APPLIED AND ENVIRONMENTAL MICROBIOLOGY, Feb. 1976, p. 286-293Copyright © 1976 American Society for Microbiology

Vol. 31, No. 2Printed in U.S.A.

Kinetics of Growth and Substrate Uptake in a BiologicalFilm System

ENRIQUE J. LA MOTFADepartment of Civil Engineering, University of Miami, Coral Gables, Florida 33124

Received for publication 25 August 1975

The rates of growth and substrate uptake in a biological film continuous-flowreactor were studied. The experiments were performed with high fluid velocitiesto bring the reactor operation to the reaction-controlled regime, thus avoidingexternal diffusional resistances. The glucose uptake experiments were per-formed with small film thicknesses so that full substrate penetration within theentire film thickness could be obtained. In this way, the catalyst effectivenessfactor was 1.0 and the observed rate was the true, or intrinsic, rate. The resultsof the experiments indicate that both the intrinsic rate of substrate uptake andthe rate of film growth are independent of the substrate concentration remain-ing in the reactor (zero-order reactions). However, the value of the initialsubstrate concentration when the film is in the early stages ofgrowth defines themagnitude of both the rate of uptake and growth. This effect of the initialsubstrate concentration follows a saturation-function pattern.

Most of the research carried out in the fieldsof applied biology and environmental engineer-ing concerning the kinetics of substrate re-moval by microorganisms has been directed tothe case of suspensions of organisms in batchand continuous cultures. The bulk of the exist-ing literature assumes that the microbial cul-ture is a homogeneous suspension, i.e., that thesubstrate consumption process is a homogene-ous reaction. However, in recent years severalinvestigators (8; C. R. Baillod, Ph.D. thesis,Univ. of Wisconsin, Madison, 1968; W. Gulev-ich, Ph.D. thesis, Johns Hopkins Univ., Balti-more, 1967; J. A. Mueller, Ph.D. thesis, Univ.of Wisconsin, Madison, 1966) have demon-strated that mass transfer resistances play animportant role in the kinetics of biological oxi-dation of organic matter.

In the case of biological films, it has beenshown that the substrate consumption reactionis affected by both external and internal diffu-sion. Several investigators (6; 9; Gulevich, the-sis, 1967; E. J. LaMotta, submitted for publica-tion, 1975) have demonstrated that fluid veloc-ity has a positive effect on the rate of substrateuptake. As presented elsewhere (LaMotta, sub-mitted for publication), the uptake rate can beincreased by increasing the fluid velocity, untilthe process becomes reaction-controlled; in thisregime, the rate is no longer affected by theliquid velocity.The effect of film thickness on the substrate

uptake rate has also been analyzed in the liter-ature (9; 14; R. C. Hoehn, Ph.D. thesis, Univ. of

Missouri, Columbia, 1970; LaMotta, submittedfor publication). LaMotta (submitted for publi-cation) has shown that, unless there is com-plete substrate penetration within the biologi-cal film and external diffusional resistanceshave been eliminated, the true rate cannot beobserved. These conditions can be satisfied byintroducing a sufficiently high fluid velocityand by dealing with small film thickness.With regard to the order of the substrate

uptake reaction, there are several differentopinions that can be found in the publishedliterature. However, most of the rate expres-sions in current use are of the zero-order type orof the Monod type.The evidence presented in the literature sup-

ports the hypothesis of zero-order uptake ratesby microorganisms. The investigation de-scribed herein also demonstrates that zero-or-der kinetics adequately explains the substraterate by biological films.The classic studies of Monod (11) on the

growth of pure batch cultures of Bacillus colihave been used extensively in the fields of ap-plied biology, fermentation engineering, andenvironmental engineering. The resultingequation,

A= (/Amax) +S

where , is the specific growth rate, /umax is theasymptote of the curve, K, is the numericalconstant, and S is the substrate concentrationat time t in a batch reactor, relates the specific

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growth rate and the concentration of substrateremaining in a batch system, when substrate Sis the limiting nutrient.Although Monod chose an equation of the

form of equation 1 because it was " ... bothconvenient and logical to adopt a hyperbolicequation ... similar to an adsorption isothermor to the Michaelis equation" (11), S. Ghosh(Ph.D. thesis, Georgia Institute of Technology,Atlanta, 1969) and F. E. Stratton (Ph.D. thesis,Stanford Univ., Stanford, Calif., 1966) have de-rived it from considerations of the Michaelis-Menten relationship for the rate of substrate-enzyme reaction.A massive experimental effort has been un-

dertaken by Gaudy and co-workers (2-5, 13, 14)to test the applicability of the equation ofMonod to both batch and continuous cultures ofmicroorganisms. Their results, obtained withboth pure and heterogeneous batch cultures ofmicroorganisms, indicate that ,u is controlled inbatch systems by the initial substrate concen-tration, S,,, and that the effect of S,, on ,u can bedescribed by an expression of the form of equa-tion 1 in which S (instantaneous) is replaced byS,, (cf. reference 3), as indicated in equation 2,

so=max K + (2)

This behavior was confirmed in several testswith a wide variety of substrates (includingsewage) and for both pure cultures and hetero-geneous populations.With regard to the effect of substrate concen-

tration on the rates of film growth and sub-strate uptake in biological film systems, simi-lar considerations to those discussed above canbe made. Regardless of the type of reactor inwhich the film is growing, film developmenttakes place under affixed conditions. This im-plies that if the reactor is a continuous-flowreactor and film growth is to start at time zero,the influent concentration of substrate will de-termine the number of permeation sites set bythe cells to attain a balanced growth accordingto the amount of substrate externally available.This fixed number of active sites for uptake willdefine the magnitude of the rates of substrateuptake and film growth. Film accumulationwill proceed at a constant rate determined bythe magnitude of the concentration of substratein the influent stream (a parameter that can bekept at a constant level), following a relation-ship similar to equation 2, i.e.,

SiM =/Zmax K, Si

in which AL is the specific film growth rate

BIOLOGICAL FILM SYSTEM 287

(reciprocal seconds), ftmax is the maximumfilm growth rate, K8 is the numerical constant,and Si is the influent substrate concentration.At each level of influent substrate concentra-

tion, the number of film active sites will remainconstant, yielding constant substrate uptakeand film growth rates. This number of filmactive sites will not be affected by decreases inreactor substrate concentration that resultfrom increasing biomass, until the concentra-tion of nutrients becomes too low and a declin-ing growth phase will take place.With these concepts in mind, the objectives of

this investigation were to (i) observe the intrin-sic rate of the substrate consumption reactionby biological films, (ii) observe the kinetics ofthe film accumulation process, and (iii) observethe affect of the influent substrate concentra-tion on the rates of substrate uptake and filmgrowth.

MATERIALS AND METHODSAll experiments were performed under aerobic

conditions in a continuous-flow laboratory reactorsimilar to that used by Kornegay and Andrews (9).It consisted of two concentric cylinders of differentdiameter, the inner one being stationary and theouter rotating about its axis. A glucose solutioncontaining sufficient mineral nutrients flowed con-tinuously through the annular space at a rate of 100ml/min. The concentration of glucose in the influentstream was given several values (from 2 to 200 mg/liter) to test the effect of this parameter on theintrinsic uptake rate and on the rate of film accumu-lation. Glucose concentration was determined ac-cording to a modified Glucostat procedure, which isdescribed in detail elsewhere (E. J. LaMotta, Ph.D.thesis, Univ. of North Carolina, Chapel Hill, 1974).

Film thickness was determined by means of tworemovable slides provided in the inner cylinder andfour plugs located at different levels on the rotatingcylinder wall. The thickness of the film adhering tothe slide (or the plug) was measured by placing thesample on the stage of a microscope provided with agraduated micrometer. By focusing first at the top ofthe film, and then at the bottom of the film, thedifference between the two readings gave the filmthickness.

Before the series of experiments was started, thereactor contents were inoculated with film samplescollected from one of the trickling filters of theChapel Hill waste water treatment plant, and con-tinuous operation was immediately started using alow flow rate (0.167 cm3/s). The operation of thesystem was continued in this fashion for severaldays; later, gradual increases in the flow rate anddecreases in the concentration of glucose were intro-duced until a flow of a 2.8 x 10-8 mol of glucosesolution per cm3 (5 mg/liter) at 1.667 cm3/s (100 ml/min) was established.The development of a stable film was a long proc-

ess; the first films obtained did not adhere well to

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the plastic walls and sloughed off as a whole. Theacclimation of the microorganisms to the adversehydraulic shearing conditions required approxi-mately 1 month, after which period the film wasextremely resistant to shear. Microscopy examina-tion of the film under these conditions revealed thepresence of free-swimming protozoa and a networkof filamentous microorganisms, both characteristicof biological films.

Measurement of the specific film growth rate.The rate of film growth on the walls of the cylinderscan be established by setting up a mass balance onthe microorganisms present in the reactor. Usingfilm volume as the parameter to measure film accu-mulation, it can be hypothesized that the rate of filmgrowth is directly proportional to film volume, i.e.,

-=,uV (4)

where ra is the rate of film accumulation (cubic centi-meters of film per second), ,u is the specific filmgrowth rate constant (reciprocal seconds), V is thefilm volume (A-8, [cubic centimeters]), 8 is the filmthickness (centimeters), and A is the film area(square centimeters).

Equation 4 assumes that film density is constant,i.e., that film mass is proportional to film volume;this assumption is supported by the observations ofKornegay and Andrews (9) from their experimentswith films using glucose as the only source of carbon.However, Hoehn and Ray (7) have recently publishedthe results of their experiments with films grown in amedium whose constituents included (per liter) 700mg of bacteriological nutrient broth, 150 mg of hy-drated sodium acetate, and 150 mg of glucose, point-ing out that film density (dry mass per unit volume) isnot constant throughout the entire film thicknessrange. Whether this discrepancy is due to differencesin the growth media used by these two investigatorsremains to be ascertained. As reported later, limitedmeasurements of concentration of adenosine 5'-tri-phosphate (ATP) per unit of film area, performed byme, revealed a direct proportionality between film-active biomass and film volume. Hence the assump-tion on which equation 4 is based is reasonable.The mass balance equation during the early stages

of film growth, i.e., when suspended microorganismscannot be detected in significant amounts, would be

Accumulation = input - output + growth

dVdt

-0- 0 + ,UV (5)

Integration of equation 5 yields

DV= ft{dt

whose solution, expressed in terms of film thickness,is

In-=,ut (6)

According to this equation, a plot of InS versus tshould yield a straight line with slope ,. This slope

can be easily determined by graphically measuringthe time required for the film to double its thickness,td, and using the following relationship derived fromEquation 6:

In 2 0.693td t(,

(7)

Once the degree of depletion of substrate and hy-draulic shear impose sufficient stress on the film as topreclude indefinite exponential growth, the film accu-mulation curve is expected to level off following thecharacteristic growth pattern observed in batch cul-tures of suspended microorganisms.

Measurement of the specific substrate uptake rate.As mentioned before, in measuring the substrate up-take rate, care must be taken to eliminate the mask-ing effects of diffusional resistances, both externaland internal. To measure the true reaction order, thesystem must be operated in the kinetic regime andthe film thickness must be maintained at levels thatyield an effectiveness factor close to 1.0. High rota-tional speeds (greater than 150 rpm) and thin filmsthat allowed complete substrate penetration through-out the entire film thickness resulted in the desiredconditions.

In these circumstances, a mass balance on sub-strate is

dS= SV, = QS, - QS,, - r,.-A-8 (8)

where VL is the volume of liquid solution in thereactor (cubic centimeters), Sb is the substrate con-centration in the bulk of the solution (moles/cubiccentimeter of solution), Si is the influent substrateconcentration (moles/cubic centimeter of solution), Seis the effluent substrate concentration (moles/cubiccentimeter of solution), t is time (seconds), r, is thetrue rate of substrate uptake per unit of film volume,(moles/second per cubic centimeter of film), A is thefilm surface area (square centimeters), 8 is film thick-ness (centimeters), and Q is the volumetric flow rateof substrate solution (cubic centimeters/second).

Under steady-state conditions, and with completemixing of the fluid (see LaMotta, thesis, 1974, for thedemonstration of the validity of these assumptions),equation 8 yields

r.= A (Si - Sd)A

(9)

Equation 9 allows the calculation of the true sub-strate uptake per unit of film volume from a knowl-edge of measurable parameters.The effect of influent substrate concentration on

the rates of growth and substrate uptake was testedby using eight different values of Si in correspond-ingly different runs.The relationship between the rate of film growth

and the substrate uptake rate can be expected tofollow the simple relation

Y (10)

288 LA MO'ITA

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BIOLOGICAL FILM SYSTEM 289

where Y is the yield coefficient (cubic centimeters offilm per mole).

RESULTSFilm growth pattern. Typical film growth

curves are shown in Fig. 1, demonstrating thatfilm accumulation throughout time follows anexponential pattern such as that described byequation 6. It can be seen that the slope of thestraight line portion of the curve, i.e., ,, in-creases as influent substrate concentration in-creases. This point is illustrated in Fig. 2, inwhich all the values of ft are plotted as afunction of Si. It can be seen that the observa-tions follow the pattern described by equation3, thus supporting the assertion made earlierthat the influent substrate concentration is the

0

%D

E

L.v

factor that determines the magnitude of therate of film growth.To find the numerical values of the parame-

ters, ftmax and K,, which would best fit equa-tion 3 to the observed data, two different statis-tical procedures can be used. First, a lineariza-tion of the graph can be obtained by using thetraditional Lineweaver-Burk plot (see Fig. 3),and a linear regression analysis would providethe values of the unknown parameters. One ofthe disadvantages of this method is that it issensitive to small errors in the low-concentra-tion range. Besides, the least-square regressionline using these scales will not necessarily pro-vide the best-fit curve in the plot f versus Si.This curve can be obtained using a second sta-tistical procedure, namely, a nonlinear regres-

Trme (hours)

FIG. 1. Typical film growth curves for several influent glucose concentration.

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I0.141

2 ZL

2

:E3t0,p

0tI41(f)

5

4

3

2

O~~~~~~~.-t max0

0-S

00

0 10 20 30 40 50 60 70 80 90 100 110

108 SI, moles / cm3

FIG. 2. Effect of influent glucose concentration onthe specific growth rate, ,u. The line of best fit wasfound by a nonlinear regression analysis yielding1m = 5.142 x 10-5 per s and K, = 4.395 x 10-8mol/cm 3.

.6 l la

_ .2 ~~~/Ima-x max S,

71.0

-.8

° .6 0

.4

2 0 _

.Ocr max I In|v R o E ~~~r,7 a 0

(I08 Si, moIes/cm3)-FIG. 3. Lineweaver-Burk plot of the same data as

Fig. 2. The linear regression analysis yields ftmax =

10.59 x 10-5 per s and K, = 15.98 x 10-8 mollCm3.

sion analysis on the data. Both types of analysiswere run by using computer programs preparedby the Department of Biostatistics of the Uni-versity of North Carolina at Chapel Hill. Giventhe form of the function, with its parametersand variables, both regression analysis pro-grams provide the numerical values of the pa-rameters, such that the sum of the squares ofthe residuals is a minimum. The linear regres-sion program is based on standard regressiontechniques, such as those described by Draperand Smith (1), whereas the nonlinear regres-sion analysis uses an iteration procedure simi-lar to that described by Nelder and Mead (12).The numerical values of the parameters, as

provided by this analysis, are the following:nonlinear regression, max = 5.142 x 10-5/sand K, = 4.395 x 10-8 (mol/cm3); linear regres-sion (Lineweaver-Burk linearization), fimax =

10.59 x 10-/s and K8 = 15.98 x 10-8 (mol/cm3).

Although more experimental data are neededin the intermediate-concentration range, ananalysis of Fig. 2 shows that a more realisticvalue of fma\ is that given by the nonlinearregression analysis. As stated before, the Line-weaver-Burk plot distorts the actual experi-mental observations, leading to erroneous esti-mates of the parameters. Therefore, the valuesprovided by the nonlinear regression analysiswill be adopted here.

Intrinsic glucose uptake rate. The values ofthe intrinsic glucose uptake rates obtainedfrom eight different influent glucose concentra-tions are presented in Fig. 4a through c; it isseen that the intrinsic rate does not show anydependence on the bulk-liquid glucose concen-tration, thus supporting the hypothesized zero-order intrinsic kinetics (r, = k ., a constant).

If attention is paid to the magnitude of theintrinsic uptake rate, k ,, it will be seen that itincreases as influent glucose concentration in-creases. This can be more clearly appreciated inFig. 5, which is a plot of k,. versus Si. Again, asaturation function, of the form of equation 11,i.e.,

k = k , K+ im-xK,+ Si (11)

provides a satisfactory fit to the data. The val-ues of k,m,ix and K8 were obtained by using thesame statistical procedures described in theprevious section, and they are: from nonlinearregression, k,. = 8.03 x 10-8 (mol/s per cm3 offilm) and K, = 1.11 X 10- (mol/cm3); fromlinear regression, k.M = 6.21 x 10-8 (mol/sper cm3 of film) and K8 = 6.8 x 10-8 (mol/cm3).Using the same considerations presented in

the case of equation 3, it may be concluded thatthe parameters estimated from the nonlinearregression analysis are more realistic thanthose estimated from linearization of the data(see Fig. 6). However, the numerical values ofk.x1X and K, cannot be considered reliable sincethe analysis was performed on limited experi-mental information.The validity of equation 10 is tested in Fig. 7,

which is a plot of the values of k,. versus ft. Itcan be seen that equation 10 provides a good fitthrough most of the points. The best-fit straightline forced through the origin of coordinates isk = 0.952 x 10- ft (mol/s per cm:' of film). Theyield coefficient, as given from equation 10,would be Y = (1.05 x 103) (cm3 of film)/(mol ofglucose). This value predicts reasonably wellthe thickness of film observed in this investiga-tion under each substrate concentration.Film activity measurements. As pointed out

previously, the theoretical zero-order model for

290 LA MOTTA

u ., z df.e . .0 .,r . t

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BIOLOGICAL FILM SYSTEM

2

0~~~~~~o ,7AP' *.

-kv =1.04 x-8 moles/secB - I.O4Xl0 cm3 film

6 - (a) Si=1.23x10-8 moles cm3

=2.2 mg/l4

0 2 3 4 5 6 7 8 9109 Se, moles/cm3

Effluent Glucose Concentration

O I I I

8 0

(b) Si= 22.5 x 10-8 moles/cm36 40.5mg/I

4 k 4.58 x I-8 moles/secv ~~~cm3 film

oaH1 I I

o 6 8 10 12 14 16 18 20 22108 Soe moles/cm3

Effluent Glucose Concentration

kv = 7.67 x 1 8 moles /sec8 acm3film

6 0

(c) Si = 1.11 x 10-6moles/cm34 = 200 mg /I

Ai| I I I I

291

.3 .4 .5 .6 .7 .8 .9 1.0 1.1

106Sel moles/cm3Effluent Glucose Concentration

FIG. 4. Zero-order relationship between intrinsic uptake and effluent glucose concentration.

glucose removal in a biological film system im-plies that substrate is depleted in the superfi-cial film layers; in this way, glucose should notbe detected at depths greater than the depth ofsubstrate penetration. However, it is possiblethat the substrate consumption reaction de-viates from zero-order kinetics at the low-con-centration range attained in the deeper layersof film; in this case the actual depth of penetra-tion will be different from the theoretical zero-

order critical depth, 8,. Furthermore, by-prod-ucts of the glucose metabolism carried out inthe superficial layers can diffuse into thedeeper film strata providing additional sources

of food. Nevertheless, it is conceivable that thebasal microorganisms will suffer from sub-strate starvation leading to the accumulation ofdead organic matter in the deep layers of film.

In this situation, it is of interest to measure

the viability of the film biomass to see if theassumption of proportionality between film vol-ume and film active mass can be supported.The method for measuring film viability con-

sisted of determining the concentration of ATPusing the JRB-ATP photometer.

Fig. 8 is a plot of the values of ATP mass per

unit of film area versus film thickness gatheredfrom several runs under Si = 200 mg/liter and

Vol. 31, 1976

1.(

%-

can EC .)

0 rf)o EE u0

acr

0

._C

a)

U)

co

0

C._o

C

C:

0

0-0

11

I1

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c 6 -

)o|60/ K s t~~~~~~~~~~~S,E k

5 moqek, Sa,0~~~

DE4

003

c 2

0

0 10 20 30 40 50 60 70 80 90 100 110 120

108 S, moles / cm3Influent Glucose ConcentrOtion

FIG. 5. Effect of influent glucose concentration on

the intrinsic uptake rate, k,.. The line of best fit wasfound by a nonlinear regression analysis yieldingkr maj, = 8.03 X 10-8 molls per cm3 of film and K, =

1.11 X 10-7 mollCm3.

12 2 3 4 5 .6 .7 8 .0

10-

0~~~~~~~~~~~

0~~~~~~~~

0

( 1Q8 S,, moles / cm3 )

FIG. 6. Lineweaver-Burk plot of the same data as

Fig. 5. The linear regression analysis yields k,. a6.21 x 10 molls per cm3 of film andK6 = 6.8 x

J-8 mol/Cm3.

a few points determined in preliminary testswhen Si = 5 mg/liter. It shows that the mass ofATP increased approximately proportionally tofilm thickness up to ca. 320 gm; for film thick-ness greater than 320 ,um, the mass ofATP perunit area remained approximately constant.These two regions would seem to indicate that,once a certain film thickness has been ex-

ceeded, a portion of the film consists of nonliv-ing organic matter.The important point shown in Fig. 8 is that

film activity remained approximately propor-

tional to film thickness in a thickness range

exceeding the theoretical zero-order depth ofglucose penetration. This finding supports theassumption made earlier that film volume isproportional to active film mass; thus, express-

ing the rate of film growth and the rate ofsubstrate consumption in terms of film volumerather than film mass is justified.

6

Q ,5E 4

2

O 2 3 4 S105 sec.

Spec t c Growth Rote

FIG. 7. Relationship between intrinsic uptake rateand specific growth rate. The regression line was

forced through the origin of coordinates, yielding k,= 0.952 x 10-3 ,U (equation 10).

E,4

E4

Q.

<t 3

2 3

100 200 300 400

FhIm th.ckness, m,crons

500 600

FIG. 8. Relationship between mass of ATP per

unit film area and film thickness.

DISCUSSIONThe results presented in the previous section

demonstrate that the specific rate of filmgrowth is independent of the concentration ofsubstrate remaining in the reactor. Referringto Fig. 1, for Si = 1.23 x 10-8 mol/cm3 (2.2 mg/liter), the effluent glucose concentration (whichis equal to the concentration in the liquid bulk)after 29 h was measured to be S, = 2.4 x 10-9mol/cm3 (0.43 mg/liter), and exponentialgrowth still was occurring. For Si = 2.25 x 10-7mol/cm3 (40.5 mg/liter), exponential growthceased after 46.5 h, namely, when the effluentglucose concentration had decreased to 3.2 x

. II70

.5

.0

'5

.0

.5

10

.5

.0

'5

0O 0 -S,. 276 1O-8-oles

5,-sr/cm3

10

0

292 LA MOTTA

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BIOLOGICAL FILM SYSTEM 293

10-8 mol/cm3 (5.8 mg/liter); in other words,within a wide range of glucose concentration inthe bulk liquid (from 40.5 to 5.8 mg/liter), expo-nential growth was not affected by glucose con-centration. These observations support the hy-pothesis presented previously that the rate offilm growth is not affected by decreases in reac-tor substrate concentration. Once the film mi-croorganisms establish an internal steady statedetermined by the initial conditions of foodavailability, the rate of growth will not changedespite significant reductions in the substrateconcentration, until critical environmental con-ditions preclude further exponential film accu-mulation.The effect of substrate concentration initially

available on the magnitude of film growth rateis clearly seen in Fig. 2. Growth rate increasesdramatically as initial substrate concentrationis increased from low values up to ca. 10 x 10-8mol/cm3. Beyond this value, the rate of filmgrowth becomes insensitive to changes in theinfluent glucose concentration. This phenome-non can be explained from the following consid-erations: first, a saturation of the cell activesites would produce a maximum substrate utili-zation that would be reflected in a maximumrate of film growth and, second, the high hy-draulic shear conditions would additionally im-pose a limit on the growth pattern.With regard to the specific substrate uptake

rate, Fig. 4 clearly demonstrates that the ratefollows a zero-order rate law. However, the con-centration of substrate initially present whenthe film is in the early stages of developmentdetermines the magnitude of the rate of sub-strate uptake, and consequently the rate of filmgrowth.

It is worthwhile emphasizing the fact thatthe measurements of the true (or intrinsic) rateof substrate uptake were performed under reac-tion-controlled conditions (obtained with highfluid velocities) and negligible internal-diffu-sion effects (obtained with small film thick-nesses). Only under these circumstances canthe true rate of uptake be observed; otherwise,the observed reaction order will deviate from

zero order, as demonstrated elsewhere (La-Motta, thesis, 1974).

ACKNOWLEDGMENTSThis research was performed at the facilities of the Waste-

water Research Center, University of North Carolina atChapel Hill, which also provided the necessary funds.

LITERATURE CITED

1. Draper, N. R., and H. Smith. 1966. Applied regressionanalysis. John Wiley & Sons, Inc., New York.

2. Gaudy, A. F., and E. T. Gaudy. 1971. Biological conceptsfor design and operation of the activated sludge proc-ess. In Water pollution control research series, 17090FQJ 09/71.

3. Gaudy, A. F., A. Obayashi, and E. T. Gaudy. 1971.Control of growth rate by initial substrate concentra-tion at values below maximum rate. Appl. Microbiol.22:1041-1047.

4. Gaudy, A. F., M. Ramanathan, and B. S. Rao. 1967.Kinetic behavior of heterogeneous populations in com-pletely mixed reactors. Biotechnol. Bioeng. 9:387-411.

5. Gaudy, A. F., P. Y. Yang, R. Bustamante, and E. T.Gaudy. 1973. Exponential growth in systems limited bysubstrate concentration. Biotechnol. Bioeng. 15:589-596.

6. Hartmann, L. 1967. Influence of turbulence on bacterialslimes. J. Water Pollut. Control Fed. 39:958-964.

7. Hoehn, R. C., and A. D. Ray. 1973. Effects of thickness onbacterial film. J. Water Pollut. Control Fed. 45:2302-2320.

8. Kobayashi, T., G. Van Dedem, and M. Moo-Young. 1973.Oxygen transfer into mycelial pellets. Biotechnol.Bioeng. 15:27-45.

9. Kornegay, B. H., and J. F. Andrews. 1969. Characteris-tics and kinetics of biological film reactors. In FederalWater Pollution Control Administration, final report.Research grant WP-01181 (Department ofEnvironmen-tal Systems Engineering, Clemson University).

10. Monod, J. 1942. Recherches sur la croissance des culturesbacteriennes. Herman et Cie, Paris.

11. Monod, J. 1949. The growth of bacteria cultures. Annu.Rev. Microbiol. 3:371-394.

12. Nelder, J. A., and R. Mead. 1965. A simplex method forfunction minimization. Comput. J. 7:308-313.

13. Peil, K. M., and A. F. Gaudy. 1971. Kinetic constants foraerobic growth of microbial populations selected withvarious single compounds and with municipal wastesas substrates. Appl. Microbiol. 21:253-256.

14. Ramanathan, M., and A. F. Gaudy. 1969. Effect of highsubstrate concentration and cell feedback on kineticbehavior of heterogeneous populations in completelymixed systems. Biotechnol. Bioeng. 11:207-237.

15. Tomlinson, T. G., and D. H. M. Snaddon. 1966. Biologicaloxidation of sewage by films of microorganisms. AirWater Pollut. Int. J. 10:865-881.

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