September, 1933
I N D U S T R I h L A N D E N G I N E E R I N G C H E M I S T R I-
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of liquid in which concentrations are moderately uniform throughout the body and in which slow reactions can be completed. The liquid stationary (or unstirred) film t'lieory is based mainly on the laws which have been found regarding the rates of solut'ion of solids and gases by batches of stirred liquids. These laws should be applied only v-ith caution to absorptions where the area of the liquid inteIface is very great compared to the liquid volume.
20
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6
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LITERATURE CITED (1) Davis and Crandall, J . Am. Chefn. SOC.,52, 3767, 3769 (193U). (2) Davis, Thomson, a n d Crandall, Ihid., 54, 2345 (1932). ( 3 ) P a y n e and Dodge, IND.ENQ.CHEM.,24, 630 (1932).
a -
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m minutes
FIGURE3. ABSORPTIONOF C l R B O N DIOXIDE BY W A T E R ( EXPERIMEXT 7)
RECEIVED February 7 , 1933. Presented before the Division of Industrial and Engineering Chemistry at the 85th Meeting of the .imerican Chemical Society, Washington, D. C., March 26 t o 31, 1933.
Studies in Agitation 11. Sand Concentration as Function of Sand Size and Agitator Speed
those trapped between 32 and The distribution of sand under the influence of 42 mesh, b e t w e e n 48 and 65 cation ( 2 ) the distribution a simple paddle ugitutor has been studied for mesh, between 65 and 100 mesh, of a roughly screened sand carious sizes and amounts of sand, f o r carious and between 80 and 150 mesh was described for various posiagitator speeds, and at various positions in the Tyler standard s c r e e n s . The tions in a tank equipped with a tank. I t is found that the curves of sand consand was w a s h e d f o l l o w i n g simple paddle agitator. It was screening. shown that uniformity of concentration 7's. agitator speed show three distinct For each sand size, tests were centration was never attained, tones, corresponding respectively to absence of run varying the agitator speed and that there was a hydraulic strong cerfical currents, marked vertical curwhile h o l d i n g c o n s t a n t the classifying action on the part rents, and marked centrifugal forces (saturation). total amount of sand and water. of the agitator, the fines tending The curves ,for earious sand sizes have been correThe agitator speeds varied from to move toward the wall of the v e l o c i t i e s so low that s a n d tank while the coarser particles lated empirically, a plot of log Q/D" against R could barely be detected in sust e n d e d to concentrate in the yielding a straight line for each of the two lower pension to speeds so high that c e n t e r of the tank under the zones. I t is suggested that the position of the there was marked coning of the paddle. This effect was deemed saturation point (that is, the agitator speed at surface of the water and slight worthy of further study, and in which maximum concentration is obtained) m a y splashing, a range of 18 t o 88 the present paper are presented r. p. m. F o r t h e 32-42 a n d data showing the influence of be used as u criterion of intensity" of" aaitation. " 05L100 m e s h f r a c t i o n s the agitator spee;d on the distribution of sand of various sizes effect of varying total amount when varying total amounts of sand are present. of sand with constant amount of water was studied a t various agitator speeds. EXPERIMENTAL PROCEDURE Results could be duplicated satisfactorily. However, The equipment used in these tests was the Lame as that checks were more difficult to obtain in the area immediately previously described ( 2 ) . Briefly, it consists of a cylindrical around the paddle than a t points nearer the wall of the tank. steel tank 4 feet 4 inches (1.32 meters) in diameter equipped Checks mere also less satisfactory a t very low sand concentrawith a simple paddle stirrer with a blade 2 feet 1 inch (0.64 tions, particularly with the larger sand sizes. meter) long. The water level in the tank was, as before, held arbitrarily a t 2 feet (0.61 meter) above the bottom of the EXPERI~C~EXTAL RESULTS tank, corresponding to 220.5 gallons (835.7 liters) of water. The paddle was kept in the low position, with its lower edge In Figure 1 are plotted curves showing the concentration of 5.25 inches (13.34 em.) from the bottom of the tank. sand in milligrams per 100 cc. of water, as a function of Sampling and analysis of samples mere carried out in the agitator speed for the sampling point 2 inches ( 5 em.) from manner previously described except that samples of about the wall and 2 inches from the bottom of the tank. For each 500 cc. were taken, to reduce sampling errors. instead of of the four sand sizes a total of 15.5 pounds (7 kg.) of sand samples of about 125 cc. The agitator was operated for was used. The curves for the finer sand sizes show three about 15 minutes before samples were taken to insure the distinct zones-one a coniparatil-elyflat curve a t low agitator attainment of a steady state. The sand used was carefully speeds indicating a rapid increase in sand concentration with screened to size, the four fractions selected for study being velocity, the second a steeper curve which terminates in the
N A PREF'IOUS communi-
1026
I N D U S T R I A L A N D E IC’ G I N E E R I N G C €1E M I S T R Y
Vol. 25, No. 9
shaped ring of sand in the plane of the paddle. It is suggested that this latter zone, in which centrifugal forcesare of importance, be called the saturation zone. While this does not imply saturation in the physico-chemical sense, 8 it does signify the maximum concentration of sand that can be obtained under any given set of conditions. The upper break, or saturation point, indicates that for 0 any given position in the tank there exists an agitator x a speed beyond which an increase in speed accomplishes ar little or no increase in sand concentration. In fact, for ? some points in the tank an increase in agitator speed beyond the saturation point results in a decrease in the amount of sand suspended. This is most marked a t points above the paddle and toward the middle of the tank, and is probably due to the coning of the liquid a t high agitator speeds. Calculation indicates that the sand concentration a t saturation is almost always less 0 i than the concentration which would be attained if all the M i SAND PCR 100 C C inATCR sand in the tank were uniformly suspended, except a t E h ~ m1. S A N D C ON CENT RATION A s FUNCTION OF AGITATOR points beneath and immediately beyond the paddle. The SPEEDFOR VARIOUSSANDSIZES sand has a marked tendency to remain under the paddle, and here very high concentrations are attained. Although Figure 1 presents data for but one position third (or vertical) zone in which increase in agitator speed has in the tank, the curves for the other positions are similar little effect on sand concentration. Hixson and Crowell (1) have noted three regimes of agita- to those given. As noted above, however, the saturation tion when solid salt is stirred with water. They report that curves for points near the middle of the tank have negative a t low agitator speeds a “passive or nonflow” regime is slopes. \J7hile no theoretical treatment seems possible a t present maintained in which the solid particles remain on the bottom of the container with little or no motion. At somewhat owing to lack of data on the velocity of the water relative to higher speeds the particles move inward toward the center of the sand in various parts of the tank, ’it has been possible to the tank, a regime characterized by these authors as curvi- correlate empirically the effect of sand size, sand concentralinear flow. At high agitator speeds tion, and agitator speed, for a given position in the tank and a the particles of salt were thrown out- given total amount of sand. If from Figure 1 are read values ward from the center, and the regime for sand concentration a t constant agitator speed for the four described as turbulent flow was estab- sizes tested, and these values are plotted on log-log paper l i s h e d . It has been suggested that against the logarithmic mean clear opening of the screens the regime a t low agitator speeds is forming the limits of the fractions chosen, a straight line one in which the vertical currents set results, as shown in Figure 2. A very limited range of agiup by the agitator are not sufficient tator speeds is available for drawing this line, for a slightly to raise the sand particles from the higher r. p. m. than the one indicated falls on the saturation bottom of the tank, while the inter- curve for the 80-150 mesh fraction. The equation of the line mediate zone is one in which verti- drawn may be represented as: cal currents are of sufficient magnilog Q = m log D log K I (1) tude to produce considerable suspenwhere Q = sand concentration, mg. _ _per 100 cc. water sion of sand, the amount thus susm = slope pended being a function of agitator D = logarithmic mean clear opening of screens forming meed. In the third. or high-speed, limits of fraction Lone centrifugal fordes a r e b e c o m i n g sufficiently strong to combat effectively the f o r c e s FIGURE 2. SAND t e n d i n g to produce a CONCENTRATION AS FUNCTION OF SAND m o r e u n i f o r m sand SIZE AT CONSTANT concentration. T h e s e AGITATOR SPEED conclusions have been confirmed by a study of the motion of sand particles in glass scale model of the apparatus used in the large-scale experiments. As the agitator speed was slowly increased, the sand began to move slowly in a spiral path across the bottom of the container, forming a cone under the center of the paddle. With increasing speed more and more sand was picked from the surface of the cone and drawn into the paddle, while the sand flowing across the bottom of the tank increased in QD’ amount. At high speeds the cone of sand enAND FIGURE3. &LATION BETWEEN SANDS I Z E , SANDCONCENTRATION, AGITATOR SPEED tirely disappeared, giving place to a doughnutY)
0
+
a
September, 1933
I K D U S T R I A L A N D E N G I N E E R I N G CHEJ'IISTRY
This equation may be rearranged to give:
The curves of Figure 1 show that the intermediate portions are nearly straight lines and are approximately parallel; hence, log Q = nR log Kt (3) where R = speed of stirrer, r. p. m. Rearraneine. = n'; = n'Q ( d6RIn ) DQ D
+
u
UI
($)
The partials of Equations 2 and 4 may be substituted in the general partial differential equation: (5)
(6)
On separating variables and integrating, it is found that log &/Dm = n'R
+ log K s
(7)
This is the equation of a straight line. From Figure 2, for the point 2 inches from the side and 2 inches from the bottom of the tank, the slope of the log Q vs. log D curve is -4.0. This is therefore the value of m to be substituted in Equation 7. Plotting the logarithm of &/D-4 (or its equivalent logarithm QO4) against R, a straight line results, from which branch perpendicularly the saturation curves for the various sand sizes. The passive-flow curves likewise form a single line, as shown in Figure 3. From this relation it should be possible to interpolate and, to a limited extent, extrapolate to determine the behavior of an untested sand size.
,
1
1027
clear opening of the screens, an approximate straight line is obtained. The slope of this line varies widely from point to point in the tank, however, being negative in some cases and positive in others. The above relation therefore seems to be a t best an approximation. In Figure 4 are plotted curves showing sand concentration as a function of agitator speed for varying amounts of sand. The left-hand curves refer to the 3 2 4 2 mesh sand, with 4.3, 8.6, and 15.5 pounds (1.96, 3.9, and 7 kg.) present. The right-hand curves refer to the 65-100 mesh sand, with 15.5 and 20.6 uounds (9.3 kg.) present. The sand concentration increases as the total a m o u n t of sand is increased. There is a strict proportionality between amount of sand and sand concentration a t saturation. This is shown in Figure 5 , in w h i c h s a n d c o n c e n t r a t i o n is plotted against total a m o u n t of s a n d p r e s e n t for e a c h sand size. At velocities b e l o w the s a t u r a t i o n point, sand concentration does not increase as rapidly as the a m o u n t of s a n d , FIGURE5. SANDCONCENTRATION AT OF TOTAL and the curves ob- SATURATION A S FUNCTION AbfOUxT O F SAND tained are roughly uarabolic. KO explanation of the increase in sand concentration with amount of sand is offered a t this time, as variables other than those studied seem to be involved. However, the density of the system does not seem to be an important factor since variations in the apparent mean density of the suspension are small in comparison to the magnitude of change in concentration with amount of sand. SATURATION POINT AS MEASURE O F AGITA4TION
1 &
IC
MG
10 50 !x SAhD PER 100 CC. WPTER
zm
TOTALAMOUNT OF SAYD FIGLTRE 4. EFFECTOF VARYIXG
If plots similar to those of Figures 1 and 2 are made for other positions in the tank, and the value of the constant m is inserted in Equation 7, new curves are obtained differing from those in Figure 3 in that they are displaced horizontally. The curves in the passive-flow region, however, show very little change in position. Fair correlation is obtained by the general use of the value -4 for the exponent m, but better results are obtained hy determining for each point its own value of this slope. The slope of the log Q/D" vs. R curve seems to be fairly constant throughout the tank. As would be expected, the transition point occurs a t different agitator speeds in various portions of the tank, since liquid velocity is not a simple function o f agitator speed. The amount of sand a t saturation (taken as the maximum concentration obtained at any given point) seems to be a power function of the sand size. By plotting the logarithm of the sand concentration against the logarithm of the mean
Since the saturation point occurs a t approximately the same agitator speed for different points in the tank if the sand size and total amount of sand are held constant, it is suggested that the agitator speed a t which saturation occurs might be used as a measure of intensity of agitation. For a given position in the tank, preferably near the wall and somewhat above the paddle, and with a standard amount of standard size sand, samples could be obtained a t various agitator speeds. The paddle giving the highest sand concentration a t the saturation point could be considered as most effective in distributing the solid. Such tests should be paralleled hy power demand studies, which, in the light of the saturation point, might be interpreted in such a way as to determine the optimum agitator, or the agitator which accomplishes the distribution of solid most effectively for a given power demand. Further work along this line is in progress. ACKNOWLEDGMENT
The authors are indebted to B. E. Lukens for assistance with the experimental work.
LITERATURE CITED (1) Hixson a n d Crowell, IND. ENG.CHEM.,23, I161 (1931). (2) White, Sumerford, Bryant, a n d Lukens, Ibid.. 24, 1160 (1932). RECEIVED February 9, 1933.
