Ind. Eng. Chem. Process Des. Dev. 1986, 25, 108-116
108
microcomputers like the Apple 11. (2) The performance of IMC is superior to PI control (with decoupling). The reason is that IMC includes implicitly both a dynamic decoupler and a dead-time compensator. On-line tuning is largely unnecessary. (3) IMC has only one on-line tuning parameter per output. With the reduction in tuneable parameters compared to PID, for example, the search for appropriate values is simpler and faster. (4) The effect of the IMC tuning parameters is transparent. The adjustment to reach an acceptable performance/robustness trade-off is therefore simple. (5) The robustness of IMC against model errors is satisfactory. Increasing the filter time constant increases the robustness if necessary.
Acknowledgment Financial support from the Department of Energy and the National Science Foundation is gratefully acknowledged.
Literature Cited Astrom, K. J.; Hagander, P.; Sternby, J. Proceedings of the 19th IEEE Conference on Decision and Control, Albuquerque, NM, 1980, p 1077. Brosiiow. C.; Zhao, G. Q.; Rao, K. C. Paper presented at the Proceedings of the Automatic Control Conference, San Diego, 1984. Canney, W. M. Project Report, Department of Chemical Engineering, University of Wisconsin, Madison, 1983.
Cutler, C. R.; Ramaker, B. L. Paper presented at the AIChE 86th National Meeting, April, 1979. Also presented at the Proceedings of the Joint Automatic Control Conference, San Francisco, 1980. Economou, C.; Mwari, M.; Paisson, B. Ind. Eng. Chem. Process Des. Dev., in press. Garcia, C. E.; Morari, M. Ind. Eng. Chem. Process Des. Dev. I98PS 21, 308. Garcia, C. E.; Morari, M. Ind. Eng. Chem. Process Des. Dev. 1985a,2 4 , 472. Garcia, C. E.; Morari. M. Ind. Eng. Chem. Process Des. Dev. 1985b,2 4 , 484. Garcia, C. E.; Morshedi, A. M.; Fitzpatrick, T. J. Paper presented at the Proceedings of the Automatic Control Conference, San Diego, 1984. Hoiiett, J. MS Thesis, Rensseiaer Polytechnic Institute, Troy, NY, 1984. Hot, 8. R.; Morari, M. Chem. Eng. Sci., 1985a,in press. Holt, B. R.; Morari, M. Chem. Eng. Sci. 1985b,4 0 , 59. Mehra, R. K.; Rouhani, R.; Eterno, J.; Richaiet, J.; Rauit, R. I n "Chemical Process Control 2"; Edgar, T. R., Seborg. D. E., Eds.; Engineering Foundation: New York, 1982; p 287. Morari. M.; Skogestad, S.;Rivera, D. F. Paper presented at the Proceedings of the Automatic Control Conference, San Dego, 1984. Morari, M. Paper presented at the Automation 5, 5th International IFAC/IMEKO Conference on Instrumentation and Automation in the Paper, Piastics and Polymerization Industries, Antwerp, Belgium, 1983. Morari, M.; Ray, W.H. Chem. Eng. Educ. 1979, 13, 180. Prett, D. M.; Giiiette, R. D. Paper presented at the AIChE 86th National Meeting, April, 1979. Also presented at the Proceedings of the Joint Automatic Control Conference, San Francisco, 1980. Ray, W. H. "Advanced Process Control"; McGraw Hili: New York, 1981; Chapter 3.
Received for review September 20, 1984 Accepted May 13, 1985
Turbulent Mixing at High Dilution Ratio in a Sulzer-Koch Static Mixer J. Goldshmld,' M. Samet, and M. Wagner Environmental Engineering and Design Company, Tel-Aviv 6 1430, Israel
Water and sewage treatment calls for the addition of small quantities of chemicals to large bodies of water. Efficient utilization of the chemical added requires fast and complete mlxing. An optimal combination of turbulent dispersion down to eddies of the Kdmogoroff scale and molecular diffusion would yield fast mixing on a mdecular scale which in turn favors the desired reactions. A new theoretical model is presented for turbulent dispersion of mutually miscible liquids in a static mixer. The model predicts the rate of increase of the solute concentration in a dilute solution, from the rate of increase of the interfacial area and molecular diffusion equations. An analogy between the turbulent cascade process and the Lagrangian dispersion of the concentrated solution serves to calculate the rate of increase of the interfacial area. The model predicts that complete mixing on a molecular scale would be achieved in the Sulzer-Koch mixer, when the smallest eddies reach the Kolmogoroff scale. Experimental results obtained during the mixing of ozone solution in water, in a Sulzer-Koch static mixer, are presented. The results compare well with the theoretical predictions.
Mixing of two or more miscible liquids at high dilution ratio is common in water and sewage treatment. Ozone, chlorine, or chlorine dioxide solution in water at the parts per million (ppm) level is obtained by mixing small quantities of concentrated aqueous solution in bulk quantities of water. Here, fast and efficient mixing on a molecular level is imperative for efficient use of the chemical, since slow and inefficient mixing increases the consumption of the chemical added, due to side reactions. It was the objective of this work to study dispersion and mixing of concentrated aqueous solution in water at high dilution ratios. The study consisted of two parts: development of a mathematical model for the mixing process and experimental verification of the model. We chose concentrated aqueous ozone solution for the concentrated solution and water for the dilute solution. A Sulzer-Koch static mixer (type SMV-16 DN-80 with six 0196-4305/86/1125-0108$01.50/0
removable elements) was used to promote mixing and minimize back-mixing. Dilution ratios of 300:l to 301 were studied.
Physical Model for the Dispersion of a Concentrated Solution in a Sulzer-Koch Static Mixer For fluids, the movement of materials between various parts of the bulk occur by a combination of three mechanisms: convection of mean flow, eddy diffusion, and molecular diffusion (Brodkey, 1966). In the Sulzer-Koch static mixer, the first mechanism is unimportant due to its tendency to develop "plug flow" conditions (Chen et al., 1974), and mixing is achieved by a combination of turbulent and molecular diffusion. The concentrated solution, injected just ahead of the static mixer, is split into individual streams when it enters 0 1985
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Ind. Eng. Chem. Process Des. Dev., Vol. 25, No. 1, 1986
the first mixing element. At each intersection, a partial quantity of the mixture is sheared off into the crossing channel. In this way, any inhomogeneity is evened out two-dimensionally in the first element and three-dimensionally in the second one which is turned at 90' (Streiff, 1970). In order to simplify the model, it was assumed that the static mixer is made up of several straight pipes with an hydraulic diameter equal to that of the channels and a number of pipes equal to the number of channels. A second assumption is that the clumps of the concentrated solution have a diameter close to that of the pipe, when they enter the mixer. The clumps have the same physical properties as the dilute solution, and they are distinguished only by the higher ozone concentration. We next assume that each clump behaves like a turbulent eddy, and it breaks down to smaller and smaller eddies in a manner described by the cascade process as it moves down the mixer, due to the turbulent nature of the flow. By applying the analogy between a clump and an eddy, one can use the Lagrangian dispersion theories to evaluate the rate of change of the concentrated solution clumps surface area with time. The concentration of the solute in the dilute solution is easily calculated from the diffusion equation. Thus, the solute concentration in the dilute solution as a function of residence time in the mixer is calculated. This can be later used, together with kinetic equations, to evaluate the reaction yield.
Mathematical Model for Ozone-Solution Dispersion in a Sulzer-Koch Static Mixer Turbulent dispersion is the act of spreading out of fluid particles by random fluid motion, which, from a macroscopic point of view, is associated with eddy structure, ranging in size from very large to very small. In general, the largest eddies transfer energy to the smallest by what is believed to be a cascase process. The cascade process is also related to the scale reduction phenomenon through eddy breakdown, followed by an increase in the interfacial area and transport of momentum and mass. Thus, one can associate the turbulent dispersion problem with the cascade process. Further, since problems involving dispersion and mixing of fluid particles are essentially Lagrangian in nature, it would be natural to derive an expression for the analytical model by using the Lagrangian approach. For the case of ozone solution dispersion in water, the concentrated solution is almost identical with the dilute one, in the sense that their physical properties, such as density, viscosity, etc., are much the same. Let the concentrated solution be injected into a flow field which, for the sake of simplicity, consists of frozen turbulence. Under these circumstances, the axial location of the reference point in the mixer is irrelevant to the model. The flow field in question is made laterally bounded by the channel walls, by assuming that immediately after injection the length scale involved in the mixing process is of the order of the channel diameter. At time To,a particle passing through the reference point is marked. (See Figure 1.) Due to its random motion, the tagged particle moves in an arbitrary path, forming a well-defied control volume. A different particle, being located at time Toat another tagging point, could have moved within the same period, in an entirely different path so that its lateral location with respect to the previous particle would probably be different. Statistically, many tagged particles move along a large number of paths. By performing an ensemble average over all those paths, one obtains a description of a material volume whose bound-
109
/ / I / I,,,
E""LLOPW' RL,
/
\
CYRIXEL ~ V m F m ~
~ R m \ C uW T H
p",
Figure 1. Dispersion model from the Lagrangian view.
aries are marked in Figure 1 by the broken line. The volume enclosed by this boundary is probably similar to the volume of an eddy formed in an Eulerian system. The Lagrangian observer, however, would interpret this limiting boundary as the farthest lateral location to which a particle may move, before it loses its correlation with the tagging point. Following Brodkey (1966),we shall assume homogeneous and stationary turbulent conditions, which is acceptable for turbulent pipe flow. This implies that the various Lagrangian quantities are independent of the particle chosen and are functions of the time difference only. For the sake of simplicity, let us denote Toas the entrance time into the channel and y(0) = 0 as the tagging point. On this basis, the mean squared displacement of particles which start at different times from the point of origin in homogeneous turbulence is given by
y(t)2 = 2 J i ( t
-
7 ) R ( 7 )d r
d/dt(yo2) = 2 J t0 R ( r ) d7 where R(7) is the Lagrangian correlation function. One may note that these equations have already been considered by Corrsin (1963) and Brodkey (1966) in connection with the decay of grid-generated turbulence and mean dispersion behind a line source, in a turbulent boundary layer. Contrary to the approach described by Brodkey, these relationships were not used to obtain the eddy diffusion coefficient for the mass transport equations; rather, they were used to evaluate the rate of increase in the clumps interfacial area, by computing the rate of scale reduction at distinct times of mixing. Equations 1and 2 are of no practical value unless R(7) is properly evaluated. To this end, we have exploited the relationship R(7) = u ' ~ R ~and ( ~ for ) RL(7) used the expression suggested by Grant (1957), namely R L ( ~=) 1 - ( 7 / t T ~+) (7/tTL) In ( 7 / t T d (3) where $. is a parameter to be evaluated later on, RL(7)is the Lagrangian correlation coefficient, and TLis the Lagrangian time scale as defined by Brodkey (1966). Due to the freezing of the flow patterns, u ', the standard root mean square (rms) value of the lateral velocity fluctuations is the only existing velocity in the lateral direction. The final forms of eq 1and 2 which express the dispersion process in accordance with the Lagrangian approach, are y(t)2= ~ ' ~ t-~(11t/18tTL) [ l + ( t / 3 [ T L )In ( t / t T L ) ]
-
(4)
d/dW)21= 2 ~ ' ~ t-t (3t/4tTJ l + ( t / 2 t T d In ( t / t T J I (5) The relative rate of change of dispersion, as defined by the right-hand side (rhs) of eq 6, can be obtained from the
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Ind. Eng. Chem. Process Des. Dev., Vol. 25, No. 1, 1986 IUITlRL EOOY
Figure 2. Cascade process in homogeneous turbulent pipe flow.
"dispersion rate" eq 5 and the square of the "Lagrangian correlation length" eq 4. The dispersion rate has the dimensions of frequency and is associated with the dominant frequency of the eddy which, from the Eulerian point of view, is assumed to be responsible for the dispersion. Using Hinze's notation (1959), we may write
CQ = d/dt{u(t)2)/y(t)2 (6) where C is a parameter and Q denotes the frequency. The dominant frequency of the eddy can be expressed in terms of the eddy wavenumber value, i.e., Q = u'k, which is similar to the form suggested by Taylor (1954). Finally, by relaticg the eddy wavenumber to the eddy wavelength: k = 1/D, and by utilizing the relationships (4) and (5), the form expressing eddy size in terms of the Lagrangian parameters is obtained
D = c( &)t
x
channel diameter and are generated by mechanical breakdown of the flow (Streiff, 1977). Further breakdown into much smaller eddies follows, due to shear stresses and pressure forces exerted upon the eddies. The Eulerian observer may detect these substantial changes in the size of the eddies if, and only if, he were to move with the mean flow from one axial location to another, i.e., each time positioning himself together with the coordinate system at another axial point farther downstream from the entrance plane. Suppose now that we wish to evaluate the size of the nth eddy which is shown in Figure 2. It is apparent that in order to obtain a nonvanishing integrand in eq 8, the variable r should vary within the boundaries r,-l and r,, since r < rnWlor r > r, are outside the nth eddy and do not correlate with points of that eddy. Following this argument, eq 8 is rewritten in the form 1 D, = u(x;O) u(x;r) dr (9)
-srn uf2
where USis the friction velocity, assumed to be constant for a given conduit and flow rate. According to Taylor (1953), the Eulerian approach relates the lateral integral length scale to the average size of an eddy. The "average size of an eddy" is considered by experimenters as the mean outcome of an averaging process over many discrete eddies, each having its own characteristic shape and boundary. Since the flow field under examination is assumed to be stationary, the only averaging procedure, which is relevant in this case, is the time averaging one. The lateral length scale is equivalent to the correlation distance between two material points, located at different spatial positions. According to Lumley et al. (1964), the integral length scale in the lateral direction is given by
where the overbar denotes time averaging, x is the longitudinal coordinate, and r is the lateral one. Yet, to be able to apply this formula in practical situations, the limits of integration and the independent variable r, have to be redefined carefully. T o this end, the scale reduction concept is used. Consider the schematic description of the cascade process as shown in Figure 2. The eddies, at the entrance to the Sulzer-Koch static mixer, have the size of the
r"-I
where r is the lateral length variable, ranging from r,-l to r,. The nth eddy, however, was chosen arbitrarily to represent a typical eddy in the model; hence, eq 9 is valid for any arbitrarily chosen eddy and not only for the nth one. One may conclude, therefore, that an Eulerian observer would be able to define the size of an eddy which he is observing, by properly applying relation 9 to that very eddy. It is likely that at some instance, both the Lagrangian and the Eulerian systems of reference will coincide at the nth eddy to yield a single length scale, in the sense that r, = (y,2(t))1/22 Under these circumstances, and at that very instant, D, of eq 7 and D, of eq 9 are equal. Each eddy in the cascade process has an arbitrary shape which, in the absence of a lateral mean velocity gradient and due to a long time-averaging procedure, is spherical. In this sense, the lateral length scale is interpreted as the mean eddy diameter (size) D, which according to the definition is identical with the Sauter mean diameter (Monin and Yaglom, 1971). Computational Procedure A numerical procedure was developed to evaluate the variables in eq 7, which when known, were used-to calculate the decrease in the Sauter mean diameter D along the mixer. There are six variables in eq 7 to be considered: v'/U., TL,E, C,t, and U.. Following Schlichting (1968) and Monin and Yaglom (1971), the ratio u'/U, was assumed to be a universal function of rld, on the basis that the flow in question is
Ind. Eng. Chem. Process Des. Dev., Vol. 25, No. 1, 1986
unidirectional and homogeneous and the Re number is sufficiently large. Furthermore, the profile of v ’/ U , can be assumed to be independent of the actual shape of the conduit, provided that the normalizing length parameter d is properly defined (Schlichting, 1968);this was done by defining d as the channel hydraulic diameter. For each eddy size (denoted by the index n),the value of r, was used as the lateral variable; thus, the ratio r / d was known. The profile of v’/U, was adopted from the data of Laufer (1954),and the value v’/V, for each n could be evaluated. Following Lumley and Panofsky (1964) as well as Monin and Yaglom (1971), the Lagrangian integral time scale is of the form TL = 2.05ld(U*/~’)~/U* (10) where u’is the longitudinal root mean square velocity. The profile of u’/U, was also adopted from Laufer (1954))and its value was obtained in a manner similar to u’/ U,. The equation for calculating U, was derived from basic concepts of hydrodynamics
U, = ( ~ Q / T D ) ( N ~ / D ~ N ) ’ / ~
(11)
where Q is the flow rate of the dilute stream, Ne is the Newton number of the Sulzer-Koch mixer, whose dependence on Re is given by the manufacturer (Schneider, 1983))D is the mixer pipe diameter, and N specifies the number of channels in the mixer. The term [ appears as a compensating factor in Grant’s equation and relates r to the decay mode of RL(r),for a given value of TL. From the experimental curve of RL(7) vs. r reported by Shlein (1971))the value of 7owas obtained at the point where RL(rO)decreases to 5% of its initial value. With r o and TLknown, [ could be computed from eq 3. C is a parameter, and the best way to determine-its value is by imposing on eq 7 an initial condition that Do should yield an eddy size which is 95% of the channel diameter. The value 0.95$ was arbitrarily chosen. Any other number which keeps Do close to d will do (see Figure 2). The parameter t in eq 7 does not represent a continuous time variable but is defined only in quants. This follows since t is defined for each eddy. So long as the tagged point correlates with a certain eddy, it is tied to a certain time t. When it correlates to a different eddy, it is tied to a different time. In other words, t according to the Lagrangian approach is analogous to r in the Eulerian approach. Therefore, for each eddy n, t should equal the corresponding Lagrangian time scale TLn. The model for dispersion and scale reduction in the static mixer was developed assuming that the flow within each channel resembles a fully developed turbulent pipe flow. In the experimental program, we faced conditions where the flow in the mixer was transitional. It became, therefore, necessary to modify the above computational procedure to meet also transitional flow conditions. For transitional flow, the following assumptions were made: The physical picture is of slugs resulting from sudden bursts at the wall, causing eddy breakdown. Thus, the process is similar to the turbulent case, only less effective. The only parameters that have changed in the transitional flow are v’/U,, u’/U*, and U,. The radial distributions of u ’/ U, and u ’/ U, for transitional pipe flow were adopted from the data of Wygnanski and Champagne (1973). To modify U,, eq 11 was reconsidered. It was noted that the only parameter which is influenced by the change in flow regime is the Newton number, Ne. This parameter was shown by the Sulzer-Koch manufacturer to be of the form
Ne = D(693/Re
111
+ 0.7)/2t2d
where t is the void fraction, D and d are defined as before, and Re is the Reynolds number. After substituting eq 12 into eq 11,the modified value of U* for transitional flow was obtained. Experimental Section In order to verify the physical model and computational procedure described earlier, an experiment was designed where a concentrated ozone solution was injected ahead of a Sulzer-Koch static mixer into a continuous flow of ozone demand free water. The experimental unit consisted of two subsystems which were combined together. Subsystem A, which included the dilute stream flow system and the mixing section, is depicted in Figure 3. Subsystem B, the concentrated ozone solution subsystem, is depicted in Figure 4. A special sampler, presented in Figure 5, was installed at the exit from the mixer. The sampler was built of two 1.2 mm 0.d. stainless steel tubings, one at the center and the other close to the pipe wall. Throughout the experimental run, liquid flowed continuously out of the system through the sample tubings and was either drained or collected in test tubes for analysis. Two markedly different ozone concentration levels, >10 ppm in the concentrated solution and 0.3, eq 19 forms a straight line on semilog paper. Different lines were obtained for different flow rates. All the lines seem to meet close to the point: t / T k = 1; (Cd/C,d;l) = 1. Thus, all the lines can be represented by the equation
The numerical value of the parameter a varies from one flow rate to the other, but in all cases, it is close to 6.24. If we now substitute for the expoqent from eq 15, the relationship between Cd/C&I and D,/q is approximated by Cd
en u b d l
II n
(23)
ufl
The upper limit for C d is CJ-I and th_elower limit for bfl is q; therefore, C d = C&l when q = D,. In other words, the concentrated solution reaches full dilution (mixing on a molecular scale) in the Sulzer-Koch static mixer, when the size of the smallest eddies are of the order of the Kolmogoroff scale. Equation 21 was replotted in Figure 12, only this time as H vs. cd/c&, with all other parameters kept constant. It is obvious from the examination of Figure 12 that for high dilution ratios, i.e., H < 5%, the concentration in the dilute solution, leaving the mixer at the operating conditions listed in the figure, will be c d N 0 . 4 C a for all values of H. However, for H 2 5 % ,the approach to complete dilution is faster and the greater H is, the faster is the approach to complete dilution. This was to be expected since the mixing problem is much more acute in systems with high dilution ratio, such as those encountered in water and sewage treatment.
Figure 13. Comparison between Cd found experimentally and Cd calculated from eq 21.
Figure 13 is a comparison between the calculated and the experimentally measured values of C d at various operating conditions. If full agreement had been achieved, all the points would have fallen on the 45O line. The data points in Figure 13 congregate about the 45O line, but deviations up to 0.07 ppm are reported. (The error due to the analytical procedure used to determine the ozone concentration is estimated at f0.0025 ppm.) Conclusions For high dilution ratios (H 5 5%), one cannot assume immediate and complete mixing, when a fully miscible concentrated solution is injected into large quantities of water. A theoretical model, based on turbulent dispersion of the concentrated solution down to the Kolmogoroff scale eddies, coupled with molecular diffusion, was developed for the Sulzer-Koch static mixer. The model predicts the rate of increase of the solute concentration in the dilute
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Ind. Eng. Chem. Process Des. Dev., Vol. 25, No. 1, 1986
solution. The model was tested experimentally by injecting concentrated ozone solution into a water stream at dilution ratios of 1/30 to 1/300. The experimentally determined concentrations of ozone in the dilute solution agreed quite well with the concentrations calculated from the dispersion equations. The model for the Sulzer-Koch static mixer predicts that complete mixing on a molecular scale would be achieved at about the same time when the smallest eddies reach the Kolmogoroff scale. Acknowledgment This research was supported by a grant from the National Council for Research and Development, Israel, and the K.F.K. Karlsruhe, Germany. Nomenclature A = cross-sectional area of the mixer a = area per unit volume C = constant C, = ozone concentration in the concentrated solution Cd = ozone concentration in the dilute solution D = pipe diameter DL = diffusivity in the liquid phase Qp = clump diameter D = mean eddy diameter; also Sauter mean diameter D, = mean eddy diameter pertaining to the nth eddy d = hydraulic diameter of the channel in the Sulzer-Koch mixer E = rate of turbulent energy dissipation fn = Darcy's friction factor g = gravitational constant H = Ll/(LI + L,) h = distance along the length of the mixer KL = overall liquid film mass-transfer coefficient kL = liquid film mass-transfer coefficient kL' = o.31(~C)-213~~.p~udg/.p~2]1'3 k = wavenumber
L, = superficial velocity of the concentrated solution L2 = superficial velocity of the dilute solution N = number of channels in the Sulzer-Koch mixer Ne = Newton number = fD/(2t2d) Q = liquid flow rate r = lateral position Re = Reynolds number ItL(') = Lagrangian correlation coefficient
Sc = Schmidt number TL = Lagrangian time scale Tk= Kolmogoroff time scale t = time v-' = root mean square of lateral velocity fluctuations U = mean longitudinal velocity U , = friction velocity u' = root mean square of axial velocity fluctuation y = lateral position Lagrangian frame of reference. Greek Letters 7 = ( t - To)
il = frequency 9 =
Kolmogoroff length scale
= void fraction u = kinematic viscosity t
dynamic viscosity of the dilute solution = density of the dilute solution 6 = coefficient (defined in eq 3)
& = Pd
Literature Cited Brodkey, R. S. in "Mixing Theory and Practice"; Uhi, V. W., Gray, J. B., Eds.; Academic Press: New York, 1966; Vol. 1, Chapter 2. Caiderbank, P. H. I n "Mixing Theory and Practice"; Uhl, V. W.. Gray, J. B., Eds.; Academic Press: New York, 1967; Vol. 2, Chapter 6. Chen, S. J.; Develion, P. D.; Bor, T. P. DECHEMA-Monogr. 1974, 74, 77. Corrsin, S. I n "Handbuch der Physik", 2nd ed.; Flugge, S.,Truesdell, E., Ed.; Springer-Verlag: Berlin, 1963; Voi. 8, Part 2. Goidshmid, J., 1st year annual report, 1983. Goldshmid, J., final report, 1984. Grant, A. M. J. Meteorol. 1057, 74 (4). 297. Hinze, J. 0. "Turbulence"; McGraw-Hill: New York, 1959. Laufer, J. National Advanced Committee on Aeronautics, Report No. 1174, 1954. Lumley, J. L.; Panofsky, M. A. "The Structure of Atmospheric Turbulence"; Interscience: New York, 1964. Monin, A. S.; Yaglom, A. M. "Statistical Fluid Mechanics"; The MIT Press: Boston, 1971;Vol. 1. Schiichting, H. "Boundary Layer Theory", 6th ed.; McGrawNill: New York, 1968. Schneider, G., personal communication, 1983. Shechter, H. Water Res. 1973, 7, 729. Shlein, 0. J. Ph.D. Dissertation, The John Hopkins University, Baltimore, MD, 1971. "Standard Methods for the Examination of Water and Wastewater", 14th ed.; APHA, AWWA, WPCF: New York, 1975. Streiff, F. Sulzer Tech. Rev. 1077, 3 , 106. Taylor, G. 1. Roc. R . SOC.London, Ser. A 1935, A751, 421. Taylor, G. I. Proc. R . SOC.London, Ser. A 1953, A Z f 9 , 186. Wygnanski, I.; Champagne, F. H. J. FluU Mech. 1973, 59, 281.
Received for review July 9, 1984 Revised manuscript received March 4, 1985 Accepted May 1, 1985
