428
Ind. Eng. Chem. Fundarn. 1982, 21 I 420-434
Characterization of Concentration Boundary Layer in Oxygen Absorption Young H. Lee" and Sydney Luk Department of Chemical Engineering, Drexel Universify, Philadelphia, Pennsylvania 79 104
Mass transfer behavior very close to an air-water interface is investigated by analyzing concentration fluctuation
data obtained with an oxygen microprobe at various radial and axial positions in the surface region of a free-surface stirred tank. The direction of the axial flow pattern is found to affect mass transfer significantly: the local mass transfer is always higher on the tank wall side where the flow direction is upward. A small amount of bovine serum albumin or yeast added causes a decrease in surface circulation velocity. The decrease in mass transfer coefficient due to added yeast is caused not by the change in eddy renewal but mainly by the change in effective diffusivity.
Introduction The introduction of oxygen microprobes in recent years enabled investigation of the mass transfer behavior very close to the gas-liquid interface. Bungay et al. (1973) first measured concentration fluctuations in the surface region of stirred water and interpreted the fluctuation frequency as the surface renewal rate. Tsao and Lee (1975) used a similar technique in studying a surfactant system. The problems with the earlier versions of microprobe are that the probe response is relatively slow (Davis and Lozano, 1979), that the current output drifts considerably and, often, in an unpredictable manner, and that the spatial resolution is poor due to a rather bulky and bare platinum tip. The microprobe was considerably improved by Lee et al. (1978) in terms of response time and spatial resolution, and it was used to investigate the hydrodynamic effect of surfactants on oxygen absorption (Lee et al., 1980). In the previous investigations, concentration data obtained a t a single location were related with the mass transfer coefficient of the system. The assumption for such studies is that the time-averaging is equivalent to areaaveraging. In this investigation, however, the local concentrations are measured at various radial and axial positions in the surface region of a free-surface stirred tank, and the results are analyzed to reveal the overall picture of mass transfer behavior close to the phase boundary. The effect of small amounts of bovine serum albumin and yeast, both of which are surface active, are also investigated. Experimental Section Absorption Cell. The absorption cell used in experimental runs consisted of an 8 cm i.d., 9 cm tall glass tank equipped with four stainless steel baffles (each 1cm wide and equally spaced) and a magnetically driven impeller as shown in Figure 1. The impeller used was a 5.0 cm diameter flat-bladed turbine with six equally spaced blades (each 1.2 cm X 1 cm). Approximately 300 mL of test solutions was used for all runs so that the impeller was located 3.5 cm below the liquid surface. Prior to absorption runs,the test solutions were deoxygenized by sparging with nitrogen. The oxygen was then absorbed from air into the liquid via the unbroken flat surface. For all runs, temperature was maintained constant at 22 f 1 "C. The impeller was driven by a feed-back controlled precision motor via magnetical coupling. The impeller speeds employed were 10, 25, and 50 rpm, which corresponds to impeller Reynolds numbers of 417, 1042, and 2083, respectively. Oxygen Microprobe. The microprobe used for detecting local dissolved oxygen concentrations is essentially
a miniaturized oxygen electrode, which was constructed in our laboratory. Details of the theory, the design, and the construction method are given elsewhere (Lee et al., 1978). Two electrodes, a platinum cathode and an Ag/ AgCl reference electrode, were used for the measurement as shown in Figure 3. The cathode consists of a glasscoated platinum needle with a tip diameter on the order of 1to 2 pm. The exposesd cathode area was made extremely small (approximately, less than the area of 1pm diameter circle) so that local concentrations can be measured with a fast response and a high spatial resolution. The exposed platinum tip, which is slightly recessed, is covered with a polystyrene membrane. The membrane applied not only improves the spatial resolution and the long-term stability in current output, but it also prevents formation of excessive meniscus around the probe when the probe is dipped in water. A schematic diagram of the cathode (microprobe) is given in Figure 2. The tip portion of the cathode was made gradually tapered so that the tipe maintains its position when dipped in an agitated liquid. The stability of the probe tip in a flow field was confirmed by visual observation through a 280X stereomicroscope. A t 50 rpm, the radial movement of the probe tip was less than 10 pm. At lower impeller speeds, the radial movements were negligible. The probe response to a step change in dissolved oxygen concentration could be approximated as first order. Probes that showed a response time of less than 10 ms were selected for this study. Since this type of microprobe requires an electrolyte in the liquid, 1.5 w t % KCl in distilled water was used as the base for making up different test solutions. The useful probe life is rather short, ranging from 10 h to 1week of intermittent use. The current output of the probe drifts somewhat (1to 10 % per hour), so frequent calibrations are required. The drift is approximately linear for a short time duration. However, since the data are taken and processed by a computer, the drift problem could easily be taken care of. Each probe was tested and calibrated before and after each experimental run. The tests involved measurements of air-saturation current, nitrogen-saturation current, stirring sensitivity, and response time. Experimental Procedure and Data Acquistion. A schematic diagram of the overall experimental setup is shown in Figure 3. The current output from the microprobe was first amplified by a fast response, low noise current amplifier (Keithly Model 427) and then sent to a computer (North Star Microcomputer) via an analog-todigital converter. The current amplifier is equipped with
0196-4313/02/1021-0428$01.25/0 0 1982 American Chemical Society
Ind. Eng. Chem. Fundam., Vol. 21, No. 4, 1982 429 I
MICROMANIPULATOR
-
AID
I'
CO H V E R T E R
KEITHLEY 427
MICROCOMPUTER ELECTRODE w
2 U.
U.
m 4
/ I
\
MAGNETMOTOR-
__ HlGH RESOLUTION CRT D I S P L A Y
Figure 3. Overall experimental setup.
Figure 1. Cutaway view of the absorption cell (8 cm i.d. X 9 cm; arrows indicate points of measurements).
(a)
(b)
Figure 2. Oxygen microprobe: (a) overall dimensions; (b) magnified view of tip.
a polarization voltage source for the microprobe (a constant value at -0.7 V was used). The sampling interval for data acquisition was varied between 0.1 and 0.003 s depending on impeller speed and the type of solution. The computer is equipped with a high-resolution CRT display for instant viewing of the data, so noisy or abnormal data can be rejected. A mini-floppy disk drive was used for on-line data storage, and a dot-matrix printer capable of high resolution graphics was used for plotting concentration data. Since the current output from the microprobe is extremely small (on the order of to A), careful grounding and shielding were necessary: besides using low-resistance coaxial cable wherever possible, the absorption cell was located in a grounded metal cage, and the magnet that drives the impeller was placed external to the cage to eliminate noise interference. The inside of the cage was fully humidified to prevent evaporation of water during experimental runs. The radial position and the depth of the probe from the water surface was adjusted by a micromanipulator capable of three-dimensional movement with the vertical direction controllable in steps of 3 pm. The radial positions where
the measurements were made were 0.7, 1.2, 1.7, 2.2, and 2.7 cm from the center of the tank, respectively. These points are indicated by arrows in Figure 1. A typical experimental run consisted of the following: (a) Place the probe at the nonstirred surface of a deoxygenated test solution and record the surface concentration. This surface value agreed well with that of air-saturated test solution. (b) Turn on the impeller to a desired speed, lower the microprobe 2 to 3 cm below the liquid surface, and record the initial bulk concentration. (c) Lift the probe out of the liquid, then gradually lower it in steps of 3 pm until a signal is detected. The probe position where the signal is first detected was considered as the surface, from which all the subsequent depths were measured. (d) Position the probe at a desired depth and record the concentration fluctuations. (e) Repeat step (d) for different depths. (f) Lower the probe to the bulk, and record the final bulk concentration. (g) Repeat step (a) to get the surface concentration at the end of a run. The surface and the bulk concentrations obtained in steps (a) and (g) were used for point-by-point normalization of concentration fluctuation data, assuming a linear change (this method is justified because the change is usually small). The dissolved oxygen concentration in the bulk of the liquid increases with time, since the absorption occurs under unsteady state. However, it is shown (Luk, 1982) that the thickness of the concentration boundary layer remains constant with time. This means that, once normalized, the concentration fluctuates around a steady mean value. This has been observed for all concentration fluctuation measurements. Since the depths are measured while the liquid is stirred, the method described above introduces a systematic error with a maximum being the ripple height of the liquid surface. However, these systematic errors were minimized by the following procedures: (a) Obtain the average concentrations from concentration fluctuation measurements at different depths. (b) Plot the average concentrations against the depth, and draw a least-square line for the data points. (c) Extrapolate the concentration profile to zero depth and check whether the extrapolated value corresponds to the surface concentration. (d) If they agree, then the depths measured are considered correct; if not, move the concentration profile slightly up or down so that the match is achieved, and correct the measured depths accordingly. In general, the corrections were minimal except the case of the base solution at 50 rpm. Approximate values of ripple height at the liquid surface at different rpm's could be estimated with the microprobe. The microprobe gives no signal when placed in air. Also, since the tip portion is hydrophobic due to the polystyrene membrane applied, water does not form an excessive meniscus when the probe tip just touches the water surface. For the ripple height measurements, the following
430
Ind. Eng. Chem. Fundam., Vol. 21, No. 4, 1982
procedures were used: (a) Lower the microprobe and locate the liquid surface under a no-agitation condition. (b) Raise the probe with the micromanipulator to a fixed position (for example, 30 pm above the surface). (c) Turn on the impeller to a desired speed. (d) Lower the probe in 3 - ~ mintervals and record the probe signal for each depth. The distance between the point where the intermittent signal first appears and the point where a continuous signal appears is considered as an approximate ripple height. For clean water (1.5 wt 5% KC1 solution), the ripple heights estimated at midpoint between the impeller shaft and the tank wall were 3, 6, and 36 pm for 10, 25, and 50 rpm, respectively. For other systems (10 ppm BSA and 1000 ppm yeast), the ripple heights were lower. Three test solutions were used: 1.5 wt % KC1 (used as the control), 10 ppm BSA, and 1000 ppm yeast solution. The bovine serum albumin (BSA) used was from Sigma Chemical, supplied as a powder which contains 96 to 99% albumin. The yeast was the Fleishman’s dried baker’s yeast deactivated with formaldehyde to prevent respiration. The overall liquid phase mass transfer coefficients were calculated from continuous measurements of the bulk oxygen concentration with a recently patented dissolved oxygen (DO) probe (Taylor, 1980) capable of measuring dissolved oxygen even in unstirred solution (the probe and the amplifier were supplied by L. G. Nester Co., Model 8000). The probe response is relatively fast: in water, it takes approximately 25 s for the probe to reach 90% of the final value to a step change in concentration. Since the agitation rates employed in our experiments were rather low (10 to 50 rpm), conventional DO probes, which require a high liquid velocity around the probe tip, were considered inadequate. Also, the use of a microprobe in bulk concentration measurement was abandoned, because for a long-term monitoring of concentrations (30 min to 1h and over) the output current drift was excessive and unpredictable. The new type of DO probe employs two cathodes instead of one to eliminate most of the flow dependency of the probe output (Taylor, 1980). In calculating the overall liquid phase mass transfer coefficient, kL,the moment method outlined by Dang et al. (1977) was used.
Experimental Results Concentration Fluctuations. Figure 4 shows the raw concentration data measured at different depths from the liquid surface at 50 rpm. The data shown are obtained at a radial position, 1.7 cm from the center of the tank. The concentrations indicated by c, and c b are the saturation and the bulk values, respectively. Since the time of measurement differs from depth to depth, it is noted that the initial bulk concentrations (marked c b in Figure 4) are somewhat lower than the bulk values at different depths: for exainple, in Figure 4a, the bulk value indicated by the flat portion of the concentration trace marked 550 pm is higher than that marked cb. For all impeller speeds used, the magnitude of surface waviness was considerably smaller than the thickness over which the concentration changed, less than 10% in the worst case (50 rpm, 1.5 wt % KCl solution). Therefore, the fluctuating concentration measured by the microprobe is not as much due to the surface waviness, but it is mainly due to convective diffusion of oxygen into the liquid. The classical theories such as Higbie’s penetration theory (1935) or Danckwert’s surface renewal theory (1951) are not convenient to explain the observed concentration fluctuations. Rather, a model based on individual eddy,
L
L
0
C
I
1
2
1
2
L
0
1 0
1
2
TIME, SEC (C)
Figure 4. Concentration fluctuation data at 50 rpm: (a) clean system; (b) with 10 ppm of BSA; (c) with 1000 ppm of yeast.
such as those of Fortescue and Pearson (1967) and Ruckenstein (1968,1971),appear to be more suitable. The latter model was used by Luk (1982) to interprete the concentration fluctuation data in terms of mass transfer in an individual eddy. The observed concentration fluctuations can be explained, qualitatively, as follows. The liquid surface is continuously swept by liquid elements or eddies having different sizes and velocities. Each eddy when passing the surface region renews itself with its own velocity causing a concentration gradient to form within the eddy. When these eddies pass by the microprobe tip located at a fixed depth from the surface, the concentration in each eddy is detected. In general, there are two circulation patterns in a stirred tank (Brauer, 1979): the primary one is the rotational or tangential movement around the shaft, and the secondary one is the vertical circulation having radial and axial components. Although most of the energy transferred from the impeller is consumed by the primary movement, it is the secondary movement that is of greater importance for mass transfer. The primary motion has to do with the frequency of the concentration fluctuations measured, because it is this flow pattern that is chiefly responsible for carrying eddies past the probe tip. In the primary flow pattern, the circulation velocity becomes faster as the center of the tank is approached. Thus, it is expected that, at a fixed impeller speed, the frequency of concentration fluctuation is higher on the shaft side compared with the tank wall side. This trend is experimentally shown in Figure 5, where the concentrations measured at different radial positions at 50 rpm are compared. The effect of adding 10 ppm of BSA or 1000 ppm of yeast (both are surface active) to the base solution (1.5 wt % KC1) is that the primary motion near the surface is considerably reduced. This is evidenced by the observed slowdown in concentration fluctuation frequency compared with a clean system (Figure 4: (a) vs. (b) and (c)). The
Ind. Eng. Chem. Fundam., Vol. 21, No. 4, 1982 431 POSITION FROM CENTER __. . 0.7
cm 0 C L E A N S’lSTEll A
in
ppa B S P
0 1000 PD’
0
YEAST
“\
Y
0
1.0
2.0
TIME. S E C
Figure 5. Slowdown in fluctuation frequency closer to tank wall (clean system, 50 rpm, depth = 120 to 150 pm). POSITION FROM CENTER
DEPTH
cion
S U R F A C E , pm
Figure 6. Change in concentration profile with radial position (clean system, 50 rpm).
reduction in primary circulation velocity near the surface region may be due to the surface pressure exerted by these surface-active materials (Ruckenstein, 1967; Davis, 1972). Concentration Profile. Figure 6 shows an example of normalized concentration profiles constructed from concentration measurements. Each point plotted represents an average of the concentration fluctuations over a sufficient time interval, such that a further increase in time of averaging does not affect the average appreciably. It is shown (Luk,1982) that only a limited time is required for such an averaging, typically, on the order of the exposure time calculated from the local mass transfer coefficient. The concentration profiles obtained at 50 rpm (Figure 6) shows that they become steeper as the tank wall side is approached this means that the local mass transfer coefficient increases with radial position. Similar trends were observed for 10 and 25 rpm. These experimental results can be explained in terms of the flow pattern in the stirred tank. In the secondary flow, the liquid swells up from below on the wall side and goes down near the shaft side. Thus, eddies near the tank wall are greatly affected by the upswelling motion which is much more effective for mass transfer (Chan and Scriven, 1970)than the downward flow near the shaft side. Note that the local mass transfer coefficient is lower on the shaft side even though the primary circulation velocity is higher in that region. This is a clear indication of the importance of the secondary flow pattern and its direction on mass transfer. The time-averaged, local concentration profile reflects the local “renewal rate” or ‘‘exposure time” of eddies at the particular position: a steeper profile indicates faster renewal by local eddies. The renewal rate of an eddy can be set equivalent to characteristic eddy velocity divided
DEPTH
FROM SURFACE.
pm
Figure 7. Effect of BSA and yeast on concentration profile (50 rpm, 1.7 cm from center).
by characteristic eddy length. In order for eddies to have fast renewal rates, they must have a combination of small size and fast velocity. Thus, the following can be deduced from the observed concentration profiles: if the distributions of eddy sizes passing through different radial positions are similar, the characteristic eddy velocity decreases as the center of the tank is approached; or, if the characteristic velocity of eddies remain more or less the same, eddy size increases as the center of the tank is approached. Since the upswelling flow has a strong effect on mass transfer and thus the characteristic eddy velocity, the former seems to be a better interpretation. Figure 7 shows the effect of adding 10 ppm of BSA or lo00 ppm of inactivated yeast on the concentration profile (50 rpm, 1.7 cm from center). In the case of BSA, the concentration profile is steeper compared with that of the base solution. Similar trends were observed with other radial positions and rpm’s. If the interpretation used in the above paragraph is employed, this means that the eddy “renewal rate” becomes faster when BSA is added. This appears somewhat contradictory to the result presented in the previous section that the primary circulation velocity in the surface region is decreased when BSA is added. Also, Davis (1972) predicts that the upswelling local eddy velocity is reduced due to the reverse-spreading force due to surfactants. In the case of 1000 ppm of yeast, the steepness of the concentration profile is more or less the same as that of the base solution. This indicates that, although the primary circulation velocity near the surface region is somewhat slowed down due to the added yeast, the local eddy renewal rate is not affected appreciably. Concentration Boundary Layer Thickness. In Figure 8, the concentration boundary layer thicknesses are plotted as a function of radial position for the three different rpm’s. The thickness was taken arbitrarily as the depth where the normalized concentration becomes 0.16. In general, the boundary layer thickness increases as the center of the tank is approached. This shows the importance of flow direction in tilting the boundary layer as explained in the previous section. It is of interest to compare the experimental observation with the prediction by the film theory d = D/RL (1) By taking the oxgen diffusivity, D,as 2.23 X cm2/s (St-Denis, 1971),and using the overall liquid mass transfer coefficients measured by the DO probe (L. G. Nester probe) for a clean system (0.2 wt 9’0KC1 solution), the boundary layer thickness predicted are 500,280, and 155 pm for 10, 25, and 50 rpm, respectively. If average
432
Ind. Eng. Chem. Fundam., Vol. 21, No. 4, 1982 P O S I T I O N FROM CENTER,
CM
P O S I T I O N FROM CENTER. 2
CM
P O S I T I O N FROM CENTER,
CP
0
I
I
I
I
I
I
I
I
i I I
I 0 CONTROL
ESA AYEAST
I
I (c)
‘b)
(a)
Figure 8. Variation of concentration boundary layer thickness with radial position: (a) at 10 rpm; (b) at 25 rpm; (c) at 50 rpm. I
I
CONTROL B S A I O ppm 0
Y E A S T 1000 ppm
-2 5
--- 10
0
400
RPH RPFt
800
D E P T H FROM S U R F A C E , p m
Figure 9. Change in concentration fluctuation amplitude with depth at 10 and 25 rpm.
thicknesses are taken from Figure 8 for the base solution (open circles), the agreement is reasonably good with the predicted values. In order for an eddy to be effective in renewing the surface, its size has to be greater than the thickness of the boundary layer. It is to be noted that small penetration assumption often used in gas absorption models in analyzing individual eddy (for example, Ruckenstein, 1968) is valid only for those eddies having a dimension much greater than the thickness of the concentration boundary layer. Figure 8 shows that the boundary layer thickness decreases with the addition of 10 ppm of BSA (closed circles), whereas, for yeast, the thickness increases somewhat (open triangles). These results indicate that the surface renewal of eddies is not affected significantly by the yeast added, while a noticeable change is observed when BSA is added. It appears that the renewal rate is increased in the case of BSA, which is rather difficult to explain. Amplitude of Fluctuation. From the time-averaged values of concentration measured, we can get information on the steady or static nature of the process, whereas in-
0
200
DEPTH FROM SURFACE,
400
pm
Figure 10. Change in concentration fluctuation amplitude with depth at 50 rpm.
formation on process dynamics can be obtained by analyzing the fluctuating component. The amplitudes of concentration fluctuation are compared by using the normalized standard deviation defined as follows
where N is the total number of points and Ciis the ith sampled concentration normalized based on the difference between the saturation concentration, cs, and the bulk concentration, c b . Figures 9 and 10 show the normalized standard deviation of concentration fluctuations as a function of the distance from the liquid surface for the three impeller speeds. In general, the amplitude of fluctuation becomes higher and the higher amplitudes concentrated closer to the interface as the impeller speed is increased. This means that the eddies become more active (in terms of surface renewal) with the increase in impeller speed. Figure 11shows that, at a fixed impeller speed, the eddies closer to the tank wall side are more active compared with the shaft side. This
Ind. Eng. Chem. Fundam., Vol. 21, No. 4, 1982 433
,
I
I
12c P O S I T I O N FROM C E N T E R
I
cm cm
u
0
0.7 1.7
0
2.7
cm
4
0
100 D: u 3 ,
4 t-
I-
80
Y r
4 0
w
KO I
I-
4
tY
V
40 V Y
U W
>
Y
4
20
0 1 0 0 0 ppm YEAST
0 0
10
25
50
IMPELLER SPEED,
RPM
Figure 12. Effect of BSA and yeast on average concentration gradient. 0
200 D E P T H FROM S U R F A C E ,
400
pm
Figure 11. Change in concentration fluctuation amplitude with radial position at 50 rpm.
result is in good agreement with the observed concentration profiles. The effect of adding 10 ppm of BSA to the base solution can be clearly observed in Figures 9 and 10: the eddy activity is considerably reduced. The effect of 1000 ppm of yeast is not as significant, indicating that the eddies are more or less as active as in the clean system. Effective Diffusivity. When the mean concentration gradient over the entire interface is known, the effective diffusivity can be estimated from the overall mass transfer coefficient measured by the bulk DO probe. The mean gradient can be obtained by graphically integrating the local gradients such as the ones shown in Figure 6. The results are shown in Figure 12. Each average gradient in the figure represents the surface renewal of eddies averaged for the entire contact area for the particular impeller speed. The effective diffusivity can be calculated by the following equation Deff = k/(dC/dy)y=o (3) where C represents the normalized concentration. The results are shown in Figure 13. For the clean system at 10 rpm, the effective diffusivity calculated is 2.4 X cm2/s, which agrees well with the literature values at 22 cm2/s: St-Denis, 1971). However, at "C (2.1 to 2.5 X higher rpm's, the effective diffusivities are significantly cm2/s for 25 rpm and 1.55 X 10" cm2/s lower: 1.4 X for 50 rpm. It is not clear why the effective diffusivities decrease as the impeller speed is increased: the error in esimating the mean concentration gradients increases somewhat a t higher rpm's (due to surface waviness as explained in the Experimental Section) but the observed decrease is beyond possible experimental error. Figure 13 also shows that the effective diffusivities become significantly lower when lo00 ppm of yeast is added: approximately 60% of the value of the clean system. This indicates that, for yeast, mass transfer is decreased not as
. Y)
2.0
N V r
-
b n
0
x
t-> I
> I v1
a
Y
; I 1.0
-
> Y tV
Y
L Y L
0 CONTROL
rn
1 0 ppm BSA
0 1000 ppm YEAST
0
0
10
25
IMPELLER SPEED,
50 RPM
Figure 13. Effect of BSA and yeast on effective diffusivity.
much due to a decrease in eddy renewal rate but mainly due to a decrease in the effective diffusivity. Because the yeast is surface active, it tends to be adsorbed at the interface, causing a decrease in the effective diffusivity. Lozowski et al. (1980) report a similar observation. For BSA, it appears that the effective diffusivity is considerably lower than that of a clean system at 10 rpm, whereas at higher rpm's, the differences are not as significant, although the trend remains. Since the results on concentration gradient and those on the amplitude of fluctuation are contradictory for BSA, a further investigation may be necessary before any conclusion can be drawn.
434
Ind. Eng. Chem. Fundam. 1902, 21 I 434-437
Summary of Experimental Results From the concentration fluctuation data obtained at various radial and axial positions very close to the gasliquid interface of a free-surface stirred tank, local values of concentration gradient, boudnary layer thickness, and fluctuation amplitude were obtained. Significant experimental results are summarized as follows. (a) It is the secondary flow pattern (axial flow) that affects mass transfer. Especially, the direction of the secondary flow has a strong influence on local eddy renewal rate: the local mass transfer rate is always higher on the tank wall side, where the flow is upward, compared with the tank center, where the flow is predominantly downward. As a result, the concentration boundary layer thickness decreases as the tank wall is approached. (b) Compared with a clean system, 10 ppm of BSA (bovine serum albumin) or 1000 ppm of yeast, both of which are surface active, reduces primary circulation velocity in the surface region. (c) Compared with a clean system, the surface renewal by eddies is not much affected by the presence of 1000 ppm yeast. The observed decrease in mass transfer is mainly due to a decrease in effective diffusivity. (d) Compared with a clean system, the concentration boundary layer thickness becomes thinner when 10 ppm of BSA is present. The BSA also reduces the amplitude of concentration fluctuation. These two results contradict each other and a further investigation is necessary. Acknowledgment The financial support of the National Science Foundation through the Grant 78-13282 is gratefully acknowledged.
Nomenclature c, = air saturation concentration, g/cm3 cb = concentration in liquid bulk, g/cm3 C = average normalized concentration, dimensionless Ci = ith sampled normalized concentration, dimensionless d = concentration boundary layer thickness, cm D = oxygen diffusivity in liquid, cm2/s kL = overall mass transfer coefficient, cm/s S = standard deviation or normalized concentration fluctuations, dimensionless y = depth from surface of liquid, cm Literature Cited Brauer, H. Adv. Biochem. Eng. 1079, 13, 87. Bungay, H. R.; Huang, M. Y.; Sanders, W. M. Am. Inst. Chem. Eng . j . 1073, 19, 373. Chan, W. C.; Scriven, L. E. Ind. Eng. Chem. Fundam. 1070, 9 , 114. Danckwerts, P. V. Ind. Eng. Chem. 1051, 43, 1460. Dang, N. D. P.; Karrer, D. A.; Dunn, I. J. Biotech. Bioeng. 1977, 19, 853. Davis, J. T. Am. Inst. Chem. Eng. J . 1072, 18, 169. Davis, J. T.; Lozano, F. J. Am. Inst. Chem. Eng. J . 1070, 25 405. Fortescue, G. E.; Pearson, J. R. A. Chem. f n g . Sei. 1987, 2 2 , 1163. Higbie, R. Trans. Am. Inst. Chem. Eng. 1035, 3 1 , 365. Lee, Y. H.;Tsao, G. T.; Wankat, P. C. Ind. Eng. Chem. Fundam. 1978, 17, 59. Lee, Y. H.; Tsao, G. T.; Wankat, P. C. Am. Inst. Chem. f n g . J . 1980, 26, 1008. Lozowski, D.; Langa, J.; Andrews, G. F.; Stroeve, P. Chem. Eng. Commun. 1980, 6 , 349. Luk, S.,W.D. Thesis, Drexel University, 1982. Ruckenstein, E. Int. Chem. Eng. 1087, 7 , 490. Ruckenstein, E. Chem. fng. Sei. 1966, 2 3 , 363. Ruckenstein, E. Chem. f n g . J . 1971, 2 , 1 St-Denis, C. E.; Feii, C. J. D. Can. J . Chem. Eng. 1071, 49, 885. Taylor, R., presented at Instrument Society of America Annual Meeting, Houston, Oct 1980 Tsao, G. T.; Lee, D. D. Am. Inst. Chem. Eng. J . 1075, 21, 979.
Received f o r reuiew September 10, 1981 Reuised manuscript received June 2, 1982 Accepted June 23, 1982
Lubrication Flows in Viscoelastic Liquids. 3. Approach of Parallel Surfaces Subject to Constant Loading Pradeep Shlrodkar' Department of Chemical Engineering, University of Massachusetts, Amherst, Massachusetts 0 1003
Alex Bravo and Stanley Middleman" Department of Applied Mechanics and Engineering Sciences, University of California, Sen Diego, La Jolia, California 92093
An isothermal incompressible viscoelastic liquid fills the space between two plane parallel rigid circular disks and retards their approach driven by a constant load. A constitutive model due to Wagner is used to predict half-times for closure of the disks. Resutts are compared to expectations for the inelastic power law model and are discussed in light of exlsting data for the phenomenon. The approach here utilizes the lubrication approximations and assumes that the normal stresses generated by the flow are of secondary importance to the transient shear stress behavior.
Lubricating fluids of unusual characteristics may be created by the addition of soluble high molecular weight polymers. There still exists considerable uncertainty over the most primitive question: does the contribution of the polymer increase or decrease the load-bearing capacity of a viscoelastic lubricating film? In a series of papers 'Mobil Oil Chemical Company, Edison, NJ.
(Shirodkar and Middleman, 1982; Shirodkar et al., 1982) and in greater detail in the thesis of Shirodkar (1981) we have performed experiments and theoretical ana yses aimed at providing an answer to this question. Of course a preexisting literature is available, reviewed thoroughly in Shirodkar (1981) and briefly in Shirodkar and Middleman (1982),also aimed toward this question. Relevant papers that provide some perspective include Brindley et al. (1976), Davies and Walters (1973), Grimm (1978),
0196-4313/82/102 1-0434$01.25/0 0 1982 American Chemical Society
f
