EDDY DIFFUSION A. A. KALINSKE AND C. L. PIEN Institute of Hydraulic Research, University of Iowa, Iowa CitJy,Iowa Experimental data are reported relating to the diffusion of mass by eddies in turbulent flow. The theory of eddy diffusion developed by G. I. Taylor is confirmed by experiments on the diffusion of foreign material in a turbulent water stream. A technique for determining directly the eddy diffusion coefficient in flowing water has been developed, and the method can be readily adapted to gases. One of the important items revealed by these studies is that the scale of the turbulence enters directly into the eddy diffusion relationship, and it must be measured or estimated if diffusion in turbulent fluids is to be predicted accurately.
clp/dl
=
2v3&‘Rdt
(2)
where Y is the distance traveled in the direction that u is taken by is the mean square of values of Y any particle in time t , and for a large number of particles. If the fluid is moving with a mean velocity, U , in the direction x , then t can be replaced by X/C. Analysis of Equation 2 in the light of what is known about the general nature of turbulence leads to several significant deductions. First, when x or t is very small, R is close t o unity, which reduces Equation 2 to
dF/dz
=
2>x/U2
(3)
This indicates that y z varies as 22. Second, as x becomes large, correlation coefficient R becomes zero; then s,” Rdx is a constant, and Equation 2 can be written
T
HE diffusion of mass and heat in turbulent fluids is a prob-
lem of vital importance in numerous chemical engineering processes. The intermixing caused by the eddies in turbulent fluids has been compared to the role of molecules in molecular diffusion. Although this analogy has been useful in helping us to understand the general process of eddy diffusion, carrying the analogy too far has led to certain misunderstandings and false interpretations of data. The principal trouble seems to be due to the fact that in molecular diffusion the scale of observation is always many times greater than the mean free path of the molecules, while in turbulent fluids the scale on which observations are made may easily be of the order of the size of the eddies or their mixing distance. Thus in natural streams or in the atmosphere where relatively large eddies are present, if the scale of the diffusion phenomena we are interested in is of the order of the eddy size, then our diffusion process is not analogous to molecular diffusion; and the relation giving the spread of matter-for instance, with distance or time-is not comparable to the ordinary molecular diffusion relation. The correct diffusion equation for such conditions was, it is believed, developed by Taylor ( 2 , 3); however, it appears that no one has tried to verify this relation in detail experimentally or to make use of it practically. Excellent work on eddy diffusion in air streams was done by Towle and Sherwood (4), but they did not fit their data into Taylor’s relation. I n order to obtain basic data on eddy diffusion and t o check Taylor’s relation, which seems to be fundamental to the entire problem of eddy diffusion, an extensive series of measurements was made of mass diffusion in water flowing in rectangular open channels. From the data a relation governing the spread of matter in a turbulent fluid is obtained, and a technique for measuring directly the eddy diffusion coefficient was developed. Furthermore, i t was apparent that diffusion experiments can give quantitative data on the intensity and scale of turbulence.
dF/dx = Z~XOO/U~
(4)
where xo = Jc;” Rdx for values of x when R has become zero. Thus the slope of the F 2 us. z curve is a constant for large values of x. When this occurs, the eddy diffusion process is analogous to that of molecular diffusion for which a diffusion coefficient is defined as
I>
=
”(“) 2 dx
(5)
By analogy Taylor defined an eddy diffusion coefficient in the same manner; however, the value of d F z / d x used is the maximum value as given by Equation 4. By introducing D into Equation 4 and integrating, for values of x when R has become zero, there is obtained: 20 y2 = - (x 2 0 ) (6)
U
-
Using Equation 6, the spread or diffusion of matter can be decan be determined termined for large values of x. Also, if for several values of x , it is possible to determine eddy diffusion coefficient D and also length factor z0, which really characterizes the scale of the turbulence as far as the diffusion process is concerned. It should also be noted that the intensity of the turbulence can be determined, if D and xo are knolvn, from the relation: ~2
=
DU/XQ
(7)
EXPERIMENTAL TECHNIQUES
From the above analyses it is apparent that the eddy diffusion coefficient can be determined in a turbulent water stream by injecting at a particular point matter that has the same density as the water, so that it follows the eddy movements exactly. One method tried was to inject through a fine tube droplets of immiscible liquid having the same density as water. A mixture of n-butyl phthalate and benzine and also of carbon tetrachloride and benzine was used. The liquid was colored dark, and the position of the droplets downstream from the injector was photographed on motion picture film. From each frame of the film it was possible to determine the position, Y , of droplets transverse to the injector for various distances downstream. By analyzing several hundred frames of the film, values of y2 could be obtained for different values of 2. By plotting F 2 against x,the maximum value of df;;l/dx could be obtained, and knowing the stream veloc-
GENERAL THEORY
The theory of continuous movements of masses of fluids in turbulent flow was developed by Taylor, and this theory is expressed in terms of a correlation coefficient, R , between the velocity, v , of a fluid particle in a specified direction at any instant and its velocity, ut, an instant later: where the bars indicate a time average, and 3 is the mean square value of the fluctuating velocity, u. Taylor then showed the following relation to exist:
220
INDUSTRIAL AND ENGINEERING CHEMISTRY
March, 1944 Figure 1.
cal
Typi-
Figure 2.
Variation of
Diffusion
Mean Velocity, of Diffu-
Obtained
sion Coefficient, and of
for Determining
Scale of Turbulence at
the
Diffusion
Center Line of a Water
Coeff i c i e n t s
Channel, One Foot Wide
Data
22 1
--
0.6
0.5
0.4
c 0
ACTUAL POINTS POINTS
0 REFLECTED
.I2 .08 -04
0
.04 D8 .I2
-12 .08 .04
0 .04 .08 .I2 2 0 -16 .I2 .08 .04 0 .04 .08 .I2 .I6 .20 .20 .I6 VERTICAL DISTANCE FROM INJECTOR IN FT.=Y
ity a t the injection point, the value of D could be calculated from Equation 5. Because this method was laborious, it was not used very much; also, it was not feasible a t high water speeds and could not be used with dirty water. Diffusion experiments in turbulent water indicated that the distribution of concentration of injected matter transverse to the line of flow follows the normal error law. Theoretical analyses also indicate this to be approximately true (4). Thus the concentration C of any matter injected at a point x = 0 is given by
b
where Y is the distance above or below a line through the injector in the direction of flow. To eliminate the constant M / d s 2 , let CObe the concentration at Y = 0:
where the logarithm is to be base 10. After some experimentation it was decided to inject into the flowing water a mixture of hydrochloric acid and alcohol, the proportions being adjusted to give the mixture a density equal to that of the water. This mixture was adopted because it is easy to prepare and stable, and bemuse the concentration of chlorides in water is simple to determine. The procedure was to inject a steady flow of the mixture into the water stream and to collect samples of the water a t various transverse distances Y,at several sections downstream from the injector. The sampling time was about 1.5 to 2 minutes, which ensured that the average chloride concentration was obtained at any point. The standard method of water analysis for determining chloride concentrations in water was used in these studies. Since the water in the chan-
'
.I2 .08 .04
0 .04 .08 .I2
.I6
nels contained some chlorides, it was necessary to take a control sample continually. The concentration ratios, C/C,, were plotted against Y for the different sections, and a mean curve was drawn through the plotted points. According t o Equation 9, if C / C , is plotted against Y on semilogarithmic paper, a straight line should result whose slope will be -4.606 F 2 ; this permits the determination of Y". Plotting the various F 2 values against 2, D can be calculated according to'Equation 5. Figure 1 shows typical data obtained for determining one eddy diffusion coefficient. The mean curve drawn through the plotted ppints follows the normal error law very well. It should be noted that theoretically, according to a t any Equation 9, only two samples are necessary to obtain section-one sample at Y = 0 and the other at some other value of Y . However, to ensure against any error and t o verify that the concentration distribution actually does follow the normal error law, eight to ten samples were taken a t each section. The work involved in taking and analyzing the water samples was not particularly great. RESULTS
The technique described made it possible to determine the eddy diffusion coefficient a t various points in a stream cross section. This was done in several sizes of open channels for various water speeds and with different boundary roughnesses. I n this manner a complete picture of the diffusion characteristics of the flow was obtained. Typical variation of values of the eddy diffusion coefficient a t the center line of an open channel are shown in Figure 2. The 20 values are also given; they indicate the variation of the scale of turbulence as far as diffusion is concerned. I n these experiments the measured values of D were then used in calculating the distribution vertically of suspended sand, and the values obtained checked actual measurements of the sediment distribution very well.
I N D U S T R I A L AND E N G I N E E R I N G CHEMISTRY
222
If it is assumed that the transfer of mass by the eddies is similar to the momentum transfer process, our measured value of D should compare to the kinematic eddy viscosity, e, which occurs in the relation between the shear, 7 , and the mean velocity gradient in fully developed turbulent flow: 7
=
(10)
epdU/dy
Figure 3. A Generalized Plotting of the Diffusion Data for Water Flowing in Open Channels
/
4.0
3.0
If the transfer of momentum and mass are physically identical, then D and e should be the same. Experimental verification is somewhat hard because of the difficulty in calculating 2.0 e accurately since assumptions must be made relating to the value of 7 ; also d U / d y cannot be determined with any great accuracy. Some calculations of e were made, and the values 1 . 0 obtained compared well with the measured values of D. I n order to develop the general diffusion equation for the spread of matter with distance or time in turbulent flow, the dimensionless param0 0 LO eter F2U/2Dxo was d o t t e d against 2/20 for all the diffusion experiments. The plotted data tended to follow a smooth curve (Figure 3). Each type of symbol represents a different run. Experiments were made a t various points in two sizes of water channels, 1.00 and 2.25 feet wide, and a t different water depths and discharges. The water speeds varied from 1.0 to 5.0 feet per second. Referring t o Taylor's analysis of the eddy diffusion process, it is apparent that the curved part of the plot in Figure 3 is where correlation coefficient R is greater than zero; the straight line is in accord with Equation 6, since
T o find R as afunction of x from the data given in Figure 3, the values of d P / d x and d 2 F ' / h 2 were carefully determined from the experimental curve, and R was calculated from the relation (Equation 2) :
This really makes R equal to the second derivative of the curve shown in Figure 3. Calculated values of R for various values of 2/20 are plotted in Figure 4; the data follow the law:
R =
Vol. 36, No. 3
(13)
c-z/zo
2.0
SO
40
intensity, d?/U, varies directly with distance or time. When x is large compared to XO, Equation 14 reduces to Equation 6, in which case F 2 varies as 2. It is thus apparent that, in order to predict diffusion in turbulent flow, it is necessary to know both the eddy diffusion coefficient and the scale of the turbulence as defined by zo. If direct measurement of D cannot be made, it should be calculated from the shear and velocity gradient relation, Equation 10. If direct measurement of xo is not available, it can be estimated by using the relation given in Equation 7, 20 = DU/v& and by making an estimate as to the value of d $ / U , the relative intensity of turbulence. Measurements in normal turbulent flow indicate that this relative intensity is of the order of 0.05 to 0.15, depending on the proximity to a boundary, NO-MENCLATURE
All quantities are in foot-pound-second units : C = concentration of foreign material, p.p.m. D = measured diffusion coefficient Q = total water discharge R = correlation coefficient between velocity of a particle a t one instant and an instant later 1.0
Figure
0.8
Coefficient as Obtained
4. Calculated
Values of Correlation
T h a t the correlation curve might be a function such as Equation 13 has been pointed out by others-for instance, Dryden ( I ) . With this function for R, the diffusion equation in turbulent fluids becomes:
When x is small compared t o
This indicates that
XO,Equation
14 reduces to
4%for a given value of relative turbulence
from the Mean Curve
March, 1944
INDUSTRIAL AND ENGINEERING CHEMISTRY
S = channel slope t = time U = mean velocity in z direction v = turbulent velocity component perpendicular to direction of mean flow w2 = mean square value of e, Y2 = mean square value of transverse spread of diffusing matter 2 = distance downstream from point a t which diffusing matter is injected xo = scale of turbulence defined by Rdx e = kinematic eddy viscosity
p T
223
= fluid density = unit shear in fluid LITERATURE CITED
(1) Dryden, H.L., IND. ENQ.CHEM.,31, 416 (1939). (2) Taylor, G. I., Proo. London Math. SOC.,20, 196 (1921). (3) Taylor, C. I., Proc. Roy. SOC.(London), ISIA, 421 (1935) (4) Towle, W.L.,and Sherwood, T. K., IND. ENO.CHEM.,31, 457 (1939). BABBD on the Ph.D. thesis of C. L. Pien, University of Iowa.
Lactic Acid Condensation
Polvmers d
PREPARATION BY BATCH AND CONTINUOUS METHODS E. M. FILACHIONE AND C. H. FISHER Eastern Regional Research Laboratory, U. S. Department of Agriculture, Philadelphia, Pa. PREVIOUS INVESTIGATIONS The preparation, properties, and reactions of condensation ONDENSATION prodpolymers of lactic acid are reviewed; batch and continuous ucts or polymers of Althou h t h e reported methods for converting lactic acid into its condensation findings o f previous workers lactic acid are imporare not in complete agreepolymers are described. Removal of water during the tant because: (a) They ocment, aqueous lactic acid dehydration or self-esterification of lactic acid is facilitated cur in all aqueous solutions solutions and dehydrated lacby relatively high temperature, reduced pressure, sulfuric of lactic acid containing aptic acid appear to consist mainly of “free lactic acid:’ acid or similar esterification catalyst, and an entraining proximately 18% lactic acid (monomeric a-hydrox agent, such as benzene or toluene. The resulting conor more. (b) They are onic .acid), water, densation polymers, which react readily with methanol, promising chemical intermecondensation or self-esteriare useful for making methyl lactate. diates. (c) The condensafication products, such as tion polymers of intermedilactyl lactic acid and lactyl lactyl lactic acid, 111: ate molecular weight can be used as such or after slight alteration as plasticizers. ( d ) The HOCH(CHs)COOCH(CHs)COOCH(CHs)COOH (111) condensation products of higher molecular weight can be converted into useful plastics by condensation with certain These three components occur in various proportions, the extreme limits bein pure water and completely polymerized lactic acid, I. vegetable oils, glycols, and other chemicals. (e) They are exLactide, If, has the same ultimate composition as completely cellent for storing and transporting lactic acid in a highly conpolymerized lactic acid, but appears to occur only in traces centrated condition; the completely polymerized linear product (3 6, 18, 13, 16, 17) in equilibrium mixtures of monomeric lactic (I) and lactide (11)are equivalent to 125% lactic acid: acid, water, and polylactic acid. The ease (1, 6, $6, 30) with which lactide, 11, is hydrolyzed to lactyl lactic acid is of interest (-OCH(CHs)CO-)z + HOCH(CHs)COOH -+ in this connection. The composition of various equilibrium mixtures of monomeric (1) lactic acid, water, and polylactic acid is shown in Figure 1 prep r e d from data taken from publications of Bezzi, Riccobon;, and ullam (6), Watson (@), and Thurmond and Edgar (39). The I I linear condensation polymers were considered as one component co CHCHI in the preparation of Figure 1 although the polymers were principally lactyl lactic acid (the dimer) in the relatively dilute solu\ ’0 tions and higher polymers (such as di- tri-, and tetralactyl lactic (11) acid) in the more concentrated solutions (6). The variation in molecular weight of the polylactic acid with concentration ( 6 ) Our interest in the production and properties of polymerized ex ressed as total acidit after hydrolysis is shown in Figure 2. lactic acid was created largely by finding that the linear condenOtter workers assumed tzat the polylactic acid occurring in lactic acid solutions was the dimer, lact 1 lactic acid. Other data resation polymers constitute an excellent starting point for making garding the composition of equiliirium mixtures of lactic acid methyl lactate (19). When methyl lactate is formed by the interare given in Table I. action of polylactic acid and methanol, probably alcoholysis as Figures 1 and 3 show that both water and polylactic acid can well as esterification is involved, and only small quantities of occur in concentrations as high as loo%, but that the highest concentration reached by monomeric lactic acid in equilibrium water are formed in the reaction. Absence of water is advantamixtures is 47 mole % or 62% by weight. The highest mole and geous because methyl lactate is readily hydrolyzed during distilweight concentrations of monomeric acid are attained when the lation when appreciable quantities of water are present. Moretotal acidit after hydrolysis is 100-105 and SO%, respectively. over, methyl lactate and water distill as an azeotropic mixture. Instead of gavlng the usual connotation of purity, “100% concentration” when applied to lactic acid designates a mixture The present paper summarizes earlier information on polylactic containing approximately 47 mole yo monomeric lactic acid, acid, much of which (3, 6) has not been readily available and de34.mole yo water, and 19 mole yo polylactic acid; the polylactic scribes both batch and continuous methods for dehydrating or acid has an average degree of polymerization (6)of 2.75. Water, polymerizing lactic acid. an important constituent of 9 0 0 % lactic acid”, occurs In ap-
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