Estimation of Mass Transfer Velocity Based onMeasured Turbulence Parameters

Johannes G. JanzenDept. of Environmental Engineering, Federal University of Rondonia,

Jardim dos Migrantes, Ji-Parana 76900-726, RO, Brazil

H. Herlina and Gerhard H. JirkaInstitute for Hydromechanics, University of Karlsruhe, Karlsruhe 76128, Germany

Harry E. SchulzNucleus of Thermal Engg. and Fluids, Dept. of Hydraulics and Sanitation, Sao Carlos School of Engineering,

University of Sao Paulo, Sao Carlos 13560-970, SP, Brazil

John S. GulliverDept. of Civil Engineering, University of Minnesota, Minneapolis, MN 55455

DOI 10.1002/aic.12123Published online December 30, 2009 in Wiley InterScience (www.interscience.wiley.com).

The aim of this study is to quantify the mass transfer velocity using turbulence pa-rameters from simultaneous measurements of oxygen concentration elds and velocityelds. The surface divergence model was considered in more detail, using dataobtained for the lower range of b (surface divergence). It is shown that the existingmodels that use the divergence concept furnish good predictions for the transfer veloc-ity also for low values of b, in the range of this study. Additionally, traditional concep-tual models, such as the lm model, the penetration-renewal model, and the largeeddy model, were tested using the simultaneous information of concentration and ve-locity elds. It is shown that the lm and the surface divergence models predicted themass transfer velocity for all the range of the equipment Reynolds number used here.The velocity measurements showed viscosity effects close to the surface, which indi-cates that the surface was contaminated with some surfactant. Considering the results,this contamination can be considered slight for the mass transfer predictions. VVC 2009American Institute of Chemical Engineers AIChE J, 56: 20052017, 2010

Keywords: laser-induced uorescence, particle image velocimetry, airwater gastransfer, grid-stirred tank, mass transfer velocity

Introduction

Gas transfer across the airwater interface plays an impor-tant role in geophysical and industrial processes. A historical

dominant interest lies in the transfer of oxygen from air to

water, because the dissolved oxygen content of a body of

water is a primary indicator of the water quality. However,

the increase of industrial activity in the last century pro-

duced, as a consequence, the emission of large amounts of

different gases and chemical compounds in the atmosphere

and in water bodies. The interest in understanding the

Correspondence concerning this article should be addressed to J. G. Janzen atjohannesjanzen@gmail.com.

VVC 2009 American Institute of Chemical Engineers

AIChE Journal 2005August 2010 Vol. 56, No. 8

underlying mechanisms of interfacial mass transfer was thendirected to quantify transfer rates of a broader range of gasesand compounds, such as carbon dioxide, methane, and vola-tile organic compounds, among others. Considering slightlysoluble gases, where the liquid phase plays the dominantrole, the study of oxygen absorption by water is convenientbecause the existing methods to measure concentrations andvelocities permit to obtain insights about the general mecha-nisms that control mass transfer. Being essential to maintainlife, the oxygen consumed in water bodies must be recov-ered, which occurs through the airwater interface. Thus,reoxygenation (or reaeration) should be carefully estimated.The mathematical models, which are used to quantify inter-facial mass transfer, involve the turbulence level to whichthe interface is subjected. Considering the absence of bub-bles and drops, the ux of gas, N, across the airwater inter-face is customarily expressed as

N KCS CB; (1)where K is gas transfer velocity, which is applied to thetransfer of volatile compounds and gases, CS and CB are thesaturated and bulk concentration of the dissolved gas,respectively. An accurate value of K is needed to effectivelypredict gas transfer across the airwater interface. Althoughnumerous conceptual theories have been proposed over theyears (i.e., Lewis and Whitman,1 Higbie,2 Danckwerts,3

Kishinevsky and Serebriansky,4 Toor and Marchello,5 Coan-tic,6 and Schulz and Schulz7), the turbulence-related para-meters in these theories are still poorly defined.The theoretical predictions of interfacial mass transfer typi-

cally consider the one-dimensional form of the Ficks rst law:

N DdCdz

Interface

; (2)

where D is the molecular diffusivity of the dissolved gas inwater and z denotes the vertical direction (with the conventionthat positive is in the downward direction).In this study, some models were chosen to evaluate the

mass transfer coefcient K (or transfer velocity). The modelsinvolve different parameters (or concepts) related to turbu-lence, permitting to compare predictions of K using these pa-rameters (or combination of parameters). The lm model byLewis and Whitman1 assumed that the transfer occurs onlyby molecular diffusion across a lm of constant thickness.Outside of this lm, the uid is well mixed. It could bededuced that the liquid lm coefcient is equal to

K Ddc

; (3)

where dc is the film thickness, also named as theconcentration boundary layer thickness, which must bequantified. Recognizing that the assumptions made for the(stagnant) film model would perhaps introduce errors in thequantification of K, Higbie2 and Danckwerts3 proposed thatpatches of the liquid surface are periodically replaced by fluidelements from the well-mixed bulk. Assuming that moleculargas transfer takes place for a certain time, after which thesurface is renewed, which occur at a mean rate s, it can beshown that

K Ds

p; (4)

where s is the mean surface renewal rate. The key problem inthe penetration-renewal models lies in the prediction of s. AfterHigbie2 and Danckwerts,3 the models were more directed torelate surface renewal directly to measurable turbulencequantities. The large eddy model (Fortescue and Pearson8)and the small eddy model (Lamont and Scott9) are perhaps thebest known of these models. In the large eddy model, forexample, s is usually replaced by the ratio U/L, where U and Lare the characteristic velocity and length of the larger eddies,respectively. The horizontal root-mean-square (rms) velocity,

u02p

, and the length macroscale at the surface are commonchoices (Calmet and Magnaudet10), which lead to Eq. 5:

K / Du02

pL

!1=2: (5)

As the dimension of s is the inverse of the time, Daviesand Lozano11 replaced s by the inverse of the time macro-scale, T1, obtained from the autocorrelation function forvelocities at the water surface.McCready et al.12 developed a technique for the quanti-

cation of K that relies upon the gradient of instantaneous ve-locity normal to the surface, b0 dw/dz, where w and z arethe velocity component and coordinate normal to the surface,respectively. They used velocity measurements by Lau13

near a solid surface, but outside of the concentration bound-ary layer, and assumed that these could apply near a freesurface. Tamburrino and Gulliver14 realized that b0 could bemeasured at a free surface by applying the continuity equa-tion (Eq. 6) and measuring u and v trough particle tracking.

b0 @u@x

@v@y

: (6)

The model permits to correlate turbulence parameters andmass transfer following a structured way. Various investiga-tors have used a measured surface divergence to determineb0 and predict K (Kumar et al.,15 Law et al.,16 Tamburrinoand Gulliver,17 Xu et al.,18 and Tamburrino et al.19). Lawand Khoo,20 using CO2 and water with different concentra-tions of glycerol (to obtain different Schmidt, Sc m/D,numbers), proposed

K 0:22bD

p: (7)

where b is the rms of b0. McKenna and McGillis21,22 suggestedthat the surface divergence may also account for the presenceof surface films. Considering contaminated surfaces, theypresented the equation

K 0:5bv

pScn

n 1=2 clean suraces2=3 surfaces behave like a rigid wall

8

where m is the kinematic viscosity. Equation 8 may be viewedas a natural extension of the clean surfaces approach forslightly contaminated surfaces. It is not valid for strong

2006 DOI 10.1002/aic Published on behalf of the AIChE August 2010 Vol. 56, No. 8 AIChE Journal

contamination, having in this case the same limitationsmentioned by Hasegawa and Kasagi,23 who studied contami-nated surfaces quantified through the Weber and theMarangoni numbers, the last expressing the effect of surfactantconcentrations on the surface tension. The authors performednumerical calculations using b(t) b cos(xt) and presentedthe results as a function of b/x. Different behaviors of themass transfer were observed for values of b /x higher or lowerthan 1. For highly contaminated surfaces, the mass transfercould not be quantified through the surface divergence,whereas clear and slightly contaminated surfaces correlatewell with the single equation

K 0:4bD

p: (9)

For clean surfaces, Eqs. 79 are equivalent except for thedifferent coefcients. It can be noted that the important pa-rameters governing the scalar transport for the clean surfacesare b and D. b is related to the hydrodynamics very close tothe interface, and D is related to the species involved ininterfacial transport. Equation 8 shows a distinction betweenclean and rigid contaminated surfaces, quantied throughthe additional parameter n. Additionally, to the effects ofcontamination on b directly, Eq. 8 proposes a correctionusing n when quantifying K. In this study, both proposalswere considered (with and without n), reecting the diverseviews on the subject found in the literature.One reason that analyses of gas transfer at a free surface

have been slow in coming is the difculty in making meas-urements near the free interface. The concentration boundarylayer is usually of the order of 10- to 100-lm thick and ismobile, which makes measurements within the concentrationboundary layer difcult, and limited in range of ow condi-tions. In the last 2 decades, laser-induced uorescence (LIF)techniques have been used in studies of oxygen (O2) transferacross airwater interfaces. A uorescent dye, dissolved inwater, uoresces when excited by laser light. The uores-cence intensity is a function of the dye concentration and ismeasured with a CCD camera. The LIF technique allows thevisualization of the concentration boundary layer, yieldingfar more detailed information than time-averaged mass trans-fer experiments (Jahne and Haussecker24). More recentattempts for the measurement of oxygen concentration eldsnear the airwater interface have been reported, for example,by Woodrow and Duke,25 Lee,26 Schladow et al.,27 Herlinaand Jirka,28 Janzen et al.,29 among others. Descriptions ofexperiments with simultaneous measurement of velocity andoxygen concentration in the near surface region are less fre-quent in the literature, which is probably related to the dif-culty in obtaining results with a good resolution for bothelds. Chu and Jirka30 presented results obtained with amicroprobe for concentration measurements and split-lmanemometry for the turbulence measurements. Atmane andGeorge31 used a microprobe and polarographic methods forthe concentration measurements and LDV for the velocitymeasurements. The authors furnish results for turbulent masstransfer, although mentioning the need of more accuratedata. Herlina,32 Janzen,33 and Herlina and Jirka34 presentedmeasurements in which the LIF technique was used to mea-sure the concentration eld, and the particle image velocime-try (PIV) technique was used to measure the velocity eld.

Such combined measurements are needed to provide anunderstanding of the positive and negative mass uxes insidethe upper water layer, as emphasized by Atmane and George.31

This article presents results of measurements of both theconcentration and the velocity elds near the airwater inter-face, using the nonintrusive LIF and PIV techniques. Theconcentration and velocity measurements were simultaneous.The dimensions of the velocity elds measured in this studywere larger to those presented by Herlina and Jirka,34 allow-ing observation of the interaction between the concentrationand the velocity elds over larger distances. A comparisonof the different concentration boundary layer thicknesses ismade, and the parameters dened in different models of Kare calculated within the lower range of b considered in thisstudy.

Experimental Methods

Oscillating grid system

Figure 1a is a view of the oscillating-grid tank used inthis study. The experiments were conducted in a tank madeof Perspex, with a 0.50 m 0.50 m square cross-sectionand 0.65 m height. A grid with 6.25-cm mesh size with a so-lidity of 36% was used, fullling the criteria of Hopngerand Toly.35 The grid was positioned 20.0 cm above the bot-tom of the tank to minimize secondary motion. The experi-ments were conducted at the Institute for Hydromechanics ofthe University of Karlsruhe, Germany, with a full descriptionof the oscillating-grid tank given by Herlina32 and Janzen.33

The grid was operated with a 5.0 cm stroke S, and the fre-quency f varied from 2.0 to 5.0 Hz. The water depth abovethe grid, h, was maintained at 28.0 cm. The mean water tem-perature was 26.5C. Data acquisition was begun 10 min af-ter the onset of oscillation because the oscillating grid turbu-lence is sensitive to initial conditions (Cheng and Law36).All the experiments were performed in a single sequence, toguarantee similar environmental and surface conditions, andto permit comparisons between the data. Only the frequencywas changed from one run to other. Table 1 presents experi-mental parameters, where Re is the Reynolds number denedfor the experiments as f S2/m.

Concentration measurements

A LIF technique was used to obtain dissolved oxygen con-centration elds at the airwater interface. Pyrene butyricacid (PBA) was used as indicator for dissolved oxygenconcentration in water (Vaughan and Weber37). The changein uorescence lifetime, s, and intensity, F (also calledquenching) is quantitatively described by the Stern-Volmerequation:

F0F s0

s 1 KSVC; (10)

where F0 and s0 are the fluorescence intensity and lifetime,respectively, in the absence of the quencher and KSV is theStern-Volmer quenching constant. Hence, the quencherconcentration (in this case dissolved oxygen) can bedetermined by measuring the intensity of emitted fluorescence(in this case from PBA).

AIChE Journal August 2010 Vol. 56, No. 8 Published on behalf of the AIChE DOI 10.1002/aic 2007

Figure 1b shows the LIF setup used in this study. Thetank was lled with 2 105 mol/L PBA. The water in thetank was bubbled with nitrogen to remove oxygen throughstripping. An initially dissolved oxygen concentration ofabout 0.7 mg/L was achieved after 20 min of bubbling. Theexperimental conditions imposed the saturation concentrationat the interface during the experiments for a low solubilitygas such as oxygen. A pulsed nitrogen laser (MNL 801)with a mean energy power of 0.4 mJ and a wavelength emis-sion of 337.1 nm was used to excite the PBA solution. Thelaser beam was guided into the center of the tank through aUV-mirror and a combination of lenses. A FlowMaster CCDcamera (1024 1280 pixels and 12 bit) with a macro-objec-tive was used to obtain images of 9.5 mm 11.9 mmfrom a distance of about 30 cm. The measurements have aresolution of 9 lm. The PBA uorescence intensity liesbetween 370 and 410 nm. An optical bandpass lter wasmounted in front of the camera to ensure that only the uo-rescent light could pass through. The maximum number ofsuccessive images was limited to 300. Nine hundred imageswere taken for each run, which were averaged to obtain thetemporally averaged eld. This eld was then horizontallyaveraged to obtain the average proles. A series of image-processing steps were performed on the raw images, includ-ing noise removal, water surface detection, correction oflaser attenuation, and correction of optical blurring near the

interface. The image processing procedure followed themethod described by Woodrow and Duke25 and Herlina andJirka.28 The standard deviation of oxygen concentrationimages with the same concentration suggests that the resolu-tion of the current setup is about 5%, as described by Her-lina,32 who used the same LIF setup of this study.

Velocity measurements

The velocity eld was measured using a PIV technique(Cheng and Law36). Figure 1b also shows the PIV setupused in this study. The light source was a 25 mJ NdYaglaser. A CCD camera with a resolution of 1024 pixels 1280 pixels was used to record the images. The eld ofview of the camera was 9.4 cm wide and 7.4 cm high. Poly-amid particles with a nominal diameter of 10 lm were usedas ow eld tracers. After capturing and storing the imagesin the computer, the PIV software DaVis was applied toeach pair of images (cross-correlation) to obtain velocityvector elds. The interrogation area chosen to evaluate thevectors was 0.23 cm 0.24 cm (32 pixel 32 pixels) witha 50% overlap. The size of the interrogation area affectsdata resolution, while the overlap of interrogation areas pro-vides inherent correlations among the adjacent vectors. Ninehundred images were taken for each experiment run. Basedon the technique described in Westerweel,38 the accuracy(systematic error) of the current PIV system was evaluatedto be less than 0.5 pixels (Herlina32).

Results and Discussion

Concentration boundary layer thicknesses

A normalized concentration, C*, is used here to describethe behavior of the mean concentration in the concentrationboundary layer, as

Table 1. Experimental Parameters

Run no. f (Hz) S (cm) Re

1 2 5 57912 3 5 86873 4 5 11,5824 5 5 14,478

Figure 1. (a) Oscillating-grid tank (Herlina32); (b) experimental setup (Herlina32).

2008 DOI 10.1002/aic Published on behalf of the AIChE August 2010 Vol. 56, No. 8 AIChE Journal

Cz Cz CBCS CB ; (11)

where C z is the mean gas concentration at distance zdownward from water surface. The concentration boundarylayer thickness, dc, is normally defined as the depth where thevalue of the normalized concentration C* attains a value of0.01, or

Cdc Cdc CBCS CB 0:01: (12)

The boundary layer is then the distance over which thegas concentration varies from the surface to 0.99 times thebulk value. Thinner than the concentration boundary layer, aregion close to the surface is usually dened, within whicheddy diffusion is slight compared to molecular diffusion.This region is called diffusive sublayer, and throughout itthe mean concentration prole evolves linearly (Magnaudetand Calmet39). Woodrow and Duke25 presented a seconddenition of the concentration boundary layer, dD, as theintercept between the prolongation of the line determined bythe concentration gradient at the water surface with the line

which corresponds to the constant value of concentration,CB:

dD CS CBd C=dzjz0: (13)

Considering an exponential prole for C* (Chu andJirka30), the position z dD implies that the concentrationattains 0.63 times the bulk liquid value (1/e). As dD isobtained considering the linear part of the concentration pro-le, it is referred here as diffusive sublayer. The instanta-neous uctuations around the mean concentration can beused to calculate rms proles. Schulz and Schulz7 showedthat a peak of less than 0.5 must exist for the normalizedrms curves of the concentration uctuations, rc*, dened asfollows:

rc Cz Cz2

qCS C1 : (14)

Lee and Luck,40 Atmane and George,31 Herlina,32 andJanzen33 presented experimental results for rc*.

Evolution of mean concentration along the depth

The nondimensional mean concentration proles areshown in Figure 2a. These proles are the result of a two-step process: (1) a mean prole for each image (or timestep) was obtained, and (2) the so obtained 900 proles wereaveraged along time. As expected, the proles for higher tur-bulence conditions have steeper gradients close to the sur-face than those for lower turbulence conditions. The prolespermitted calculation of the boundary layer thickness foreach run, as dened by Eq. 12, and the diffusive sublayerthickness, as dened by Eq. 13. A display of the concentra-tion boundary layer thickness against Reynolds number isfurnished in Figure 2b, showing that the boundary layerthickness is thinner for higher turbulent conditions. Figure2b additionally shows the diffusive sublayer thicknessagainst the Reynolds number, which is also thinner forhigher agitation. This is a direct consequence of the steepergradients at higher Reynolds number. For mean of the pres-ent data, dD 0.36dc. All values are presented in Table 2.

Evolution of standard deviation along the depth

Concentration. Figure 3 presents the normalized rmsconcentration proles, rc*, as dened by Eq. 14. The proleshighlight two facts: (a) the curves reach a maximum of z* z/dc between 0.27 and 0.30, with an average value of 0.29;and (b) the normalized maximum of rc* ranges between 0.13and 0.17. These values are similar to results obtained inother works (Lee and Luck40 obtained maximum rc*between about 0.06 and 0.3, Atmane and George31 obtainedvalues between 0.2 and 0.3, Magnaudet and Calmet39

obtained values between 0.28 and 0.35 for different Schmidtnumbers, and Herlina32 obtained values between 0.14 and0.18) and are lower than the theoretical limit of 0.5.The peak position, here denoted by dr, is close to the sur-

face, and numerical simulations performed by Magnaudet

Figure 2. (a) Mean concentration proles; (b) concen-tration boundary layer thickness and diffusivesublayer thickness vs. Reynolds number.

AIChE Journal August 2010 Vol. 56, No. 8 Published on behalf of the AIChE DOI 10.1002/aic 2009

and Calmet39 showed that it occurs near the edge of the socalled outer diffusive layer. The space between the sur-face and the peak position may be considered as the regionwhere the free surface has a stronger inuence, whereas thespace under the peak position may be considered as theregion where turbulence of the bulk liquid has a strongerinuence. The mean position of the peak of rc* is dr 0.29dc, which is close to dD 0.36dc. For this set of experi-ments dr 0.80dD, on average (the coefcient is in therange of 0.770.86). It is possible to use dr as indicator ofthe distance along which diffusive effects are important.This may be more representative, because dr represents theeffect of turbulent uctuations and concentration gradient.All values are presented in Table 2.

Velocity. Figure 4 shows the variation of rms turbulenthorizontal and vertical velocities with depth. The horizontalvelocity, urms, shows characteristics similar to the results ofBrumley and Jirka,41 with some proles forming a bulgein the surface inuenced region and indicating a viscousboundary layer. The increase in the horizontal intensity out-side of the inuence of the viscous layer, while the verticalintensity, wrms, decreases rapidly, is a result of the redistribu-tion of the kinetic energy from the vertical motion to thehorizontal motion, as shown experimentally by Law et al.16

and theoretically by Hunt and Graham42 and Magnaudet.43

The horizontal and vertical velocities for the Reynolds num-

bers 8687 and 11,582 have almost the same value near theinterface. A possible reason for this result may be the forma-tion of circulation regions (secondary circulations or meanow) whose geometrical characteristics changed from oneagitation condition to the other, affecting the velocity valuesin the measurement region. Secondary circulations are pres-ent in oscillating grid-stirred turbulence, as pointed byMcKenna and McGillis.44

At the surface (z 0) the vertical rms velocity valuesmust equal zero, relative to the location of the free surface.

Figure 3. Variation of rms (root-mean-square) concen-tration with depth.

Figure 4. Variation of rms (root-mean-square) turbulent(a) horizontal and (b) vertical velocities withdepth.

Table 2. Measured Parameters

Re 5791 8687 11,582 14,478Boundary layer thickness, dc (m) (103) 1.359 1.137 1.099 0.9973Diffusive layer thickness (Eq. 13), dD (m) (104) 4.730 4.040 4.000 3.630Diffusive layer thickness (peak of Eq. 14), dr (m) (104) 4.082 3.108 3.108 2.922Distance over which

w02

pis linear, K (m) (103) 9.86 9.86 8.70 8.70

Horizontal rms velocity,u02

p(m/s) (103) 1.59 3.26 3.76 5.06

Vertical rms velocity difference Dw02

p(m/s) (103) 1.48 2.66 2.26 3.31

Surface divergence, b Dw02

p .K (s1) 0.15 0.27 0.26 0.38

Undisturbed length macroscale, Lu (m) (102) 3.37 3.92 3.31 3.63Disturbed length macroscale, Ld (m) (102) 6.65 5.31 5.70 4.93Time macroscale, T (s) 9.24 5.08 1.89 6.08

2010 DOI 10.1002/aic Published on behalf of the AIChE August 2010 Vol. 56, No. 8 AIChE Journal

The values different from zero shown in Figure 4b are dueto the resolution of the PIV measurements, in which veloc-ities are evaluated within an interrogation area. In this case,the obtained surface velocities include distances untilabout 2.4 mm under the surface, where vertical velocitycomponents still exist. Despite this bias at the surface, theproles of the vertical rms velocity show a quasi-linearrange until a distance of about 10 mm from the surface,which agrees with the hypothesis of McCready et al.12 in thesurface divergence models. Furthermore, the linear portionof the rms vertical velocity corresponds to the viscousboundary layer, which following the estimation from Brum-ley and Jirka,41 should range between z/zs 0.4 and 0.6(where zs is the distance from the center of the stroke towardthe water surface, or the average grid depth). In this study, itcorresponds to z/zs 0.4. The linear regression R is greaterthan 0.99. Values of the rms velocities at the surface andalong the region of linear variation are important in different

models that predict K. The values of the rms velocitiesu02

p(horizontal rms value at the surface) and D

w02

p(difference

between the extreme values of the vertical rms values in the linear

region) are shown in Table 2, together with K (the distance alongwhich the linear evolution of the vertical rms velocity holds) and b,whichwas calculated directly from b D

w02

p .K.

Nondimensional rms proles of the horizontal and verticalvelocity uctuations are shown in Figure 5. The measured

rms velocities are scaled with their values at z 28 mm,while z is normalized with L1 (L1 0.1zs). Brumley andJirka41 found that within a distance of 10% of zs from thesurface, the velocity uctuations are inuenced by the sur-face. In this study, it corresponds to the used value of28 mm. It can be observed that the normalized vertical uc-tuation proles superimpose along all the depth, whereas thehorizontal uctuation proles are differently inuenced bythe surface, spreading out close to the surface. The solidlines, in Figure 5, are theoretical proles proposed by Huntand Graham,42 obtained with parameters shown by Brumleyand Jirka.41 The shape of the measured and theoretical pro-les agrees well for the vertical velocities. However, for thehorizontal velocities, the observed behavior deviates fromthe theoretical curves close to the surface, with the experi-mental velocity values being lower than the theoretical val-ues. The results of Brumley and Jirka41 and Herlina andJirka34 showed similar behaviors.

Evolution of integral length scales along the depth

Velocity Integral Length Scales. Length scales of turbu-lence are important to understand the change in shape of atypical eddy as it interacts with the free surface. To obtainthe integral length scale, the correlation functions must becalculated. For two dimensional data, the correlations func-tions in the horizontal and vertical directions follow thesame denition:

Rki r

vki

0x:vki0x r

vki x vki x r

; (15)

where vi0 is the fluctuation either of u or w (horizontal and

vertical velocity components), vi is the rms value either of u0 or

w0, i and k are the directions (x or z), and r is a distance vector.Once Ri is calculated, the correspondent length scale iscalculated as follows:

Lki

Z 10

Rki rdrk: (16)

Because there are two integral scales for each direction (xand z), a total integral length scale was calculated for eachdirection:

Lx 2Lxx 2 Lxz 2

q(17)

Lz 2Lzx 2 Lzz 2

q: (18)

The total horizontal and vertical integral length scales areplotted against the distance z in Figure 6, which shows thatthe vertical integral length scale decreases toward the inter-face, whereas the horizontal integral length scale increasestoward the interface. This presents eddies that atten outundergoing a reduction in their vertical extent and anincrease in their horizontal extent. This reduction-increasingbehavior is quantied, in this study, through factors in therange 1.72.7, which are close to the values of 2.03.0 foundby Handler et al.45

Figure 5. Vertical turbulence uctuations near the inter-face.

(a) Vertical uctuation and (b) horizontal uctuation.

AIChE Journal August 2010 Vol. 56, No. 8 Published on behalf of the AIChE DOI 10.1002/aic 2011

The integral length scales are not constant within theregion of measurement. Furthermore, there are two mainscales to consider: horizontal and vertical. Therefore, it isdifcult to dene a characteristic value for a length macro-scale close to the surface. Within the region shown in Fig-ure 6, the vertical scale, L[z], for each run has a mean valuebetween 2 and 3 cm, whereas the horizontal scale, L[x], has amean value between 3 and 5 cm. As the effect of the surfaceon the horizontal scale is clearly shown (the abrupt distortionat the upper part of all runs shown in Figure 6), it seems rea-sonable to register the length just before the atten outeffect as an undisturbed characteristic horizontal lengthscale, Lu, as also adopted by Hunt and Graham

42 and Calmetand Maganudet.46 In this case, this characteristic value fallswithin the interval from 3.4 to 4.0 cm (Figure 6). Addition-ally, the width of the attened eddies at the surface may beused as a measure of the disturbed horizontal length scale,Ld. For the present set of experiments, this disturbedlength falls within the interval from 4.9 to 6.7 cm. Figure 6also shows straight lines tting the points of the disturbedregions, which allows linear extrapolation to the surface. Allvalues of undisturbed and disturbed length scales are shownin Table 2. Brumley and Jirka41 suggested, on the basis ofthe Hunt and Graham42 theory, that a purely kinematic effectof the surface is expected to extend over one integral lengthscale, L1 0.1zs from the surface, which corresponds to thesurface inuenced layer. The same undisturbed lengthscale was also adopted by Calmet and Magnaudet.46 In thepresent set of experiments, the average grid depth was main-tained constant at 28 cm, which results in L1 2.8 cm. Itis interesting to see that the undisturbed horizontal lengthscale oscillates between the limits of 3.32 and 3.92 cm(mean value of 3.56 cm), which may be compared to L1and to the values of (0.80.96)L1 found by Calmet andMagnaudet.46

The macroscales permit calculation of the turbulenceReynolds number at the surface, obtained with the horizon-tal rms velocity scale and the disturbed length scale

ReT u02

pLd=v

. Figure 7 shows that ReT is proportional

to the equipment Reynolds number, Re, permitting the useof the latter as a measure of turbulence at the interface,where Re 47ReT on average.The linkage of surface turbulence to the bulk turbulence

of course depends on the equipment used to generate turbu-lence, a reason to directly measure supercial parameters toquantify mass transfer. The proportionality presented in Fig-ure 7, however, shows that oscillating grid systems allow agood control of the supercial turbulence conditions.

Concentration Integral Time Scale. The concentrationtime scale is a measure of the average time during which theconcentration persists with the same characteristics at a pointof the space. The correlation function of concentration withrespect of time is dened as follows:

Figure 7. Proportionality between ReT and Re.

Figure 8. Proles of concentration time scale (z1 5 z/dD).

Figure 6. Proles of velocity integral length scales,with linear curve t of the expansion of L[x]

near the surface.

(a) Re 5791, (b) Re 8687, (c) Re 11,582, and (d) Re 14,478.

2012 DOI 10.1002/aic Published on behalf of the AIChE August 2010 Vol. 56, No. 8 AIChE Journal

Rcn ci0t ci0t ncit cit n ; (19)

where c0 is the fluctuation of concentration, c is the rmsconcentration, and n is a delay time. Once Rc is calculated, thecorrespondent time scale is calculated as follows:

Tz Z H0

Rctdnjz; (20)

where H is the measurement time. Rc always reached thevalue 0 before attaining H. The measured concentrationfields permit calculation of the time macroscale for eachposition along the z axis, which generates a function T(z), asshown by Eq. 20. Figure 8 shows T(z) along the normalizeddistance z z/dD. All profiles indicate a high andapproximately constant T(z) in most of the diffusivesublayer. Figure 8 also shows a horizontal line, whichindicates the position z 0.80dD. It can be seen that theintegral time scale is relatively constant above this line,which corresponds to the mean value of dr. Figure 8incorporates three independent means of obtaining lengthscales related to the diffusive phenomena: (a) the verticalaxis is presented in nondimensional form, using dD, obtained

from the mean concentration profiles; (b) the horizontal lineshows the mean value of dr, obtained from the rms profilesfor concentration fluctuations; and (c) the length of theregion, where T is nearly constant, is close to both dD and dr,and is obtained integrating autocorrelation functions (Eqs.19 and 20). The present measurements therefore show awell-defined layer, consistently reproduced using the inter-section of the concentration gradient, dD, the peak of therelative concentrations fluctuations, dr, and the integrationof the autocorrelation functions. Farther from the surface,the concentration time scale tends to decrease.The time scale close to the surface decreases from 9.24 to

1.89 s for Reynolds numbers increasing in the range from5791 to 11,582. An unexpected behavior is observed for theReynolds number of 14,478, which shows a time scale of6.08 s close to the surface. Considering the trend obtainedfor the lower Reynolds numbers, a value less than 1.89 swas expected for Re 14,478. As the uctuating velocitiesfor this Reynolds number show a coherent behavior of therms values (Figure 4), a possible reason would be changes inthe conditions of the mean ows, such as secondary circula-tion. To observe the mean ows in detail, a larger region ofmeasurement would be necessary. All values of the timemacroscales are shown in Table 2.

Figure 9. Sequence of instantaneous uctuating oxygen concentration elds and related uctuating vertical veloc-ity elds.

The images were recorded at 4 Hz. In this gure, however, only every second recorded image is shown, thus the time interval betweenthe shown images is 0.50 s. The reported elds are typically 12 mm 6 mm.

AIChE Journal August 2010 Vol. 56, No. 8 Published on behalf of the AIChE DOI 10.1002/aic 2013

Simultaneous visualization of velocity andconcentration elds

Figure 9 presents examples of instantaneous concentrationand vertical velocity elds, allowing visualization of thetransport of oxygen at the water side of the interface. Theorigin of the vertical axis, z, is located at the water surface.Note that portions of the surface layer with high oxygenconcentration are peeled off by a turbulent structure andthen transported into the bulk region. The resolution of theimages for the concentration elds is better than that for thevelocity elds. Notwithstanding this resolution difference, itis seen that the evolution of the concentration elds isclosely related to the vertical velocity elds. In other words,the observed distortions of the concentration spots arecoupled with the observed movements of the liquid.The concentration elds evolution in Figure 9 indicates a

clear horizontal movement directed from right to left, whichcoincides with the direction indicated by the velocity vectors.It is known that the vertical velocity is zero at the surface, butthe resolution used for the velocity elds imposed a higher ve-locity value at the surface. Considering pictures c, f, h, and i,the vertical velocity components at the surface (z 0) aremainly directed from the surface to the bottom. Pictures a, b,c, f, h, and i show a clear descending movement in the leftside, also seen in the lower left quarter of the remaining pic-tures, which transports the parcels of liquid with higher oxygenconcentration deeper into the liquid. Pictures g, h, and i showan ascendant movement on the right side, which implies in athinning of the surface region with higher oxygen concentra-tion, as expected. It is seen that the general form of the con-centration records, with a deeper penetration into the liquid onthe left side of the pictures, and a horizontal movement fromright to left, is consistent with the velocity elds recorded atthe same time. However, a complete description of the owand the concentration elds can only be obtained with a three-dimensional visualization of the subsupercial region.

Using measured parameters to quantify thegas transfer velocity

Combining Eqs. 1 and 2, the mass transfer coefcient K(or transfer velocity) can be directly estimated in this study,following the equation

Kref D dCdz

z0

CS CB D

dD: (21)

The gradient, dC/dz|z 0, is a linear t based on the maxi-mum slope of the concentration prole near the interface.The transfer velocities estimated using Eq. 21 are shown incolumn 1 of Table 3 (reference K). The measured turbulenceparameters (Table 2) were used to compare predictions of Kfrom different models with the reference K, from Eq. 21.The molecular diffusivity D and water kinematic viscosity mat 26.5C were adopted as 2.47 109 and 8.63 107 m2/s, respectively (Janzen33).The results of b (Table 2) and Kref (column 1, Table 3)

permit comparison of the present data with the surface diver-gence models. Figure 10 follows the graphs of Xu et al.,18

where K(Db)1/2 is plotted against b. The shadowed areascorrespond to the data of McKenna and McGillis22 and thoseused by Xu et al.18 The data of Tamburrino and Gulliver,17

Tamburrino et al.,19 Tamburrino and Aravena,47 Herlina andJirka,34 and Xu et al.48 are plotted together with the presentresults. It can be observed that the data from Herlina andJirka34 are comparable to the present data, although thetransfer velocity was determined through long-time oxygenconcentration measurements in the bulk liquid. For the pres-ent set of data, the mean coefcient of Eq. 7 is 0.26, whichis close to the coefcient 0.22 of Law and Khoo.20 The pre-dictions obtained with Eq. 7 are shown in column 6 of Table3. As Figure 4a indicates the presence of viscous effects

Table 3. Predictions of the Mass Transfer Velocity (m/s)

Column 1 2 3 4 5 6 7

Re Kref, Eq. 21

Film model,Eq. 3 using

Penetration-renewal model

(Eq. 4 using s T1,after Daviesand Lozano11)

Large eddymodel, Eq. 5

Divergence model using

dc dr Eq. 7 Eq. 8, n 0.6135791 5.58* 1.81 6.05 16.3 7.77 4.23 4.978687 6.54 2.16 7.95 22.1 12.3 5.68 6.6611,582 6.55 2.24 7.95 36.2 12.8 5.58 6.5414,478 7.23 2.46 8.45 20.2 15.9 6.74 7.90

*Mass transfer velocity values must be multiplied by 106.

Figure 10. K(Db)21/2 plotted against b, evidencing dataobtained for the lower range of b.

2014 DOI 10.1002/aic Published on behalf of the AIChE August 2010 Vol. 56, No. 8 AIChE Journal

near the surface, the observed maintenance of values closeto predictions of Eq. 7 suggests that the contamination of thesurface is slight, in the sense described by Hasegawa andKasagi.23 Equation 8 (McKenna and McGillis22) needs thevalue of n to permit predictions. The use of Eq. 8 alsoimplies that the experimental value of n must be betweenthe limits of 1/2 and 2/3. A best t procedure was followed(least square method) between the predictions of Eq. 8 andthe reference K, and the value n 0.613 was obtained,which satises the suggested limits. The exponent n 0.613found to t Eq. 8 suggests that the surface is contaminated,especially when compared with the results of McKenna andMcGillis.22 This is consistent with the reduction of the hori-zontal rms velocity displayed in Figures 4 and 5, and is apossible reason for the lower values of the rms concentrationnormalized maximum presented in Figure 3, when comparedwith the results of Magnaudet and Calmet.39

The predictions of Eq. 8 using the best t value of n areshown in column 7 of Table 3. Figure 11 shows measuredand predicted results of K, using the divergence models.It is known that dc dc(D), which would change the

exponent of D in the stagnant lm model and introduce newvariables. However, measured values of vertical scales forthe concentration eld already consider the effects of D intheir results, so the lm model may be used with the differ-ent vertical scales measured here. Columns 2 and 3 of Table3 show the results obtained for K using dc and dr, respec-tively. Column 2 evidences the fact that the usual denitionof boundary layers (like Eq. 12) overestimates the lmthickness used in Eq. 3. Column 3 shows agreement withcolumn 1, a consequence of the similarity between dr anddD. Considering Eq. 4 and the Davies and Lozano

11 model(time macroscale at the surface replaces s1), the values ofK reported in column 5 of Table 3 are obtained. The meas-ured time scales are too small to t the reference K valueswithout a proportionality coefcient. A similar conclusion

was also reported by Davies and Lozano.11 Considering Eq.5 with the horizontal scales of velocity and length right atthe surface (Calmet and Magnaudet10), the values of Kreported in column 6 are obtained. In this case, Eq. 5 alreadypredicts the need of a proportionality coefcient.If predicted and measured K are proportional, the ratio K/

Kref will be constant for the different agitation conditions.Figure 12 shows K/Kref obtained from Table 3 against theReynolds number. Different behaviors are observed for dif-ferent predictions. The predictions of Eq. 4 oscillatestrongly. The predictions of Eq. 5 behave smoother thanthose of Eq. 4, but K/Kref increases signicantly with theReynolds number, which implies that probably not all detailsof the mass transfer phenomena are captured by this model.The predictions of Eq. 3 and the length scales dc and drshow K/Kref nearly constant, suggesting that both scales canbe used to predict K, provided that the proportionality coef-cient is known. The predictions obtained with Eqs. 8 and 9,the surface divergence models, show small variations in theratio K/Kref with the Reynolds number, also pointing to theconvenience of their use to quantify K. In this case, Eq. 8was already adjusted to the data (n 0.613), which leads toK/Kref 1. The mean values of K/Kref obtained with thepresent set of data are given in Table 4, which may be usedin the respective predictive equations of K.

Figure 11. Comparison of the predictions obtainedfrom Eqs. 8 (McKenna and McGillis22) and 9(Law and Khoo20) with measured data.

Equation 8 was used with n 0.613.

Figure 12. Ratio K/Kref against Re, showing that pre-dictions which consider the lm thicknessand the supercial divergence maintain aconstant proportionality with Kref.

Equation 8 was used with n 0.613.

Table 4. Mean Values of K/Kref

Equation referred in thetext and in Figure 12 Observed K/Kref

Eq. 3 (dc) 0.335Eq. 3 (dr) 1.176Eq. 4 (s T1) No constant

K/Kref observedEq. 5 No constant

K/Kref observedEq. 8 (n 0.613) 1.0Eq. 9 0.864

AIChE Journal August 2010 Vol. 56, No. 8 Published on behalf of the AIChE DOI 10.1002/aic 2015

Conclusions

Simultaneous PIV and LIF measurements were used toinvestigate the interaction between airwater gas transfer andturbulence generated in an oscillating-grid tank. The rms ve-locity proles showed the usual damping of the verticalmotion in interfacial regions, while the horizontal motiontends to increase. In this study, viscous effects wereobserved close to the surface, also partially damping the hor-izontal velocity uctuations. This indicates that the surfacewas lightly contaminated by some surfactant. The velocityintegral length scales reveal a signicant attening of a typi-cal eddy near the airwater interface. This attening is evi-denced by a notably larger horizontal scale and a commen-surately smaller vertical scale.Considering the concentration measurements, vertical

mean and rms concentration proles were obtained from theconcentration elds. The peak of the concentration uctua-tion intensity proles could be very well registered. The con-centration time scale proles showed an approximately con-stant value of the time macroscale in the diffusive sublayer.It was shown that the diffusive sublayer may be computedfollowing different ways: using the mean concentrationprole, the prole for the rms value of the concentrationuctuations, or the integral time scale, obtaining similarthicknesses.A comparison was made between mass transfer velocities

obtained directly from its denition (Eq. 21, used as refer-ence) and using different models. The transfer velocitieswere quantied using turbulence parameters from the simul-taneous concentration and velocity measurements. The sur-face divergence model was considered in more detail, withmeasured b values ranging from 0.15 to 0.38 (lower rangeof b). It is shown that the predictions of the equation of Lawand Khoo20 are close to the reference value of K; while avalue of n 0.613 permits the use of the equation ofMcKenna and McGillis,22 where n satises the limits 1/2\ n\ 2/3. The comparisons show that, for the present experi-ments, predictions close to the reference mass transfer veloc-ity were obtained using either the lm model with the diffu-sive sublayer (dD, dr) or the surface divergence models. Theconstancy of K/Kref for the Reynolds number ranging from5791 to 14,478 points to the good predictions of these mod-els. From the obtained results, the contamination of the sur-face can be considered slight for the mass transfer predic-tions.

Acknowledgments

The authors are indebted to the Brazilian research support foundationsCAPES (trough process 2201/06-2), CNPq, and FAPESP, for the supportof different stages of this study, and to the German Science Founda-tion (DFG Grant No. Ji18/7-1).

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Manuscript received Dec. 8, 2008, and revision received Oct. 17, 2009.

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