroxide and calcium carbonate systems, the yield stress values are largely consistent between off-line and in-line data, and follow the expected τ y / ϕ 3 dependence (see Equation (14)), although off-line measurements for magnesium hydroxide are slightly below those measured in-line. This difference may simply be due to the different qualities of fits for yield stress values between the techniques, given the low levels of yield stress overall in these systems (and thus higher degree of fit uncertainty). The discrepancy for the barytes yield stress data is larger, with the in-line measurement for the ϕ = 10% system being around an order of magnitude larger than the off-line value for the ϕ = 15% suspension.
It is noted again that no off-line data were available for the ϕ = 10% systems, as clear evidence of sedimentation was present in the rheometer cell. Although the τ 0 = 2 Pa value determined from the in-line data may suggest the suspension is above the gel-point, it may be that due to the highly dense nature of barytes, the lithostatic load is great enough to overcome the correspondingly low yield strengths, leading to compressional collapse at this concentration. Also, the very low yield stress apparent in the off-line data at ϕ = 15% may indicate that some degree of settling or network consolidation occurred in this system (and may explain some of the hysteresis in these data; see Figure S6 in Data S1).
Nonetheless, there is no strong evidence for significant settling from the off-line infinite-shear viscosity data (Figure 7b) where all systems display the expected dependence (see Equation (15)), which are given for completeness and to illustrate trends. Any considerable settling within certain systems (e.g., barytes at 15%) would lead to a reduction in reported viscosity values. In addition, the viscosity of magnesium hydroxide and calcite systems were compared between off-line and in-line techniques at a shear rate˙γ = 10 s −1 ; see Table S5. Again, the correlation between the two techniques is close, especially for the calcite suspensions, highlighting the performance of the in-line method for accurately measuring flow behavior and secondarily giving confidence of the data quality for off-line measurements.
Broadly, both the yield stress and viscosity results confirm that aggregation plays a dominant role in the rheological behavior of these suspensions. With the exception of the in-line measurements for magnesium hydroxide and barium sulfate, which are similar in magnitude, the trends are for both yield stress (particularly in the off-line measurements) and effective viscosity to increase strongly with a tendency to aggregate: magnesium hydroxide is most strongly aggregated, then calcium carbonate, then barium sulfate. This trend can also be viewed from the perspective of solids packing fraction.
Samples were taken from the mixing tank (approximately 60 ml each) and allowed to consolidate under gravity for 7 days. Mean values of the mean bed packing fraction, ϕ m , in the settled (wet) samples are given in Table S1 in Data S1, where it is noted that the value for magnesium hydroxide (ϕ m = 8.4%) is very close to that measured by Johnson et al 4 for the same material using two methods (ϕ m = 7.5 and 7.8%); different measurement methods accounting for the difference.
With reference to the rheometric results described above, there is a strong inverse correlation between bed packing fraction and measured rheological parameters in the off-line results. The species with the lowest packing fraction (magnesium hydroxide) has the highest yield stress and viscosity at a given solids loading. We note that the more likely a particle species is to form an aggregate network, the lower its bed packing fraction, 4,11,29,32,33,49 although this conclusion requires further testing.
The values of the fit parameters k 1 and k 2 (for the yield stress and relative viscosity, via Equations (14) and (15), respectively) are listed in Table S6 in Data S1. The calculated values of k 1 and k 2 were used to  (15) derive the fit parameters a 1 , b 1 (for yield stress via Equation (14)), a 2 and b 2 (for relative viscosity via Equation (15)) for each choice of the particle size measure d p ; in particular, d p was chosen to be one of the following: μ (i.e., m 1 ), d 50 , d [2,3] and d [3,4], which are listed in Table S2 in Data S1. The resulting fits are shown in Figure 8 for relative viscosity and Figure S7 in Data S1 for yield stress (d p = d [2,3] and [3,4] not shown, for clarity of visualization); the fit parameters are listed in Table 1.
It is clear from these fits that the dependences on particle size and solids volume fraction assumed by Thomas,16,17 indicated by dashed-dotted lines in Figures S7 and 8 and given in Equations (12) and (13) While some studies also predict a floc-size (rather than individual particle-size) dependence, 63 yield stress often increases with the surface area to volume ratio, as an inverse to particle size, as found elsewhere. 16 For the suspensions considered here, the complication of aggregation and increasing size distribution is present, and more explicit knowledge of the relative aggregate size distributions of the suspensions (rather than the disperse primary particle sizes) may aid understanding. However, it is also interesting that the correlation between relative viscosity and particle size is stronger than that for yield stress, and is in line with expectations, that is, that smaller particles at a given volume fraction, which therefore have a greater number density, will increase the overall fluid resistance to flow to a greater degree than larger particles.
Nevertheless, it is clear from the above discussion that particle size alone-when represented by a single measure, d p -is not sufficient when used to model yield stress and relative viscosity, in so far as different choices for d p yield entirely different dependences. A particular dependence may shed light on which physical process is dominant, but the observed ambiguity makes such an identification impossible. Such ambiguity was also found by Li et al, 71 depending on whether number-, surface-or volume-weighted mean particle sizes were used. However, the effective particle size-from a rheological point of view-depends strongly on the aggregation state, which in general depends strongly on the solids concentration. Moreover, the method used for particle sizing may differ from the aggregate size at high solids loadings. For example, both sedimentation (i.e., wet sample) and electron micrography (i.e., dry sample) were used for particle sizing by Thomas 16 and in the present study a very dilute suspension was used, albeit in a wet cell of a laser diffraction sizing instrument.
It is suggested that, as well as some measure of particle size in whatever form (e.g., d 50 ), at least two other properties of the solid phase be given equal consideration: (a) the width of the particle size  Table 1 and Table S6 in Data S1 [Color figure can be viewed at wileyonlinelibrary.com] T A B L E 1 Calculated values of fit parameters a 1 and b 1 (Equation (14)) and a 2 and b 2 (Equation (15)); μ (i.e., m 1 ), d 50 , d [2,3] and d [3,4] given in Table S2 Parameter Thomas 16,17 Choice for d p as offering greater assurance that, in the absence of a concentration gradient across the pipe diameter, the measured rheological properties are accurate. The ability of two models to predict the flow properties of complex suspensions-specifically the Fanning friction factor-was assessed with the HB and HBE models. It was found that, of the two models for predicting the friction factor that were tested, that of Madlener et al 41 performed better and is recommended in general on that basis and because it was derived analytically.
The influence of solids volume fraction and particle size on the rheology of complex solid-liquid suspensions was investigated. By combining the in-line method with off-line measurements, it was possible to reassess the dependence of yield stress and relative viscosity on particle size put forward by Thomas 16,17 and it was found that particle size as a single measure was not sufficient to account for the observations.
In particular, the yield stress of suspensions was found to increase with particle size, and the effective suspension viscosity to decrease with particle size; a per-species investigation of the influence of particle size on rheology would be a valuable topic in any future study. Agreement with other studies was mixed as there is no clear picture of the mechanisms through which rheology is influenced by particle size and aggregation state. These unanswered questions, central to understanding the rheology of complex suspensions, are left as subjects for future study.
However, it is suggested that, as well as mean particle size, the width of the particle size distribution and the divergence from ideal, hardsphere behavior or tendency to aggregate/flocculate (e.g., via the maximum packing fraction or effective viscosity) must be considered.