prediction of just suspended speed for mixed slurries at

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chemical engineering research and design 9 1 ( 2 0 1 3 ) 227–233

Contents lists available at SciVerse ScienceDirect

Chemical Engineering Research and Design

j ourna l ho me page: www.elsev ier .com/ locate /cherd

rediction of just suspended speed for mixed slurries atigh solids loadings

nci Ayrancia,∗, Theodore Nga, Arthur W. Etchells III b, Suzanne M. Krestaa

Department of Chemical and Materials Engineering, University of Alberta, Edmonton, Canada T6G 2V4Rowan University, Glassboro, NJ, United States

a b s t r a c t

One design heuristic used to determine the just suspended speed, Njs, for mixed slurries assumes that the mixture

Njs is dominated by the particle phase with the maximum Njs. This approach does not incorporate the effect of the

second solid phase. Two new models are proposed to predict the mixture Njs: the power model and the momentum

model. These models determine the mixture Njs using the sum of the power or the sum of the momentum required

to suspend the individual solid phases. The models were tested using experimental data for two impellers, a Lightnin

A310 impeller and a 45◦ pitched-blade turbine. A range of off-bottom clearances, and six mixtures of solids up to

27 wt% solids loading completed the data set. The power model accurately predicts mixture Njs for both impellers

over the full range of clearances and up to 27 wt% mixtures.

© 2012 The Institution of Chemical Engineers. Published by Elsevier B.V. All rights reserved.

Keywords: Just suspended speed; Stirred tank; Power model; Momentum model; Mixture; High solids loading

�L D0.85

. Introduction

n many solid–liquid mixing operations the main objectives mass transfer between the two phases. To maximize the

ass transfer the entire surface area of the solids should bexposed. This can be achieved by operating at the completeff-bottom suspension condition. The key operating parame-er for this condition is the impeller speed, which is called theust suspended speed (Njs). Njs is defined as the impeller speedt which no solids remain stationary at the bottom of the tankor more than 1 or 2 s (Zwietering, 1958). Solid–liquid mixings a power intensive operation, so accurate prediction of Njs ismportant. Current correlations are limited to unimodal slur-ies at low solids loadings, but many industrial slurries areomposed of mixtures of solids with varying densities and par-icle sizes at high concentrations. The gap between researchnd industry is vast, and the need for an accurate design modelor mixed slurry Njs is clear.

Current correlations have significant limitations because
any parameters play an active role in solids suspension.
Abbreviations: A310, axial impeller provided by Lightnin; B, bronze;

xchange resin; S, sand; SG, small glass beads or specific gravity; UF, ur∗ Corresponding author at: Department of Chemical and Materials Engdmonton, Alberta, Canada T6G 2V4. Tel.: +1 780 492 9221; fax: +1 780 4

E-mail address: [email protected] (I. Ayranci).Received 8 February 2012; Received in revised form 10 July 2012; Accep

263-8762/$ – see front matter © 2012 The Institution of Chemical Engittp://dx.doi.org/10.1016/j.cherd.2012.08.002

Geometry is by far the most important factor. The effects ofimpeller and tank diameter, impeller type, off-bottom clear-ance of the impeller, shape of the tank bottom, and thepresence, shape, and clearance of the baffles have been stud-ied by many authors (Baldi et al., 1978; Ibrahim and Nienow,1996; Myers and Fasano, 1992; Armenante and Nagamine,1998). Njs is also a function of particle and liquid properties,such as the particle density, particle diameter and shape, andliquid density and viscosity (Nienow, 1968; Baldi et al., 1978).The behavior of the particles is different when there are manyother particles around them; therefore, solids loading is alsovery important (Ayranci and Kresta, 2011).

The large number of parameters affecting Njs makes itdifficult to determine a robust design correlation. The first cor-relation was suggested by Zwietering (1958) and it is still thecorrelation that is most often used in calculations.

Njs = S

(g(�s − �L)

)0.45d0.2

p �0.1X0.13

(1)

LG, large glass beads; Ni, nickel; PBT, pitched blade turbine; R, ionea formaldehyde; wt%, weight percent.ineering, University of Alberta, 7th Floor ECERF, 9107-116 Street,92 2881.

ted 1 August 2012neers. Published by Elsevier B.V. All rights reserved.

http://www.sciencedirect.com/science/journal/02638762

www.elsevier.com/locate/cherd

mailto:[email protected]

dx.doi.org/10.1016/j.cherd.2012.08.002

228 chemical engineering research and design 9 1 ( 2 0 1 3 ) 227–233

Nomenclature

Roman charactersC off-bottom clearance (m)D impeller diameter (m)dp particle diameter (m)g acceleration due to gravity (m/s2)H liquid height (m)M momentum (kg m/s2)Mjs momentum at just suspended conditions (kg

m/s2)Mjs,1 Mjs for particle one (kg m/s2)Mjs,2 Mjs for particle two (kg m/s2)Mjs,mix Mjs for mixture (kg m/s2)Mo momentum numberN impeller rotational speed (rps or rpm)Njs just suspended speed (rps or rpm)Njs,1 Njs for particle 1 (rps or rpm)Njs,2 Njs for particle 2 (rps or rpm)Njs,max Njs maximum (rps or rpm)Njs,mix mixture Njs (rps or rpm)Np power numberPjs power consumption at just suspended speed

conditions (W)Pjs,1 Pjs for particle 1 (W)Pjs,2 Pjs for particle 2 (W)Pjs,mix Pjs for mixture (W)r radius (m)S Zwietering’s Njs constantT tank diameter (m)Vz velocity in the axial directionWb baffle width (m)xS mass fraction of the solids in the slurryxL mass fraction of the liquid in the slurryX Zwietering’s mass ratio percent (mass of

solid/mass of liquid × 100)

Greek characters� kinematic viscosity (m2/s)�L liquid density (kg/m3)�S solid density (kg/m3)�sl slurry density (kg/m3)�sl,1 unimodal slurry density for particle 1 (kg/m3)�sl,2 unimodal slurry density for particle 2 (kg/m3)�sl,mix mixture slurry density (kg/m3)

1

Some of the parameters that affect Njs are included in thiscorrelation but the accuracy of the exponents has been ques-tioned by many authors. Kasat and Pandit (2005) compiled thedifferent exponents on the common parameters suggestedby various authors. Their comparison showed that the Zwi-etering correlation is still the one that predicts the data mostclosely. The Zwietering correlation, however, does not providean answer for mixed slurry Njs.

The literature on mixed slurry suspension is only beginningto be developed, and initial studies focused on dilute slurries.Baldi et al. (1978) studied a mixture of glass beads with twoparticle sizes and found that Njs can be predicted using anaverage particle size, at low solids loadings. Montante andMagelli (2007) did a computational study on the distribution
of solids for dilute slurries with two solid phases which havedifferent densities but same particle sizes. They showed that
the two solids phases are not affected by each other. RecentlyAyranci and Kresta (2011) reported results for a wide vari-ety of binary mixtures at high solids loadings (up to 56 wt%).Their study showed that the presence of a second solid phasemay significantly affect the mixture Njs. This effect is ampli-fied for mixtures above 20 wt% solids, because at that pointthe particle–particle interactions start to dominate. The par-ticle sizes and the densities of the two solid phases play animportant role in the mixture Njs.

The current design heuristic for mixed slurries is to assumethat the mixture is composed of only the particle fraction thatis hardest to suspend. The Njs for that fraction is predictedusing the Zwietering correlation, and treated as the mixtureNjs. This design heuristic has many flaws, some of which wereshown by Ayranci and Kresta (2011). Of the five mixtures theytested, only one mixture followed the design heuristic up tohigh solids loadings, and a second mixture followed it up to13 wt%, but then failed. The other mixtures did not follow thedesign heuristic. The ratio of the particle size, the particle den-sity, and the solids loadings of the two solid phases all had aneffect on mixture Njs. A more robust and physically realisticmodel for predicting mixture Njs is needed.

In this study we propose and test two models that are basedon the total power and the total momentum required to sus-pend solids in a stirred tank.

2. Model development

2.1. Current design heuristic

The current design heuristic is based on the maximum uni-modal Njs in a mixture:

Njs,mix = max(Njs,1, Njs,2) (2)

For example, if a mixture Njs needed to be determined fora mixture of 1.5 wt% SG with 1.5 wt% B, the Njs of the uni-modal slurries of the two particles should be calculated andthe maximum value should be used as the mixture Njs. Theunimodal slurry Njs is predicted from the Zwietering correla-tion (Eq. (1)). In the example the unimodal slurry Njs is 318 rpmfor 1.5 wt% SG and 1142 rpm for 1.5 wt% B. The mixture Njs isthe maximum of the two values, which is 1142 rpm.

2.2. Power model

The power model is proposed based on a hypothesis that thepower required to suspend a mixture is the sum of the powerrequired to suspend each of the solid phases in the mixture.

Pjs,mix = Pjs,1 + Pjs,2 (3)

where Pjs,mix is the power required to suspend the mixture,and Pjs,1 and Pjs,2 are the power required to suspend the firstand the second solid phases, respectively. The power requiredto suspend each solid phase is calculated at the just suspendedcondition based on the unimodal slurry density:

Pjs = �slNjs3D5Np (4)

�sl =(xs/�s) + (xL/�L)

(5)

chemical engineering research and design 9 1 ( 2 0 1 3 ) 227–233 229

dsaudotcftpeaeapnsstm

as

N

r

aabs

2

Apr

M

wmsMn

M

c

M

tc

Fig. 1 – The experimental setup with a PBT impeller. Njs is

In combining Eqs. (3) and (4) to find Njs,mix, the impelleriameter term, D, cancels out. Current practice is to use thelurry density to correct for the presence of the solids andssume that the power number is constant, which also allowss to eliminate Np. This assumption has some uncertaintyue to the presence of a low concentration layer at the topf the vessel which will increase the solids concentration inhe bottom of the vessel, and the possibility of a lower solidsoncentration in the vicinity of the impeller due to centrifugalorces. Micheletti et al. (2003) and Jafari et al. (2012) inves-igated whether there is an effect of solids concentration,article size, and particle type on the power number. Jafarit al. (2012) found that in general the power number decreasest high solids loadings by of the order of 20%, while Michelettit al. (2003) found that it either stays the same, or increases bybout 20%. Our measurements of Np at varying Re for singlehase and 25 wt% small glass beads showed that the powerumber for the single phase and the slurry are almost theame. The assumption that the power number remains theame for each slurry (Pjs,mix, Pjs,1, and Pjs,2) was applied, withhe understanding that this may introduce some error into the

odel.When D and Np are cancelled out the mixture Njs becomes

function of the densities of the mixed and the unimodallurries and the Njs’s of the unimodal slurries.

js,mix =(

�sl,1N3js,1 + �sl,2N3

js,2

�sl,mix

)1/3

(6)

In Eq. (6), Njs,1 and Njs,2 can be calculated using Eq. (1), oreplaced with the experimental values.

It should be noted that the power model does not includeny terms to take the particle–particle interactions intoccount; therefore, it is very likely that the mixture Njs will note accurately predicted when particle–particle interactions aretrong.

.3. Momentum model

second hypothesis is that the momentum required to sus-end a mixture is equivalent to the sum of the momentumequired to suspend each of the solid phases in the mixture.

js,mix = Mjs,1 + Mjs,2 (7)

here Mjs,mix is the momentum required to suspend theixture, and Mjs,1 and Mjs,2 are the momentum required to

uspend each individual unimodal slurry. The momentum,, can be calculated through the dimensionless momentumumber (Mo) (Machado et al., 2011):

o =∫ D/2

0�LV2

z 2�rdr

�LN2D4= M

�LN2D4(8)

The momentum required to suspend each solid phase isalculated at just suspended conditions:

js = Mo�slN2jsD4 (9)

In combining Eqs. (7) and (9) to find Njs,mix, the momen-
um number and impeller diameter are constant, so the termsancel out. Like the power model, the mixture Njs is thus a
determined by visual observation below the tank bottom.

function of the mixed and unimodal slurry densities and theNjs of the unimodal slurries, this time to the power of two:

Njs,mix =(

�sl,1N2js,1 + �sl,2N2

js,2

�sl,mix

)1/2

(10)

3. Experimental procedure

Fig. 1 shows the experimental setup. A fully baffled (Wb = T/10)cylindrical plexiglass tank with an inner diameter of 24 cmwas used for the measurements. The cylindrical tank wasplaced inside a square tank to prevent optical distortion. Inorder to maintain stability at high impeller speeds, both tankswere bolted to a steel frame. The just suspended speed wasobserved visually from the bottom of the tank.

A Lightnin A310 impeller and a four bladed 45◦ downpumping PBT both with a diameter of D = T/3 were used. Theimpellers were attached to a shaft with a diameter of 1.27 mm(T/20). The off-bottom clearance was defined as the distancebetween the bottom of the impeller hub and the bottom ofthe tank. The blades were flush with the bottom of the hubfor both impellers. The dimensionless off-bottom clearance,C/T, was varied from 0.15 to 0.33. Water was used as the liquidphase for all the experiments.

Seven different particles with various physical properties
were tested: nickel (Ni), small glass (SG), urea formalde-hyde (UF), bronze (B), sand (S), large glass (LG), and ion


Table 1 – Particle properties.

Type Size (�m) Density(kg/m3)

Vt (m/s)

Nickel (Ni) 61–104 8900 0.139Small glass beads (SG) 74–125 2500 0.066Urea formaldehyde (UF) 150–250 1323 0.044Bronze (B) 150–297 8855 0.225Sand (S) 350–500 2656 0.144Large glass beads (LG) 595–841 2500 0.177Ion exchange resin (R) 677 1370 0.086

0

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200 0a

b

0 20 0 400 60 0 800 10 00 120 0 14 00 1600 180 0 2000

Njs

, max

(rpm

)

Njs, measure d (rpm)

SG+B C/T =0.1 5SG+B C/T =0.2 5SG+B C/T =0.3 3SG+Ni C/T=0.15SG+Ni C/T=0.25SG+Ni C/T=0.33R+B C/T=0. 15R+B C/T=0. 25R+B C/T=0. 33LG+B C/T =0.15LG+B C/T =0.25LG+B C/T =0.33R+LG C/T =0.15R+LG C/T =0.25R+LG C/T =0.33

Standard deviation : 10 %PBT - Current design heur istic

=

0

200

400

600

800

100 0

120 0

140 0

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180 0

200 0

0 20 0 40 0 60 0 80 0 1000 1200 140 0 160 0 180 0 200 0

Njs

, max

(rpm

)

Njs, measure d (rpm)

SG+B C/T=0.15SG+B C/T=0.25SG+B C/T=0.325SG+Ni C/T=0.15SG+Ni C/T=0.25SG+Ni C/T=0.325R+B C/T =0.15R+B C/T =0.25R+B C/T =0.32 5LG+B C/T=0.15LG+B C/T=0.25R+LG C/T=0.15R+LG C/T=0.25R+LG C/T=0.325UF+S C/T=0.25

Standard deviation : 12.6%

A310 - Curr ent design heur isti c

=

Fig. 2 – The parity plot between the current design heuristicand the experimental data. The current design heuristicuses the maximum Njs in the mixture, calculated using the

exchange resin (R). The particle properties are given in Table 1.The particles were chosen to give a wide range of densi-ties (1.3 < SG < 8.9) and particle sizes (61 �m < dp < 841 �m). Themixtures tested are given in Table 2 along with the ranges ofsolids loadings. For each data set the mass of the more denseparticles was kept constant while the mass of the less denseparticles increased. A set of experiments where the mass ofthe dense particle is higher than the mass of the less denseparticle was also conducted to validate the models tested forall cases. This set of experiments was for the mixture of R withLG at C/T = 0.25.

At the beginning of every experiment, the tank was filledwith water and particles were weighed and poured into it.The liquid height was then adjusted to give H = T. The shaftwas attached to the motor, and the off-bottom clearancewas adjusted. Once the desired clearance was achieved, theimpeller was started. The impeller speed was increased insteps and the system was left for 2 min to reach steady state.After steady state was reached, the particle behavior at thebottom of the tank was observed for 15–45 s to determinewhether Njs was reached. The particles behind the baffleswere consistently the last particles to be suspended. The justsuspended speed was reached when no particles remainedstationary at the bottom of the tank for more than 1 or 2 s(Zwietering, 1958). After that the motor was switched off andthe off-bottom clearance was adjusted for the new measure-ment. More details about the experimental setup and theprocedure are given in Ayranci and Kresta (2011).

4. Results and discussion

First the current design heuristic results are presented to pro-vide a baseline for comparison. Next the performance of thepower and momentum models is tested for all six mixtures ofsolids, two impellers, and varying off-bottom clearances, andthe performance of the two models is compared. The powerand the momentum models require the use of unimodal slurryNjs’s. Initially the unimodal slurry Njs’s are calculated from theZwietering correlation, and then the experimental values are
used. The performance of the power model is compared forthe two cases.
Table 2 – Particle mixtures and solids loadings.

Less dense particles (wt%) Denser particles (wt%) To

wt%

SG (1.5–26) B (1.5–1.3) 3–27.LG (1.5–26) B (1.5–1.3) 3–27.R (1.5–26) B (1.5) 3–27.R (1.5–25) LG (1.5–1.4) 3–26.SG (1.5–26) Ni (1.5–1.3) 3–27.UF (1–10) S (1–5) 2–15

Zwietering correlation. (a) PBT and (b) A310.

4.1. Test of current design heuristic

Fig. 2a and b compares the prediction of Njs,max using theZwietering correlation to the experimental mixture Njs. In theZwietering correlation, S is a function of impeller and tankgeometry and particle properties. Fig. 2a and b represents thebest possible predictions using the current form of the Zwi-etering correlation since the S values were obtained for thespecific particles and the geometries used here (Ayranci andKresta, 2011). For each mixture the wt% of the more diffi-cult to suspend solids remains constant as the wt% of theeasier to suspend solids increases. The prediction of mix-ture Njs is constant because the Njs of the easier to suspend
solids does not change enough with increasing concentrationto overtake Njs,max. In Fig. 2a the mixture Njs for R with LG at
tal solids loading Density ratio Particle size ratio

vol%

3 0.8–12 ∼1:3.5 ∼1:23 0.8–12 ∼1:3.5 ∼3:15 1.3–21.8 ∼1:6.5 ∼3:14 1.7–20 ∼1:1.8 ∼1:13 0.77–14.9 ∼1:3.6 ∼1:1

1.1–9.1 ∼1:2 ∼1:2


Table 3 – The list of S values used in the calculations forunimodal slurry Njs.

Impeller D C Sa

A310 T/30.15 6.840.25 7.540.325 7.97

PBT T/30.15 5.40.25 6.180.33 7.15

a The S values were taken from Ayranci and Kresta (2011).

CRNtbcteiabTi

FeZ

should, however, be noted that these predictions use the Zwi-etering correlation for unimodal slurry Njs’s. The Zwietering

/T = 0.15 remains constant even though the concentration of changes at each experimental point and the experimental

js does in fact increase. The predicted mixture Njs is similaro the experimental data at the lowest solids concentration,ut is consistently lower than the experimental Njs when theoncentration increases. The current design heuristic failso capture the physics behind mixed solids suspension. Theffect of the presence of both solid phases must be includedn the model. The standard deviation between the measurednd the predicted values for all of the mixtures at varying off-ottom clearances is 10% for the PBT and 12.6% for the A310.he fact that the trend does not follow the experimental data

s of greater concern.

0

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2000a

b

0 200 400 600 800 1000 1200 1400 1600 1800 2000

Njs

, pre

dict

ed(r

pm)

Njs, me asured (rpm)

SG+B C/T=0.15SG+B C/T=0.25SG+B C/T=0.33SG+Ni C/T=0.15SG+Ni C/T=0.25SG+Ni C/T=0.33R+B C/T=0.15R+B C/T=0.25R+B C/T=0.33LG+B C/T=0.15LG+B C/T=0.25LG+B C/T=0.33R+LG C/T=0.15R+LG C/T=0.25R+LG C/T=0.33

Standard deviation: 8.3%

PBT - Power Model(using Zwietering for unimodal Njs )

= +

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800

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1400

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1800

2000

0 200 400 600 800 1000 1200 1400 1600 1800 2000

Njs

, pre

dict

ed (r

pm)

Njs, measured (rpm)

SG+B C/T=0.15SG+B C/T=0.25SG+B C/T=0.32 5SG+Ni C/T=0.15SG+Ni C/T=0.25SG+Ni C/T=0.32 5R+B C/T=0.15R+B C/T=0.25R+B C/T=0.325LG+B C/T=0.15LG+B C/T=0.25R+LG C/T=0.15R+LG C/T=0.25R+LG C/T=0.325UF+S C/T=0.25


A310 - Power Model(using Zwietering for unimodal Njs )

ig. 3 – The prediction of mixture Njs without anyxperimental data using the power model and thewietering correlation with the (a) PBT and (b) A310.

a

b

4.2. Power model and momentum model usingZwietering unimodal Njs

To find the mixture Njs through the power or momentum mod-els, the slurry densities are first calculated from Eq. (5). Theunimodal slurry Njs’s are calculated from the Zwietering cor-relation (Eq. (1)). The S values used in the calculations are givenin Table 3. The mixture Njs is then calculated from Eq. (6) forthe power model, and from Eq. (10) for the momentum model.

Fig. 3a and b shows the power model parity plots for the PBTand the A310. While some data points are on the parity line, asimilar trend to the current design heuristic (Fig. 2a and b) isseen: the data flattens out as the solids loading is increased.A comparison of the power model (Fig. 3a and b) with the cur-rent design heuristic (Fig. 2a and b) shows that there is nosignificant improvement from the current design heuristic tothe power model prediction. The standard deviation is 8.3%for the PBT and 9.7% for the A310. The low standard devia-tion does not give information about the trend, and the trendshows that the physics is not captured.

Fig. 4a and b shows the momentum model parity plotsfor the PBT and the A310 impellers. The trend in these fig-ures is very similar to the power model, and also the currentdesign heuristic. This may indicate that there is no signifi-cant difference between the two models and the heuristic. It

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, pre

dict

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Njs, measured (rpm)



PBT - Momentum model(using Zwietering for unimoldal Njs )

0

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0 200 400 600 800 1000 1200 1400 1600 1800 2000

Njs

, pre

dict

ed (r

pm)

Njs, measured (rpm)

SG+B C/T=0.15SG+B C/T=0.25SG+B C/T=0.325SG+Ni C/T=0.15SG+Ni C/T=0.25SG+Ni C/T=0.325R+B C/T=0.15R+B C/T=0.25R+B C/T=0.325LG+B C/T=0.15LG+B C/T=0.25R+LG C/T=0.15R+LG C/T=0.25R+LG C/T=0.325UF+S C/T=0.25


A310 - Momentum mode l(using Zwietering for unimodal Njs )

Fig. 4 – The prediction of mixture Njs without anyexperimental data using the momentum model and theZwietering correlation with the (a) PBT and (b) A310.


0

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1000

1200

1400

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1800

2000a

b

0 200 400 600 800 1000 1200 1400 1600 1800 2000

Njs

, pre

dict

ed(r

pm)

Njs, measured (rpm)


Standard deviation: 17.3%PBT - Momentum model

= +

800

1000

1200

1400

1600

1800

2000

js, p

redi

cted

(rpm

)

SG+B C/T=0.15SG+B C/T=0.25SG+B C/T=0.325SG+Ni C/T=0.15SG+Ni C/T=0.25SG+Ni C/T=0.325R+B C/T=0.15R+B C/T=0.25R+B C/T=0.325LG+B C/T=0.15LG+B C/T=0.25

Standard deviation: 15.7%A310 - Momentum model

correlation is known to have limited accuracy above 10 wt%solids, so above this concentration the predictions are not veryreliable. In order to test the true strength of the power andmomentum models, the experimental unimodal slurry Njs’sare used in a second test.

4.3. Power model and momentum model usingexperimental data

Fig. 5a and b shows the mixture Njs predicted from powermodel where the unimodal slurry Njs is obtained from experi-ments. These plots show a completely different trend than thepredictions using Zwietering unimodal slurry Njs’s. The pre-dicted Njs’s follow the parity line closely. Most of the data iswithin ±20% of the parity line. This indicates that the physicsof the solids suspension is captured up to 20 wt% solids forall mixtures. Beyond 20 wt% solids, particle–particle interac-tions can become quite strong. For the LG + B, R + B, and R + LGmixtures Njs increases with increasing solids and the modelcaptures Njs up to the highest loading tested, 27 wt% solids.The data for UF + S goes up to only 10 wt% solids. For the SG + Bmixture there is an unexpected drop in Njs above 20 wt% SG(Ayranci and Kresta, 2011) and for SG + Ni Njs is constant. Thepower model cannot predict these effects because there are noterms for particle–particle interactions. Based on this informa-tion the data points above 20 wt% SG for both SG + B and SG + Ni
mixtures were eliminated from the data set shown in Fig. 5aand b and from subsequent analysis. The resulting standard
0

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, pre

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Njs, measured (rpm)


Standard deviation: 9.6%PBT - Power model

= +

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1400

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0 200 400 600 800 1000 1200 1400 1600 1800 2000

Njs

, pre

dict

ed (r

pm)

Njs, measured (rpm)

SG+B C/T=0.15SG+B C/T=0.25SG+B C/T=0.325SG+Ni C/T=0.15SG+Ni C/T=0.25SG+Ni C/T=0.325R+B C/T=0.15R+B C/T=0.25R+B C/T=0.325LG+B C/T=0.15LG+B C/T=0.25R+LG C/T=0.15R+LG C/T=0.25R+LG C/T=0.325UF+S C/T=0.25

Standard deviation: 9%A310 - Power model

= +

Fig. 5 – The parity plot for the power model at varyingclearances for all mixtures with the (a) PBT and (b) A310.

0

200

400

600

0 200 400 600 800 1000 1200 1400 1600 1800 2000

N

Njs, measured (rpm)

R+LG C/T=0.15R+LG C/T=0.25R+LG C/T=0.325UF+S C/T=0.25= +

Fig. 6 – The parity plot for the momentum model at varying
clearances for all mixtures with the (a) PBT and (b) A310.
error is 9.6% for the PBT and 9% for the A310. The mixture Njs

can be predicted accurately with the power model up to 27 wt%solids for a range of off-bottom clearances, with two separateimpellers, in the absence of particle–particle interactions.

Fig. 6a and b shows the momentum model results for exper-imental Njs. The momentum model captures the physics, butover-predicts the mixture Njs, leaving more data points outsidethe ±20% range. The standard error of the momentum modelprediction is 17.3% for the PBT and 15.7% with the A310. InFig. 7 comparison of the power and momentum models withboth the PBT and the A310 shows that the momentum modelconsistently over-predicts mixture Njs. The standard devia-tion between the two models is 6.4% when the data for bothimpellers is combined. We conclude that the power modelprovides a better prediction of mixture Njs.

The performance of the power model has been analyzed interms of parity plots up to this point. This allows the compari-son of model accuracy for different data sets, but prevents thevisibility of some details, such as the effect of solids loading. Acloser look at the raw data is required to observe these details.Fig. 8 compares the power model predictions to experimentaldata and the current design heuristic for two representativecases at C/T = 0.25. As the solids loadings increase the mixtureNjs increases. At low solids loadings the power model predic-tions fall on top of the experimental data for both mixtures.As the solids loadings are increased the mixture Njs increases
both with the power model predictions and the experimen-tal data, and the power model starts to give moderately


0

200

400

600

800

1000

1200

1400

1600

1800

2000

0 200 400 600 800 1000 1200 1400 1600 1800 2000

Njs

-Mom

entu

m M

odel

(rpm

)

Njs - Power Model (rpm)

PBTA310


PBT and A310Momentum vs Power model

Fig. 7 – Comparison of the momentum model and thepower model at varying clearances for all particles with thePBT and A310.

400

600

800

1000

1200

1400

1600

0 5 10 15 20 25 30

Njs

, mix

(rpm

)

Xw

LG+B Experimental

LG+B Power Model

LG+B Heuristic

R+LG Experimental

R+LG Power Model

R+LG Heuristic

PBT - Power Model and Heuristic

Fig. 8 – Comparison of mixture Njs obtained fromexperiments, power model, and current design heuristic forLG + B and R + LG mixtures with PBT at C/T = 0.25.

otidcmb

bciw

5

Teect

Zwietering, Th.N., 1958. Suspending of solid particles in liquid byagitators. Chem. Eng. Sci. 8, 244–253.

ver-predicted Njs,mix. This figure shows that the power modelends to over-predict mixture Njs, but provides a significantmprovement over the current design heuristic, shown as theashed lines. If the heuristic was used for design, a signifi-ant number of particles would not be suspended. The powerodel, in all cases, provides conservative design but never

eyond 20% error.Reviewing the data, the current design heuristic is rejected

ecause the trends are not captured. The momentum modellearly over-predicts Njs,mix, based on Fig. 7. The power models recommended with use of experimental unimodal datahere possible.

. Conclusions

he objective of this study was to propose and test two mod-ls to accurately predict mixed slurry Njs. Analysis of thexperimental data for several mixtures at varying off-bottomlearances and solids loadings for two impeller geometries ledo the following conclusions:

• The current design heuristic is inadequate for the predictionof mixture Njs since it ignores the addition of a second solidphase, and cannot predict the basic trend.

• The momentum model consistently over-predicts Njs and isrejected.

• When the Zwietering correlation is used for unimodal slurryNjs the power and momentum models lose strength andshow similar behavior to the current design heuristic. Theauthors recommend use of experimental unimodal slurryNjs if possible.

• The power model, as given below, predicts mixture Njs accu-rately for both the PBT and the A310 impellers up to 27 wt%solids over a range of off-bottom clearances when unimodalslurry Njs’s are obtained from experiments.

Njs,mix =(

�sl,1N3js,1 + �sl,2N3

js,2

�sl,mix

)1/3

Acknowledgments

The authors would like to thank Lightnin and NSERC for fund-ing, Sherritt Metals Inc. for providing nickel particles, andMaria Garcia from Rowan University for sharing data for sandand urea formaldehyde.

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