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 a = empirical exponent for breakage probability
p = empirical exponent for breakage probability 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 e =
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
1610 Ind. Eng. Chem. Res., Vol. 26, No. 8, 1987
the fluid velocity at the boundary is unknown. Either the wall slip is known or it is expressed as a function of process variables such as wall shear stress and interfacial energy. The presence of slip is desirable for easy flow since resistance to flow is reduced. In hollow-fiber artificial kidneys, the slip velocities at the fiber wall have been estimated to be as high as 50% of the maximum velocity (Kried and Goldstein, 1974). Ponter et al. (1976) reported improved mass-transfer efficiency in a packed column with mixed packing of nonwetting PTFE Teflon and wetting glass rasching rings. An additional advantage of the presence of wall slip is the reduced rate of viscous heat generation. The presence of slip can also increase convective heat transfer at a boundary (Kraynik, 1976). This paper examines the validity of the no-slip condition when the flowing liquid does not wet the surface, using two different experimental systems. Flow experiments have been conducted in two geometries: (1)tube flow and (2) packed bed flow. The surface characteristics in each set of experiments were varied from wetting (static contact angle = 0") to various degrees of nonwetting (static contact angle = 52", 68", 75O, and 110'). Measurements of the velocity profile and the packed bed friction factor were obtained and compared for the wetting and nonwetting cases.
Background Experimental observations of wall slip have been observed in the flow of wetting non-Newtonian liquids such as paint and paste (Green, 1920; Scott Blair and Crowther, 1929), polymeric solutions (Astarita et al., 1964; Carreau et al., 1979),lubricants (Vignogradov et al., 1978),hydraulic fracturing fluids (Allen and Roberts, 1982; Pilehvari, 1984), biological fluids (Bugliarello et al., 1964; Kried and Goldstein, 1974), and emulsions (Cohen and Chang, 1984) and in the flow of stratified two-phase fluids (Russel and Charles, 1959). It has been reported that in nonhomogeneous fluids, wall slip occurs due to the formation of a solvent skin layer near the wall and that the bulk fluid slips on top of this skin layer (Metzner et al., 1979). A theoretical framework has been reported by Cheng (1974), Metzner (1977), Metzner et al. (1979), and Janssen (1980) for polymer solutions and melts. These studies explain the formation of the skin layer by the migration of macromolecules from regions of high stress level (the wall) to regions of low stress level (the bulk liquid) when dissolved macromolecules become aligned and stretched during the flow process. Various mechanisms have been suggested for the formation of such a skin layer: (1)radial migration of particles (Maude, 1959; Serge and Siberberg, 1962; Karmis et al., 1966; Sheshadri and Sutera, 1972); (2) hydrodynamic interaction (Maude and Whitmore, 1956; Cox and Brenner, 1968); (3) entrance exclusion (Bell, 1969); (4) mechanical exclusion (Maude and Whitmore, 1956; Brunn, 1981). Despite the enormous effort to analyze the wall-slip phenomena in wetting fluid-flow situations, quantitative predictions of slip velocities and identification of proper experimental conditions under which wall slip occurs are still elusive. To date, it appears that the effect of nonwetting surfaces on the flow of single-phase Newtonian liquid has not been investigated. It is generally accepted that wetting Newtonian fluids do not slip at the wall (for example see: Goldstein, 1969). At present, there is no single convincing explanation of the slip phenomena in the vicinity of a solid wall. On a molecular level, slip can be caused by either adhesive failure or cohesive failure in macromolecular systems (Blyler and Hart, 1970; Pearson and Petrie, 1968). For adhesive failure, one expects the slip to be strongly
dependent on molecular structure, molecular weight, and distribution of molecular weight. The latter is not a plausible mechanism in aqueous solutions. This study explores the existence of wall slip during the flow of a nonwetting single-phase Newtonian liquid.
Experimental Section The experimental approach consisted of comparing the pressure drop-flow rate relationship of distilled water flowing in a tube and a packed bed having surfaces which were either wetting or nonwetting. In both experiments the solid surfaces were chemically treated to yield a nonwetting surface with respect to water. The degree of nonwetting was characterized by the air-water contact angle. Both experiments were conducted in a constanttemperature room set at 22 "C. A. Chemical Treatment of Glass Surfaces. Precision-bored Pyrex glass tubes were used in the tube flow experiments, and solid borosilicate glass spheres of various diameters were used in the packed column experiments. The surface energy of the glass surface was controlled by chemisorption of organochlorosilanes,chlorotriphenylsilane (Alfa Products), using the method outlined in Menawat et al. (1984). Chemical adsorption of organochlorosilanes on glass results in a stable, homogeneous surface. Direct evaluation of the degree of wetting is difficult on cyclindrical or spherical surfaces. The water-air contact angle measured through water on a flat glass plate, which was given the same identical chemical treatment as the tubes and spheres, was used as a measure of the degree of nonwetting. The contact angle on the flat surface was measured by the sessile drop method (Adamson, 1976). The surface treatment involved two procedures: (1) cleaning and (2) reaction. In the cleaning procedure, the glass surfaces were scrubbed gently in detergent and rinsed with distilled water 4 times. Subsequently they were washed with methyl ethyl ketone (Fisher Scientific) to remove any organic adsorbed materials. They were then rinsed 4 times with distilled water (HPLC grade). The glass surfaces were then boiled in concentrated HN03 (assay 70% HNOJ for 20 min and then rinsed 4 times with distilled water (HPLC grade). They were then outgassed at 115 "C under vacuum for 24 h in an ultraclean vacuum oven to desorb the physically adsorbed water. All glasswares used were cleaned by the same procedure. Triphenylchlorosilane (Alfa Products) was dissolved in Teflon beakers with 500 mL of n-hexane (Fisher Scientific). The concentration of triphenylchlorosilane controlled the surface energy of the glass surface. It was found that the water-air contact angles of 52", 68", and 75" were achieved by reaction with triphenylchlorosilane solutions at concentrations of 0.05,0.5, and 4 mg/mL, respectively. The reactions were carried out in a clean stainless steel cylindrical vessel. The glass tubes or spheres were placed in the vessel, which was then filled to the top with the triphenylchlorosilane solution. The reaction was allowed to proceed for 24 h. The vessel was then drained and rinsed 4 times with n-hexane to remove any physically adsorbed molecules of triphenylchlorosilane. The glass surfaces were then outgassed at 115 "C under vacuum to desorb the physically adsorbed n-hexane for 24 h. To check the stability of the treated surface, each of the three treated flat glass plates was placed in a closed container, containing air and water, and stirred at 50 rpm. The air-water contact angles were measured on these plates after periods of 2, 24, and 48 h. No significant changes in the contact angle measurements were found. B. Tube Flow Experiments. The experiments were carried out by using precision-bored Pyrex tubes. The
m-
Ind. Eng. Chem. Res., Vol. 26, No. 8, 1987 1611
SUPPLY
J
SUPPLY TANK
CIRCULAR PYREX TUB F R E W E N C Y METER TRACKER UNIT
w PRE-AMPLIFIER
4 PHOTO MULTIPLIER
A
OPTICAL BENCH
GLASS COLUMN
1
DIxk880N ADAPTER
j
PACKED MEDIA
1
F i g u r e 1. Schematic apparatus used for the tube flow studies.
wettability of the tube surfaces were characterized by the air-water contact angles given above. The inside tube diameters were 8.8900 f 0.0005 and 1O.oooO f 0.0005 mm. A schematic diagram of the experimental setup is shown in Figure 1. The flow system consisted of a 50-gal supply tank filled with distilled water, the Pyrex glass tube, a recycle tank, and a centrifugal pump. The flow system was enclosed so that the influx of dust particles was minimized. The pressure head was constant. The desired flow rate in the glass tube was achieved by adjusting the outlet needle valve. The Reynolds number range achievable in the apparatus was between 50 and 12000. Local velocity profiles in the glass tube were measured by a Laser Doppler anemometer system (DISA Type 55L). The LDA was used in the differential-Doppler, forward scattering mode. The laser source is a 5-mW He-Ne Laser (Spectra-Physics). The measuring volume formed is an ellipsoid of the dimensions, 80.5, 82, and 422 bm, in the x , y, and z directions, respectively. For this configuration, the Doppler frequency, f ~ was , 600 kHz for a velocity reading of 1m/s. Seeding of particles in the solution was provided by adding 0.1 cm3of homogenized milk per liter of distilled water. The particle concentration in the flow field is approximately lo8 particles/cm3, and the particle size was approximately 0.3 pm (George and Lumley, 1973). The LDA system was mounted on an optical bench and a precision milling table. This arrangement allowed the entire system to move in all three directions within an accuracy of 0.2 mm. Local velocities in the tube were measured at normalized radial positions of r / R = 0.12, 0.4, 0.66, 0.8, and 0.94. Attempts to measure the local velocities at positions closer to the tube wall surface were not successful due to an inherent limitation of the LDA system. C. Packed Bed Experiments. A schematic diagram of the experimental setup for the packed bed flow experiments is shown in Figure 2. The flow system consisted of a 50-gal supply tank, a packed bed, a recycle tank, and a centrifugal pump. The pressure head was kept constant by adjusting the recycle rate in the pump.
ROTAMETER
_.
RECYCLE TANK
Figure 2. Schematic apparatus used for the packed bed flow studies.
The packed bed was made of a Pyrex tube (100-cm long, 5-cm i.d.) with cone-shaped connectors at the inlet and outlet connecting to the rest of the flow system via Tygon tubing (1.2-cm id.). The packing used was spherical precision solid glass balls or identical Teflon (PTFE) balls of 3- and 6-mm diameter. The bed was supported by a stainless steel wire mesh located at 55 cm from the flow inlet so that end effects were negligible. The bed is 10.16-cm long, and its porosity was determined to be 0.38 and 0.39 for the 3- and 6-mm balls, respectively. The pressure drop across the packed bed was measured by using a U-type manometer filled with mercury for high-flow-rate runs. At extremely low flow rates, the sensitivity of ordinary liquid-filled U-type or inclined-type manometers was insufficient for measuring pressure differences across the bed with accuracy. A micromanometer with an air bubble was used to measure the low-pressure differentials (Pinkava, 1970). The micromanometer is filled with chloroform as the manometer fluid. The sensitivity of the micromanometerincreased with a decreasing difference between the liquid densities; if however there is a small density difference, experimentallyit was difficult to establish a stable liquid interface. The densities of both chloroform and water were measured after mixing and subsequent separation. Tabulated values must not be used since there is always some mutual solubility, and even trace impurities may cause considerable errors. In this investigation, the measuring range was from 1.5 X to 2 mm of water; the accuracy was fly0 and f0.25% at the lower and higher ends of the range, respectively. The response time was found to be approximately45 s. The overall error in the pressure drop measurements was estimated to be about f l %. Flow rates were determined from precalibrated rotameters. The estimated error of these measurements is f l 7 0 . The wetting characteristics of each glass-ball packing was
1612 Ind. Eng. Chem. Res., Vol. 26, No. 8, 1987 I
1 0 1 0 9
0 8
0 7
06 0 5 0 4
c 3 3 2 01 0 0
0 5
0 4
0 3
02 0 1 0 0 0 1 0 2 Radial Distance From Center 01 Tube r c71
03
C 4
3 5
-05
0 4
-03
-02 -01 00 01 02 Radial Distance From Center 01 Tube I, cm
0 3
04
0 5
0 0
10,
I
" " ,
0 5
-04
-03
-02 .01 0 0 0 1 0 2 Radial Distance From Cenler 01 Tube cw
03
04
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k
>
05
1
06
0 1
00 0 5
0 4
0 3
02 0 1 00 0 1 02 Radial Dislance From Center Of Tube r cm
03
0 4
0 5
Figure 3. Effect of contact angles on velocity profiles in 10-mm tube a t Reynolds number of 42.
altered by varying the contact angles from 0" to 75' by the chemical treatment described above. To achieve an even higher degree of nonwetting, solid Teflon (PTFE) balls of identical dimensions were also used. They gave rise to contact angles of llOo. Results and Discussion A. Tube Flow. For untreated glass tubes, the contact angle is Oo, Le., the water fully wets the glass surface. In a steady, fully developed flow through an untreated glass tube, one expects a parabolic profile for water at Re < 2100, provided that there is no wall slip. Figure 3 shows the variation of the local velocity of the water as a function of the radial position in a 10.000-mm Pyrex tube at Re = 42. Similar experiments were carried out in a 8.890-mm
tube to investigate the effect of the tube diameter. The results obtained in the 10.000-mmtube are similar to those from the 8.8900-mm tube. (Data in tabular form are available in supplementary material.) For the wetting case (static contact angle = O O ) , the local velocity profile is indeed parabolic. There is good fit between the experimental data and the parabolic profile (within 1-2%). From the local velocity profile, two important parameters can be estimated: (i) the velocity at the wall (at r = R-) and (ii) the wall shear stress (at r = E ) . The velocity and shear-stress values near the wall were estimated by extrapolating the profiles to the wall. For the wetting case, the velocity at the wall was found to be zero. The accuracy of the extrapolation is assumed to be as good as the fit of the data to the parabolic profile, which is within 1-2%. Thus, wall shear stress can be determined in tube flow experiments by two methods, namely (i) from the slope of the local velocity profile at the wall and (ii) from pressure drop data (7, = D(AP)/4L). Both methods give the same wall-shear-stress value (within an error range of f570) for the wetting case. Compared with the known value of viscosity of water at constant temperature of 22 "C, the ratio of T,/(du/dy) is within an error range of *El%
*
For the nonwetting cases (0 = 52", 68",and 7 5 " ) , as shown in Figure 3, the local velocity profile deviated significantly from a parabolic profile near the wall region. The local velocities for the nonwetting cases were much higher than those of the wetting case. This deviation increased with an increase in the degree of nonwetting (increasing static contact angle). This implies that as the wall became increasingly nonwetting, the frictional resistance to flow decreased. Extrapolating the velocity profile to the wall, we obtained a finite value of velocity, which is termed the slip velocity, Us. The experimental data clearly show that the no-slip condition at the wall to be invalid for nonwetting surfaces at low flow rates (Re = 42). Farther away from the wall, the velocity profiles followed a near parabolic shape. Moreover, the flow rates in the nonwetting cases were 5-1570 higher than the wetting cases. Since the available pressure drop for the flow was constant, these observations can be caused only by slip at the wall. Carreau et al. (1979) have found that experimental flow rate exceeded the predicted value by a factor as great as 4 when wall slip occurred in his study of polymer flow on inclined planes. The local velocity profiles for Re = 50, 100, and 140 are shown in Figures 4, 5, and 6, respectively. The results show that the magnitude of wall slip decreased significantly at higher Re (>140) and that there was virtually no difference in the velocity profiles for the wetting and nonwetting cases. For both the transient and turbulent regimes, or even at the upper end of the laminar regime, no experimental evidence of wall slip was found. The exact mechanism responsible for such an observation is not clear. The effect of increasing the Reynolds number on the slip velocity for various degrees of nonwetting surfaces is shown in Figure 7. The maximum slip velocity occurs at a Reynolds number of 100 for all three nonwetting surfaces. The existence of a maximum slip velocity is due to the two limiting boundaries: (1) as the Reynolds number approaches zero, the slip velocity must approach zero because the flow velocity itself approaches zero; (ii) as the Reynolds number becomes large (in this study 140), the slip velocity is found to approach zero. Thus, the maximum slip velocity lies between these two limits. As shown in Figure 7 , the slip velocity can be as high as 0.25 cm/s at a Reynolds number of 100 for the nonwetting case (static contact
-
Ind. Eng. Chem. Res., Vol. 26, NO. 8, 1987 1613 11,
0 01 O
~
-0 5
-0 4
-0 3
.O 2 -0.1 0.0 0.1 0.2 Radial Distance From Center 01 Tube r, cm
03
0 4
0.5
Y
00 i
1 1 ,
05
-04
-03
-02 -01 00 0 1 02 03 Radial Distance Frcm Center 01 Tube r cm
04
Figure 5. Effect of contact angles on velocity profiles in IO-" a t Reynolds number of 100.
0 5
tube
75" 2 5
O0 10
..
\\\"
0
.0.5
-0.4
.O 3
-0.2 .0.1 0 0 0 1 0 2 Radial Distance From Center 01 Tube r, cm
03
0.4
0.5
/
1 1
7
05
-04
-02
-03
-01
00
0 1
03
0 2
Radial Distance From Center 01 Tube r
04
05
cm
Figure 6. Effect of contact angles on velocity profiles in 10-mm tube a t Reynolds number of 140.
00
-1
1
J
-0.5
-0 4
.O 3
-0.2 -0.1 0 0 0 1 0 2 Radial Distance From Center Of Tube r, cm
03
0 4
0.5
75"
03
11
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66O
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r
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z
.% 0
-2" -.m
2
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-0 4
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.O 2
-0.1 0 0 0.1 02 Radial Distance From Center 01 Tube r, cm
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120
140
160
Figure 7. Slip velocity a t tube wall vs. Reynolds number for static contact angles Oo, 52", 6 8 O , and 75' in 10-mm tube. n~
Figure 4. Effect of contact angles on velocity profies in 10-mm tube a t Reynolds number of 50.
angle = 7 5 O ) . Astarita et al. (1964) also observed a slip velocity of 35% of the film velocity when slip occurred in a steady laminar gravity flow of a 2% CMC solution down a flat plate. Figure 8 shows the shear stress, 7,, as a function of Reynolds number for different degrees of wetting. For full wetting case (e = OO), the plot of 7 , vs. Re is a straight line with a slope of 8 p 2 / p D 2 , as expected from the HagenPoiseuille equation; however, as the degree of nonwetting increases, there is a significant reduction in the shear stress which is no longer a linear function of Reynolds number. At low flow rates (low Re), the reduction in wall shear stress due to a nonwetting surface (T,,,) can reach 60% (at Re = 42), as shown in Figure 9. The significance of the wall slip on the reduction in wall shear stress at various
60 100 Reynclds Number
f i
0 14 wE 0 1 2
x 010
1
2
006 006
c
2
s
004
0 02 0 00
0
20
40
80
80 100 120 Reynolds Number
140
160
180
200
Figure 8. Wall shear stress vs. Reynolds number for static contact angles Oo, 52', 68', and 75' in 10-mm tube.
Re is evident in Figure 9. Figure 9 also shows that there exists a critical degree of nonwetting before the reduction
1614 Ind. Eng. Chem. Res., Vol. 26, No. 8, 1987 0.8
00 0
20
40
80
100 120 Reynolds Number
60
140
160
180
200
Figure 9. Ratio of wall shear stress for nonwetting surfaces to wall shear stress for full wetting ( 7 , , / ~ ~vs.) Reynolds number for 10mm tube. 18
'T
I-
I
0.0004y 0.001
I
0.01
0.1
1
Vb(CMiS)
Figure 11. Pressure drop (AP)vs. bulk velocity (V,) for 3-mm packing.
0
20
40
60
80 100 Reynolds Number
120
140
'60
180
Figure 10. Slip coefficient (0) vs. Reynolds number for 10-mm tube.
in 7, becomes significant. At low degrees of nonwetting (0 I5 2 O ) , the reduction in 7, is minimal. The wall-slip effect is commonly lumped into a single term called the slip coefficient, /3. This procedure is used in the now generdy accepted relationship proposed originally by Mooney (19311, where p is a constant defined by
where /3 = Us/ 7,. This has been experimentally verified in slurry-flow situations (Cheremisinoff, 1981). Figure 10 shows the plot of vs. Re for the various wetting and nonwetting cases studied in this investigation. For the nonwetting surface, @ varies nonlinearly with Re and static contact angle. It is apparent that the mechanism for the wall slip observed in the present work is more complex than those observed in non-Newtonian wetting flow studies. B. Packed Bed Flow. The pressure drop due to flow of a liquid in a packed bed has been extensively studied, and several correlations exist for estimating the pressure drop and the friction factor. Most of the investigations have been conducted with easily wetted packings. The pressure drop characteristics and packed bed friction factors have been experimentally determined for both wetting and nonwetting packings. The pressure drop characteristics at various velocities, using 3-mm glass beads (both wetting and nonwetting), are shown in Figure 11. The results for the wetting glass packing (e'= Oo) agree well with Ergun's correlation. A t low flow rates, AP is a linear function of the bulk velocity, Vk For the nonwetting cases (0 = 6 8 O , 7 5 O , and 1 l O O ) and at low velocities, the pressure drop is significantly lower than that of the wetting packing (8 = OO). At a higher bulk
0.00f
0.1
0.01
1.0
Vb(CMIS)
Figure 12. Pressure drop (AP)vs. bulk velocity (V,) for 6-mm packing.
velocity (V, > 0.025 cm/s), there is no significant difference in AF' between the wetting and nonwetting cases. A similar trend is observed for an identical bed but with a larger packing bead size of 6 mm, as shown in Figure 12. In both figures, the Teflon PTFE (0 = l l O o ) packing shows a large reduction in AP. As the degree of nonwetting decreases, the deviation of AP from the wetting packing (0 = 0') decreases. Hence, the reduction of AP is a function of the degree of wetting. In other words, at a constant AP,a low-energy surface, such as PTFE or treated glass beads, promotes a higher flow rate than a high-energy surface (untreated glass, 0 = OO). This behavior is a confirmation of the slip effect of the nonwetting packing surface, which was also observed in the tube-flow experiments. As in the tube-flow experiments, the slip effect diminishes at higher flow rates. The presence of wall slip on the nonwetting packing surfaces will reduce the packed bed friction factor. In
Ind. Eng. Chem. Res., Vol. 26, No. 8, 1987 1615 1000
1
. *
110~
I
75' 110'
DP = e mm
'K",
A
I L I O
,
1 1 01
10
1
8
8
I
1
100
1000
2
4
6 8 10 Packed Bed Reynolds NuTbei
12
14
Figure 15. Ratio of friction factor for wetting surfaces over nonvs. Reynolds number for 6-mm packing. wetting surface (fw/fnw)
R.P
Figure 13. Packed bed friction factor (f,) vs. Reynolds number (Re,) for 6-mm packing. 1
2 6 ,
--
, 00
02
04
06
08
10
-
, 1 2
1 4
Packed Bed Reynolds Number
Figure 14. Ratio of friction factor for wetting surfaces over nonwetting surface (fw/fnw)vs. Reynolds number for 3-mm packing.
Figure 13, the packed bed friction factor, f,, is presented as a function of the packed bed Reynolds number for D, = 6 mm. For the wetting glass beads (6 = O O ) , the experimental data are in excellent agreement with the theoretical prediction given by Ergun's equation: at Re < 10, f ,= 150/Re, at Re
> 1000,
f , = 1.75
For the nonwetting case, there is a significant reduction in f , at Re, < 10, due to slip at the packing surface. As shown in Figure 14, the reduction in f ,increases with an increasing degree of nonwetting. Maximum reduction in f , is achieved by the PTFE packings, which provide surfaces having the lowest energy. In other words, the lower energy surfaces offer a lower resistance to flow. These reductions in AP and f ,are not due to factors such as variations in porosity or possible compressibility of the PTFE packing. The experimental results presented here were reproducible within an error of 1370,which is significantly smaller than the 60% reduction in AP observed in the nonwetting beds. Similar results were observed for the silanated glass beads, whose diameters were identical with the untreated glass beads. The reductions in AF' and f p can only be explained by invoking the wall-slip hypothesis. To show the significance of the slip effect on the friction-factor reduction, the ratio of f p for the wetting surface, f,, to that of nonwetting surface, f,,, is plotted against Reynolds number for various degrees of nonwetting. This is shown for 3-mm and 6-mm beads in Figures 14 and 15, respectively. The shapes of these curves are quite similar to the slip velocity vs. Reynolds number plots given in
Figure 7. For the 3-mm beads, the maximum value of 2.4 and occurs around Re, = 0.3-0.5, whereas for the 6-mm beads the ratio is 3.4 and occurs at a Re, = 2. The magnitude of the reduction in f p is clearly a function of the hydrophobic quality of the surface. It is also noted that the reduction in f,, at t9 = 52O, is very small, suggesting that a critical degree of nonwetting may be required for the slip effect to become operative. Note that similar observations were made in the tube-flow experiments (see Figures 8 and 9).
f w / f n w is
Conclusions 1. On the basis of velocity measurements near the wall region and the pressure drop in packed bed flow, we believe that wall slip exists at low bulk velocities and at low pressure drops when the solid surface is nonwetting to the liquid phase. 2. In the tube-flow experiments, it was found that the velocity profile at the wall region significantly deviates from the normal parabolic profile when slip occurs. The magnitude of slip is proportional to the degree of nonwetting of the surface. The slip coefficient, us/^,, is found to vary strongly with Reynolds number and the static contact angle of the surface. There seems to exist a critical wall shear-stress value (7, = 0.2 dyn/cm2), above which the slip effect totally disappears. 3. For packed bed flow, it was found that at a constant flow rate the required pressure head, AP,decreases as the surfaces become more and more nonwetting. It has been shown that Ergun's equation is not valid when the liquid phase does not wet the packing at low Reynolds numbers. 4. The friction factor, f,, is reduced by as much as 33% below that of a wetting packing surface. A critical pressure drop (0.01 in. of H 2 0 for D, = 3 and 6 mm) seems to exist at which the slip effect disappears. The nonwetting surface must have a static contact angle greater than 52' for the slip effect to be significant. Acknowledgment This study is supported by an NSF grant (CPE-83-1133-0). Nomenclature
D = tube diameter D, = particle diameter in packed bed f = packed bed friction factor = friction factor for nonwetting packing surface f, = friction factor for wetting packing surface L = length of packed bed AP = total pressure drop across packed bed or along tube
lw surface
Q = volume rate of flow
Ind. E n g . Chem. Res 1987,26, 1616-1621
1616
r = radial distance from center of tube R = radius of tube R- = very close to the tube wall Re,, = packed bed Reynolds number U, = slip velocity V , = bulk fluid velocity Greek Symbols @ = slip coefficient defined as U s / ~ w 8 = static contact angle p = viscosity p = density T = shear stress 7, T~~
7,
= wall shear stress = wall shear stress measured for nonwetting surface = wall shear stress measured for wetting surface
S u p p l e m e n t a r y Material Available: Tables of velocity profiles and pressure drop data at different Reynolds numbers and grain sizes, (6 pages). Ordering information is given o n a n y masthead page.
Literature Cited Adamson, A. W. Physical Chemistry of Surface, 3rd ed.; Wiley: New York, 1976. Apelian, D.; Luk, S.; Piccone, T.; Mutharasan, R. “Removal of Liquid and Solid Inclusions from Steel Melt”, Proceedings of the 5th International Iron and Steel Congress, Washington, DC, April 1986. Allen, T. 0.;Roberts, A. P. Production Operation, 2nd ed.; Oil & Gas Consultants: Tulsa, OK, 1982. Astarita, G.; Marruci, F.; Palumbo, G. Ind. Eng. Chem. Fundam. 1964, 3, 333. Bell, J. J . Comp. Mat. 1969, 3, 244. Blyler, L. L., Jr.; Hart, A. C., Jr. Polym. Eng. Sci. 1970, 10, 193. Bugliarello, G.; Kaupur, C.; Hsiao, G. Proceedings of the International Congress on Rheology; Lee, E. H., Copley, A. L. Eds.; Interscience: New York, 1964; Vol. 4, p 351. Brunn, P. Int. J . Multiphase Flow 1981, 7, 229. Carreau, P. J.; Bui, W. H.; Leroux, P. Rheol. Acta 1979, 18, 600. Cheng, D. C. H. Ind. Eng. Chem. Fundam. 1974,13, 394. Cheremisinoff, N. P. Fluid Flow-Pumps, Pipes and Channels; Ann Arbor Science: Ann Arbor, MI, 1981.
Cohen, Y.; Chang, C. Chem. Eng. Commun. 1984, 28, 73. Cox, R.; Brenner, H. Chem. Eng. Sci. 1968, 23, 147. Den Otter, J. L. Rheol. Acta 1965, 14, 329. George, W. K.; Lumley, J. L. J . Fluid Mech. 1973, 60, 321. Goldstein, S. Annual Review of Fluid Mechanics; Seans, W. R., Van Dyke, M. Eds.; Annual Reviews: Palo Alto, CA, 1969; Vol. 1, p 1. Test. Mat. 1920, 20, 451. Green, H. Proc. Am. SOC. Janssen, L. P. B. M. Rheol. Acta 1980, 19, 32. Karmis, A.; Goldsmith, H.; Mason, S. Can. J . Chem. Eng. 1966,44, 181.
Kraynik, A. M. Ph.D. Dissertation, Princeton University, Princeton, NJ, 1976. Kried, D. K.; Goldstein, R. J. Technical Report, 1974; University of Minnesota, Minneapolis. Maude, A. Br. J . Appl. Phys. 1959, 10, 371. Maude, A.; Whitmore, R. Br. J. Appl. Phys. 1956, 7, 98. Menawat, A.; Henry, J., Jr.; Siriwardne, R. J . Colloid Interface Sci. 1984, 101, 110. Metzner, A. B. Improved Oil Recovery by Surfactant and Polymer Flooding; Shah, D. D., Schechter, R. S., Eds.; Acadamic: New York, 1977. Metzner, A. B.; Cohen, Y.; Rangel-Nafaile, C. J . Non-Newtonian Fluid Mech. 1979, 5, 449. Mooney, M. J . Rheology 1931,2, 210. Pearson, J. T. A.; Petrie, C. J. S. Polymer Systems, Deformation and Flow, Proceedings of the 1966 Annual Conference of the British Society of Rheology; Wetton, R. E., Wholow, R. W., Eds.; Macmillan: London, 1968. Pilehvari, A. A. Ph.D. Dissertation, University of Tulsa, Tulsa, OK, 1984. Pinkava, J. Handbook of Laboratory Unit Operations for Chemists and Chemical Engineers; Bryant, J., Transl.; Gordon & Breach: New York, 1970. Ponter, A. B.; Taymour, N.; Dankyi, S. 0. Chem.-hg.-Tech. 1976, 48, 636. Russel, T. W. F.; Charles, M. E. Can. J . Chem. Eng. 1959, 18, 120. Scott Blair, G. W.; Crowther, E. M. J . Phys. Chem. 1929, 33, 321. Serge, G.; Siberberg, A. J . Fluid Mech. 1962, 14, 136. Rheol. 1972, 14, 351. Sheshadri, V.; Sutera, S. Trans. SOC. Vignogradov, G. V.; Froishteter, G. B.; Trilisky, K. K. Rheol. Acta 1978, 17, 156. Received for review October 15, 1985 Revised manuscript received January 18, 1987 Accepted April 15, 1987
Impact of the Catalytic Activity Profile on Observed Multiplicity Features: CO Oxidation on Pt/Al,O, Michael P. Haroldt and Dan Luss* Department of Chemical Engineering, University of Houston, Houston, Texas 77004
The observed multiplicity features of a catalytic pellet in which an isothermal Langmuir-Hinshelwood reaction occurs are sensitive to the distribution of the catalytic components. Specifically, surface migration of the catalytic components may change the range of reactant concentrations or temperatures over which rate multiplicity occurs or shift the maximal rate outside the concentration region in which multiple solutions occur’. These changes in the multiplicity features may be used to detect qualitative changes in the metal concentration profile without having to carry out a destructive test. The supported metal profile of many industrial precious metal catalysts is nonuniform. This catalytic a c t i v i t y profile may affect considerably the observed catalytic activity. Previous studies have shown that the p e r f o r m a n c e may be optimized by the use of a nonuniform catalytic activitv mofile. For Dositive-order reactions. the catalvtic -
A
* To whom correspondence should be addressed. ‘Presently at the Department of Chemical Engineering, University of Massachusetts, Amherst, MA 01003.
activity should be concentrated near the s u r f a c e to minimize the detrimental influence of intraparticle diffusional limitations ( C a r b e r r y and Minhas, 1969). A number of investigators have studied the interaction between a reaction rate which exhibits a m a x i m u m for an intermediate c o n c e n t r a t i o n (ex.. bimolecular Langmuir-Hinshelwood kinetics), pore digusion, and nonunifgrm catalytic activity ( W e i and Becker, 1974; Villadsen, 1976; Becker and Wei, 1976, 1977; Hegedus et al., 1977, 1979; Morbidelli et al., 1982; Morbidelli and Varma, 1982; W o n g and Szepe, 1982).
0888-5885/87/2626-1616$01.50/0 0 1987 American Chemical Society
