-
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 [email protected].
VVC 2009 American Institute of Chemical Engineers
AIChE Journal 2005August 2010 Vol. 56, No. 8
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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
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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
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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).
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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
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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
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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
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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.
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2010 Vol. 56, No. 8 AIChE Journal
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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
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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
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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
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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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the AIChE DOI 10.1002/aic 2017