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 fluctuating velocitv i n the ith u direction dissipation function of stream Reynolds number kinematic viscosity fluid density
SUBSCRIPTS X
= x
Y
= =
2
direction (roughly radial)
y direction (roughly tangential) z direction (vertical)
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
,
rnco
111.) 1720.
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 a n d Marshall, 1952; Steinberger a n d Treybal, 1960). An attractive scheme for correlating such mass transfer in stirred tanks is to predict the particle slip velocity a n d 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 a n d scale of turbulence (Comings et al., 1948; Galloway a n d Sage, 1964; van der Hegge-Zijnen, 1958; Maisel a n d 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
~~
~~
Differential equations describing particle motion in a turbulent fluid were applied to a flow model for turbulent stirred tanks, and solutions obtained for particle slip, over-all, and fluctuating velocities a t various particle sizes and densities, agitation speeds, viscosities, Lagrangean integral scales, and drag coefficient levels. Particle velocities were measured by streak photography and compared with the calculated values. The particle motion followed the fluid motion very closely; settling velocities were markedly lower than in still fluid. This behavior appears to be due to abnormally high drag. Because of the intense turbulence, the threshold of detection for slip velocity was too high to permit other than order of magnitude verification of the calculated slip velocities. The calculated slip velocities indicate that, particle-to-fluid mass transfer in stirred tanks cannot be explained in terms of a simple slip velocity model.
not available for stirred tanks. However, the fluid velocity data presented by Schwartzberg and Treybal (1968) provide a basis for a n attempt on the stirred tank slip velocity problem. Theory
Tchen (1947; Hinze, 1959) has developed a force balance equation for small spherical particles entrained in a turbulent fluid. We have somewhat simplified and generalized this equation and use it in the analysis to follow.
I1
I
IV
I11
v
Analysis of term I1 in the light of the observed phase lags between the fluctuating velocity components indicates that the instantaneous value of cdlU - VI can be replaced by V],,, and that IU - Virms can be equated to 1U - VI with in term 11 with small error. We therefore replace CdIU - VI,,, where the value of C, is that corresponding to IU - VIrm8. This effectively converts Equation 1 into a linear equation, albeit one in which the value of CdlU - VI,,, and the variable ( U , - Vi) are interdependent. Using this assumption, Equation 1 can be rearranged as follows :
cdiu -
cdju
where
I is the net vertical field force acting on the particle; 11, the drag force; 111, the pressure gradient force producing fluid acceleration; IV, the added mass correction (accounting for the relative fluid acceleration necessary to accommodate a relative particle acceleration) ; and V, the force required to accelerate the particle itself. I n the absence of strong relative accelerations caused by external forces, the Basset term has been neglected. We assume that any abnormal drag effects produced by relative acceleration can be incorporated in C d . Because the wavelength of the turbulence is large, forces arising from viscous shear stress,gradients have been neglected. In research on sediment transport (Rouse, 1 9 3 9 ) it has been shown that upward particle diffusion will oppose particle settling in turbulent fluids. Assuming that equivalent kinematic and dynamic warys exist for describing diffusion (Einstein, 1905), particle d i f b i o n can be regarded as being due to a diffusional force. We can take account of this upward diffusional force in Equation 1 by defining F, as:
3Pf
q=-
2PP
(5)
+ Pf
and
f=
2dPP 2Pi7
- P/)g
+ Pf
T o solve Equation 3 a fluid velocity model is needed. Kim and Manning’s (1964) measurements of the spectrum of turbulence in the discharge stream of turbine impellers indicate that the turbulence intensity decreases extremely rapidly as frequency increases. If these measurements are valid and applicable to the bulk of the fluid, the high frequency components of turbulence can be neglected in solving Equation 3. Therefore a suitable model for U iin a stirred tank might be:
4 2
where q is the fraction of the net gravitational force which is not counterbalanced by the diffusional force. Since, as pointed out below, particle settling velocities in stirred tanks range about 4070 of their still fluid value, q may be as low as 0.4. Because of term 11, Equation 1 is nonlinear except in the Stokes law range, and is therefore somewhat intractable. However, for our purposes a solution yielding reasonable average values of 1U - VI would be acceptable even though it might be somewhat in error with respect to the instantaneous values of (Ut - Vi).
T h e factor above is the ratio of the peak amplitude to the root mean square amplitude, u t ’ , for a sinusoidal fluctuation. T h e frequency w is defined as rUT/2L because the Lagrangean integral scale L is for sinusoidal fluctuations a formal measure of the distance traversed in a quarter period. After substituting for Ut it is a relatively simple matter to show that the nontransient solution of Equation 3 is:
where $ is the phase shift between ui’ and
3w2(1
VOL. 7
vi’.
Further:
- q)2u,,,’2
NO. 1 F E B R U A R Y 1 9 6 8
7
Simpler asymptotic solutions exist for large and for very small particles. T h e solution of Equations 8 and 9 requires the proper value of b , which is a function of the U - V(,,, value being sought. This involves trial and error. A graphical procedure (Schwartzberg, 1965) using a log CdRe, log Re, plot can facilitate such solutions. Using this procedure, solutions were obtained for Equation 9 for various agitation conditions and fluid and particle properties. T h e necessary fluid behavior parameters were obtained from the stirred tank fluid velocity correlations developed by Schwartzberg and Treybal (1 968)-i.e.,
-
u,,,'
=
3 w
Y
0.57ND2 ( TZH)lia
~
0.4 0.4 1.0 0.4
2 .5ND2 L ( T2H)lIa
= -___
T h e value of L needed to specify w is probably proportional to the impeller diameter, 0 , but its exact value is not known. Figure 1 presents 'U - V us. d, for assumed values of L equal to (1/3)(0.08D), (0.080),and 3(0.080), where 0.080 is the a\.erage value determined by Cutter (1960) for the Eulerian integral scale in the discharge stream of turbine impellers. Figure 1 also presents :U - V ' us. d, for a variety of assumptions concerning the relative magnitudes of 7 and C., Particularly at small values of d,, the slip velocity curves are sensitive to the assumed values of 7, Cd, and L . Slip velocity us. d, curves with stirring speed, particle density, and fluid viscosity as parameters are presented in Figures 2 and 3 for q = 0.4, C, normal, and L = 0.080. Figures 1, 2, 3, 6, and 7 are based on results for a fluid density of 1.0 gram per cc. and for a 17.3-inch diameter baffled tank agitated by a 6-inch diameter, six-bladed turbine impeller. Experimental
Vertical-plane suspended particle velocities were measured using the streak photography technique described by Schwartzberg and Treybal (1968). T h e particle properties are presented in Table I. T h e particle velocities were compared on a local basis with the corresponding local vertical-plane fluid velocities in terms of: the relative magnitudes of the average velocities and the velocity fluctuations; the difference between the vertical velocity components, the relative settling velocity; and the vector difference between the velocities, the slip velocity. The local values of these quantities were averaged over the whole sampling zone.
.ooI
1X 1X 3X 1X I
.oI
PARTICLE DIAMETER (CMs Figure 1.
Calculated slip velocity vs. particle size
Parameters.
v, cd,
and
I.
N
= 500 r.p.m.
In the presence of velocity fluctuations, slip velocity can exist even when there is no difference between the average velocity of the fluid and the particle. T o help account for the effect of fluctuation on slip, each local fluid and particle velocity vector for a given set of operating conditions was broken down into a group of four vectors representing the velocity at the average angular and magnitude deviations from the mean velocity. Vector differences between the corresponding members of the local fluid and particle vector quartets were calculated and averaged locally in magnitude, and then these local averages were averaged over the whole sampling zone. This procedure accounts for the effect of fluctuation magnitude on the average slip velocity. Phase differences between the fluid and particle velocity fluctuations are not accounted for, because they are experimentally indeterminate. Results
T h e average relative settling velocities for the test particles in the agitated fluid are presented in Table I. These velocities, with one exception, are averages for all runs with a given type
-
l&EC FUNDAMENTALS
I
0.1
Table I. Particle Properties, Settling Velocities, and Velocity Magnitude Differences Velocities, IncheslSec. Ejective S,, still W, Av. Size, S, agitated Density, Particle GramslCc. Inches settling settling vu u, s/s, 0.2 0.507 0.15 0.40 0.0215 IRC 50, ion exchange resin 1 .lo8 Lucite diamond tablets 1.175 0.09 X 0.10 1.5 2.62 1.7 0.57 x 0.10 0.7 0.28 0.231 1.9 6.75 Lucite spheres 1.175 1.1 2.40 1.5 0.46 0.13 X 0.13 Nylon square tablets 1,128 X 0.085 2.1 4.88 1.9 0.43 0.044 2.85 Marble 1 .o 2.85 0.4 0.35 0.165 PVC cubes0 1.319 a I n 5-cp. sugar solution.
8
1/3(0.08D) 3(0.08D) 0.08D 0.08D
N o . o f Test Conditions 5 13
3 4 3
4
+ 0
w
?IOP 0
v
> t 0
s2 1.0 n
-1
v)
0.1 .001 1
0.1 PARTICLE DIAMETER (CM. )
PARTICLE DIAMETER (CM. ) Figure 2.
Calculated slip velocity vs. particle size Parametmer.
N.
pp = 1.5 g./cc.
of particle. Because of the large velocity fluctuations (up to three to 30 times greater than the mean settling velocities), these average settling velocities do not have much more than one-figure accuracy. Because the discrepancy between settling velocity and fluctuating velocity was particularly bad a t high impeller speeds, settling velocity figures for the slowest settling particles a t these high speeds were not used in Table I. I n spite of the low accuracy of the data, it is clear that S, the agitated fluid settling velocity, is substantially less than S,, the still fluid settling velocity, and that S/S,does not range far from 0.3 to 0.5 for all particle sizes and densities tested. I t appeared that the settling velocity reduction might be due to upward particle diffusion. Particle concentrations, measured photographically, were found to decay exponentially in the vertical direction. Such exponential decay is usually a n indication of upward particle diffusion. Using the method of‘ Kalinske and Pien (1944), particle diffusivities were estimated from our fluctuating and mean velocity data and Cutter’s (1960) data on the scale of turbulence in stirred tanks. I n terms of these diffusivities and the measured concentration gradients, upward particle diffusion could not possibly have caused reductions in settling velocity of the magnitude noted. Enhanced drag due to the turbulence appears to be the cause for most of the reduction in settling velocity. Upward particle diffusion and nonlinearities in the drag law appear to cause a relatively small fraction of the total reduction in settling velocity. One of the authors (Schwartzberg, 1967) recently reviewed the theory and available data for sediment suspension in natural streams. It appears that settling velocity reduction due to enhanced drag may also occur to a certain extent in this situation and that existing sediment suspension theory, which is based solely on upward particle diffusion, may therefore require modification. Fluctuating Velocitie,~. Experimentally determined values of v’/u’, the ratio of the particle and fluid velocity fluctuations, are listed in Table I1 for the various test particles. Since these ratios are close to 1.0, it appears that the particles follow the
Figure 3.
Calculated slip velocity vs. particle size
Parameters.
fi and pp.
N = 500 r.p.m.
fluid velocity fluctuations fairly closely. u ’ / u ‘ can be obtained from Equation 8.
An expression for
The v’/u’ value obtained from Equation 12 depends on 6, which in turn depends on C, and 7. O u r settling velocity data can be accounted for formally either by assuming 7 = 1.0 and Cd three times its normal value, or by assuming 7 = 0.4 and Cd normal. Predicted u ’ / u t values were calculated for each of these assumptions (Table 11). T h e possible range of variation of v’/u’ is limited by q, and a critical test of these alternative assumptions exists only when q is small. For the particles tested q is small only for the marble chips, and here the ut/.’ data are correlated better if the enhanced drag assumption holds. This result and the settling velocity result strongly suggest that such enhanced drag does exist. Further improvement in agreement for u t / u ’ would be obtained if w were based on a n L value larger than the 0.080 used in calculating the predicted results in Table 11. For example, for L equal to 3 ( 0 . 0 8 0 ) , three times the Eulerian integral scale, the predicted v’/u’ for the marble chips would be 0.98 under the enhanced drag assumption. This suggests that the Lagrangean scale is large relative to the Eulerian scale
Table II. Velocity Fluctuation Ratio ( v ’ / u ’ ) Estimated Ratios for 7J
Particle IRC 50 0.09-inch Lucite 0.23-inch Lucite Nylon Marble PVC (in sugar solution)
= 0.4,
= 1.0, Cd3 x
Experimental Ratio 0,996 0.997 0.917 0.917 0.979
normal Cd 3 L = 0.080 0.973 0.902 0.897 0.925 0.605
L = 0.08D 0.995 0.925 0.905 0.934 0.868
0.928
0.927
0.940
VOL 7
NO, 1
normal,
FEBRUARY 1968
9
in stirred tanks-a suggestion that is supported by the eddy sizes found in the streak photographs. Particle Velocity. Equation 8 and the correlations in our fluid motion paper imply that on the average in the sampling zone:
where ?V is of the order of magnitude of S, the agitated fluid settling velocity of the particle. Since the sampling zone is largely a region of downflow, the sampled mean particle velocity will be greater than the corresponding fluid velocity, The values of TV corresponding to the best fit of Equation 13 for the entire sampling zone and the whole flow velocity range are presented in Table I for the various particles tested. When these values of TV are used, the average deviation between the experimental and predicted values of lp,l is less than 8% for each particle tested. The average vertical plane velocity of the 0.09-inch Lucite particles is plotted CIS. A'D2/(T2H)1'3 in Figure 4. The points for the 9-inch impeller are low because downflow occurred over a smaller area with this large impeller. Slip Velocities. Apparent vertical-plane slip velocities computed from corresponding pairs of particle and fluid and in Figure 5. fluid velocity data are plotted us. ;L'D2/(TZH)1/3 Also plotted in Figure 5 are dashed lines indicating the apparent slip velocities that on the average would be computed from pairs of fluid velocity data for the same operating conditions. When a limited number of data are used t o calculate local average velocities, they are inherently imprecise because of the
d
4I
8I
12 I
I
I
16
turbulent velocity fluctuations. Because of this lack of precision, velocity differences will be computed even when such differences should not exist-e.g., when two groups of fluid velocity data are compared for the same operating conditions. When larger sets of data are used, the precision is improved and the magnitude of these erroneous slip velocities is reduced. The dashed lines in Figure 5 take account of the size of the velocity data sample. Apparent slip velocities were calculated from actual pairs of fluid velocity data and then statistically adjusted for sample size. Thus the dashed lines correspond to the slip velocity that on the average would be computed from pairs of fluid velocity data compiled from 100, 200, and 400 streak images each. These constitute the threshold of detection for slip velocities computed from fluid-particle data pairs obtained from the corresponding number of streak images. With the possible exception of the dense marble particles and the large 0.23-inch Lucite spheres, it is doubtful whether the apparent slip velocities are significantly greater than the appropriate threshold of detection. Attempts to detect the real magnitude of the slip velocities by greatly increasing the number of measurements failed. As more streaks were measured, the calculated slip velocities appeared to decrease just as rapidly as the threshold of detection. The 0.23-inch Lucite spheres and the marble particles appear to have roughly constant vertical-plane slip velocities of about 3 to 3.5 inches per second. These velocities check roughly with those obtained by solving Equation 9. The agreement is best and occurs for the same reasons in the experimental and predicted cases when the enhanced drag, long Lagrangean scale assumptions are used in solving Equation 9. Since the remaining particles are smaller or less dense than the 0.23-inch Lucite spheres or the marble particles, their slip velocities should all be less than 3.5 inches per second. Moreover, the vector difference between U , and V , can never be less than either or S. Therefore the slip velocities for the particles should lie between W or S, whichever is greater, and 3.5 inches per second.
.I
1
I
N D2/(T2H)I/3 (Inches per Sec.) Figure 4.
0.09-inch Lucite particles in water in baffled tanks agitated b y six-bladed turbine impellers
10
l&EC FUNDAMENTALS
IO
40
IO0
(T2H) l/3 (INCHES PER SECOND)
Vertical-plane particle velocity vs. ND2/
(T2H)'i3
4
Figure 5.
Apparent slip velocity vs. N D 2 / ( T Z H ) ' I 3
Experimental d a t a for baffled tanks agitated b y six-bladed turbine impellers
I
I
$-
\
a
-.--
5
' 10 . .001 Figure 6. diameter
I
I
I
1
.01 0.1 PARTICLE DIAMETER (CM.
I 1.0
Calculated mass-transfer coefficients vs. particle Parameters.
I.(
and p p .
N
=
PARTICLE DIAMETER (CM. 1 Figure 7. diameter
Parameters.
500 r.p.m.
Mass Transfer Implications. In calculating slip velocities from Equation 9 the assumptions 7 = 0.4, C, is normal, and L = 0.080 are liberal in the sense that they yield larger slip velocities at small particle sizes than the probably more realistic enhanced drag, long Lagrangean scale assumption. Slip velocities obtained using these liberal assumptions were used to calculate particle Reynolds numbers, which in turn were substituted into Steinberger and Treybal's (1960) correlation for particle-liquid mass-transfer coefficients. These calculations were carried out o'ver a wide range of particle sizes and densities a t various agitation conditions. The resulting values of the predicted mass-transfer coefficient, k, are plotted us. d, in Figures 6 and 7. T h a e curves are qualitatively similar to those developed by Harriot (1962) for mass transfer to particles moving a t their still fluid settling velocity. These curves agree Xvith existing mass-transfer data-e.g., those of Barker and Treybal (1960) and Harriot (1962)-in predicting a small effect of d,, on k for particles of moderately large size. Apart from this, the various experimentally based masstransfer correlations (Barker, 1959 ; Calderbank and Moo Young, 1961; Mack and Marriner, 1949; Marangozis and Johnson, 1962; Nagataet al., 1960; Oyama and Endoh, 1956) vary so widely that it is (difficult to compare experimental performance trends for stirred tanks meaningfully with those predicted by our curves. In general our predicted mass transfer coefficients appear to be too lo\v. At high impeller speeds and large particle sizes it is suspected that the discrepancy may occur because attrition causes excessively high ex:perimental values for k. T h e most significant disagreement between the experimental and our predicted values of k occurs for very small particles such as those irivestigatmed by Harriot. Here the predicted values are only one third io one fourth of those expected, and attrition could not possibly be a factor. Considering the liberal assumptions of the present calculation, it appears that slip velocity alone cannot explain the experiment ally observed behavior of k. Discrepancies are particularly bad whenever the slip velocity tends to become van-
Calculated mass transfer coefficient vs. particle
N
p and p p .
= 150 r.p.m.
ishingly small-e.g., a t very low particle-fluid density differences, or very small particle sizes. Even after allowing for mitigating factors, such as attrition, the simple slip velocity particle Reynolds number mechanism does not appear adequate for dealing with particle-fluid mass transfer in stirred tanks. Harriot previously reached a somewhat similar conclusion. Acknowledgment
The authors are grateful to the National Science Foundation for its support in the form of Cooperative Fellowships during which this work was completed. Nomenclature
b Cd
dP
D
F,
f
= = = = =
=
k
= = =
L
=
g
H
.v
=
+
1.5 pjCd , U - Vlmsi[dn(2pp P/)I drag coefficient particle diameter impeller diameter net field force acting on particle in vertical direction 2ab, - P/)d(2P, Pf) gravitational acceleration depth of liquid fill in tank mass transfer coefficient Lagrangean integral scale impeller rotary speed, revolutions per unit time
+
+
Q S S"
= 3P//QP,
T
= tank diameter= time
t
U
J U- VI,,,
u - VI
I_V U'
V
v U'
FV
P/) = particle settling velocity in agitated fluid
= particle settling velocity in still fluid = fluid velocity
root mean square slip velocity absolute value of slip velocity average fluid velocity root mean square fluid fluctuating velocity particle velocity = average particle velocity = rpot m e p square particle fluctuating velocity = V u - U , (difference in average vertical plane speed of particle and fluid in sampling zone)
= = = = =
VOL. 7
NO. 1
FEBRUARY 1968
11
GREEKLETTERS 6,s = Kronecker delta 7 = 6FZ/[743(P, - P f k l Pf = fluid density PP = particle density 4 = fluctuation phase lag between particle velocity and fluid velocity W = T U 2L (frequency of fluid velocity fluctuations) SUBSCRIPTS
i
= zth component = vertical component = vertical plane
2
U
literature Cited Barker, 3. J., D. Eng. Sc. thesis, New York University, New York, 1959. Barker, J. J., Treybal, R. E., A.I.Ch.E. J . 6, 289 (1960). Calderbank, P. H., Moo Young, M. B., Chem. Eng. Sci. 16, 39 ( 1 9 h- l-’/i . \
-
Harriot, P., Chem. Eng. Sci. 17, 149 (1962). Hegge-Zijnen, B. G. van der, Appl. Sci. Res., Sect. A7, 205 (1958). Hinze, J. O., “Turbulence,” p. 352, McGraw-Hill, New York, 1050
K&ske, .\. A , Pien, C. L., Ind. Eng. Chem. 36, 220 (1944). Kim, \l‘. J., Manning, F. S., A.I.Ch.E. J . 10, 747 (1964). Mack, D. E., Marriner, R. .\., Chem. Eng. Progr. 45, 545 (1949). Maisel, D. S., Sherwood, T. K., Chem. Eng. Progr. 46, 172 (1950). Marancozis. J.. Johnson. .A. I.. Can. J . Chem. E n e . 40. 231 (1962). Nagat< S., ’et a’l., M e m . Fac. E&. K2oto Univ. 22,“86 (i960). Oyama, Y., Endoh, K., Kagaku Kogaku 20, 576 (1956). Ranz, \Y.E., Marshall, \Y. R., Jr., Chem. Eng. Progr. 48, 141, 173 ,
I
/,ncn\
(lY>L/.
Rouse, H., Proceedings, Fifth International Congress for Applied Mechanics, p. 550, Ij’iley, New York, 1939. Schwartzberg, H . G., Ph.D. thesis, New York University, New York. 1965. Schwarizberg, H. G., Proposal to Federal \l’ater Pollution Control Administration, 1967. Schwartzberg, H. G., Treybal, R. E., IND.ENG. CHEM.FUNDAMENTALS 7; 1 (1968). Steinberger, R. L., Treybal, R. E., A.I.Ch.E. J . 6, 227 (1960). Tchen, C. M., Ph.D. thesis, Delft, 1947. ’
~
Comings, E. \Y.,Clapp, J. T., Taylor, J. F., Ind. Eng. Chem. 40, 1076 (1948). Cutter, L. .
