Environ. Scl. Technol. 109 1, 25, 1294- 1300
Calvert, C. C.; Smith, L. W. J . Dairy Sci. 1972, 55, 706. Ham, W. E.; Kline, E. A,; Ensminger, M. E. Am. J . Vet. Res. 1949, 10, 150. Sahli, B.P. Vet. Hum. Toxicol. 1982, 24, 173. National Research Council. In Mineral Tolerance of Domestic Animals; National Academy of Sciences: Washington, DC, 1980. Dickinson, J. 0. Am. J . Vet. Res. 1972, 33, 1889. Sharma, R. P.; Street, J. C. J . Am. Vet. Med. Assoc. 1980, 177, 149. Murthy, G. K.; Reha, U. J . Dairy Sci. 1968, 51, 610. Schroeder, H. A,; Balassa, J. J. J . Chronic Dis. 1961, 14, 236.
Van Bruwaene, R.; Gerber, G. B.; Kirchmann, R.; Colard, J. Int. J . Enuiron. Stud. 1982, 190, 47. Powell, G. W.; Miller, W. J.; Morton, J. D.; Clifton, C. M. J. Nutr. 1964, 84, 205. Smart, G. A.; Sherlock,J. C. Food Addit. Contam. 1985, 2, 139.
(34) White, W. B.; Clifford, P. A.; Calvery, H. 0. J. Am. Vet. Med. Assoc. 1943, 102, 292. (35) Murthy, G. K.; Rhea, U.; Peeler, J. T. J . Dairy Sci. 1967, 50, 651. (36) Dorn, C. R.; Pierce, cJ. O., 11; Chase, G. R.; Phillips, P. E.
In Truce Substances in Environmental Health; Hemphill, D. D., Ed.; University of Missouri Press: Columbia,MO, 1973. (37) O'Dell, G. D.; Miller, W. J.; King, W. A.; Ellers, J. C.; Jurecek, H. J . Dairy Sci. 1970, 53, 1545. (38) Archibald, J. G. J . Dairy Sci. 1949, 32, 877. (39) Fitzgerald, P. R.; Peterson, J.; Lue-Hing, C. Am. J . Vet. Res. 1985, 46, 703. (40) Altman, P. L.; Dittner, D. S. In Biology Data Book, 2nd ed.; Federation of American Societies for Experimental Biology: Bethesda, MD, 1972; Vol. 1.
Received for review May 11, 1990. Revised manuscript received March 13, 1991. Accepted March 27, 1991.
Prediction of Gas/Water Mass Transport Coefficients by a Surface Renewal Model Wllllam E. Ashert and James F. Pankow"
Department of Environmental Science and Engineering, Oregon Graduate Institute, 19600 N. W. Von Neumann Drive, Beaverton, Oregon 97006 Aqueous-phase carbon dioxide concentration fluctuation time scales measured at carbon dioxide/water interfaces were used in a surface renewal model to calculate the aqueous-phase mass-transport coefficient kL for a range of turbulence conditions for cleaned and film-covered water surfaces. The calculated kL values were compared to measured kL values from a separate set of experiments over the same range of turbulence and interfacial conditions. This test of a surface renewal model with directly measured parameters has shown that kL may be accurately estimated by the model for clean interfaces and high turbulence intensities. The data also show that surface renewal models are not appropriate for calculating mass fluxes at film-covered water surfaces.
Introduction Due to the importance of gas transport in environmental science, geophysics, and chemical engineering (11,the role of turbulence in driving the exchange of gases across a gas/liquid interface has been the focus of much research. The rate of transport of a sparingly soluble, nonreactive gas into an agitated liquid is controlled by the turbulence in the liquid phase (2). However, it is not known how the transport rate depends on the turbulence conditions present for all situations of interest. Knowledge of this relation or conditions under which gas/liquid masstransport models such as surface renewal are applicable would enable more accurate modeling of the exchange rate. Despite their age and simplicity, surface renewal models are of particular interest because of their continued use in predicting gas fluxes (37.3,and recent experimental evidence suggests that under certain conditions they may provide an accurate physical description of the transport process (6). In a gas/liquid exchange process with liquid-phase rate control and a well-mixed and homogeneous bulk phase, the 'Present address: Battelle Marine Sciences Laboratory, 439 W. Sequim Bay Rd., Sequim, WA 98382. 1294
Environ. Sci. Technol., Vol. 25, No. 7, 1991
totalflux F (mol cm-2 s-l) of the gas into or out of the bulk fluid is generally described by (2) F = kL(C, - C,) (1) where C, (mol ~ m - is~ the ) liquid-phase saturation concentration as determined by Henry's law, Cb(mol ~ m - ~ ) is the concentration of the exchanging gas in the bulk phase of the liquid, and kL (cm s-l) is the liquid-phase mass-transfer coefficient (Le., the transport rate constant). The hydrodynamical dependence of the exchange process is contained in kL. There are several models currently available that describe this dependence (7-13). As formulated by Higbie (12)and extended by Danckwerts (13), surface renewal theory models the gas/liquid transfer process by assuming that patches of the liquid surface are periodically replaced by fluid elements from the well-mixed bulk phase below. In this model, the fluid surface is envisioned to be a mosaic of parcels of water with varying residence times and surface concentration profiles as shown in Figure 1. The parcels of bulk water that replace the surface are turbulence eddies that come into direct contact with the gas/liquid interface. When F # 0, the concentration of gas in the bulk water will not be in equilibrium with the gas phase. Therefore, while these water parcels are at the surface, gas will be transferred into or out of them by molecular diffusion across the gas/liquid interface. As shown in Figure 2 for F > 0 (i.e., invasion), each renewal event will increase the magnitude of the local liquid-phase concentration gradient of the solute gas at the interface. Because the rate of molecular diffusion is proportional to the concentration gradient, this increase in the gradient will increase the local flux of the gas into the fluid surface. As the solute gas accumulates in the interfacial layer of liquid, the local flux will decrease because the concentration gradient at the surface decreases. Replacement of this surface element by bulk fluid with lower solute concentration through surface renewal will carry accumulated solute into the bulk phase and increase the local concentration gradient. These two effects greatly increase the flux above the flux due to molecular diffusion
0013-936X/91/0925-1294$02.50/0
0 1991 American Chemlcal Society
non-renewed surface, Cs
renewed surface, Cb
eddies FburO 1.
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Schemtlc diagram of surface renewal. non-renewed renewed surface surface interface
eddy
concentration ‘b
Figure 2.
‘s
‘b
InstantaneOuS surface Concentration profiles before and
after a surface renewal event.
through a stagnant layer of liquid at the interface. Surface renewal theory, as first formulated by Highie, makes the assumption that all fluid elements have equal residence time T ( 8 ) at the interface. kL may then be written as (12)
kL = 1 . 1 3 ( D / ~ ) ~ / * (2) where D (cm28-l) is the molecular diffusion coefficient of the gas in the liquid. Because it is assumed that T is determined by the time scales of the turbulence eddies, T determines the dependence of k, on the liquid-phase hydrodynamics. Explicit mathematical forms for this dependence are suggested by solution of the advectiondiffusion equations using an assumed near-surface turbulence eddy velocity profile (7,8).In general, these results show that kL may be defined as = K(DV/L)’/2 (3) where V and L are turbulence velocity and length scales, respectively, and K is a constant that depends on the choice of turbulence scales used. Recognizing that the quantity L / V defines a turbulence time scale shows that eq 3 is mathematically identical with eq 2. However, because the advection-diffusion equations used to derive eq 3 were not solved in terms of surface renewal, the relation presented in eq 3 does not provide proof that eq 2 may be used to predict kL. The assumption made by Higbie (12) of a constant residence time is physically unrealistic in light of the stochastic nature of turbulence eddy time scales that control T . Furthermore, it should be recognized that defining the complex hydrodynamics associated with the interaction of aqueous-phase turbulence with a free surface in tern of a single parameter is not rigorously correct. For example, Asher and Pankow (6) have shown that there is a distribution of surface concentration fluctuation time scales. Therefore, when T is used in eq 2, it is more properly defined as a statistical parameter of the prohability distribution of surface element lifetimes such as the kL
mean or, if the distribution is nonnormal, the mode. Surface renewal theory has been hypothesized to provide a reasonable description of the hydrodynamics associated with liquid-phase rate-controlled transport at a clean interface (14). In support of this, the data of Asher and Pankow (6)suggest that surface renewal may be observed at a clean liquid surface. In addition, the results of Asher and Pankow (15)suggest that eq 3 as defined by Fortescue and Pearson (7) or Lamont and Scott (8) describes the correlation of kL with turbulence intensity and length scale for clean surfaces. However, other more recent models of gas/liquid mass transport adopt the view that complete surface renewal cannot occur even at a clean gas/liquid interface (9-11). Rather, they assume that turbulence eddies can only approach an interface to a depth defined by a diffusiveor concentration boundary layer postulated to exist very close to the liquid surface. Whiie these recent models are more sophisticated than stagnant film models ( 2 ) ,they share the feature that there exists a layer close to a clean gas/liquid surface where turbulence motions are damped out by viscous stresses. In support of boundary layer type models, Lee and Luk (16),Luk and Lee (17), and Jirka and Ho (18) reported measurements of oxygen concentration fluctuations near an air/water interface that show the existence of a layer of water near the interface in which surface renewal does not occur. However, Asher and Pankow (6)have provided evidence suggesting that these layers either do not exist at clean water surfaces or are of much thinner dimensions than assumed in the boundary layer models. Furthermore, the results of Lee and Luk (16),Luk and Lee (17). and Jirka and Ho (18)must be viewed with caution because surface films may have been present and boundary layers are known to exist in the presence of surface films and at liquid/solid interfaces. As discussed above, because the existence of concentration fluctuations with varying time scales is fundamental in surface renewal, the model could he tested further if those time scales could be measured accurately under conditions of known interfacial cleanliness and turbulence intensity. Measurement of the time scales would allow comparison of kL values predicted from a surface renewal model with experimental kL values. Such a comparison could elucidate the conditions where the surface renewal model is applicable. In this study, the laser-induced fluorescence/2’,7’-dichlorofluorescein (LIF/DCFS) technique of Asher and Pankow (6,15)was used to observe fluctuationsin the total carbon dioxide concentration ( C , mol cn-7 in the aqueous surface microlayer as a function of (1)depth in the microlayer, (2) mechanically generated aqueous-phase turbulence intensity, (3) length scale of the turbulence, and (4) interfacial cleanliness. Because the time scales of the fluctuations in CT can be related to the surface element time scales, they can be used to predict kL by use of eq 2. The results will he compared with experimental k, values obtained by Asher and Pankow (15) under a range of turbulence intensities and length scales for both clean and film-covered surfaces.
Experimental Methods The apparatus used has been described in detail elsewhere (6,15). Briefly, it consisted of a 25 cm X 25 em X 35 cm tank of water with a headspace for the transporting gas and a vertically oscillating grid of bars with square cross section. All gases were prepurified and introduced into the tank headspace through 0.4-pm HEPA filters to reduce particulate contamination of the interface. As discussed by Asher and Pankow (6,15)and Asher (19),use Environ. Scl. Technol.. Vol. 25, No. 7. 1991 1295
of a vertically oscillatinggrid allows calculation of the bulk fluid turbulence velocity and length scales by use of the relations developed by Hopfinger and Toly (20). However, because Brumley and Jirka (21)and Hannoun et al. (22) have shown that these relations are not strictly applicable close to a gas/liquid interface, the calculated turbulence scales are at best rough estimates of the true values near the water surface. The LIF/DCFS concentration fluctuation measurement technique has been described by Asher and Pankow (6,15). It relies on the pH-dependent fluorescence emissions of DCFS and l-hydroxypyrenetrisulfonicacid (HOPSA) dyes to track C02 fluctuations in the aqueous surface microlayer. The excitation light source was the 488-nm line of an argon ion laser and the incident laser beam was perpendicular to the plane of the water surface. The fluorescence was detected through the wall of the tank by using a photodiode and narrow bandpass interference filter at an angle of approximately 45" with the surface normal. Bulk-phase fluorescence from the DCFS and HOPSA dyes was blocked by the addition of the dye orange-G (OG) to the aqueous phase. OG is a nonfluorescing dye which strongly absorbs the incident excitation light at 488 nm. This prevented fluorescence from being observed below an easily calculated fluorescence cutoff depth, z, (pm). As discussed by Asher and Pankow (61, z, is defined to be the depth at which F(z,) = F(0)/100 (4) where F ( z ) is the fluorescence intensity observed from depth z , where z is measured in the positive direction downward from the interface. Because z, is determined by the amount of light absorbed in the surface microlayer, the value of z, can be decreased by increasing the concentration of OG. In fact, because F ( z ) is directly proportional to the 488-nm light intensity, F ( z ) is determined by Beer's law (6) and z, can be calculated for a known concentration of OG. F(z) is a nonlinear function very near the aqueous surface (19). The totaJ fluorescence FT observed at a given pH is given by
FT = J m F ( z ) dz
(5)
As described by Asher (19),we can also define the fraction of FT from a surface layer with depth zd, Le., F T , z , as
FT,= ~ F T - ~J Z0 d F ( { )d{
(6)
where { is a dummy integration variable. Figure 3 gives
FTSz for the case when z, = 300 pm. Because a large fraction of FT will come from well above zc, concentration fluctuations from very close to the CO,/H,O interface may be observed. Use of a laser to induce fluorescence allows a very small area of the surface to be sampled and causes minimal spatial averaging of the fluorescence fluctuation times. In this work, the sampling area was approximately 0.1 cm in diameter, which was as small as the smallest calculated turbulence length scale (6,19). The small sampling area caused a temperature increase due to local optical heating of the aqueous surface layer of -4 "C 8-l (6). This was negligible compared to the relatively rapid turnover of the water surface due to turbulence. Although technically possible, smaller sampling areas were not used due to excessive heating. Surface-active contaminants were removed from the bulk water by bubbling with helium gas as suggested by Scott (23). This was followed by removal of the surface 1296 Envlron. Scl. Technol., Vol. 25, No. 7, 1991
0'75
3
/
0
Figure 3. Depth integrated fluorescence intensity F T J ,plotted vs depth, z , showing fraction of total fluorescence intensity coming from the aqueous surface layer.
water using sheets of 100% rayon lens paper followed by surface vacuuming using a glass Pasteur pipet and peristaltic pump. Tests of surface cleanliness on an interface that had been rayon/vacuum cleaned (RVC) did not indicate that contamination was present (6,15,19). Purging the headspace of the tank with filtered gas kept the surface clean for the duration of a gas-exchange experiment. In addition to the RVC surface experiments, concentration fluctuation time scales were also measured with a l-octadecanol (1-OD) film present on the water surface. Typically, a 1-OD film was formed by pipeting 0.07 cm3 of an 8 X lo4 mol cmS solution of 1-OD in n-pentane onto an RVC water surface. The mass of 1-OD used was calculated to be sufficient to form a surface monolayer. The pentane was evaporated by using dry, filtered nitrogen gas. While the mass of 1-OD used was sufficient to form a monolayer, no determinations were made concerning the state of the film formed on the water surface. However, its effect on concentration fluctuations in the surface microlayer may be seen in the data presented below.
Results and Discussion Examples of the raw fluorescence fluctuation time series data for CO, invasion through RVC and 1-OD monolayer-covered C02/H20interfaces are shown in Figures 4 and 5, respectively. The aperiodic decreases in the total fluorescence intensity, FT, are caused by the water at the surface absorbing CO, with the concomitant decrease in pH causing a decrease in fluorescence intensity. Because the C 0 2 partitrl pressure in the gas phase was always greater than the in situ partial pressure of COP in the aqueous phase as defined by Henry's law, transport of COz was always into the water. Therefore, the aperiodic increases in FT were not caused by desorption of C02by the surface. Rather, they were caused by replacement of the water at the surface by bulk water with lower C02 concentration and, therefore, higher pH. The net increase in CT caused the pH of the bulk tank water to decrease and led to the generally decreasing trends in peak fluorescence intensities observed in Figures 4 and 5 . The CT fluctuation time scales can be estimated from the temporal widths of fluorescence fluctuation peaks such as are present in Figures 4 and 5. Asher and Pankow (6)
1 .oo
Table I. Calculated Turbulence Time Scale T,,, Concentration Fluctuation Time Scale Distribution Mode TM,Measured Transport Coefficient kL,,,,, Predicted Transport Coefficient kL,,, Estimated Concentration Boundary Layer Thickness bo, and Diffusive Length Scale tid for Rayon/Vacuum-Cleaned Interfaces with z, = 150,300, and 450 jtm and for 1-Octadecanol Monolayer Covered Interfaces with z, = 300 jtma
0.75
T,,,
FT 0.50
T,, s
s
iO4k~,,, cm s-l
io4kL,p% cms-
pm bc,
pm 6dv
z, = 150 pm, Rayon/Vacuum-Cleaned Interface
' I
0.19 0.47 0.50 0.51 1.2 1.3
0.22 0.66 0.75 0.86 2.6 2.4
93 47 42 43 15 12
99 57 50 54 29 30
18 35 39 40 112 140
19 33 38 35 67 64
z, = 300 pm, Rayon/Vacuum-Cleaned Interface 0.00
~ , , , , , , , , , , ' ' 1 ' 1 , ' ' ' 1 1 ' ' ' ~ ' 0
5
15 20 TIME (s)
10
25
30
35
Figure 4. Surface fluorescence fluctuation intensity F, time series for COPinvasion through an aqueous surface cleaned by the rayonlvacuum procedure. The data have been normalized to give FT = 1.0 at time zero. T , and z , for this time series were 0.33 s and 300 pm, respectively.
1'0°
0.75
0.21 0.27 0.33 0.46 0.49 0.59 0.80
0.33 0.36 0.60 0.73 0.74 0.95 1.2
81 78 60 55 54 48 42
20 23 28 37 33 45 62
24 25 32 35 35 40 46
z, = 450 pm, Rayon/Vacuum-Cleaned Surface
0.19 0.24 0.44 0.50 0.59 0.83 1.1 1.3 1.7
0.30 0.39 0.54 0.79 0.85 1.4 1.8 1.8 3.2
93 82 46 43 37 25 17 12 7.0
85 75 63 52 51 40 35 35 26
18 21 37 40 46 67 98 141 244
23 26 30 37 38 48 56 55 74
I
I\
z, = 300 pm, 1-Octadecanol Film-Covered Surface
0.19 0.31 0.48 0.80
0.40 0.68 0.97 1.3
'Exdanation
0.00
87 75 61 46 42 37 27
kr 5 I
0
I
I
I
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I
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15
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Figure 5. Surface fluorescence fluctuation intensity F, time series for COP invasion through an aqueous surface covered by a l-octadecanoi film. The data have been normalized to give F, = 1.0 at time zero. T , and z , for this time series were 0.40 s and 300 pm, respectively.
measured the time scales of the fluorescence fluctuation peaks for a range of turbulence intensities and z, for both 1-ODfilm-covered and RVC interfaces. They found that an average concentration time scale TA(s) and, because the distribution was nonnormal, a distinct mode TM (s) could be calculated from the distribution of concentration time scales (6). TAfrom an individual experiment has been shown to be correlated with the calculated turbulence macro time scales (19). This result, and the fact that fluorescence fluctuations such as shown in Figures 4 and 5 only occurred when there was a net flux of C02 into the water column, show that the fluorescence fluctuations were caused by turbulence in the aqueous surface layer. Furthermore, because numerical modeling of the hydration kinetics of COPhas shown the production rate of HzC03was faster than the
17 13 9.6 6.9
74 57 47 41
99 132 177 246
26 34 41 47
of all Darameters is Drovided in the text.
calculated turbulence Kolmogorov time scale (19),the concentration fluctuations were able to track small-scale turbulence motions down to the viscous-diffusive subrange. This implies that because the dominant mass-transport time scale will be the most frequently observed surface element time scale and because the concentration fluctuations were able to track the turbulence time scales at the water surface, the most probable concentration fluctuation time scale defined by TM is the appropriate parameter for use in prediction of ItL using eq 2. Because surface renewal theory was originally formulated in terms of a turbulence time scale, strictly speaking, that type of time scale and not a concentration fluctuation time scale should be used in eq 2 to predict kL. Although TMis determined by the turbulence time scales as discussed above, McCready and Hanratty (24) and Stewart (25) have shown that the fastest turbulence fluctuations will not be detected by monitoring concentration fluctuations due to the effects of molecular diffusion. However, because McCready and Hanratty (24) and Stewart (25) have shown that the fastest turbulence fluctuations are not important in terms of promoting mass transport, it is not clear that substitution of turbulence time scales for T in eq 2 will increase the predictive accuracy of the Higbie model relative to the use of TM. Values of TM determined here were used to predict kL by use of the Higbie (12)model for the 1-OD film and RVC data by substituting TM for T in eq 2. Table I lists TMand Environ. Sci. Technol., Vol. 25, No. 7, 1991
1297
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40
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kLm (xlO' crn s-') Figure 6. Application of the Hlgbie surface renewal model. Plot of k , calculated by substitution of T, for T in eq 2 vs k , measured under identical turbulence conditions. Key for RVC interface data: (0)z, = 150 pm, (A)2 , = 300 pm, ( 0 )z , = 450 pm. Key for the 1-OD z , = 300 pm. Solld line is slope of fllmtovered interface data: (0) 1 with zero intercept.
predicted kLvalues (kL,p)for a RVC surface with z, = 150, 300, and 450 pm and for a 1-OD film-covered surface with z, = 300 pm. Also included in Table I are measured kL values ( k L , m ) obtained under identical turbulence and interfacial conditions by Asher and Pankow (15). Increases in the level of near-surface aqueous-phase turbulence are accompanied in Table I by decreases in the calculated turbulence Kolmogorov time scale, T,,. Also listed in Table I are values for the concentration boundary layer thickness, 6,, and the diffusive length scale, ad. These parameters are discussed in detail below. Figure 6 is a plot of kL,pvs kL,, for both RVC and 1-OD film-covered surfaces for the data listed in Table I. Use of this surface renewal model to predict kL for a 1-OD film-covered interface results in very large discrepancies between measured and predicted k L values over the range of turbulence conditions studied here. As with the 1-OD film-covered data, the low turbulence intensity (ie., large 7,,), RVC data plotted in Figure 6 show that use of T Min eq 2 leads to an overprediction of kL. As will be discussed below, the overprediction of kL at the lower turbulence intensities for the RVC surfaces and for the 1-OD filmcovered surfaces may have been caused by incomplete renewal of the water surface by the turbulence eddies. In contrast to the 1-OD film-covered surfaces and lower turbulence intensity RVC surfaces, use of TM in eq 2 for the higher turbulence intensity (i.e., small T,,), RVC interfaces results in good prediction of kL. This result suggests that TM is an appropriate time scale for use in the Higbie (12) model for a clean surface and high turbulence intensities. This conclusion is supported by the time series of FT shown in Figure 4 for C 0 2 transport through a RVC interface. For a RVC interface, F T aperiodically returns to its initial value of 1. Because COz invasion decreased the pH of the surface water and the observed fluorescence intensity, the return of F T to 1 implies that the sampling volume defined by z, was replaced by bulk fluid with a pH at or near the starting pH. Furthermore, because of the functionality of FT,*,complete replacement of fluid at the surface was necessary to achieve an F T of 1. Because the high turbulence intensity RVC 1298 Envlron. Sci. Technol., Vol. 25, No. 7, 1991
interface was completely renewed by the turbulence, the Higbie (12) surface renewal model was able to predict kL accurately. The overprediction of kL observed for RVC interfaces at the lower turbulence intensities was most likely caused by incomplete renewal of the aqueous surface layer by the turbulence eddies. This was demonstrated by inspection of the peak heights in the raw fluorescence fluctuation time series at the lower turbulence intensities. These data showed that fewer eddies completely renewed the aqueous surface at the lower turbulence levels than at higher turbulence intensities. Because there were fewer events that completely renewed the C02/H20interface at the lower turbulence intensities, use of the Higbie surface renewal model resulted in overprediction of kL. Explanation of the poor prediction of the measured transport rates for the 1-OD film data is provided by examination of the FT vs time data shown in Figure 5. In contrast to Figure 4, Figure 5 shows that, after its initial decay, F T never rose above 0.6. This implies that the aqueous surface microlayer was never completely renewed by the turbulence eddies. Surface renewal models would not be expected to apply for a film-covered interface as seen in the 1-OD film covered interface data shown in Figure 6. Furthermore, it would be expected that a surface renewal model would overpredict kL because Asher and Pankow (15),Jahne et al. (26), and Frew et al. (27) have shown that the presence of an interfacial film or surfaceactive material decreases the rate of gas/liquid mass transport. In addition to the analysis of the raw fluorescence fluctuation data, the conditions for which surface renewal is applicable can also be seen by examining the relative magnitudes of the mass-transport resistances for the total process and each relevant subprocess. Here, it is assumed that the two major sources of resistance to gas exchange are a resistance due to surface renewal and a resistance from the concentration boundary layer. Mass-transport resistances, defined by Liss and Merlivat (28) as the reciprocal of the transport coefficient, allow comparison of the relative importance of the two different transport mechanisms. Because the effects of surface renewal and boundary layers on gas exchange may be thought of as resistances in series (28), the total resistance to mass transport & (s cm-') may be written as (7) Rq- = Rs + RB where Rq- is given by
Rr = l / k L , m
(8)
Rs (s cm-l) is the resistance due to surface renewal and is defined by RS = 0 . 8 8 6 2 ( T ~ / D ) '=/ ~l / k ~ , ~ (9) and RB (s cm-l) is the resistance from the concentration boundary layer. Equation 7 shows that if >> RB,surface renewal will provide the dominant resistance to transport and kL,, will be equal to eq 2. However, if RB >> Rs, the concentration boundary layer resistance will control & and use of eq 2 would result in inaccurate prediction of kL,m* Equations 7-9 show that RB is equal to (10) RB = & - RS = l/kL,m - l / k ~ , p Therefore, the resistance to transport due to the existence of a boundary layer at the water surface may be estimated from the difference in the reciprocal of the measured transport velocity and the reciprocal of the transport velocity due to surface renewal as calculated from eq 2 and
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Flgure 7. Plot of the effective resistance due to a concentration boundary layer RB vs the concentration fluctuation time scale mode T , for clean and fllm-covered water surfaces. R B is defined In eq 10 to be the difference between total resistance to mass transfer R, and the resistance due to surface renewal R,. Data key Is identical with Figure 6. The horizontal solid line shows R, = 0.
TM, Figure 7 shows a plot of RB (calculated by using eq 10) vs TMfor RVC and 1-OD film-covered interfaces. For RVC surfaces, RB is very small for TM less than 1.0 s. This suggests that for clean water surfaces with relatively intense turbulence, the major source of resistance to gas transport is due to surface renewal and not to the presence of a concentration boundary layer. This implies that a surface renewal model may be able to predict kL for TM less than 1.0 s. For larger T M however, , RBwas much greater than zero, which suggests that as the turbulence intensity decreases, a concentration boundary layer forms and its resistance controls the transport velocity. Therefore, it would be expected that a surface renewal model would do a poor job of predicting kL for TMgreater than 1.0 s, as observed in Table I and Figure 6. As shown in Figure 7,RB for the 1-OD monolayer-covered interfaces was much greater than zero regardless of TM. This suggests that for film-covered interfaces, the major resistance to gas transport is due to the presence of a concentration boundary layer. Under these conditions, surface renewal models would not be expected to be able to predict kL. The data presented in Table I and Figure 6 also support this conclusion. The increase in RBfor decreasing turbulence intensity at RVC interfaces and the behavior of RB at 1-OD filmcovered surfaces can also be interpreted in terms of the relative magnitude of the concentration boundary layer depth 6, defined by
6, = D/kL,rn
and the diffusive length scale 6d
6d
(11)
defined by
= (2DT,)'f2
(12)
Physically, 6, represents the mean concentration boundary layer thickness at the surface (11). Therefore, it is an empirical measure of the mean depth of the concentration gradient shown in Figure 2. For turbulence and interfacial conditions where the eddies do not completely renew the surface, 6, may be thought of as the distance of closest approach to the gas/liquid interface of the turbulence eddies. The diffusive length scale 6d is the mean distance
Flgure 8. Plot of difference between estimated COncentrabion boundary layer thickness 6, and diffusive length scale 6, vs the concentration fluctuation time scale mode Tu. Data key is identical with Flgure 6 and the horizontal solid line shows 6, - 6, = 0.
the gas will diffuse into the water surface over the most probable concentration fluctuation lifetime. If 6, is smaller than ad, then the gas will be able to diffuse directly into the turbulence eddy when it is at the surface. Because this condition is the basic hypothesis of surface renewal, it is expected that when 6, Iad, the Higbie model will be able to predict kb If 6, is large compared to 6d, the gas will be unable to diffuse directly into the turbulence eddy during the eddy lifetime at the interface. Rather, it must pass through a stagnant layer before it reaches an area of active turbulence in the fluid. Surface renewal theory as formulated by Higbie (12)makes no allowance for the existence of a stagnant layer. Therefore, when 6, > 6d, it is expected that surface renewal theory in general, and the Higbie model in particular, would be unable to accurately predict kL. Figure 8 is a plot of 6, - 6d vs TMfor RVC and 1-OD film-covered water surfaces. For the experiments with an RVC surface, 6, - 6d is very close to zero for TM less than 1.0 s. As discussed above, this implies that surface renewal was occurring under these conditions. For experiments with an RVC surface and TMgreater than 1.0 s and for experiments with a 1-OD film-covered interface, 6, - 6! is greater than zero, which implies that a concentration boundary layer existed. This suggests that surface renewal did not occur under these conditions. Therefore, the analysis of Figure 8 for the RVC surface data agrees with the results presented in Figures 6 and 7. Figures 7 and 8 show that surface renewal should not be used to predict kL at clean water surfaces that have surface element time scales greater than 1 s. By use of eq 2 with T equal to 1 s and D = 1.7 X cm2 s-l (corresponding to COz),kL is calculated to be 4.65 X cm s-l. By use of the empirical correlation of kL with wind speed developed by Liss and Merlivat (%), this kL value is found to correspond to a wind speed of 9.3 m s-l. This suggests that surface renewal models should not be used to predict kL at wind speeds below 10 m s-l, especially when surface films are present. Because the films formed by 1-octadecanol are easily dispersed by wind and wave motions (29) and are different from those produced biologically (27,30),the film-covered interface results presented here are not directly applicable to the Ocean or lakes. However, Frew et al. (27)have shown Envlron. Scl. Technol., Vol. 25, No. 7, 1991
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that phytoplankton-generated surfactant material will also reduce kL. Furthennore, it was postulated that this decrease in k L was due to the films damping turbulence in the surface layer (27). Because the majority of these naturally occurring surface-active compounds are soluble surfactants, which are not readily dispersed by wind or waves (27, 30),and because both soluble and insoluble surfactants have similar effects on the hydrodynamics governing air/water gas transport, the data presented here suggest that if films are present on an air/water interface, surface renewal models should not be used to predict kL.
Conclusions This work has shown that use of the Higbie (12) surface renewal model with directly measured interfacial concentration fluctuation time scales allows accurate prediction of measured transport velocities for a cleaned gas/liquid interface and high turbulence intensities. Therefore, the concentration fluctuation time scales may be considered to be a surrogate for the theoretical surface renewal time scales. However, it has also been shown that the Higbie model (12) does not accurately predict transport rates through film-covered interfaces. Although it is not definitively known if the ocean surface is better modeled with a clean or dirty surface, laboratory studies by Frew et al. (27) have shown that biogenically produced surface-active compounds can drastically reduce kL. Furthermore, recent research shows that naturally occurring surfaca-active organic compounds may be found at the ocean surface under a range of conditions (30). Although there is less known concerning surface-active material in freshwater, it is reasonable to assume that fiis are also present on lake surfaces. Given the results discussed here, we can conclude that extreme caution must be used when applying surface renewal models to predict air/water gas-exchange rates in the natural environment. Acknowledgments We thank Donald Mackay for helpful discussions concerning the conditions under which surface renewal models are applicable to gas/liquid transport.
Literature Cited Brutsaert, W., Jirka, G. H., Eds. Gas-Transferat Water Surfaces; D. Reidel: Hingham, M 4 , 1984. Danckwerts, P. V. Gas-Liquid Reactions; McGraw-Hill: New York, 1970. Gulliver, J. S.; Halverson, M. J. Water Resour. Res. 1989, 25, 1783-1793. Komori, S.; Nagaosa, R.; Murakami, Y. AZChE J . 1990,36, 957-960.
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(5) Davies, J. T.; Lozano, F. J. AIChE J. 1979, 25, 405-427. (6) Asher, W. E.; Pankow, J. F. Chem. Eng. Sci. 1989, 44, 1451-1455. (7) Fortescue, G. F.; Pearson, J. R. A. Chem. Eng. Sci. 1967, 22, 1163-1176. (8) Lamont, J. C.; Scott, D. S. AZChE J. 1970, 16, 513-519. (9) Kitaigorodskii, S. A. J. Phys. Oceanogr. 1984,14,960-972. (10) McCready, T. J.; Vassiliadou, E.; Hanratty, T. J. AZChE J . 1986, 32, 1108-1115. (11) Brumley, B. H.; Jirka, 6.H. Physicochem. Hydrodyn. 1988, 10, 295-319. (12) Higbie, R. Trans. Am. Znst. Chem. Eng. 1935,35,365-389. (13) Danckwerts, P. V. Znd. Eng. Chem. Process Des. Deu. 1951, 43, 1460-1467. (14) Davies, J. T.; Lozano, F. J. AZChE J. 1984, 30, 502-504. (15) Asher, W. E.; Pankow, J. F. Tellus 1986, 38B, 305-318. (16) Lee, Y. H.; Luk, S. Znd. Eng. Chem. Fundam. 1982, 21, 428-434. (17) Luk, S.; Lee, Y. H. AZChE J. 1986, 32, 1546-1554. (18) Jirka, G. H.; Ho, A., H.-W. J. Hydraul. Eng. 1990, 116, 835-847. (19) Asher, W. E. The effect of mechanically generated turbulence and interfacial films on a liquid phase rate controlled gas/liquid mass transport process. Ph.D. Thesis, Department of Environmental Science and Engineering, Oregon Graduate Institute, Beaverton, OR, 1987. (20) Hopfinger, E. J.; Toly, J. A. J. Fluid Mech. 1976, 78, 155-175. (21) Brumley, B. H.; Jirka, G. H. J . Fluid Mech. 1987, 183, 235-263. (22) Hannoun, I. A.; Fernando, H. J. S.; List, E. J. J. Fluid Mech. 1988, 189, 189-209. (23) Scott, J. C. J. Fluid Mech. 1975, 69, 339-351. (24) McCready, T. J.; Hanratty, T. J. AZChE J. 1984, 30, 816-81 7. (25) Stewart, W. E. AZChE J. 1987, 33, 2008-2016. (26) Jahne, B.; Huber, W.; Dutzi, A.; Wais, T.; Ilmberger, J. In Gas-transfer at water surfaces; Brutsaert, W., Jirka, G. H., Eds.; D. Reidel: Hingham, MA, 1984; pp 303-310. (27) Frew, N. M.; Goldman, J. C.; Dennett, M. R.; Johnson, A. J. J. Geophys. Res. 1990, 95C, 3337-3352. (28) Liss, P. S.; Merlivat, L. In The Role of Air-Sea Exchange in Geochemical Cycling; Buat-Menard, P., Ed.; D. Reidel: Hingham, MA, 1986; pp 113-127. (29) Goldman, J. C.; Frew, N. M.; Dennett, M. R. Deep Sea Res. 1988, 35, 1953-1970. (30) Williams, P. M.; Carlucci, A. F.; Henrichs, S. M.; Van Vleet, E. S.; Horrigan, S. G.; Reid, F. M. H.; Robertson, K. J. Mar. Chem. 1986, 19, 17-98. Received for review April 3, 1990. Revised manuscript received March 4, 1991. Accepted March 12, 1991. Financial support f o r this work was provided by the Donors of the Petroleum Research Fund, administered by the American Chemical Society. W.E.A. also acknowledges support by the US.Department of Energy under Contract DE-AC06-RL01830-12682.
