1604
Ind. Eng. Chem. Res. 1987,26, 1604-1609
Simulation of Aggregate Growth and Breakage in Stirred Tanks Yong H. Kim and Larry A. Glasgow* Department of Chemical Engineering, Kansas State University, Manhattan, Kansas 66506
Successful modeling of the dynamic behavior of the particle-size distribution in coagulation and wastewater treatment processes is necessary for improved understanding and design of flocculation operations. At present, both probabilistic and stochastic modeling efforts are constrained by the lack of accurate information regarding the breakage mode and fragment-size distribution and the interaction between these quantities and hydrodynamics. Indeed, many prior modeling efforts have been based upon breakage mechanisms that are quite erroneous for inhomogeneous turbulent flow fields. The objectives of this work are to demonstrate the true nature of floc breakage in turbulence, to show that deterministic models are quite inappropriate, and to provide a means for the simulation of batch flocculation experiments. The significance of floc breakage to the operation of treatment processes appears to have been appreciated as early as 1960; now any operator familar with lost solids over the weir in bioflocculation or shortened filtration runs in domestic water production can describe the symptoms of floc disintegration. As investigators began to look at the problem, it was natural to start with kinetic descriptions that simply reversed Smoluchowski’s (1917) orthokinetic model for the rate of removal of primary colloidal particles; for example, Healy and La Mer (1964) suggested that floc-size reduction occurred by primary particle erosion as a consequence of surface shear. In contrast, Thomas (1964) advocated bulgy pressure deformation and rupture in analogy with existing theories of droplet breakage in emulsification. The idea that a definite relation must exist between parent aggregation size and the scale by turbulent eddies responsible for disintegration was put forward by Parker et al. (1972). Matters were further complicated when Ham and Christman (1969) offered experimental evidence of breakage caused by floc-floc collisions. In retrospect, any theory of floc disintegration based upon primary particle liberation should have been viewed with suspicion; since Michaels and Bolger (1962) and Vold (1963) had described the process of floc-floc collisions leading to what is generally called “multiple-level” aggregation, it was apparent that breakage would produce a broad spectrum of fragment sizes. Furthermore, the turbulent energy necessary for disintegration is simply not available at small scales. Eddies comparable to the Kolmogorov microscale are not capable of removing individual particles from the small floc structures except under the most peculiar circumstances. The first direct evidence of the true nature of floc disintegration was probably the experimental investigation reported by Glasgow and Hsu (1982). In this work, photographs of individual floc-breakage events caused by a turbulent jet showed a very broad distribution of daughter particle sizes; it was also shown that the disintegrations could generally be classified as either “thorough” or “erosive”. Binary breakage, ever popular with theorists, was not found to occur to any significant extent under any hydrodynamic conditions. This study made it clear that the interaction of nonlinear, stochastic hydrodynamics with the products of a random growth process could not be satisfactorily described by deterministic differential equations. *To whom all correspondence should be addressed.
Of course, this fact has been appreciated recently by a number of investigators who have examined the population balance for this purpose: Domilovskii et al. (19811, Pandya and Spielman (1982,1983), and Hsu and Glasgow (1983, 1985) have considered this approach. The population balance has undeniable appeal for this purpose; however, a problem common to most of the studies of this type is the need for a posteriori parameter determination to provide adequate representation of size-distribution behavior. This is necessary because the breakage rate, the breakage mode, and the daughter particle-size distribution are not known quantities, although their form has been both assumed and derived. In extreme cases, such as the Pandya-Spielman study (their model has nine parameters), this leads to the question of uniqueness, as it has been noted that quite different parameter selections can lead to equivalent fits. We have therefore taken a different approach to this problem, one in which more complete hydrodynamic information has been gathered for the flow field of interest, a stirred, baffled tank.
Tank Hydrodynamics Laboratory-scale flocculation experiments have traditionally been carried out in either cylindrical Couette apparatuses or in stirred, baffled tanks. The latter is of practical interest since virtually all flocculation processes are conducted in inhomogeneous turbulent-flow fields. Obvious examples include paddle-flocculation basins, upflow clarifiers, and aeration tanks in bioflocculation. The stirred, baffled tank has been studied by Holmes et al. (1964), Cutter (1966), Rao and Brodkey (1972), Okamoto et al. (1981), Placek and Tavlarides (1985), and Glasgow and Kim (1986) among others. The impeller stream is of primary interest for it has been shown that the dissipation rate there can exceed the mean tank value by as much as 2 orders of magnitude. It is likely that floc-breakageevents in STR’s are largely confined to the impeller stream. Similarly, in paddle flocculators one would expect disintegration principally in the vicinity of the blade tips. In our investigation, an acrylic plastic tank was constructed with a diameter of 24 cm and a height of 22.5 cm. Four 2.5-cm baffles were installed at equal spacing on the inner wall of the tank, and agitation was provided by a six-bladed turbine-type impeller with a diameter of 11 cm. The norma1 operating capacity of the tank was 8750 cm3, and the impeller was driven with a variable-speed motor capable of any rotational speed between 20 and 300 rpm. Mean power inputs were determined through measure-
0888-5885/87/2626-1604$01.50/0 0 1987 American Chemical Society
Ind. Eng. Chem. Res., Vol. 26, No. 8, 1987 1605 ment of the applied torque and the angular velocity of the impeller. Laser-Doppler velocimetry was used to obtain instantaneous point-velocity measurements in the impeller stream. The instrumentation consisted of TSI optics and frequency tracker with a Hughes He-Ne laser; this system was used with a 50-mm beam expansion in the on-axis backscatter mode of operation, and the analog signal was sampled with a Nicolet 4094A digital storage oscilloscope. Sample records were read into an IBM PC via an RS-232 interface for the necessary off-line processing. A principal goal of this part of the work was the determination of the local dissipation rate per unit mass, t. This is normally achieved with either the Taylor equation for the large-scale dynamics, as used by Cutter (1966), ~
~
3
1
1
(1)
or the isotropic relation, E
= 15vu2/X2
(2)
used by Rao and Brodkey (1972) where X is the Taylor microscale. With spatial measurements, the integral length, LE,and the microscale, A, could be determined from the integral and the curvature, respectively, of the correlation coefficient graph. The dissipation rate can also be obtained by integration of the one-dimensional energy spectrum: t
= 15VJmKlzE1(K1) dK1
(3)
When spatial measurements cannot be made, Taylor’s hypothesis is used to relate wave number to frequency: K~ = 2an/U1 (4) This technique was employed by Okamoto et al. (1981), for example. Since time-series data were collected in our case, the temporal microscale, 7x, was determined from the curvature of the autocorrelation coefficient, ~(71, as described by Tennekes and Lumley (1972). Some of these data have been previously presented in Glasgow and Kim (1986). To provide a comparison for the earlier estimates, E was also calculated using velocity data obtained from direct analysis of the frequencies of the Doppler bursts. By relating the integral length scale, 1, to the characteristic vertical size of the impeller stream at the measurement point, it was possible to use Taylor’s inviscid relation, eq 1. A number of cumulative probability functions for point velocity are presented in Figure 1. Notice that on the impeller stream center line at a distance 10 mm from the blade tips, the mean velocity is approximately 56 cm/s at 182 rpm but only 20 cm/s at 36 rpm. Typical relative intensities on the center line at 22 mm vary from about 40% to 60% over most of the practical range of impeller speeds. The Eulerian integral time scale at the same position is inversely related to speed as expected, ranging from about 13 ms at 100 rpm to 112 ms at 38 rpm. Frequency spectra were also determined at various positions in the impeller stream. The spectra exhibit rapid fall-off between 10 and 100 Hz with approximate slopes of - 3 / 2 (log-log), but they were of little value for the central question since it is impossible to calculate the critical perturbing frequency for a given floc structure. The probability distributions for velocity shown here indicate the likelihood that breakage events occur primarily in the impeller stream and not at remote locations in the tank. This is a valuable hypothesis because it leads to a physically useful definition of the time step in a stochastic simulation of batch-breakage experiments. Holmes et al. (1964) conducted a study of stirred, baffled tanks in
O
o
~
T
7
5
d
6
d
T
-
-
3
Z
3
0
CUnULRTIUE PERCENT
Figure 1. Cumulative probability distributions for point velocity measured on the impeller stream center line with laser-Doppler velocimetry. The leaf-type turbine impeller diameter was 11 cm; therefore, an impeller speed of 60 rpm corresponds to an impeller Reynolds number of 76 027.
which an ionic tracer pulse was injected at the center of the impeller; a conductivity cell placed in the impeller stream revealed sinusoidal behavior of constant period, identifiable for about five cycles. These data led them to define a characteristic circulation time, t,, based upon the average recirculation loop length and the mean velocity
t, (D/d)2/w (5) where D and d are the tank and impeller diameters, respectively. We feel that t , is a rational choice for the size of the time step in breakage-simulation procedure since it is inversely related to the frequency of entity exposure to the impeller stream. Floc-Breakage Events and Simulation In efforts to obtain a clearer picture of the true nature of the disintegration process, we conducted a series of experiments in which individual aggregates were introduced into both a turbulent jet and the impeller stream of the stirred tank described above and the resulting disintegrations were recorded photographically. As these experiments have been described elsewhere, we will not review them in detail but simply summarize findings essential to the task at hand. The jet experiments have demonstrated clearly that kaolin-Fe3+ flocs were much more fragile than kaolin-polymer flocs formed with extensive interparticle bridging. In fact, our work showed that kaolin-Fe3+ aggregates would suffer extensive disintegration at low dissipation rates, certainly for E < 75 cm2/s3. We were surprised to observe breakage events in the stirred tank well out into the impeller stream and not just in the immediate vicinity of the blade tips. It has been reported by Ali et al. (1981), among others, that droplet breakage occurs frequently in the vortical fluid motions trailing the blade tips. Despite the 800 or so observations made in both the jet and tank experiments, we were unable reliably to quantify the aggregate breakage mode as a function of parent particle size and local dissipation rate. The recorded disintegrations were simply too variable in severity, probably because of both the structural differences found in floc of similar size and the role played by transient and energetic vortical structures associated with both jet and impeller stream peripheries. For these reasons we conducted a series of batch-breakage experiments with
1606 Ind. Eng. Chem. Res., Vol. 26, No. 8, 1987
D D
D
m
0.00
~
0.0
Figure 2. Mean diameter of kaolin-polymer and kaolin-Fe3+ flocs (in pm) as a function of mean vessel dissipation rate as determined from power input. These data have been taken from numerous batch-breakage experiments performed in our laboratory under alkaline conditions (Glasgow and Hsu, 1984). The Kolmogorov microscale, 7. is also shown.
both kaolin-Fe3+ and kaolin-polymer flocs in which the mean particle size was recorded as a function of mean dissipation rate. Although the experiments displayed some expected variability, they were remarkably reproducible. Some of the data from this extensive series of experiments are illustrated in Figure 2. The complexity of the floc-breakage phenomena has been referred to above. The intractable nature of disintegration events stems from the large number of variables in play. An incomplete list would include the physicochemical conditions present during flocculation, such as pH, ionic strength, and mixing intensity; the nature of the coagulant; the level of aggregation; the aggregate age and history; and the nature of the disruptive hydrodynamic forces. Obviously, random and nonlinear stochastic processes are interacting in breakage phenomena, and we are unlikely to be in a position in the near future to characterize an individual breakage event adequately. For these reasons, we sought a procedure that would allow us to simulate batch-breakage experiments and produce reasonable dynamic particle-size-distribution behavior, but without parametric overkill. It seems likely that the mean particle-size variation indicated in Figure 2 reflects the dependence of the metastable floc size upon agitation intensity for the particular apparatus and flocculation environment studied. On this basis, we decided to relate the probability of breakage upon exposure to the impeller stream to fixed cumulative volume percentages; it follows that an expression of the form p[breakage upon exposure to impeller stream] = ae'dfl (6)
can be evaluated by minimization of the s u m of the squares of the deviations. This is precisely what we have done for the data of the type shown in Figure 2, resulting in p ( € , d )=
2 . 2 ~ 3 3 ~ 3 0
(7)
for kaolin-Fe3+ aggregates and p ( & ) = 10.8~0.69d1~97
0.4
0.0
1.2
1.6
(8)
for kaolin-polymer flocs. Equations 7 and 8 are obviously unphysical since the breakage probabilities should approach 1 asymptotically.
2.4
2.0
F L O C DIADETER,
D I S S I P A T I O N RATE PER U N I T flASS
A - T z
mm
Figure 3. Variation of the probability of breakage per exposure to the impeller stream for kaolin-polymer flocs as a function of aggregate size.
An alternative with some underlying physical justification can be written
where x: N 0 . 1 0 8 ~ ~ .for ~ ~kaolin-polymer ~d~.~ flocs. The behavior of this probability expression is illustrated in Figure 3. Before any of the probability equations given above can be used for the simulation of floc breakage, it is necessary to adopt a scheme for assigning fragment sizes. The simplest method, and the one we have been using to date, is to assign fragment sizes by using uniform random numbers until an appropriate volumetric condition is satisfied. However it should be remembered that certain very likely fragment sizes will undoubtedly arise in laboratory experiments due to the distribution of turbulent energy and nature of large aggregate formation. The breakage simulqtiop procedure can be easily summarized: a random number is selected for each parent entity and compared to p ( ~ , d )if; breakage is found to occur, daughter-particle sizes are assigned with additional random numbers until the chosen volumetric conditions is met; when every aggregate in the original set has been treated, the stage is complete and the simulation time is incremented. The simulation procedure described above was run repeatedly for the kaolin-Fe3+ case with E = 30 cm2/s3. The computed stochastic mean number of particles and the mean diameter are shown in Figure 4 as functions of the simulation stage. The standard deviation of the mean particle diameter for all realizations of the simulation increased in the early stages and decreased steadily thereafter; this is in keeping with our experimental observations of batch breakage in which the mean particle diameter tended to stabilize under elevated dissipation rates at about 100-300 bm. Figure 5 illustrates application of the breakage simulation to an assumed distribution of kaolin-polymer flocs. The indicated response is typical for a step ipcrease in dissipation rate, say from 200 to 800 cm2/s3. Under corresponding experimental conditions, the response is very rapid, with most of the change being accomplished in the first 60 s. It should be noted that the simulation does not indicate the persistence of large flocs that was evident in experiments. Some kaolin-polymer flocs were exceedingly strong, and it is clear that eq 9 is more likely to allow their survival. Although this has been
Ind. Eng. Chem. Res., Vol. 26, No. 8, 1987 1607
o
f 3000
.Y
2000 C
I' 100
.
0
I
2
3
4
5
6
7
8
9
IO
Simulation slope
Figure 4. Results of repeated simulations of batch breakage of large kaolin-Fe3+ flocs a t a mean dissipation rate of 30 cm2/s3.
flocculation process; this model was then solved repeatedly using stochastic inputs for parameters such as flow rate and influent particle concentration. We reject this approach, noting that a process that rapidly consumed primary particles but produced only 20 pm floc would not provide adequate solid-liquid separation. More recently, Pearson et al. (1984) have described a Monte Carlo simulation of coagulation in which the effects of various collisional mechanisms upon particle growth are assessed. They list three collision-rate functions of possible importance in coagulation practice: isotropic turbulent shear, turbulent inertia (likely to be important only in cases where entity and medium densities are significantly different), and differential sedimentation. The first of these, developed by Saffman and Turner (1956), is attractive because of its simplicity but is in question because floccuation environments tend to be inhomogeneous and anisotropic. Thus, we are confronted by the problem of identifying the eddies responsible for the relative particle motion in coagulating systems. It would be rare to find the integral-scale Reynolds number in flocculation processes large enough to produce an extensive inertial subrange, and it is likely that relative motion among small particles is produced by dissipation-range eddies. A dissipation rate of 30 cm2/s3in water, for example, will produce a Kolmogorov microscale (7 = ( v ~ / E ) of~ /about ~ ) 135 pm. Thus, there is some justification for assuming that relative motion of small flocs is induced by turbulence that is at least locally isotropic. It is apparent, however, that agglomeration in full-scale flocculation basins will also occur by differential gravity settling. This is perhaps less likely in laboratory-scale stirred tanks, except at locations remote from the impeller. Our initial efforts at simulation of coagulation therefore employed the following expression for the probability of interparticle contact by collision: p[collision by isotropic turbulence] = 0.288(di + d j ) 3 N ( ~ / v ) 1(10) /2
CUMULATIVE VOLUME PERCENT
Figure 5. Example of the performance of the breakage simulation with an arbitrarily assumed initial distribution of kaolin-polymer flocs.
done and the simulation stage interpreted in terms of the circulation time, the technique is suited only for lean, batch-breakage processes where the reaggregation of fragments is effectively minimized; it will therefore be of little use for the design, control, and simulation of generalized flocculation processes until floc growth is effectively treated.
Aggregation Simulation We have accordingly undertaken the development of a model for particle growth or agglomeration. Since agglomeration is contingent upon interparticle collision, it is necessary to express the likelihood of such events. Smoluchowski's (1917) description of orthokinetic flocculation has been tested repeatedly over the years, and his rate expression remains unquestioned for monodisperse conditions. The increasing number of small aggregates, however, leads to floc-floc encounters; hence, this part of the problem cannot be treated strictly in terms of the primary particle concentration. Glasgow and Hsu (1984) have presented adequate experimental support for this statement. Cockerham and Himmelblau (1974) have formulated a number-balance upon primary particles for a flow-through
Note, however, that in a well-mixed tank where time is interpreted in terms of recirculation, the probability of interaction may be dominated by the random stagewise redistribution of particles. Furthermore, it is likely that higher level (larger) aggregates are formed primarily by differential settling, suggesting that the exact mechanism of collision may be less important than particle-pair proximity. Therefore, the modified aggregation simulation algorithm functions in the following manner: An initial set of particles is randomly distributed throughout a spatial array measuring (100 X 100 X loo), and a check is then performed to determine the number and identity of all pairs of nearby neighbors, i.e., particle pairs that could produce affiliation. Of course there are at least three mechanisms by which interparticle contact can occur: Brownian motion, relative motion induced by turbulence, and differential sedimentation. This is an undesirable complication since the turbulent fields in floccualtion basins and laboratory-scale stirred tanks are quite inhomogeneous. Conceivably, one might devise a simulation procedure using different probability expressions suitable for the principal regions of the tank, the region surrounding the blade tips, the fluid streaming from the impeller, and the vessel periphery. We doubt that such compartmentalization is presently warranted, given the uncertainties concerning floc structures and, in the case of macromolecular coagulants, the time-dependent conformation of the absorbed polymer. For these reasons we have based our recent efforts on random redistribution at the beginning of each evolutionary step, followed by determination
1608 Ind. Eng. Chem. Res., Vol. 26, No. 8, 1987
nERN DISSIPATION RRTE, cm2/sec3
Figure 7. Comparison of simulation performance with experimental results obtained in our laboratory from numerous aggregationbreakage experiments conducted with the kaolin-polymer system a t pH's ranging from 9.7 to 9.9 (Glasgow and Hsu, 1984). The mean particle diameter is reported as a function of the mean vessel dissipation rate, Z. W1
j t . ( 1 I 1 I I 1 . I l l 0
I
2
3
4
5
6
7
B
9
10
I1
12
13
Simulotion stage
Figure 6. Illustration of the maximum and mean diameters obtained in repeated realizations of the aggregation simulation using an interaction distance (dimensionless) of 10 and an initially monodisperse collection of 30-pm particles.
of particlepair proximity. Such a simple approach appears to have the advantage of reproducing the observed behavior of the particle-size distribution without attention to the causes of interparticle contact. If a pair is found to be within the specified interaction distance, an aggregate is formed and the size of the new particle is determined, assuming a fixed increase in floc volume to account for the incorporation of interstitial water. It would be possible to vary the interaction distance with simulation stage to account for changes in adsorbed polymer configuration, but we have not yet done this. Figure 6 illustrates the typical performance of a large number of realizations of the aggregation simulation, showing both the maximum and the average floc diameters as functions of the simulation stage. More than one dozen aggregation-breakageexperiments have been performed to obtain particle-size distributions during the early stages of coagulation; a high-molecular weight anionic polyelectrolyte was used as the coagulant for kaolin dispersions at modest ionic strengths and alkaline pH's. These conditions produced extremely rapid aggregation and very large floc structures. A comparison of data from these experiments with results from the combined simulation procedure is provided in Figure 7. The experimental data points, shown as mean floc diameters, are represented by open triangles. Typical equilibrium sizes obtained from simulation runs are shown as filled-in squares. These data indicate that the combined simulation procedure is capable of realistically portraying batch aggregation-breakage behavior for kaolin-polymer flocs in stirred tanks. Figure 8 shows a comparison between the simulation and typical behavior obtained in a single aggregation-breakage experiment. In the experiment, the mean dissipation rate was increased from 90 to 650 cm2/s3at t = 600 s. The ensuing breakage caused the mean floc diameter to fall from 450 pm to about 250 Fm in just 60 s. The performance of a single realization of the combined aggregation and breakage algorithms is shown as a solid line and is quite remarkable. Comparable per-
D
3,,rT-+-o
I
0
100
200
,
300
I
1
I
400
TIflE,
Experiments1 Data
, , 600, , 730, ,
500
I
800
,
90:
sec
Figure 8. Single realization of the combined, batch aggregationbreakage simulation compared to an experiment in which the mean vessel dissipation rate was instantaneously increased a t t = 600 s.
formance has been obtained under a variety of hydrodynamic conditions. /
Discussion and Conclusions Separate simulation algorithms have been developed for agglomeration by interparticle collision and disintegration by disruptive hydrodynamic forces. We have shown that the simulation procedures yield qualitatively correct behavior. The breakage simulation does not utilize ill-defined parameters with weak physical interpretations; it requires only batch breakage data for a dilute dispersion of the flocs of interest. The aggregation simulation does not distinguish between particle growth by turbulence-induced collision or differential sedimentation, yet it produced size-distribution behavior very similar to that seen in experiments. We have combined the two simulation procedures for the purpose of modeling dynamic particle-size distribution behavior in laboratory-scale equipment, and results obtained thus far are encouraging. We believe that our simulation procedure is fairly robust based upon comparison with droplet data obtained from a limited number of emulsification experiments. This work has demonstrated the utter hopelessness of deterministic models of dispersed-phase breakage and
Ind. Eng. Chem. Res. 1 9 8 7 , 2 6 , 1609-1616
agglomeration. I t has also shown that it is possible to obtain realistic simulations of batch floc breakage, using only easily determined mean particle size-dissipation rate data to select the probability expression. It remains to be demonstrated that the combined procedures can yield realistic results for any proposed change in hydrodynamic environment, given a particular floc history.
Acknowledgment We are grateful to the National Science Foundation for partial support of this work under CPE-8411911.
Nomenclature a = empircal constant for breakage probability A = empirical constant for estimate of inviscid dissipation rate d = impeller and particle diameters, cm and pm D = tank diameter, cm 1 = integral length scale, cm LE = Eulerian integral length scale, cm N = number density of particles, cm-3 p = probabilities of interparticle collision or floc breakage t , = circuation time for stirred, baffled tank, s u = characteristic velocity of integral-scale eddies, cm s-l Greek Symbols
empirical exponent for breakage probability p = empirical exponent for breakage probability e = dissipation rate per unit mass, cm2 X = Taylor microscale, cm K~ = wavenumber, cm-’ v = kinematic viscosity, cm2 s-l p( 7) = temporal autocorrelation coefficient 7 = Taylor temporal microscale, s w = angular velocity of impeller, s-l 7 = Kolmogorov microscale, cm a=
1609
Literature Cited Ali, A. M.; Yuan, H. H. S.; Dickey, D. S.; Tatterson, G. B. Chem. Eng. Commun. 1981,10,205. Cockerham, P. W.; Himmelblau, D. M. J. Environ. Eng. Diu. (Am. SOC.Civ. Eng.) 1974, 100, 279. Cutter, L. A. AIChE J. 1966, 12, 35. Domilovskii, E. R.; Lushnikov, A. A.; Piskunov, V. N. J. Appl. Math. Mech. (Engl. Traml.) 1981, 44, 491. Glasgow, L. A,; Hsu, J. P. AIChE J . 1982, 28, 779. Glasgow L. A.; Hsu, J. P. Part. Sci. Technol. 1984, 2, 285. Glasgow, L. A.; Kim, Y. H. J. Environ. Eng. Diu. (Am. SOC.Civ. Eng.) 1986,112, 1158. Ham, R. K.; Christman, R. F. J . Sanit. Eng. Diu. (Am. SOC.Ciu. Eng.) 1969, 95, 481. Healy, T. W.; LaMer, V. K. J . Colloid Sci. 1964, 19, 323. Holmes, D. B.; Voncken, R. M.; Dekker, J. A. Chem. Eng. Sci. 1964, 19, 201. Hsu, J. P.; Glasgow, L. A. Part. Sci. Technol. 1983, I , 205. Hsu, J. P.; Glasgow, L. A. J . Chin. Inst. Chem. Eng. 1985, 16, 251. Michaels, A. S.; Bolger, J. C. Ind. Eng. Chem. Fundam. 1962, I, 153. Okamoto, Y.; Nishikawa, M.; Hashimoto, K. Inter. Chem. Eng. 1981, 21, 88. Pandya, J. D.; Spielman, L. A. J . Colloid Inter. Sci. 1982, 90, 517. Pandya, J. D.; Spielman, L. A. Chem. Eng. Sci. 1983, 38, 1983. Parker, D. S.; Kaufman, W. J.; Jenkins, D. J. Sanit. Eng. Diu. (Am. SOC.Civ. Eng.) 1972, 98,’79. Pearson, H. J.; Valioulis, I. A.; List, E. J. J. Fluid Mech. 1984,143, 367. Placek, J.; Tavlarides, L. L. AIChE J . 1985, 31, 1113. Rao, M. A.; Brodkey, R. S. Chem. Eng. Sci. 1972,27, 137. Saffman, P. G.; Turner, J. S. J . Fluid Mech. 1956, I , 16. Smoluchowski, M. 2.Phys. Chem. 1917, 92, 129. Tennekes, H.; Lumley, J. L. A First Course in Turbulence; M I T 1972. Thomas, D. G. AIChE J. 1964,10, 517. Vold, M. J. J. Colloid Sci. 1963, 18, 684.
Received for review August 19, 1985 Revised manuscript received January 27, 1987 Accepted April 1 3 , 1 9 8 7
Experimental Observations of Wall Slip: Tube and Packed Bed Flow Sydney Luk,?Raj Mutharasan,*t and Diran Apelianff College of Engineering, Drexel University, Philadelphia, Pennsylvania 19104
T h e validity of “no-slip” condition when a Newtonian liquid does not wet the solid surface is experimentally investigated. Velocity profiles in tubes and friction factors in packed beds were measured and compared with literature values for full wetting and various degrees of nonwetting cases. When tube surface is nonwetting, wall slip does exist at low Reynolds numbers and the velocity profile in the wall region deviates significantly from the parabolic profile. The magnitude of slip is found to vary strongly with the Reynolds number and the static contact angle of the surface. In packed bed flow, the friction factor is 60% lower than that predicted by Ergun’s equation when the packing surface is nonwetting. When a flowing liquid wets the solid surface with which it is in contact, the liquid molecules adhere to the solid surface, resulting in zero velocity at the interface. Stoke’s “no-slip” hypothesis at the liquid-solid interface for Newtonian fluids has been verified experimentally, and it is well established. The classical fluid-mechanical relationships, such as Poiseuille’s and Ergun’s equations, are based on theoretical and experimental studies in which the no-slip boundary condition has been invoked. Consider the case when the liquid phase does not wet the solid ‘Research Associate. t Professor f
of Chemical Engineering. Professor and Head of Materials Engineering. 0888-5885/87/2626-1609$01.50/0
surface with which it is in contact. The question which is being posed is whether it is still valid to use the no-slip boundary condition. Use of the no-slip boundary condition in situations where the flowing liquid does not wet the surface it contacts leads to differences between experimentally observed and theoretically predicted values. It has been observed that at a constant pressure head, the flow rate of steel melts through refractory porous media is higher than those predicted by Poiseuille’s equation (Apelian et al., 1986). In materials processing involving fluid motion coupled with heat and mass transfer, the consequences of wall slip are significant. If there is slip at the wall, the equation for the transport process becomes indeterminate because 0 1987 American Chemical Society
