Determination of minimum speed required for solids suspension in stirred vessels using pressure measurements.
The minimum speed required for complete suspension ([N.sub.js]) is a major parameter for solids suspension in stirred tanks. Micale et al. 7th UK Conference on Mixing (2002) determined [N.sub.js] by using a pressure gauge technique in a model vessel. In the present work [N.sub.js] was measured by the same later technique in a more practical vessel with varying C/T ratio and the impeller direction of rotation. The results were compared to those obtained by Zwietering Chem. Eng. Sci. 8, 244-253, (1958) correlation. Agreement was found between results from present work and predictions by Zwietering's correlation with maximum difference not exceeding 17%.

La vitesse minimale requise pour une suspension complete ([N.sub.js]) est un parametre fondamental pour la suspension de solides en reservoir agite. Micale et al. 7th UK Conference on Mixing (2002) ont determine [N.sub.js] par une technique de jauge de pression dans un reservoir modele. Dans le present travail, [N.sub.js] a ete mesuree par cette meme technique dans un reservoir plus pratique en faisant varier le rapport C/T et la direction de rotation de la turbine. Les resultats ont ete compares a ceux obtenus par la correlation de Zwietering, Chem. Eng. Sci. 8, 244-253, (1958). Un accord a ete trouve entre les resultats du present travail et les predictions par la correlation de Zwietering avec une difference maximale ne depassant pas 17%.

Keywords: stirred vessels, mixing, solids suspension, pressure measurements

Selima, Y.S.
Fangary, Y.S.
Mahmoud, N.A.
Pub Date:
Name: Canadian Journal of Chemical Engineering Publisher: Chemical Institute of Canada Audience: Academic Format: Magazine/Journal Subject: Engineering and manufacturing industries Copyright: COPYRIGHT 2008 Chemical Institute of Canada ISSN: 0008-4034
Date: August, 2008 Source Volume: 86 Source Issue: 4
Accession Number:
Full Text:

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] [d.sup.0.2.sub.p][(g[DELTA][rho]).sup.0.45] [B.sup.0.13]/[D.sup.0.85][[rho].sup.1.45.sub.L] (1)

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 this region.

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 mathematically derived:

[DELTA]p = [M.sub.s](1 - [[rho].sub.L]/[[rho].sub.s])g/[A.sub.b] (2)

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 obtained.

They correlated their data in order to find the sufficient speed [] which is the speed at which 98% of solids are suspended (which may be considered as nearly equivalent to [N.sub.js]) was given

[] = 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.


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 hole.

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.


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 = 0.22.


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 pressure (4)

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].


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 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 sand glass.

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 material.





Table 1 compares between results from the present work and Zwietering's correlation for C/T ratio = 0.4 and the impeller was pumping upward.

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 impeller speed:

X = P - [P.sub.i]/[P.sub.js] - [P.sub.i] (5)


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 suspension.

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 pressure.






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 12.

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 (1958).

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 particles.


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 resulted.

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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Y. S. Selima, Y. S. Fangary * and N. A. Mahmoud Faculty of Engineering, Mechanical Power Department, Ain Shams University, Cairo, Egypt

* 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)
[]     sufficient suspension speed = [N.sub.min] +
               2[N.sub.span] (rpm)
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

Greek Symbols

[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

                                                [N.sub.js] (rpm)
                          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
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