The suspension of solid particles in stirred tanks is involved in
many industrial applications such as slurry flow, catalytic reactions,
adsorption, crystallization, dissolution, and many other industrial
applications. Due to its industrial importance the subject has been
extensively studied. Zwietering (1958) expressed the value of the
minimum speed required for complete suspension "[N.sub.js]" as
a function of the vessel dimensions, solids concentration and density,
particles size, position of the stirrer, liquid viscosity, and liquid
density. He identified "[N.sub.js]" as the speed at which the
particles remain at the bottom of the tank for no more than 1 or 2 s,
which was called one-second criterion and it was widely used to
ascertain the achievement of complete suspension conditions. [N.sub.js]
was expressed by:
[N.sub.js] = S [[mu].sup.0.1.sub.L]
The one-second criterion was used by Nienow (1968), Raghava et al.
(1988), Myers and Julian (1992) and Myers et al. (1994). Armenante et
al. (1992) studied the effect of using multiple impellers on the minimum
speed required for complete suspension and it was observed that in case
of using disc turbine (DT) impeller, the power as a function of
[N.sub.js] varied when changing the number of impellers. Myers et al.
(1996) studied experimentally the effect of impeller type and impeller
clearance ratio C/T on flow patterns and agitator performance in
turbulent flow using coarse solid particles.
Armenante et al. (1998) tested the effects of impeller clearance
and diameter as well as other variables on minimum speed required for
complete suspension, [N.sub.js], and the results were close to
Zwietering's correlation. Sharma and Shaikh (2002) studied the
effect of C/T and D/T on the critical speed required for complete
suspension and the corresponding power. They identified three regions
clearly. First region when C/T < 0.1, [N.sub.js] and the
corresponding power consumption remained constant with C/T. The second
region when 0.1 < C/T < 0.35, [N.sub.js] and the corresponding
power consumption became a function of impeller clearance with
increasing impeller height; the induced secondary loop which was almost
stagnant got wider, so it needed extra energy to force particles out of
It was noticed that the flow pattern remained as single eight in
the first and second regions. For the third region, when C/T > 0.35,
the slope of [N.sub.js] was higher than that of the second region. And
the flow pattern changes to double eight (typical as the flow pattern
generated by a radial flow impeller).
Micale et al. (2002) measured [N.sub.js] by means of a pressure
gauge. This technique depends on the fact that when solids are suspended
in the liquid the density of the mixture will be higher than that for
pure liquid. Therefore, if the static pressure was measured at the
bottom of the tank the more the solids are suspended the higher the
pressure recorded. A relation between the increase in the static head
due to solid suspension and the mass of suspended solids was
[DELTA]p = [M.sub.s](1 - [[rho].sub.L]/[[rho].sub.s])g/[A.sub.b]
They suggested that by dividing the static head, P, by [P.sub.js],
the fraction of solids suspended ([M.sub.s]/[M.sub.tot]) can be
They correlated their data in order to find the sufficient speed
[N.sub.ss] which is the speed at which 98% of solids are suspended
(which may be considered as nearly equivalent to [N.sub.js]) was given
[N.sub.ss] = 24.1[d.sup.0.428.sub.p][B.sup.0.13] (3)
It can be seen that the method presented by Micale et al. (2002)
was important in measuring [N.sub.js] but it was carried out on a model
type vessel used for this purpose.
In the present work the method of Micale et al. (2002) to determine
[N.sub.js] was applied in a conical bottom tank which is widely used in
slurry transportation systems. Also, real ores as phosphate and glass
sand were used as solid particles due to their importance in mining
industry instead of using model particles as glass balloteni.
TEST RIG AND EXPERIMENTS
Experiments were carried out using conical bottom tank with
dimensions (T = 0.97 m, H = 1.34 m, [L.sub.c] = 0.3 m and cone angle is
60[degrees]) as shown in Figure 1. Four baffles were fixed 90[degrees]
apart on the tank wall to prevent vortices. A hole was opened at the
bottom of the tank (at cone apex) and was connected to an inclined
manometer filled with water. Slurry was prevented from entering the
manometer by using steal net which was welded at the entrance of the
The agitation was maintained by a standard four pitched blade
impeller with diameter D = 0.3 m. The clearance ratio C/T was either
0.22 or 0.4, by using different shafts.
Phosphate fine particles ([d.sub.p] < 74 [micro]m), phosphate
coarse particles ([d.sub.50] = 300 [micro]m) and sand glass ([d.sub.50]
= 300 [micro]m) were used as solid particles with concentrations 6 % and
10 by weight. Water was used as the carrier liquid in all experiments.
All the solids used had an average density [rho] = 2650 kg/[m.sup.3].
The impeller was rotated by variable speed motor from zero to a
value higher than [N.sub.js] calculated from Zwietering's
correlation and after steady state (steady manometer reading) the
pressure was measured.
[FIGURE 1 OMITTED]
It must be noted here that the value of S parameter in
Zwietering's correlation was 7 for C/T = 0.4 and 7.5 for C/T =
RESULTS AND DISCUSSION
Before presenting the experimental results the theory of the
technique to determine [N.sub.js] using pressure measurements employed
in the present work must be clarified.
The solid particles sedimented at the bottom of the tank behave
like a porous medium, therefore, the level of the liquid in the tank and
the level in the manometer tube are the same at unagitated condition.
However, when the impeller starts rotating, solid particles will be
gradually suspended and the density of suspension will start increasing.
It must be born in mind that due to vortex formation a negative dynamic
head is present and therefore the total pressure measured by the
manometer will be the summation of hydrostatic pressure and dynamic
pressure as given by the following equation:
Manometer read pressure = hydrostatic pressure [+ or -] dynamic
In Equation (4) the [+ or -] sign was put to show that the dynamic
pressure can be either added or subtracted from the hydrostatic pressure
according to the direction of rotation of the impeller as will be shown
in the results.
Figure 2 shows how to derive [N.sub.js] from measured pressures at
the tank bottom. The hydrostatic pressure which will result if all
particles are in suspension ([P.sub.js]) is first plotted. The data
points obtained (pressures measured at different speeds), curve A are
put on the same graph. In order to consider the effect of the dynamic
head, the last two points of the measured pressure data together with
the point of [P.sub.js] at N = 0 are fitted to a curve B. The difference
between [P.sub.js] and curve B will be added to the curve fitted to the
data points, curve A, resulting in a curve C which at a certain point
will be tangential to [P.sub.js] line. The point of tangency will be
considered as [N.sub.js].
[FIGURE 2 OMITTED]
For sand glass with average weight concentration 6%, Figure 3 shows
the results of mixture pressure while varying the impeller speed. From
Figure 3, increasing the impeller speed results in a decrease in the
measured pressure. These results are contradicting with Micale et al.
(2002) results. This can be explained by the difference in tank
geometries especially with regard to the pressure measurement method.
Micale et al. (2002) used a hole in the tank bottom and thus eliminating
the weight of settled solids from readings. Also the pressure
transmitted was the static pressure only. In the present work the
manometer senses the weight of unsuspended solids and the dynamic head.
Back to Figure 3, the squares represent the values of measured
pressure. If these values were added to the dynamic head of pure water
(not shown in Figure 3) the result will be the solid squares curve which
is lower than the value of [P.sub.js] (also found earlier by Micale et
al. (2002)). For this reason, following the same reasoning by Micale et
al. (2002) the difference in values between measured data points at high
impeller speeds and [P.sub.js] will represent the true dynamic head of
the mixture. At each point summation of the difference between
[P.sub.js] and the measured P to the measured head results in the curve
denoted by solid triangles. This curve reaches a plateau at a value of
speed equal to [N.sub.js] using the same reasoning by Micale et al.
(2002) It must be noted that the shape of the curve in the present work
did not attain a typical S shape due to the narrow speed range before
[N.sub.js] was obtained.
[FIGURE 3 OMITTED]
Figure 4 shows the results obtained for sand glass with average
weight concentration of 10%. The curves show similar trend to that of
Figure 3 but the value at which [N.sub.js] reached a plateau is
different. Figure 5 shows results for fine phosphate with average weight
concentration of 10%. From this figure similar trends are noticed with
variation of [N.sub.js] value obtained. Figures 6 and 7 show the results
obtained from experiments carried out on coarse phosphate with average
weight concentrations of 6% and 10%, respectively. The curves in these
two figures show similar trends to those found before and to each other
with different values of [N.sub.js] obtained.
From results obtained for sand glass in Figures 3 and 4 and results
obtained for phosphate (fine and coarse) in Figures 5 to 7 it is noticed
from comparing these sets of curves for the two different materials that
the relation between the pressure P and the speed N is steep for sand
glass and shallow for phosphate material. This can be explained to be
due to the variation of rheology as a result of the presence of an
appreciable amount of fine particles in phosphate compared to sand glass
despite that the coarse phosphate has similar average size to that of
The presence of fine particles decreases the effect of dynamic head
due to the increase of apparent viscosity and this effect increases by
the increase of the amount of solids suspended. The values of the
pressure are expected to vary as a function of fines percentage in the
[FIGURE 4 OMITTED]
[FIGURE 5 OMITTED]
[FIGURE 6 OMITTED]
[FIGURE 7 OMITTED]
Table 1 compares between results from the present work and
Zwietering's correlation for C/T ratio = 0.4 and the impeller was
The table shows that the maximum difference between results from
present work and those by Zwietering did not exceed 17% for fine
phosphate. For sand glass and coarse phosphate, the maximum difference
did not exceed 7.14%.
A graph can be constructed to assess the amount of solids suspended
as a ratio of P/[P.sub.js] as early suggested by Micale et al. (2002),
but in the present case there is an initial pressure read by the
manometers due to the weight of settled solids at N = 0. Therefore, the
value of this initial pressure, [P.sub.i], must be subtracted from all
readings and from [P.sub.js] to be able to use the same reasoning in
order to determine the fraction of suspended solids as a function of
X = P - [P.sub.i]/[P.sub.js] - [P.sub.i] (5)
[FIGURE 8 OMITTED]
Figure 8 shows the results of the fraction of solids suspension as
a function of impeller speed for all materials used in the experiments
in the present work. From this figure the materials which contain fine
particles are suspended more easily (phosphate material). Also, it was
found that increasing the solids concentration retards the process of
The technique of [N.sub.js] determination using pressure
measurements was used to study the effect of impeller clearance and
direction of rotation on such important parameter.
Experiments were carried out using two clearances C/T = 0.22 and
0.4. The direction of rotation was varied in order that the impeller is
either pumping upward or downward.
The minimum speed required for solids suspension was determined
using the pressure gauge technique. Figure 9 shows that [N.sub.js] for
the impeller with C/T = 0.22 and pumping upward is 288 rpm. When the
impeller rotational direction was reversed (i.e., pumping downward), for
the same clearance, Figure 10 shows that [N.sub.js] decreased to 260
rpm. It must be noticed from comparing Figures 9 and 10 that the trend
of measured pressure had changed. In Figure 9 the pressure tends to
decrease when the impeller speed was increased. The opposite was found
as shown in Figure 10. This can be explained due to the change of
dynamic head effect as a function of the direction of rotational speed.
When the impeller was pumping upward the fluid velocities near the
pressure tap increased thus reducing the pressure readings as the
impeller speed was increased. When the impeller was pumping downward the
fluid is directed towards the pressure tap where it impinges the tank
bottom and converts the velocity into pressure thus increasing the read
[FIGURE 9 OMITTED]
[FIGURE 10 OMITTED]
[FIGURE 11 OMITTED]
[FIGURE 12 OMITTED]
[FIGURE 13 OMITTED]
Figure 11 shows that [N.sub.js] for impeller with C/T = 0.4 and
pumping up is 295 rpm. For the same clearance and for the impeller
pumping downward [N.sub.js] was reduced to 280 rpm as shown from Figure
The same trend of pressure as a function of rotational speed was
found as discussed earlier.
Table 2 shows a comparison between the obtained results of
[N.sub.js] and [N.sub.js] calculated from Zwietering's correlation
In order to test the validity of the results obtained from the
present work, [N.sub.js] values were compared to those calculated by
Zwietering's correlation. Result of this comparison is shown in
Figure 13. From this figure, good agreement between results from present
work and those calculated by Zwietering's correlation are found.
The least agreement was found for fine phosphate which can be explained
as due to the lack of experimental results for Zwietering for such fine
The method of using pressure measurement to determine [N.sub.js]
reported by Micale et al. (2002) for model tank was extended to use in
real tanks successfully in the present study.
The rheology of the mixture was found to affect the value of
dynamic head. Yet, this effect does not affect the final [N.sub.js]
values which were compared to Zwietering's correlation and
agreement within 17% was found.
The amount of suspended solids as a function of impeller speed
depends on the presence of fine particles in the suspended material. The
more the fines are present the higher the amount of suspended solids are
For the same clearance the direction of rotation changed the
[N.sub.js] with an average value of 8%. For a given direction of
impeller rotation the clearance ratio changed [N.sub.js] with an average
value of 5 % for C/T = 0.4 and 2 % for C/T = 0.22.
The authors would like to thank the mixing groups in the academic
years 2005/2006 and 2006/2007 for their effort in collecting the data
for this work as a part of their B.Sc. graduation projects. Y. S. Selima
would like to acknowledge the support of Ain Shams university for
sponsoring his M.Sc. project.
Manuscript received March 6, 2007; revised Manuscript received
September 15, 2007; accepted for publication November 2, 2007.
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Minimum Agitation Speed to Attain the Just Dispersed State in
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Suspension Conditions in Stirred Vessels by Means of Pressure Gauge
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Off-Bottom Clearance on the Solids Suspension Performance of Pitched
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Y. S. Selima, Y. S. Fangary * and N. A. Mahmoud Faculty of
Engineering, Mechanical Power Department, Ain Shams University, Cairo,
* Author to whom correspondence may be addressed. E-mail address:
[A.sub.b] tank bottom area ([m.sup.2])
B concentration by weight (percent)
C clearance of the impeller from tank bottom (m)
D impeller diameter (m)
[d.sub.50] particle diameter using 50% sieve analysis (m)
[d.sub.p] particle diameter (m)
g gravitational acceleration (m/[s.sup.2])
h the average distance between the stirrer and the top
surface of fillets where no particles are suspended (m)
H tank height (m)
[L.sub.c] cone height (m)
[M.sub.s] mass of suspended solids (kg)
[M.sub.tot] total mass of mixture (kg)
[N.sub.js] minimum speed required for complete suspension (rpm)
[N.sub.min] the value at which suspension starts (rpm)
[N.sub.span] twice its value gives the range of N at which most of
suspension takes place, the value of X = 0.982 (rpm)
[N.sub.ss] sufficient suspension speed = [N.sub.min] +
P measured pressure (Pa)
[P.sub.i] pressure at N = 0 (Pa)
[P.sub.js] the pressure when all solids are suspended (Pa)
S constant in Zwietering's correlation
T tank diameter (m)
X fraction of solids suspended
[DELTA]p the increase in the static head due to solid suspension
[DELTA][rho] difference in density between solids and liquid
[[mu].sub.L] liquid dynamic viscosity (Pa s)
[[rho].sub.L] liquid density (kg/m3)
[[rho].sub.s] solid density (kg/m3)
Table 1. Comparison between results of [N.sub.js] from present work and
that predicted by Zwietering's correlation
concentration Present work Zwietering
Phosphate coarse grade 6 260 252
Phosphate coarse grade 10 265 272
Phosphate fine grade 10 240 205
Sand glass 6 270 252
Sand glass 10 278 272
Phosphate coarse grade 3.17
Phosphate coarse grade 2.57
Phosphate fine grade 17.07
Sand glass 7.14
Sand glass 2.20
Table 2. Comparison between results of [N.sub.js] from present work and
Zwietering's (1958) correlation to consider the effect of clearance
ratio and impeller direction of rotation on [N.sub.js]
CT ratio [N.sub.js] [N.sub.js] Difference
(present (Zwietering) (percent)
work) (rpm) (rpm)
0.22 (pumping up) 288 273 5.50
0.4 (pumping up) 295 293 0.68
0.22 (pumping down) 260 273 4.80
0.4 (pumping down) 280 293 4.43