This document describes two new models for predicting the just suspended speed (Njs) for mixtures of solids in stirred tanks: the power model and the momentum model. The power model predicts the mixture Njs based on the sum of the power required to suspend each individual solid phase. The momentum model predicts the mixture Njs based on the sum of the momentum required to suspend each individual solid phase. Both models were developed because the current design heuristic, which assumes the mixture Njs is the maximum Njs of the individual phases, does not adequately capture the effects of particle interactions in mixtures. The models were tested using experimental data for mixtures of various solids up to 27% solids loading in a stirred tank.

1. chemical engineering research and design 9 1 ( 2 0 1 3 ) 227–233
Contents lists available at SciVerse ScienceDirect
Chemical Engineering Research and Design
journal homepage: www.elsevier.com/locate/cherd
Prediction of just suspended speed for mixed slurries at
high solids loadings
Inci Ayrancia,∗
, Theodore Nga
, Arthur W. Etchells IIIb
, Suzanne M. Krestaa
a Department of Chemical and Materials Engineering, University of Alberta, Edmonton, Canada T6G 2V4
b Rowan 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
1. Introduction
In many solid–liquid mixing operations the main objective
is mass transfer between the two phases. To maximize the
mass transfer the entire surface area of the solids should be
exposed. This can be achieved by operating at the complete
off-bottom suspension condition. The key operating parame-
ter for this condition is the impeller speed, which is called the
just suspended speed (Njs). Njs is defined as the impeller speed
at which no solids remain stationary at the bottom of the tank
for more than 1 or 2 s (Zwietering, 1958). Solid–liquid mixing
is a power intensive operation, so accurate prediction of Njs is
important. Current correlations are limited to unimodal slur-
ries at low solids loadings, but many industrial slurries are
composed of mixtures of solids with varying densities and par-
ticle sizes at high concentrations. The gap between research
and industry is vast, and the need for an accurate design model
for mixed slurry Njs is clear.
Current correlations have significant limitations because
many parameters play an active role in solids suspension.
Abbreviations: A310, axial impeller provided by Lightnin; B, bronze; LG, large glass beads; Ni, nickel; PBT, pitched blade turbine; R, ion
exchange resin; S, sand; SG, small glass beads or specific gravity; UF, urea formaldehyde; wt%, weight percent.
∗
Corresponding author at: Department of Chemical and Materials Engineering, University of Alberta, 7th Floor ECERF, 9107-116 Street,
Edmonton, Alberta, Canada T6G 2V4. Tel.: +1 780 492 9221; fax: +1 780 492 2881.
E-mail address: iayranci@ualberta.ca (I. Ayranci).
Received 8 February 2012; Received in revised form 10 July 2012; Accepted 1 August 2012
Geometry is by far the most important factor. The effects of
impeller and tank diameter, impeller type, off-bottom clear-
ance of the impeller, shape of the tank bottom, and the
presence, 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, and
liquid density and viscosity (Nienow, 1968; Baldi et al., 1978).
The behavior of the particles is different when there are many
other particles around them; therefore, solids loading is also
very important (Ayranci and Kresta, 2011).
The large number of parameters affecting Njs makes it
difficult to determine a robust design correlation. The first cor-
relation was suggested by Zwietering (1958) and it is still the
correlation that is most often used in calculations.
Njs = S
g( s − L)
L
0.45
d0.2
p
0.1X0.13
D0.85
(1)
0263-8762/$ – see front matter © 2012 The Institution of Chemical Engineers. Published by Elsevier B.V. All rights reserved.
http://dx.doi.org/10.1016/j.cherd.2012.08.002

2. 228 chemical engineering research and design 9 1 ( 2 0 1 3 ) 227–233
Nomenclature
Roman characters
C 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 number
N 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 number
Pjs 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 constant
T tank diameter (m)
Vz velocity in the axial direction
Wb baffle width (m)
xS mass fraction of the solids in the slurry
xL mass fraction of the liquid in the slurry
X 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)
Some of the parameters that affect Njs are included in this
correlation but the accuracy of the exponents has been ques-
tioned by many authors. Kasat and Pandit (2005) compiled the
different exponents on the common parameters suggested
by various authors. Their comparison showed that the Zwi-
etering correlation is still the one that predicts the data most
closely. The Zwietering correlation, however, does not provide
an answer for mixed slurry Njs.
The literature on mixed slurry suspension is only beginning
to be developed, and initial studies focused on dilute slurries.
Baldi et al. (1978) studied a mixture of glass beads with two
particle sizes and found that Njs can be predicted using an
average particle size, at low solids loadings. Montante and
Magelli (2007) did a computational study on the distribution
of solids for dilute slurries with two solid phases which have
different densities but same particle sizes. They showed that
the two solids phases are not affected by each other. Recently
Ayranci 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 phase
may significantly affect the mixture Njs. This effect is ampli-
fied for mixtures above 20 wt% solids, because at that point
the particle–particle interactions start to dominate. The par-
ticle sizes and the densities of the two solid phases play an
important role in the mixture Njs.
The current design heuristic for mixed slurries is to assume
that the mixture is composed of only the particle fraction that
is hardest to suspend. The Njs for that fraction is predicted
using the Zwietering correlation, and treated as the mixture
Njs. This design heuristic has many flaws, some of which were
shown by Ayranci and Kresta (2011). Of the five mixtures they
tested, only one mixture followed the design heuristic up to
high solids loadings, and a second mixture followed it up to
13 wt%, but then failed. The other mixtures did not follow the
design heuristic. The ratio of the particle size, the particle den-
sity, and the solids loadings of the two solid phases all had an
effect on mixture Njs. A more robust and physically realistic
model for predicting mixture Njs is needed.
In this study we propose and test two models that are based
on 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 for
a 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 and
the maximum value should be used as the mixture Njs. The
unimodal slurry Njs is predicted from the Zwietering correla-
tion (Eq. (1)). In the example the unimodal slurry Njs is 318 rpm
for 1.5 wt% SG and 1142 rpm for 1.5 wt% B. The mixture Njs is
the maximum of the two values, which is 1142 rpm.
2.2. Power model
The power model is proposed based on a hypothesis that the
power required to suspend a mixture is the sum of the power
required 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 first
and the second solid phases, respectively. The power required
to suspend each solid phase is calculated at the just suspended
condition based on the unimodal slurry density:
Pjs = slNjs
3
D5
Np (4)
sl =
1
(xs/ s) + (xL/ L)
(5)

3. chemical engineering research and design 9 1 ( 2 0 1 3 ) 227–233 229
In combining Eqs. (3) and (4) to find Njs,mix, the impeller
diameter term, D, cancels out. Current practice is to use the
slurry density to correct for the presence of the solids and
assume that the power number is constant, which also allows
us to eliminate Np. This assumption has some uncertainty
due to the presence of a low concentration layer at the top
of the vessel which will increase the solids concentration in
the bottom of the vessel, and the possibility of a lower solids
concentration in the vicinity of the impeller due to centrifugal
forces. Micheletti et al. (2003) and Jafari et al. (2012) inves-
tigated whether there is an effect of solids concentration,
particle size, and particle type on the power number. Jafari
et al. (2012) found that in general the power number decreases
at high solids loadings by of the order of 20%, while Micheletti
et al. (2003) found that it either stays the same, or increases by
about 20%. Our measurements of Np at varying Re for single
phase and 25 wt% small glass beads showed that the power
number for the single phase and the slurry are almost the
same. The assumption that the power number remains the
same for each slurry (Pjs,mix, Pjs,1, and Pjs,2) was applied, with
the understanding that this may introduce some error into the
model.
When D and Np are cancelled out the mixture Njs becomes
a function of the densities of the mixed and the unimodal
slurries and the Njs’s of the unimodal slurries.
Njs,mix =
sl,1N3
js,1
+ sl,2N3
js,2
sl,mix
1/3
(6)
In Eq. (6), Njs,1 and Njs,2 can be calculated using Eq. (1), or
replaced with the experimental values.
It should be noted that the power model does not include
any terms to take the particle–particle interactions into
account; therefore, it is very likely that the mixture Njs will not
be accurately predicted when particle–particle interactions are
strong.
2.3. Momentum model
A second hypothesis is that the momentum required to sus-
pend a mixture is equivalent to the sum of the momentum
required to suspend each of the solid phases in the mixture.
Mjs,mix = Mjs,1 + Mjs,2 (7)
where Mjs,mix is the momentum required to suspend the
mixture, and Mjs,1 and Mjs,2 are the momentum required to
suspend each individual unimodal slurry. The momentum,
M, can be calculated through the dimensionless momentum
number (Mo) (Machado et al., 2011):
Mo =
D/2
0 LV2
z 2 rdr
LN2D4
=
M
LN2D4
(8)
The momentum required to suspend each solid phase is
calculated at just suspended conditions:
Mjs = Mo slN2
jsD4
(9)
In combining Eqs. (7) and (9) to find Njs,mix, the momen-
tum number and impeller diameter are constant, so the terms
cancel out. Like the power model, the mixture Njs is thus a
Fig. 1 – The experimental setup with a PBT impeller. Njs is
determined by visual observation below the tank bottom.
function of the mixed and unimodal slurry densities and the
Njs of the unimodal slurries, this time to the power of two:
Njs,mix =
sl,1N2
js,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 cm
was used for the measurements. The cylindrical tank was
placed inside a square tank to prevent optical distortion. In
order to maintain stability at high impeller speeds, both tanks
were bolted to a steel frame. The just suspended speed was
observed visually from the bottom of the tank.
A Lightnin A310 impeller and a four bladed 45◦ down
pumping PBT both with a diameter of D = T/3 were used. The
impellers were attached to a shaft with a diameter of 1.27 mm
(T/20). The off-bottom clearance was defined as the distance
between the bottom of the impeller hub and the bottom of
the tank. The blades were flush with the bottom of the hub
for both impellers. The dimensionless off-bottom clearance,
C/T, was varied from 0.15 to 0.33. Water was used as the liquid
phase 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

4. 230 chemical engineering research and design 9 1 ( 2 0 1 3 ) 227–233
Table 1 – Particle properties.
Type Size (␮m) Density
(kg/m3)
Vt (m/s)
Nickel (Ni) 61–104 8900 0.139
Small glass beads (SG) 74–125 2500 0.066
Urea formaldehyde (UF) 150–250 1323 0.044
Bronze (B) 150–297 8855 0.225
Sand (S) 350–500 2656 0.144
Large glass beads (LG) 595–841 2500 0.177
Ion exchange resin (R) 677 1370 0.086
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). The
mixtures tested are given in Table 2 along with the ranges of
solids loadings. For each data set the mass of the more dense
particles was kept constant while the mass of the less dense
particles increased. A set of experiments where the mass of
the dense particle is higher than the mass of the less dense
particle was also conducted to validate the models tested for
all cases. This set of experiments was for the mixture of R with
LG at C/T = 0.25.
At the beginning of every experiment, the tank was filled
with water and particles were weighed and poured into it.
The liquid height was then adjusted to give H = T. The shaft
was attached to the motor, and the off-bottom clearance
was adjusted. Once the desired clearance was achieved, the
impeller was started. The impeller speed was increased in
steps and the system was left for 2 min to reach steady state.
After steady state was reached, the particle behavior at the
bottom of the tank was observed for 15–45 s to determine
whether Njs was reached. The particles behind the baffles
were consistently the last particles to be suspended. The just
suspended speed was reached when no particles remained
stationary at the bottom of the tank for more than 1 or 2 s
(Zwietering, 1958). After that the motor was switched off and
the off-bottom clearance was adjusted for the new measure-
ment. More details about the experimental setup and the
procedure 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 the
power and momentum models is tested for all six mixtures of
solids, two impellers, and varying off-bottom clearances, and
the performance of the two models is compared. The power
and the momentum models require the use of unimodal slurry
Njs’s. Initially the unimodal slurry Njs’s are calculated from the
Zwietering correlation, and then the experimental values are
used. The performance of the power model is compared for
the two cases.
0
200
400
600
800
1000
1200
1400
1600
1800
2000a
b
0 200 400 600 800 1000 1200 1400 1600 1800 2000
Njs,max(rpm)
Njs, measured (rpm)
SG+B C/T=0.15
SG+B C/T=0.25
SG+B C/T=0.33
SG+Ni C/T=0.15
SG+Ni C/T=0.25
SG+Ni C/T=0.33
R+B C/T=0.15
R+B C/T=0.25
R+B C/T=0.33
LG+B C/T=0.15
LG+B C/T=0.25
LG+B C/T=0.33
R+LG C/T=0.15
R+LG C/T=0.25
R+LG C/T=0.33
Standard deviation: 10%
PBT - Current design heuristic
=
0
200
400
600
800
1000
1200
1400
1600
1800
2000
0 200 400 600 800 1000 1200 1400 1600 1800 2000
Njs,max(rpm)
Njs, measured (rpm)
SG+B C/T=0.15
SG+B C/T=0.25
SG+B C/T=0.325
SG+Ni C/T=0.15
SG+Ni C/T=0.25
SG+Ni C/T=0.325
R+B C/T=0.15
R+B C/T=0.25
R+B C/T=0.325
LG+B C/T=0.15
LG+B C/T=0.25
R+LG C/T=0.15
R+LG C/T=0.25
R+LG C/T=0.325
UF+S C/T=0.25
Standard deviation: 12.6%
A310 - Current design heuristic
=
Fig. 2 – The parity plot between the current design heuristic
and the experimental data. The current design heuristic
uses the maximum Njs in the mixture, calculated using the
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 the
Zwietering correlation to the experimental mixture Njs. In the
Zwietering correlation, S is a function of impeller and tank
geometry and particle properties. Fig. 2a and b represents the
best possible predictions using the current form of the Zwi-
etering correlation since the S values were obtained for the
specific particles and the geometries used here (Ayranci and
Kresta, 2011). For each mixture the wt% of the more diffi-
cult to suspend solids remains constant as the wt% of the
easier 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 concentration
to overtake Njs,max. In Fig. 2a the mixture Njs for R with LG at
Table 2 – Particle mixtures and solids loadings.
Less dense particles (wt%) Denser particles (wt%) Total solids loading Density ratio Particle size ratio
wt% vol%
SG (1.5–26) B (1.5–1.3) 3–27.3 0.8–12 ∼1:3.5 ∼1:2
LG (1.5–26) B (1.5–1.3) 3–27.3 0.8–12 ∼1:3.5 ∼3:1
R (1.5–26) B (1.5) 3–27.5 1.3–21.8 ∼1:6.5 ∼3:1
R (1.5–25) LG (1.5–1.4) 3–26.4 1.7–20 ∼1:1.8 ∼1:1
SG (1.5–26) Ni (1.5–1.3) 3–27.3 0.77–14.9 ∼1:3.6 ∼1:1
UF (1–10) S (1–5) 2–15 1.1–9.1 ∼1:2 ∼1:2

5. chemical engineering research and design 9 1 ( 2 0 1 3 ) 227–233 231
Table 3 – The list of S values used in the calculations for
unimodal slurry Njs.
Impeller D C Sa
A310 T/3
0.15 6.84
0.25 7.54
0.325 7.97
PBT T/3
0.15 5.4
0.25 6.18
0.33 7.15
a
The S values were taken from Ayranci and Kresta (2011).
C/T = 0.15 remains constant even though the concentration of
R changes at each experimental point and the experimental
Njs does in fact increase. The predicted mixture Njs is similar
to the experimental data at the lowest solids concentration,
but is consistently lower than the experimental Njs when the
concentration increases. The current design heuristic fails
to capture the physics behind mixed solids suspension. The
effect of the presence of both solid phases must be included
in the model. The standard deviation between the measured
and the predicted values for all of the mixtures at varying off-
bottom clearances is 10% for the PBT and 12.6% for the A310.
The fact that the trend does not follow the experimental data
is of greater concern.
0
200
400
600
800
1000
1200
1400
1600
1800
2000a
b
0 200 400 600 800 1000 1200 1400 1600 1800 2000
Njs,predicted(rpm)
Njs, measured (rpm)
SG+B C/T=0.15
SG+B C/T=0.25
SG+B C/T=0.33
SG+Ni C/T=0.15
SG+Ni C/T=0.25
SG+Ni C/T=0.33
R+B C/T=0.15
R+B C/T=0.25
R+B C/T=0.33
LG+B C/T=0.15
LG+B C/T=0.25
LG+B C/T=0.33
R+LG C/T=0.15
R+LG C/T=0.25
R+LG C/T=0.33
Standard deviation: 8.3%
PBT - Power Model
(using Zwietering for unimodal Njs)
= +
0
200
400
600
800
1000
1200
1400
1600
1800
2000
0 200 400 600 800 1000 1200 1400 1600 1800 2000
Njs,predicted(rpm)
Njs, measured (rpm)
SG+B C/T=0.15
SG+B C/T=0.25
SG+B C/T=0.325
SG+Ni C/T=0.15
SG+Ni C/T=0.25
SG+Ni C/T=0.325
R+B C/T=0.15
R+B C/T=0.25
R+B C/T=0.325
LG+B C/T=0.15
LG+B C/T=0.25
R+LG C/T=0.15
R+LG C/T=0.25
R+LG C/T=0.325
UF+S C/T=0.25
Standard deviation: 9.7%
A310 - Power Model
(using Zwietering for unimodal Njs)
Fig. 3 – The prediction of mixture Njs without any
experimental data using the power model and the
Zwietering correlation with the (a) PBT and (b) A310.
4.2. Power model and momentum model using
Zwietering unimodal Njs
To find the mixture Njs through the power or momentum mod-
els, the slurry densities are first calculated from Eq. (5). The
unimodal slurry Njs’s are calculated from the Zwietering cor-
relation (Eq. (1)). The S values used in the calculations are given
in Table 3. The mixture Njs is then calculated from Eq. (6) for
the power model, and from Eq. (10) for the momentum model.
Fig. 3a and b shows the power model parity plots for the PBT
and the A310. While some data points are on the parity line, a
similar trend to the current design heuristic (Fig. 2a and b) is
seen: 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 no
significant improvement from the current design heuristic to
the 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 trend
shows that the physics is not captured.
Fig. 4a and b shows the momentum model parity plots
for the PBT and the A310 impellers. The trend in these fig-
ures is very similar to the power model, and also the current
design heuristic. This may indicate that there is no signifi-
cant difference between the two models and the heuristic. It
should, however, be noted that these predictions use the Zwi-
etering correlation for unimodal slurry Njs’s. The Zwietering
0
200
400
600
800
1000
1200
1400
1600
1800
200a
b
0
0 200 400 600 800 1000 1200 1400 1600 1800 2000
Njs,predicted(rpm)
Njs, measured (rpm)
SG+B C/T=0.15
SG+B C/T=0.25
SG+B C/T=0.33
SG+Ni C/T=0.15
SG+Ni C/T=0.25
SG+Ni C/T=0.33
R+B C/T=0.15
R+B C/T=0.25
R+B C/T=0.33
LG+B C/T=0.15
LG+B C/T=0.25
LG+B C/T=0.33
R+LG C/T=0.15
R+LG C/T=0.25
R+LG C/T=0.33
Standard deviation: 8.8%
PBT - Momentum model
(using Zwietering for unimoldal Njs)
0
200
400
600
800
1000
1200
1400
1600
1800
2000
0 200 400 600 800 1000 1200 1400 1600 1800 2000
Njs,predicted(rpm)
Njs, measured (rpm)
SG+B C/T=0.15
SG+B C/T=0.25
SG+B C/T=0.325
SG+Ni C/T=0.15
SG+Ni C/T=0.25
SG+Ni C/T=0.325
R+B C/T=0.15
R+B C/T=0.25
R+B C/T=0.325
LG+B C/T=0.15
LG+B C/T=0.25
R+LG C/T=0.15
R+LG C/T=0.25
R+LG C/T=0.325
UF+S C/T=0.25
Standard deviation: 8.8%
A310 - Momentum model
(using Zwietering for unimodal Njs)
Fig. 4 – The prediction of mixture Njs without any
experimental data using the momentum model and the
Zwietering correlation with the (a) PBT and (b) A310.

6. 232 chemical engineering research and design 9 1 ( 2 0 1 3 ) 227–233
correlation is known to have limited accuracy above 10 wt%
solids, so above this concentration the predictions are not very
reliable. In order to test the true strength of the power and
momentum models, the experimental unimodal slurry Njs’s
are used in a second test.
4.3. Power model and momentum model using
experimental data
Fig. 5a and b shows the mixture Njs predicted from power
model where the unimodal slurry Njs is obtained from experi-
ments. These plots show a completely different trend than the
predictions using Zwietering unimodal slurry Njs’s. The pre-
dicted Njs’s follow the parity line closely. Most of the data is
within ±20% of the parity line. This indicates that the physics
of the solids suspension is captured up to 20 wt% solids for
all mixtures. Beyond 20 wt% solids, particle–particle interac-
tions can become quite strong. For the LG + B, R + B, and R + LG
mixtures Njs increases with increasing solids and the model
captures 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 + B
mixture there is an unexpected drop in Njs above 20 wt% SG
(Ayranci and Kresta, 2011) and for SG + Ni Njs is constant. The
power model cannot predict these effects because there are no
terms 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. 5a
and b and from subsequent analysis. The resulting standard
0
200
400
600
800
1000
1200
1400
1600
1800
2000a
b
0 200 400 600 800 1000 1200 1400 1600 1800 2000
Njs,predicted(rpm)
Njs, measured (rpm)
SG+B C/T=0.15
SG+B C/T=0.25
SG+B C/T=0.33
SG+Ni C/T=0.15
SG+Ni C/T=0.25
SG+Ni C/T=0.33
R+B C/T=0.15
R+B C/T=0.25
R+B C/T=0.33
LG+B C/T=0.15
LG+B C/T=0.25
LG+B C/T=0.33
R+LG C/T=0.15
R+LG C/T=0.25
R+LG C/T=0.33
Standard deviation: 9.6%
PBT - Power model
= +
0
200
400
600
800
1000
1200
1400
1600
1800
2000
0 200 400 600 800 1000 1200 1400 1600 1800 2000
Njs,predicted(rpm)
Njs, measured (rpm)
SG+B C/T=0.15
SG+B C/T=0.25
SG+B C/T=0.325
SG+Ni C/T=0.15
SG+Ni C/T=0.25
SG+Ni C/T=0.325
R+B C/T=0.15
R+B C/T=0.25
R+B C/T=0.325
LG+B C/T=0.15
LG+B C/T=0.25
R+LG C/T=0.15
R+LG C/T=0.25
R+LG C/T=0.325
UF+S C/T=0.25
Standard deviation: 9%
A310 - Power model
= +
Fig. 5 – The parity plot for the power model at varying
clearances for all mixtures with the (a) PBT and (b) A310.
0
200
400
600
800
1000
1200
1400
1600
1800
2000a
b
0 200 400 600 800 1000 1200 1400 1600 1800 2000
Njs,predicted(rpm)
Njs, measured (rpm)
SG+B C/T=0.15
SG+B C/T=0.25
SG+B C/T=0.33
SG+Ni C/T=0.15
SG+Ni C/T=0.25
SG+Ni C/T=0.33
R+B C/T=0.15
R+B C/T=0.25
R+B C/T=0.33
LG+B C/T=0.15
LG+B C/T=0.25
LG+B C/T=0.33
R+LG C/T=0.15
R+LG C/T=0.25
R+LG C/T=0.33
Standard deviation: 17.3%
PBT - Momentum model
= +
0
200
400
600
800
1000
1200
1400
1600
1800
2000
0 200 400 600 800 1000 1200 1400 1600 1800 2000
Njs,predicted(rpm)
Njs, measured (rpm)
SG+B C/T=0.15
SG+B C/T=0.25
SG+B C/T=0.325
SG+Ni C/T=0.15
SG+Ni C/T=0.25
SG+Ni C/T=0.325
R+B C/T=0.15
R+B C/T=0.25
R+B C/T=0.325
LG+B C/T=0.15
LG+B C/T=0.25
R+LG C/T=0.15
R+LG C/T=0.25
R+LG C/T=0.325
UF+S C/T=0.25
Standard deviation: 15.7%
A310 - Momentum model
= +
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 separate
impellers, 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, but
over-predicts the mixture Njs, leaving more data points outside
the ±20% range. The standard error of the momentum model
prediction is 17.3% for the PBT and 15.7% with the A310. In
Fig. 7 comparison of the power and momentum models with
both the PBT and the A310 shows that the momentum model
consistently over-predicts mixture Njs. The standard devia-
tion between the two models is 6.4% when the data for both
impellers is combined. We conclude that the power model
provides a better prediction of mixture Njs.
The performance of the power model has been analyzed in
terms of parity plots up to this point. This allows the compari-
son of model accuracy for different data sets, but prevents the
visibility of some details, such as the effect of solids loading. A
closer look at the raw data is required to observe these details.
Fig. 8 compares the power model predictions to experimental
data and the current design heuristic for two representative
cases at C/T = 0.25. As the solids loadings increase the mixture
Njs 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

7. chemical engineering research and design 9 1 ( 2 0 1 3 ) 227–233 233
0
200
400
600
800
1000
1200
1400
1600
1800
2000
0 200 400 600 800 1000 1200 1400 1600 1800 2000
Njs-MomentumModel(rpm)
Njs - Power Model (rpm)
PBT
A310
Standard deviation: 6.4%
PBT and A310
Momentum vs Power model
Fig. 7 – Comparison of the momentum model and the
power model at varying clearances for all particles with the
PBT 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 from
experiments, power model, and current design heuristic for
LG + B and R + LG mixtures with PBT at C/T = 0.25.
over-predicted Njs,mix. This figure shows that the power model
tends to over-predict mixture Njs, but provides a significant
improvement over the current design heuristic, shown as the
dashed lines. If the heuristic was used for design, a signifi-
cant number of particles would not be suspended. The power
model, in all cases, provides conservative design but never
beyond 20% error.
Reviewing the data, the current design heuristic is rejected
because the trends are not captured. The momentum model
clearly over-predicts Njs,mix, based on Fig. 7. The power model
is recommended with use of experimental unimodal data
where possible.
5. Conclusions
The objective of this study was to propose and test two mod-
els to accurately predict mixed slurry Njs. Analysis of the
experimental data for several mixtures at varying off-bottom
clearances and solids loadings for two impeller geometries led
to the following conclusions:
• The current design heuristic is inadequate for the prediction
of mixture Njs since it ignores the addition of a second solid
phase, and cannot predict the basic trend.
• The momentum model consistently over-predicts Njs and is
rejected.
• When the Zwietering correlation is used for unimodal slurry
Njs the power and momentum models lose strength and
show similar behavior to the current design heuristic. The
authors recommend use of experimental unimodal slurry
Njs 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 unimodal
slurry Njs’s are obtained from experiments.
Njs,mix =
sl,1N3
js,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, and
Maria Garcia from Rowan University for sharing data for sand
and urea formaldehyde.
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