FLUID AND PARTICLE MOTION IN TURBULENT STIRRED TANKS Fluid Motion HENRY G . SCHWARTZBERG AND ,Vew York UniversiQ, >Vew York, A’. Y.
R O B E R T E. T R E Y B A L
Using streak photography, fluid velocities were measured in baffled tanks turbulently agitated by sixbladed turbiine impellers. Outside of the impeller discharge zone (in most of the tank) the average and r.m.s. fluctuating velocities were found to be proportional to the rotational speed of the impeller and its diameter squared, and inversely proportional to the cube root of the tank volume. This relationship was verified over a wide range of impeller and tank sizes, fluid properties, and agitator speeds, and agrees in form with predictions based on dimensional analysis of the power dissipation in such tanks. The similarity of the turbulent power-dissipation correlation-Le., constancy of the power number-for a large number of impeller-tank arrangements suggests that the present fluid velocity correlation may have broad applicability, may b e a suitable scale-up criterion in many instances. and that maintaining a constant value of ND2/(T2H)*la Further, this fluid velocity correlation helps explain various anomalies in correlations which have been proposed for heot and mass transfer in stirred tanks.
WIDE
range of mastransfer correlations have been pre-
A sented for particle’s suspended in stirred tanks (Aksel’rud, 1955; Calderbank and M o o Young, 1961 ; Hixson and Baum, 1941, 1942 ; Hixson arid Wilkens, 1933 ; Humphrey and Van Ness, 1957; Kneule, 1056; Mack and Marriner, 1949; Mattern et a / . , 1957; Nagata et a / . , 1960a; Nagata and Yamaguchi, 1960; Oyama and Endoh, 1956). Barker and Treybal (1960) and Harriot (1962) have critically reviewed many of these correlations and presenr. a good picture of the rather discordant state of knowledge in this area. The present work, a study of fluid and particle motion in turbulently agitated baffled vessels, represents a n attempt to correlate the hydrodynamic factors influencing such mass transfer. This study has been confined to the turbulent regime in fully baffled, cylindrical tanks agitated by six-bladed turbine impellers. I t thereby covers the range of practical interest in a commonly used type of equipment for which geometrically similar components are readily available, and for which good power requirement correlations (Rushton e t al., 1950) and a large body of mass-trarxfer data exist. Further, Cutter (1960), Nielsen (1958), and Kim and Manning (1964) have recently provided information as to the scale, intensity, and spectral energy distribution of turbulence in such vessels. Though the velocity characteristics of the “jet” stream discharged by turbine impellers have been widely studied and adequately correlated (Cutter, 1960; Kim and Manning, 1964; Nagata et al., 1959b, 1960b; Nielsen, 1958; Rushton et al., 1948, 1950; Rushton and Oldshue, 1953; Sachs, 1952; Sachs and Rushton, 1954; Tennant, 1952), relatively little is known about the fluid velocity in the remainder of the tank, which usually represents more than 90% of the active volume. T h e few studies which have been made of this bulk velocity (Aiba, 1958; Metzner and Taylor, 1960; Nagata e t al., 1959a, 1959b, 1960c; Taylor, 1955) have been either largely qualitative, or confined to the !streamline regime or to a single or very limited range of operating conditions. I t has been commonly assumed, either explicitly or implicitly, that the average bulk velocity is proportional to the impeller tip speed, ND. In the present work this assumption is shown to be unfounded.
Holmes et a / . (1964) have correlated circulation times in turbine-agitated baffled vessels. From their correlation it can be inferred that the characteristic bulk velocity is proportional to N D 2 / T , a result which is in substantial agreement with the present work. Theory
For geometrically similar flow configurations, it can be shown by dimensional analysis that P / Q , the time rate of fluid energy dissipation per unit volume, is governed by a relationship of the following type:
where U is a characteristic stream velocity, L a characteristic stream dimension-e.g., its diameter-and @(Re,) is some function of the stream Reynolds number. T h e nature of @(Re,) depends on the type of flow geometry involved and the particular characteristic velocity employed. T h e characteristic velocity used might be the average velocity, G, or the r.m.s. fluctuating velocity, u ’. For example, Batchelor (1959) has demonstrated that the decay of isotropic, grid-generated turbulence is governed by the following equation :
where the constant 1.65, equal to @(Res), is known only approximately, and Le is the Eulerian integral scale, a measure of the size of main energy-containing eddies. Though proof is lacking, it is felt that Equation 2 may be universal in natureLe., independent of geometry and the large scale turbulence generating mechanisms-at least for isotropic turbulence a t high Reynolds numbers. In other cases-for example, when the familiar Fanning equation is converted into power dissipation form-the characteristic velocity used is the average velocity, 6. Using the well established power number correlations for VOL. 7
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fully baffled tanks agitated by turbine impellers, it can be shown that, for turbulent conditions, the pawer dissipation per unit volume is independent of the impeller Reynolds number, Re,, and is given by the equation:
- _- 7'9 p f N 3 D s Q
Re,
&T2H
> 10,000
0)
The right-hand side of Equation 1 can be equated to the righthand side of Equation 3. Since Equation 3 is independent of Re,, for the same situation @(Re,) should be a constant. Thus: QpfU3 P 7.9 p,N3DS ___=-= (4)
Q
g,L
goPH
Any reasonable, consistent stream dimension can be used for L, provided Q is adjusted for the exact dimension used. the order of The width of the imoeller discharm stream is and proportional to the impeller-blade width, D / 5 . Substituting D / 5 for L in Equation 4 and solving for U, there is obtained:
U s -
CND2 ( T2H)113
Rei
> 10,000
(5)
where all the constant terms have been lumped into C. If Batchelor's turbulence dissipation equation is applicable, a quantitative prediction can be made for the fluctuating velocities. Cutter (1960) has shown that for turbine impellers La is approximately 0.4 D / 5 . When this value is substituted far L, in Equation 2 and the right-hand sides of Equations 2 and 4 are equated, one obtains upon solving far us': q' =
intense plane-collimated beam of light was used to illuminate either a vertical or a horizontal 1-inch wide slice of the tank. This illumination was pulsed by a shutter rotating a t one of several known measured speeds. Suspended iiluminated by the beam were photographed, yielding dashed streak imagcs. Typical photographs are shown in Figures 2 and3. The Cartesian component lengths of short segments (about 0.5 inch long) of these images, when projected to a known scale, were measured and divided by the time interval corresponding to the number of light pulses in the measured length, thus yielding Cartesian component velocities. Roughly 6000 photographs were taken. From a selected portion of these, some 20,000 streak images were measured. T h e resultant data, classified according to operating conditions and streak location, were entered on punched cards which are available for future use. Fluid velocities were measured by using 0.03-inch diameter white polystyrene tracer particles close to neutral buoyancy. An averaxe tracer particle settling velocity of 0.05 inch per second or less in still fluid was achieved by density fractionation of the particles and adjustment of the fluid density to about 1.033 grams per cc. by the addition of salt and a wetting agent. White Lucite particles were used as tracers in the viscous liquids. Sugar solution, adjustcd to the density of the Lucite, provided a medium with a viscosity of about 5 cp. A variety of impeller and tank sizes were tested. The tank sizes are given in Table I, and the impeller configuration and dimensions in Figure 4. I n all cases, four baffles of width 0.1 T were used.
Tank 1 2A 2B 3
0.73 N D 2 ~
(T'H)*I*
Table I. Tank Dimensions Diem&,, Fill &@I, Incher h h a 17.3 18.2 11.3 17.0 11.3 9.4 9.4 9.2
Where the turbulence is not isotropic it should he anticipated from the derivation that the predicted value of ui' will be the cube root of the mean cubed value of the various u*' components. By a process analogous to that used in deriving Equation 5 it can be shown that for the viscous regime in stirred tanks the average velocity is given by:
-
U
=
CND2 D
-(E) T
'1'
Re,
' iVD2
(SCY
These correlations have been criticized (Harriot, 1962) as being theoretically unsound because T i s a tank dimension and not a dimension of the particles undergoing mass transfer. In these correlations the exponent n is frequently close to 1.0 (Barker, 1959; Hixson and Baum, 1941, 1942; Hixson and LVilkens, 1933). By dividing both sides of Equation 9 by T and multiplying by d p there is obtained:
As a result of the present work, the group ,VD2/T may be recognized as proporticlnal to the characteristic velocity in tanks for which T = H (the most commonly used configuration) or in which T/IY is maintained constant. Further,
6 ND2/(T2H)'/3
I
12
3
Inches per Second
Figure 8. Average fluctuating velocities (in sampling zone) vs. N D 2 / ( T 2 H ) 1 1 3
since n is close to 1.0, the group (a',/ T ) ( l - n hardly ) changes; its effect Lvould prove difficult to substantiate. In fact, we may really be dealin5 u,ith a correlation of the form:
Since n is close to 1 .O. the form of Equation 11 is in accord with the observation that for moderately large particles-Le., larger than 30-mesh--k is not a strong function of d,. The preceding argument strongly suggests that Equation 9 appears to have experimental validity not because T is a meaningful characteristic dimension, but because of the way in which it affects the characteristic velocity. A similar argument could be advanced for a number of heat transfer correlations (Chapman and Holland, 1965) that have been proposed for stirred tanks. Conversely, the existence and apparent validity of equations such as Equation 9 and its analogs tend to confirm the validity of the group ,VD2,'T [or M here more appropriate. .VD2/( T2H)1'3]as a measure of the characteristic fluid velocity in stirred tanks. A commonly used rule of thumb for scaling up agitated systems is to maintain equal polver per unit volume. In systems where the power number is a constant, the power per unit volume is proportional to ,V3D5/(T2H). The cube root of the power per unit volume is ND5/3(T2H)"3,which is very close to the term ArD2,'(T2H)'/3. This suggests that the applicability of the equal power per unit volume rule may stem from the fact that its use leads to nearly equal local velocities. I n conjunction with the preceding comments on heat and mass transfer, this further suggests that maintaining equal values of A r D / ( T2H)'/3may be a valid criterion for scale-up in turbulent agitated vessels. VOL. 7
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Ac knowledgment
T h e authors are grateful to the National Science Foundation for support in the form of a Cooperative Fellowship. Nomenclature
A
c 4 D
a> Pc
H L Le m n
n.
P
Q
Rei Re, sc
T U
=
coefficient in mass transfer correlation
= constant of proportionality in fluid velocity correla= = = = = = = = = = =
= = = = = =
r7 Dv
=
Ut
=
=
@(Re,) = Y
=
p,
=
tions particle size tank diameter molecular diffusivity gravitational conversion constant liquid fill height of tank characteristic width of fluid stream Eulerian integral scale Schmidt number exponent Reynolds number exponent impeller rotary speed, revolutions per unit time power dissipation volume impeller Reynolds number stream Reynolds number Schmidt number tank diameter characteristic fluid velocity average fluid velocity average velocity i n vertical plane root mean sauare fluctuatingu velocitv i n the ith direction dissipation function of stream Reynolds number kinematic viscosity fluid density
SUBSCRIPTS direction (roughly radial) y direction (roughly tangential) z direction (vertical)
X
= x
Y
= =
2
literature Cited
Aiha, S., A.I.Ch.E. J . 4, 485 (1958). Aksel’rud, G. A., ‘Vauchn. Zap. Lrovsk. Polytekn. Inst., Ser. Khim. Tekhnol. 29 (l), 63 (1955). Barker, J. J., D. Eng. Sc. thesis, New York University, Kew York, April 1959. Barker, J. J., Treybal, R. E., A.I.Ch.E. J . 6 , 289 (1960). Batchelor, G., “Theory of Homogeneous Turbulence,” p. 103, Cambridge University Press, Cambridge, 1959.
Calderbank, P. H., Moo Young, M. B., Chem. Eng. S i . 16, 39 (1961). Chapman, F. S., Holland, F. X., Chem. Eng. 72, 153 (Jan. 18, 1965); 72,175 (Feb. 15, 1965). Cutter, L. A., Ph.D. thesis, Columbia University, New York, N. Y., 1960. Harriot, P., A.I.Ch.E. J . 8, 93 (1962). Hixson, A . TV., Baum, S. J., Ind. Eng. Chem. 33, 1433 (1941). Hixson, A. \\‘., Baum, S. J., Ind. Eng. Chem. 34, 120 (1942). Hixson, A. FV., TVilkens, G. A,, Ind. Eng. Chem. 25, 1196 (1933). Holmes, D. B., Voncken, R. M., Decker, J. A., Chem. Eng. SCZ. 19,201 (1964). Humphrey, D. Tk‘., Van Ness, H. C., A.I.Ch. E. J . 3, 283 (1957). Kim, \V. J., Manning, F. S., A.I.Ch.E. J . 10, 747 (1964). Kneule. F.. Chem. Ine. Tech. 28. 221 (1956). Laufer,’ J.’, Natl. :Advisory ’Comm. Aeronaut., NXCA Tech. Rept. 1174 (1954). Mack, D. E,, Marriner, R. A., Chem. Eng. Progr. 45, 545 (1949). Marangozis, J., Johnson, .A. I., Can. J . Chem. Eng. 40, 231 (1962). Mattern, R. V., Bilous, O., Piret, E. L., A.I.Ch.E. J . 3, 497 (1957). Metzner, A. B., Taylor, J. J., A.I.Ch.E. J . 6 , 109 (1960). Nagata, S., Yamaguchi, J., Kagaku Kogaku 24, 726 (1960). Sagata, S., Yamaguchi, J., Yabuta, Seizo, Harada, Makoto, .$fern. Fac. Eng. Kyoto Cnio. 22, 86 (1960a). Nagata, S., Yamamoto, K., et ai., Kagaku Kogaku 23, 595 (1959a). Nagata, S., Yamamoto, K., et ai., Kagaku Kogaku 24, 99 (1960b). Nagata, S., Yamamoto, K., Hashimoto, K., Naruse, Y., h f e m . Fac. Eng. Kyoto L‘nic. 21, 260 (1959b). Napata. S.. Yamaxnoto. K.. Hashimoto., K.., Naruse. Y.. M e m . Fac. Eng.’Kyoto Unic. 22, 68 ( 1 9 6 0 ~ ) . Sielsen, H., Ph.D. thesis, Illinois Institute of Technology, Chicago, I
Tll
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rnco
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Oyama, Y., Endoh, K., Kagaku Kogaku 20, 576 (1956). Rushton, J. H., Costich, E. TV., Everett, H. J., Chem. Eng. Progr. 46. 395. 467 11950). Rusgton, J. H.,’Lichtman, R. S., Mahoney, L. H., Ind. Eng. Chem. 40, 1082 (1948). Rushton, J. H., Oldshue, J. Y., Chem. Eng. Progr. 49, 161, 267 11941)
Sachs, J. P., Ph.D. thesis, Illinois Institute of Technology, Chicago, Ill., 1952. Sachs, J. P., Rushton, J. H., Chem. Eng. Progr. 50, 597 (1954). Schwartzberg, H. G., Ph.D. thesis, New York University, New York, October 1965. Taylor, J. S., M.S. thesis, University of Delaware, Newark, Del., 1955. Tennant, B. \V,, M.S. thesis, Illinois Institute of Technology, Chicago, Ill., 1952. RECEIVED for review .4ugust 29, 1966 ACCEPTED July 26, 1961 Division of Industrial and Engineering Chemistry, 152nd Meeting, ACS, New York, X. Y . , September 1966. Condensation of a thesis submitted by H. G. Schwartzherg in partial fulfillment of the requirements for the degree of doctor of philosophy at New York University.
FLUID AND PARTICLE MOTION IN TURBULENT STIRRED TANKS Particle Motion H E N R Y G. S C H W A R T Z B E R G A N D AVew York University, ‘Yew York, ,V. Y .
ARTICLE to
fluid mass transfer is often correlated in terms of
P a single motion-related parameter, the particle Reynolds number (Froessling, 1938; Ranz and Marshall, 1952; Steinberger and Treybal, 1960). An attractive scheme for correlating such mass transfer in stirred tanks is to predict the particle slip velocity and from this calculate the particle Reynolds number and corresponding mass transfer coefficient. This scheme is perhaps overly simple since, in intense turbulence, mass 6
l&EC FUNDAMENTALS
R O B E R T E. T R E Y B A L
10453
transfer correlation may require added parameters based on the intensity and scale of turbulence (Comings et al., 1948; Galloway and Sage, 1964; van der Hegge-Zijnen, 1958; Maisel and Sherwood, 1950). Even if added parameters are required, the particle Reynolds number and hence the slip velocity are still important. A detailed knowledge of the fluid velocity behavior is needed to predict slip velocities, Until recently such knowledge was
