Relationship Between Mass Transfer
Coefficient and Liquid Flow Velocity in
Heterogenous Biofilms Using
Microelectrodes and
Confocal Microscopy
Paul Stoodley,1 Shunong Yang,2 Hilary Lappin-Scott,1
Zbigniew Lewandowski3.
1University of Exeter, Department of Biological Sciences, Hatherly
Laboratories, Exeter, Devon EX4 4PS, United Kingdom; telephone: (44)
1392 264348; fax: (44) 1392 263700, e-mail: P.Stoodley@exeter.ac.uk
2NALCO Chemical Co., Napierville, Illinois
3Center for Biofilm Engineering, Montana State University,
Bozeman, Montana 59717-3980
Received 20 November 1996; accepted 10 May 1997
Abstract: The relationship between local mass transfer
coefficient and fluid velocity in heterogenous biofilms
was investigated by combining microelectrodes and con-
focal scanning laser microscopy (CSLM). The biofilms
were grown for up to 7 days and consisted of cell clusters
separated by interstitial channels. Mass transfer coeffi-
cient depth profiles were measured at specific locations
in the cell clusters and channels at average flow veloci-
ties of 2.3 and 4.0 cm/s. Liquid flow velocity profiles were
measured in the same locations using a particle tracking
technique. The velocity profiles showed that flow in the
open channel was laminar. There was no flow at the top
surface of the biofilm cell clusters but the mass transfer
coefficient was 0.01 cm/s. At the same depth in a biofilm
channel, the flow velocity was 0.3 cm/s and the mass
transfer coefficient was 0.017 cm/s. The mass transfer
coefficient profiles in the channels were not influenced
by the surrounding cell clusters. Local flow velocities
were correlated with local mass transfer coefficients us-
ing a semi-theoretical mass transfer equation. The rela-
tionship between the Sherwood number (Sh,) the Reyn-
olds number (Re,) and the Schmidt number (Sc) was
found using the experimental data to find the dimension-
less empirical constants (n1, n2, and m) in the equation
Sh = n1 + n2 Re
m Sc1/3. The values of the constants ranged
from 1.45 to 2.0 for n1, 0.22 to 0.28 for n2, and 0.21 to 0.60
for m. These values were similar to literature values for
mass transfer in porous media. The Sherwood number
for the entire flow cell was 10 when the bulk flow velocity
was 2.3 cm/s and 11 when the bulk flow velocity was 4.0
cm/s. © 1997 John Wiley & Sons, Inc. Biotechnol Bioeng 56:
681–688, 1997.
Keywords: biofilm; confocal scanning laser microscopy;
laminar flow; liquid flow velocity; mass transfer coeffi-
cient; microelectrodes; Reynolds number; Sherwood
number
INTRODUCTION
Although it has long been noted that biofilms may be het-
erogenous and have considerable surface roughness, they
were generally conceptually thought of, and mathematically
modeled, as planar structures (Picaloglou, et al. 1980; Sie-
grist and Gujer, 1985). These models assumed that diffusion
was the main mass transfer mechanism inside the biofilm,
convection was dominant in the bulk liquid outside the bio-
film, and that mass fluxes were one dimensional, i.e., per-
pendicular to the substratum (Rittman and Manem, 1992;
Siegrist and Gujer, 1985).
The recent application of confocal microscopy to biofilm
research has led to a rethinking of the impact of structural
heterogeneity on mass transfer and hydrodynamics (deBeer
et al., 1994a,b; Gjaltema et al., 1994; Lawrence et al., 1991;
Massol-Deya et al., 1994; Stoodley et al., 1994). It has been
demonstrated, using Nuclear Magnetic Resonance Imaging,
that water moves through channels in the biofilm (Lewan-
dowski et al., 1994). This phenomena was quantified and
the profiles of flow velocity measured using particle track-
ing in conjunction with confocal laser microscopy (deBeer
et al., 1994b; Stoodley et al., 1994). Water movement inside
the biofilm implies that the local mass transfer coefficient
(k) may vary spatially. This was demonstrated by Yang and
Lewandowski (1995) using a mass transfer coefficient mi-
croelectrode. They measured vertical and horizontal k pro-
files at specific locations in a biofilm by guiding the elec-
trode microscopically. They found that the k varied from
place to place in the biofilm, and concluded that this was
caused by the structural heterogeneity of the biofilm. They
also found a first-order relationship between local k in a
biofilm void and the average velocity (u), but the coefficient
of proportionality varied with depth.
Quantifying the relationship between k and u is an im-
portant step towards prediction of the macro-scale k based
Correspondence to: Paul Stoodley
Contract grant sponsor: National Science Foundation
Contract grant number: EEC-8907039
Contract grant sponsor: Montana State University
© 1997 John Wiley & Sons, Inc. CCC 0006-3592/97/060681-08
on microscale measurements. Attempts to describe the re-
lationship between k and u are not new in biofilm research
but are normally measured over whole systems without re-
gard for biofilm structure. Gantzer et al. (1988) found a
logarithmic relationship between k and Reynolds number
for chemical oxygen demand (COD) to an artificial stream-
bed biofilm using a modification of a mass transfer equation
for forced convection around a single sphere (Bird et al.,
1960; Welty et al., 1969):
Sh 4 n1 + n2 Rem Sc1/3 (1)
where Sh is the Sherwood number, Re is the Reynolds num-
ber, Sc is the Schmidt number, and n1, n2, and m are con-
stants. Similar forms of Equation (1) have been applied to
flow through packed beds (Kennedy and Lennox, 1997).
This mass transfer equation is particularly useful at low Re,
when the n1 term becomes significant and accounts for the
limiting Sh that occurs when there is no flow. A more de-
tailed review of mass transfer coefficient relationships can
be found in a publication by Kennedy and Lennox (1997).
The goal of this work was to relate the local mass transfer
coefficient to local liquid velocity in a heterogenous biofilm
under laminar flow conditions. To do this, we combined the
k microelectrode developed by Yang and Lewandowski
(1995) with the particle tracking technique developed by
Stoodley et al. (1994). This allowed us to compare the re-
lationship between the mass transfer coefficient with liquid
velocity on a scale relevant to the biofilm heterogeneity
with literature values determined over entire reactor sys-
tems.
MATERIALS AND METHODS
Biofilm Reactor System
The flow cell was a closed channel (1 cm wide, 1 cm deep,
and 45 cm long) with observation windows, 31 cm from the
entrance, on the top and bottom surfaces (Stoodley et al.,
1994). The windows were coverslips (60 × 24 mm), which
were sealed in place by a rubber gasket and aluminum
flange. The flow cell was placed in a recycle loop with a
mixing chamber which had nutrient and dilution water in-
fluent streams delivered by peristaltic pumps (Masterflex,
Cole-Parmer, Niles, IL). The mixing chamber was also aer-
ated and had an overflow effluent line. The volume (V) of
the mixing chamber and the recycle loop, including the flow
cell, was approximately 140 mL. The flow cell could be
placed on the stage of an inverted microscope (Olympus
IMT-2) attached to a Bio-Rad MRC600 confocal scanning
laser microscope (CSLM) without interrupting the flow
conditions. This arrangement allowed the biofilm to be ob-
served through the bottom window while at the same time,
the top coverslip could be removed for microelectrode in-
sertion (Fig. 1). The recycle flow rate (QR) was controlled
using two vane head pumps (Masterflex, Cole-Parmer,
Niles, IL), one at the inlet of the flow cell and the other at
the outlet. The flow rate was monitored with an in-line flow
meter (McMillan Flo-sensor model 101T supplied by Cole-
Parmer, Niles, IL) or by volumetric displacement. During
biofilm accumulation, the flow cell was operated as a closed
channel with a flow rate of 3.5 cm3/s (average flow velocity
(u(ave)) 4 QR/cross sectional area (CSA) 4 3.5 cm/s). Dur-
ing the microelectrode and velocity measurements, the cov-
erslip was removed and the flow cell became open channel.
Measurements were taken at QR of 1.4 and 2.4 cm3/s. At
these flow rates, the water depth was 0.6 cm so that u(ave)
were 2.3 and 4.0 cm/s, respectively.
The Re based on the channel geometry were 258 and 448
calculated using:
Re =
u~ave!l
y
(2)
where y was the measured kinematic viscosity of the elec-
trolyte solution [0.972271 × 10−6 m2/s at 20°C (Yang,
1995)] 1 was the characteristic length, which in this case,
was the hydraulic diameter of the flow cell based on the
wetted perimeter and cross-sectional area.
Nutrients were mixed with the dilution water in a ratio of
1:10 (0.7 mL/min:6.3 mL/min) for a final concentration of:
glucose 40 ppm, potassium phosphate monobasic (KH2PO4)
70 ppm, potassium phosphate dibasic (K2HPO4) 30 ppm,
ammonium sulfate [(NH4)2SO4] 10 ppm, and magnesium
sulfate (MgSO4 z 7H2O) 1 ppm. The total nutrient influent
flow rate (QN) was set at 7 mL/min to achieve a residence
time (u 4 V/QN) of 20 min so that suspended cells would be
washed out and biofilm growth favored.
The reactor and nutrient feed was sterilized by autoclav-
ing at 121°C for 15 min. The dilution water was tap water
Figure 1. Reactor system showing the arrangement of the microelec-
trode, the inverted microscope, and the flow cell. During biofilm growth,
the flow cell was operated as a closed channel and only the inlet pump
used. When the microelectrode measurements were conducted, the top
observation window of the flow cell was removed (as shown), and the
outlet pump maintained at a slightly higher flow rate than the inlet pump
to prevent overflowing. The dashed lines in the flow cell represent particle
tracks used for the velocity measurement.
682 BIOTECHNOLOGY AND BIOENGINEERING, VOL. 56, NO. 6, DECEMBER 20, 1997
sterilized with in line capsule filters (1.0 mm prefilter and
0.1 mm filter), chlorine was removed by sparging with air.
All experiments were performed at 20 ± 1°C.
Inoculum
Pseudomonas aeruginosa, Pseudomonas fluorescens, and
Klebsiella pneumoniae were used to inoculate the reactor.
The selection process for this consortium has been de-
scribed earlier (Stoodley et al., 1994). Frozen stock cultures
(1 mL) of each of the species were allowed to thaw, and
then injected into the mixing chamber through a septum.
The reactor was run as a batch culture for the first 24 h to
allow the cells to attach before switching to continuous
culture. Experiments were conducted on biofilms that were
between 5 and 7 days old.
Biofilm Thickness Measurement and
Depth Correction
The biofilm thickness (dimension in the plane perpendicular
to the substratum) is defined in this study as the distance
between the substratum and the peaks of the highest cell
clusters. Using this definition, the channels (void fraction)
that surround the cell clusters and streamers (biomass frac-
tion) are considered integral components of the biofilm
(Fig. 2). The height of the biofilm cell clusters were mea-
sured microscopically by focusing on the substratum (glass
coverslip), and then moving the stage a known distance with
a stepper motor until the surface of the cell cluster came into
focus. Calibration of the stage stepper motor was required
because the distance traveled by the stage was not the same
as the distance between corresponding focal planes in the
flow cell due to refraction effects of the air, glass, and water
interfaces. Two methods were used to make the calibration:
(1) the microscope was focused on the lower channel wall
then refocused on the upper channel wall (an actual distance
of 1 cm) using the stepper motor and the distance of travel
was recorded—the ratio of the actual distance to the trav-
eled distance was used to determine the correction factor;
and (2) the microscope was focused on the microelectrode
which was then moved up or down an assigned distance.
The microscope was next refocused on the microelectrode
and the distance traveled by the stage was noted. The first
method yielded a correction factor of 1.359, and the second
1.360. The distance traveled by the stage relative to the
glass substratum was multiplied by 1.36 to find the depth of
the focal plane in the flow cell. This was used to calculate
the biofilm thickness and determine the depth position for
the velocity profiles.
Measurement of Biofilm Surface Area Coverage
and Length Dimensions
Image analysis was used to measure biofilm length dimen-
sions (in the plane parallel to the substratum) and surface
coverage. Processing was done on a Macintosh 7200/90
computer using the public domain NIH-Image 1.59 pro-
gram (developed at the National Institutes of Health and
available from the Internet by anonymous FTP from zip-
py.nimh.nih.gov or a floppy disk from the National Tech-
nical Information Service, Springfield, Virginia, part num-
ber PB95-500195GEI). A threshold was applied so that the
biofilm cell clusters were black and the surrounding chan-
nels, white. The relative surface coverage of the biofilm was
the proportion of black to total area. At the magnification
used in this study, only the biofilm cell clusters, not the
single cells on the glass surface, were included in the mea-
surement. Length dimensions were measured using the
‘‘line tool’’ function. Length and area measurements were
calibrated using a 1mm graticule with 10 mm divisions (Ref.
# CS990, Graticules Ltd., Tonbridge, Kent, UK).
Mass Transfer Coefficient Measurements
The local k was measured in the flow cell with a microelec-
trode using the limiting current technique (LCT). The con-
struction of the microelectrode, the LCT, and the experi-
mental set-up as applied to biofilms has been described in
detail by Yang and Lewandowski (1995). The tip diameter
of each microelectrode was approximately 10 mm and was
accurately measured microscopically. Because the micro-
electrode was mobile, k could be measured at specific lo-
cations in the biofilm under microscopic guidance by
CSLM. Profiles of k in biofilm channels and clusters were
made using a stepper motor. The method required that a
redox couple was used in conjunction with the microelec-
trode. For these experiments, we used a reaction solution of
potassium ferricyanide (K3Fe(CN)6, 25 mM) to provide the
redox couple; Fe(CN)6+++/Fe(CN)6++, and potassium chlo-
ride (KCl, 0.5M) as a supporting electrolyte to suppress
electromigration effects. Potassium ferricyanide was chosen
because it is commonly used for electrochemical mass
transfer measurements and does not react with the biofilm
over the duration of the measurements (Selman and Tobias,
1978; Yang, 1995). Therefore, the ferricyanide concentra-
tion remains constant throughout the flow cell except at the
tip of the electrode where it is consumed. Just before the k
measurements were made, the nutrient solution was ex-
Figure 2. The biofilm thickness (lf) is defined for this study as the dis-
tance between the substratum and the peaks of the highest cell clusters. The
height (h) of the individual cell clusters is also indicated.
STOODLEY ET. AL.: MASS TRANSFER AND FLOW VELOCITY IN BIOFILMS 683
changed for the reaction solution which was added for 2 h
(6 residence times) so that the reaction solution was essen-
tially at 100% strength in the reactor. It has been shown that
biofilm cells remain viable during exposure to the solution
over the time taken to conduct the measurements (Yang and
Lewandowski, 1995). The microelectrode was polarized ca-
thodically with a counter calomel electrode (Model 13-620-
51, Fisher Scientific, Pittsburgh, PA) to the limiting current
value of −0.8 V. A Hewlett Packard 4140B multimeter was
used as voltage source and picoammeter to measure the
current (I) from the reduction of ferricyanide at the elec-
trode surface. The mass transfer coefficient can be found
from:
k =
I
nAFCb
(3)
in which n 4 the mole number of electrons transferred in
the reaction (in this case 1), A is the electrode sensing area,
F is Faraday’s constant, and Cb is the bulk concentration of
ferricyanide (Selman and Tobias, 1978).
The Sherwood number was calculated using:
Sh =
kl
Dc
(4)
in which the characteristic length, l, was taken as the diam-
eter of the microelectrode and Dc is the diffusion coefficient
of ferricyanide in KCl solution. A Dc value of 7.3 × 10−10
m2/s was reported by Konopka and McDuffie (1970) for
ferricyanide (0.61 to 6.36 mM) in 1M KCl at 25°C and this
value was used in this article, after correcting for the tem-
perature difference using the Einstein Stokes equation. The
Schmidt number was 1350.
Because of possible interference to the microelectrode
from the wall of the flow cell, measurements were not made
closer than one tip diameter from the wall.
Velocity Measurements
Liquid velocity measurements were made using a particle
tracking technique described more fully elsewhere (Stood-
ley et al., 1994). Neutral density fluorescent latex spheres
(Molecular Probes, Eugene, Oregon, density 20°C 4 1055
kg/m3, ex 580 nm/em 605 nm, diameter 4 0.282 mm, 1.7 ×
1012 spheres/mL) were added to the reactor to achieve a
final concentration of about 1 × 107 particles/mL. When
imaging with the CSLM, particles traveling across the field
of view appeared as dashed lines on the screen. By mea-
suring the length of these tracks (distance traveled) and the
time taken to create the track, the velocity can be calculated.
Velocity profiles in the flow cell were obtained by capturing
images at various focal depths by raising or lowering the
motorized stage. The actual depth was calculated from the
apparent depth using the correction factor described in the
preceding ‘‘Biofilm Thickness Measurement’’ section. The
local Reynolds number was calculated using Equation (2)
for correlation with local Sh. However, in this case the
microelectrode diameter was used for the characteristic
length so that the scaling factor would be the same for the
two dimensionless groups.
Combined Mass Transfer Coefficient and
Velocity Measurements
Because the particle tracking technique for measuring flow
velocity (u) is non-intrusive and did not interfere with the k
measurement, the two techniques were readily combined
using CSLM. The electrode tip could be positioned at a
specific point (in X,Y, and Z co-ordinates) in the biofilm
and viewed with the CSLM. The particle tracking technique
was then used to measure the liquid velocity at the same
point. To avoid interference from the microelectrode on the
liquid velocity, the k profile was measured before the u
profile, although both were measured at the same X–Y lo-
cation. It took between 2–5 min to measure a k profile and
about 5 min to capture the sequence of images required to
construct a u profile. Combined u and k profiles were mea-
sured in biofilm clusters and channels.
Although k and u profiles were measured at the same
X–Y location, the depth (Z) positions did not always coin-
cide, because they were determined by different methods,
i.e., the stepper motor for the k and the focus motor for the
u. To view the data graphically and correlate Sh with Re, a
k value and corresponding u value was required at the same
depth. To achieve this, the velocity profiles were fitted to a
curve with TableCurve 2D for Windows (Jandel Scientific,
San Rafael, CA), and the resulting polynomial equations
used to find the corresponding u at the same depth as the
measured k value.
Correlations and Statistics
The Sh was related to the Re according to Equation (1) using
a linearization method described by Gantzer et al. (1988) to
find the constants n1, n3, and m. The goodness of fit of the
experimental data to the model data was estimated by linear
regression. A sensitivity analysis of Equation (1) was per-
formed using a representative set of constants at a Re of 0.1
to determine the relative contributions of each constant to
the overall prediction value of the model. Each constant was
individually varied, and the absolute percent change of the
predicted value calculated.
RESULTS
Biofilm Accumulation
After 5 days, the biofilms consisted of cell clusters sepa-
rated by surrounding water channels (Fig. 3). The structure
was similar to that reported previously (deBeer et al.
1994a,b; Stoodley et al. 1994). The largest cell clusters were
close to hemispherical in shape and ranged in height from
175–225 mm. The diameter of the cell clusters at the base of
684 BIOTECHNOLOGY AND BIOENGINEERING, VOL. 56, NO. 6, DECEMBER 20, 1997
the biofilm ranged from 10 to 500 mm. The width of the
water channels were of the same order. The surface cover-
age of the cell clusters was 41.7% (standard error 4 5.6, n
4 3). Some single cells were observed on the substratum in
the biofilm void areas. There was no appreciable difference
in the structure of the 7 day-old biofilms. Replacement of
the nutrients with potassium ferricyanide and potassium
chloride solution for the k measurement had no affect on the
biofilm structure.
Mass Transfer Coefficient Profiles
u(ave) = 4.0 cm/s
Mass transfer coefficient profiles were measured in three
biofilm channels at this velocity. The profiles were para-
bolic as k decreased from around 0.027 cm/s at a depth of
1.5 mm to 0.012 cm/s at a depth of 10 mm (Fig. 4). The rate
of decrease of k became greater as the substratum was ap-
proached. The k at the top of the biofilm was approximately
0.020 cm/s. There was no perturbation to the profiles in the
region of the surrounding cell clusters.
u(ave) = 2.3 cm/s
At this flow velocity, profiles were measured in both chan-
nels and cell clusters. In the void, the profile was similar to
those measured at the higher flow rate, and went from ap-
proximately 0.021 cm/s at a depth of 2 mm to 0.014 cm/s at
a depth of 10 mm (Fig. 5a) in a smooth parabola. However,
in the cluster, the curve was perturbed near the surface of
the cell cluster, at a depth of 170 mm (Fig. 5b). The k at the
surface of the cluster was 0.01 cm/s, then just below the
surface of the cluster there was a slight increase to 0.012
cm/s at 150 mm, before again rapidly decreasing at the
substratum. The k at the same depth in the channel was
0.017 cm/s. The k decreased at a similar rate as the surface
of the cell cluster was approached as that when approaching
the substratum in a void.
Velocity Profiles
Velocity profiles were measured in cell clusters and void
areas of the biofilm. Above the cell cluster, there was a
Figure 4. Mass transfer coefficient profiles in three different biofilm
channels at a bulk liquid average flow velocity of 4.0 cm/s (flow cell Re 4
448). Data points for each set are distinguished by symbol type. The
position of the top of the surrounding biofilm cell clusters is indicated by
the dashed line. The substratum is at depth 4 0.
Figure 5. Velocity (u) and mass transfer coefficient (k) profiles measured
in a biofilm void (a), and a biofilm cluster (b), at an average flow rate of
2.3 mL/s (flow cell Re 4 250). In the void, both curves were parabolic but
the velocity profile was much steeper (note the different scales). The sur-
rounding biofilm clusters did not appear to effect the profile. In the cell
cluster, the velocity went to zero at the top of the cluster, which was at
depth of about 170 mm. The mass transfer coefficient increased slightly at
the top of the cluster, indicating that there was a slight increase in mixing
in this region.
Figure 3. Plan view of the biofilm composed of cell clusters (white) and
surrounding channels (black). The white dashes in the channels are tracks
made by fluorescent particles used for the velocity measurements (a rep-
resentative track is shown by the arrow). The bulk flow was from right to
left. Mass transfer coefficient and velocity profiles were measured in the
large void area in the center of the image. The scale bar is 100 mm (bottom
left corner).
STOODLEY ET. AL.: MASS TRANSFER AND FLOW VELOCITY IN BIOFILMS 685
shadow region extending for about 1 mm from the top of the
cluster in which the particles could not be seen. It is not
known why this occurred. However, we were able to con-
struct a complete profile at the edge of one of the clusters,
in addition, three profiles were constructed in void areas.
The velocity profiles were parabolic indicating laminar
flow, as expected from the Re based on the flow cell ge-
ometry (Figs. 5a,b). In the channels, there was liquid flow
down to the glass substratum. At u(ave) of 2.3 cm/s, local
flow in the void increased from 0 cm/s at the substratum to
about 0.3 cm/s at the top of the biofilm. The velocity then
continued to increase to about 3 cm/s at a depth of 3.5 mm.
There was a slight break in the profile at the top of the
biofilm, indicating a lower shear rate through the biofilm
than immediately outside it. The liquid flow went to zero at
the top surface of the cluster which was about 170 mm
above the substratum.
The three velocity profiles measured at u(ave) of 4.0 cm/s
(data not shown) used for correlation with the k profiles
were also parabolic, and were all fitted with TableCurve to
fifth-order polynomial equations with r2 correlations greater
than 0.998.
Relationship Between u and k
The curve fitting equations of the velocity profiles were
used to pair u and k values at equal depths in the flow cell.
For these correlations the dimensionless groups, the Reyn-
olds number, and Sherwood number were used respectively.
The curves were generally parabolic, and the rate of in-
crease in Sh decreased as Re increased. A representative
data set and the corresponding model curve are shown in
Figure 6. The predicted values of Sh from the model were
compared with the experimental data using linear regres-
sion. Table I shows the solution for the constants and the
results from the regression analysis for the three data sets.
A sensitivity analysis was performed on data set 1 to
determine the effect of each of the constants on the pre-
dicted Sh. Only one constant was varied at a time while the
other two constants were kept at their original values. The nl
constant had the largest influence on the resulting Sh, and on
m the least influence, although all three were of the same
order (Table II).
DISCUSSION
Mass Transfer Coefficient Depth Profiles
Measured in Biofilm Channels
The profiles ran smoothly from the bulk liquid to the sub-
stratum in the void areas of a flow cell colonized with a 200
mm thick biofilm. Because the surrounding biofilm clusters
did not have a significant influence on the profiles, it may be
assumed that there was also little effect on mixing in the
channels. In this case, one dimensional models may ad-
equately describe the mechanism of mass transfer. This
finding is consistent with earlier work by deBeer et al.
(1996), which found that biofilm channels only facilitated
the mass transfer of nutrients when the average velocity
approached 11.5 cm/s (Re 4 800).
Although we used a microelectrode to investigate the
effect of local heterogeneity on mass transfer for practical
purposes, it is more useful to determine an overall k to
describe substrate removal. We used the k measured by the
microelectrode near the surface of the cell clusters for com-
parison with other results. Although the microelectrode is
measuring k from the bulk fluid to the tip of the electrode,
it is reasonable to expect that the k measured near the sur-
face of the biofilm would be similar to that for a substrate
being consumed by a biofilm under transport limitation,
because the transport conditions would also be similar. The
k values that we measured at the top of the biofilm (1.0–2.0
× 10−2 cm/s at Re of 448) were about half an order of
Figure 6. Variation of local Sherwood number with local Reynolds num-
ber measured in a biofilm void at an average flow velocity of 4.0 cm/s. The
measured data (circles) were used to determine the constants in the model
equation indicated on the graph. The curve generated by the model is
shown by solid line. The inset shows greater detail of the intercept region.
Table I. Solutions for the constants n1, n2, and m in the mass transfer
equation [Eq. (1)] for the three data sets shown in Figure 4. The regression
correlation coefficient (r2) is shown with the corresponding standard error
(SE); there were 100 data points for each data set.
n1 n2 m r
2 SE
Data set 1 1.80 0.22 0.22 0.96 0.00159
Data set 2 1.45 0.27 0.21 0.96 0.00239
Data set 3 2.30 0.28 0.60 0.97 0.00153
Table II. Percent change on the predicted Sh when half (50%) and twice
(200%) the original value of the empirical constants n1, n2, and m were
varied in the mass transfer equation [Eq. (1)].
% of original
constant value n1 n2 m
50 27 23 13
200 55 45 18
686 BIOTECHNOLOGY AND BIOENGINEERING, VOL. 56, NO. 6, DECEMBER 20, 1997
magnitude greater than those measured by Siegrist and Gu-
jer (1987) (2.3 × 10−3 cm/s for oxygen in a biofilm colo-
nized trickling filter under laminar conditions) and Gantzer
et al. (1988) (0.58 × 10−3 cm/s for COD removal by a gravel
bed biofilm at a shear Re of 441). The reaction rate of
bacterial cells in a biofilm may be nutrient limited as well as
transport limited, whereas the reaction rate at the tip of the
electrode is only transport limited. As such, it may be ex-
pected that the k value from the microelectrode is a maxi-
mum for a certain flow condition. An overall Sh of 11 was
calculated using Equation (1) with the constants from data
set 1 at a Re of 448 based on the flow cell geometry. When
the Re was 258, the Sh was 10. From these Sh, an overall k
can be calculated, but it is not clear what should be used as
the characteristic length in this case. When the hydraulic
diameter of the flow cell is used as the characteristic length,
a k value of 7.3 × 10−5 cm/s is obtained. However, if the
electrode diameter is used, then k becomes 8.0 × 10−2 cm/s.
Clearly, care must be taken in the choice of the character-
istic length, and further experiments are required to inves-
tigate relationships between measurements made at the mi-
cro scale and overall mass transfer coefficients. Possibly, a
characteristic length based on channel width would be more
appropriate because nutrients are transported to the surface
of the cell clusters from the channels.
Combined Mass Transfer Coefficient and Velocity
Depth Profiles
We were able to measure k and u profiles almost simulta-
neously by combining the electrode and particle tracking
techniques. In addition, use of the confocal microscope al-
lowed us to measure these profiles at specific locations in
the biofilm. Although the velocity went to zero at the sur-
face of the cell clusters, there was a slight increase in the
local k. A similar increase at the cell cluster–bulk liquid
interface has been previously reported (Yang, 1995). Al-
though this small increase would probably not have any
significant effect on the overall k, it may indicate mixing in
this region, possibly from vibration of the biofilm. At higher
flow rates, this effect may become more pronounced due to
vortex shedding from the cell clusters with significant con-
sequences for momentum, heat and mass transfer (Lewan-
dowski and Stoodley, 1995). It is also possible that the
increase was an artifact induced by the contact of the mi-
croelectrode with the surface of the cell cluster. Inside the
cluster, the k continued to slowly decrease as the substratum
was approached, indicating that mass transfer inside the
cluster was similar to that in the surrounding channels. It is
possible that this mixing in the clusters was caused by con-
vection in microchannels, as suggested by Yang and Le-
wandowski (1995).
The curve fitting technique applied to the mass transfer
equation [Eq. (1)] determined that Sh was proportional to Re
raised to a power ranging from 0.21 to 0.60. Literature
values show that this exponent ranges from 0.33 at Re 4 0
to 0.66 as Re tends to infinity (Kennedy and Lennox, 1997).
The constant n2 ranged between 0.22 to 0.28. Literature
values of between 0.552 and 1.1 are reported for this con-
stant (Bird et al., 1960; Kennedy and Lennox, 1997). The
dimensionless n1 ranged from 1.45 to 2.30 with an average
value of 1.85. When there is no convection, Re 4 0 and Sh
4 n1. Our value is close to literature values which show that
when there is no flow, Sh 4 2 (Bird et al., 1960; Kennedy
and Lennox, 1997). The n1 constant had the greatest influ-
ence on the predicted Sh.
Regression analysis showed that there was a good corre-
lation between the mass transfer equation [Eq. (1)] using the
solved constants and the experimental data. The r2 values
for the three data sets were all greater than 0.96. The mass
transfer equation used in this study was initially formulated
to describe heat transfer from the surface of a sphere in an
infinite medium. However, it has since been adapted to
mass transfer using the Chilton–Colburn analogy (Bird et
al., 1960), and has been applied to low Re flow through
packed beds (Kennedy and Lennox, 1997). Heterogenous
biofilms have similarities to porous media (the channels
may be considered the void fraction), and water flow
through the biofilm will often be at very low Reynolds
numbers (the characteristic length of the channels is small
as is the liquid velocity because the biofilm is usually in the
hydrodynamic boundary layer). Our data indicate that Equa-
tion (1) may also be successfully applied to describe mass
transfer processes occurring in heterogenous biofilms.
CONCLUSIONS
1. Measurement of mass transfer coefficient by microelec-
trode can be combined with the particle tracking tech-
nique and CSLM to study the relationships between
mass transfer, liquid convective flow, and biofilm struc-
ture on the microscale.
2. The velocity and mass transfer coefficient profiles were
parabolic, approaching a plateau in the bulk flow and
becoming asymptotic to zero at the wall.
3. At the applied flow velocities the heterogenous biofilm
structure had no significant effect on the local mass
transfer coefficient in the biofilm channels.
4. The local Sherwood number could be related to the local
Reynolds number using the equation Sh 4 n1 + n2 Rem
Sc1/3 where n1, n2, and m are constants found from lin-
earization curve fitting.
From Montana State University, we thank Fuhu Xia and Kjetil
Rasmussen for their experimental help, and Peg Dirckx for edi-
torial assistance. From Exeter University, we thank John Boyle
and Mick Wheelan for their technical advice.
NOMENCLATURE
Cs concentration of reacting electrolyte at the electrode surface
(Mole/m3)
d depth (L)
DC diffusion coefficient of ferricyanide in KCl solution (L2/T)
De effective diffusion coefficient (L2/T)
STOODLEY ET. AL.: MASS TRANSFER AND FLOW VELOCITY IN BIOFILMS 687
Dw diffusion coefficient in water (L2/T)
F Faraday constant (9600 coulombs/mole)
h height (L)
I current (ampere or coulombs/second)
k mass transfer coefficient (L/T)
l characteristic length (L)
lf biofilm thickness
m empirical constant (dimensionless)
n number of moles of electrons transferred (dimensionless)
n1,n2 empirical constants (dimensionless)
QN nutrient flow rate (L3/T)
QR recycle flow rate (L3/T)
r2 regression correlation coefficient (dimensionless)
Re Reynolds number (dimensionless)
Sc Schmidt number (dimensionless)
Sh Sherwood number (dimensionless)
u flow velocity (L/T)
u(ave) average flow velocity (L/T)
V volume (L3)
u hydraulic residence time (T)
y kinematic viscosity (L2/T)
References
Bird, R. B., Stewart, W. E., Lightfoot, E. N. 1960. Transport phenomena,
pp. 407–415, 642–648. Wiley, New York.
deBeer, D., Stoodley, P., Roe, F., Lewandowski, Z. 1994a. Effects of
biofilm structures on oxygen distribution and mass transfer. Biotech-
nol. Bioeng. 43: 1131–1138.
deBeer, D., Stoodley, P., Lewandowski, Z. 1994b. Liquid flow in het-
erogenous biofilms. Biotechnol. Bioeng. 44: 636–641.
deBeer, D., Stoodley, P., Lewandowski, Z. 1996. Liquid flow and mass
transfer in heterogeneous biofilms. Water Res. 30: 2761–2765.
Gantzer, C. J., Rittman, B. E., Herricks, E. E. 1988. Mass transfer to
streambed biofilms. Water Res. 22: 709–722.
Gjaltema, A., Arts, P. A. M., van Loosdrecht, M. C. M., Kuenen, J. G.,
Heijnen, J. J. 1994. Heterogeneity of biofilms in rotating annular re-
actors: Occurrence, structure, and consequences. Biotechnol. Bioeng.
44: 192–204.
Kennedy, C. A., Lennox, W. C. 1997. A pore-scale investigation of mass
transport from dissolving DNAPL droplets. Journal of Contam. Hy-
drol. 24: 221–246.
Konopka, S., McDuffie, B. 1970. Diffusion coefficients of ferri-and fer-
rocyanide ions in aqueous media, using twin-electrode thin-layer elec-
trochemistry. Anal. Chem. 42: 1741–1746.
Lawrence, J. R., Korber, D. R., Hoyle, B. D., Costerton, J. W., Caldwell,
D. E. 1991. Optical sectioning of microbial biofilms. J. Bacteriol. 173:
6558–6567.
Lewandowski, Z., Stoodley, P., Altobelli, S., Fukushima, E. 1994. Hydro-
dynamics and kinetics in biofilm systems—Recent advances and new
problems. Water Sci. Technol. 29: 223–229.
Lewandowski, Z., Stoodley, P. 1995. Flow induced vibrations, drag force,
and pressure drop in conduits covered with biofilm. Water Sci. Tech-
nol. 32: 19–26.
Massol-Deya, A. A., Whallon, J., Hickey, R. F., Tiedje, J. M. 1994. Chan-
nel structures in aerobic biofilms of fixed-film reactors treating con-
taminated groundwater. Appl. Environ. Microbiol. 61: 769–777.
Picologlou, B. F., Zelver, N., Characklis, W. G. 1980. Biofilm growth and
hydraulic performance. J. Hydraul. Div. Am. Soc. Civ. Eng. 106:
733–746.
Rittman, B. E., Manem, J. A. 1992. Development and experimental evalu-
ation of a steady state, multispecies biofilm model. Biotechnol. Bio-
eng. 38: 914–922.
Selman, J. R., Tobias, C. W. 1978. Mass transfer measurements by the
limiting-current technique. pp. 211–315. In: T. B. Drew, G. R. Coke-
let, J. W. Hoopes, and T. Vermeulen (eds.). Advances in chemical
engineering, vol 10. Academic Press, New York.
Siegrist, H., Gujer, W. 1985. Mass transfer mechanisms in a heterotrophic
biofilm. Water Res. 19: 1369–1378.
Siegrist, H., Gujer, W. 1987. Demonstration of mass transfer and pH ef-
fects in a nitrifying biofilm. Water Res. 21: 1481–1487.
Stoodley, P., deBeer, D., Lewandowski, Z. 1994. Liquid flow in biofilm
systems. App. Env. Micro. 60: 2711–2716.
Welty, J. R., Wilson, R. E., Wicks, C. E. 1969. Fundamentals of momen-
tum, heat and mass transfer, pp. 578–591. Wiley, New York.
Yang, S. 1995. Microelectrode measurement of local mass transfer coef-
ficient in biofilms. M.S. thesis, Montana State University, Bozeman,
MT.
Yang, S., Lewandowski, Z. 1995. Measurement of local mass transfer
coefficient in biofilms. Biotech. Bioeng. 48: 737–744.
688 BIOTECHNOLOGY AND BIOENGINEERING, VOL. 56, NO. 6, DECEMBER 20, 1997