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Criteria establishing biofilm-kinetic types
Wat. Res. Vol. 21, No. 4, pp. 491-498, 1987 Printed in Great Britain.All rights reserved
0043-1354/87 $3.00+0.00 Copyright © 1987PergamonJournals Ltd
CRITERIA ESTABLISHING BIOFILM-KINETIC TYPES MAKRAM T. SUIDAN, BRUCE E. RITTMANNand ULRICH K. TRAEGNER Department of Civil Engineering, University of Illinois at Urbana-Champaign, 208 North Romine, Urbana, IL 61801, U.S.A.
(Received June 1986) Abstraet--A graphical procedure for assessing the nature of substrate penetration within a biofilm is developed. This procedure, which is based on a measure of the effectiveness of substrate utilization by biofilm microorganisms, is used to determine when simple analytical solutions of biofilm models can be used accurately. These analytical solutions correspond to extreme values of bulk substrate concentration and cases where the biofilm is fully penetrated or deep. Minimum values of the effectivenessare related to biofilm thickness.
Key word~--biofilms, attached growth, kinetics, effectiveness factors, penetration, Monod kinetics
NOMENCLATURE
Df= Substrate molecular diffusivity in the biofilm (L2T -|) Dw= Substrate molecular diffusivity in stagnant liquidlayer (L2T -I) J = Flux of substrate into biofilm (ML -2 T -l) J* = Dimensionless substrate flux into biofilm K, -- Monod half-velocity coefficient (ML -3) k = Maximum specific rate of substrate utilization (T -I) L = Thickness of stagnant liquid-layer (L) L*= Dimensionless thickness of stagnant liquid-layer Lf = Thickness of biofilm (L) L7 = Dimensionless thickness of biofilm Q = Dimensionless rate modulus Sb= Bulk liquid phase substrate concentration (ML -3) S* = Dimensionless substrate concentration in bulk liquid S/-- Snbstrate concentration within biofilm (ML -3) S~ -- Dimensionless substrate concentration within biofilm Ss = Substrate concentration at liquid-biofilm interface (ML -3) S*--Dimensionless substrate concentration at liquidbiofilm interface Sw= Substrate concentration at biofilm-solid interface (ML -3) S~*= Dimensionless substrate concentration at biofilmsolid interface X/-- Biolilm microbial density (ML -3) Z = Distance normal to hioiilm surface (L) Z* = Dimensionless distance normal to biofilm surface v = Substrate concentration gradient within biofilm. INTRODUCTION Previous research (e.g. Atkinson and Daoud, 1968; Williamson and McCarty, 1976a, b; Harremo~s, 1978; Rittmann and McCarty, 1980a, b, 1981) has demonstrated clearly that the substrate-utilization kinetics within biofilms can be described accurately by a model that contains reaction with diffusion. Systems of reaction with diffusion create substrateconcentration profiles within the biofilm, as illustrated in Fig. I. Having decreasing concentration
profiles into the biofilm means that the reaction rate per unit biomass declines as the biomass is located farther away from the outer biofilm surface. Figure 1 also illustrates that external mass transport resistance, which can be modeled with a diffusion layer, affects the substrate concentration in the biofilm. Rittmann and McCarty (1981) defined three characteristic substrate-concentration profiles. Figure 1 illustrates that the fully penetrated case has a negligible decrease from the surface concentration (S,), while the deep case has the maximum decrease, because the substrate concentration reaches zero at or before the attaching surface. The shallow case is intermediate, having a concentration at the attaching surface (Sw) between 0 and S~. A priori knowledge that a biofilm's substrateconcentration profile is deep or fully penetrated offers significant advantage, because these regimes are limiting cases for which the mathematics simplify, in comparison to the general case. Although Atkinson and Davies (1974), Rittmann and McCarty (1981), and Suidan and Wang (1985) presented general-case solutions for substrate flux into a biofilm of known thickness, Ly, these solutions are somewhat cumbersome for hand calculation and, when combined into an overall reactor mass balance, do not allow for an analytical calculation of substrate concentration throughout the biofilm reactor. The simpler solutions for fully penetrated and deep biofilms can allow direct analytical calculation of biofilm-reactor performance. Rittmann and McCarty (1978) and Suidan (1986) demonstrated the direct calculations for deep biofilms. If the advantages of the simpler deep and fully penetrated kinetics are to be obtained, criteria that define when a biofilm has deep or fully penetrated kinetics must be clearly established. Rittmann and McCarty (1981) analyzed the effectiveness factor for substrate flux to establish a set of criteria; however, the criteria were only approximations designed to
491
MAKRAMT. SUIDANet al.
492
X
T Bulk Liquid
Diffusion Layer
----Z
Attaching Surface
Biofilm
Fully
enet rated
Sb
.o
~Sw @ O C O rO
L
Lf
Distance Normal to the Surface of the 8iofilm Fig. 1. Characteristic substrate concentration profiles within a biofilm. ensure that flux was not over-predicted. In addition, the phenomena underlying the criteria were not fully explored. The main goal of this paper is to clearly define the criteria for having deep or fully penetrated kinetics. Accomplishing the goal is advanced through the introduction of a new dimensionless rate modulus, Q. MODEL
Sr = S ~
dSr
d-Z=0
at
z=0
(2)
at
z = L r.
(3)
External mass-transport resistance is modeled with an effective diffusion layer (Rittmann and McCarty, 1981),
DEVELOPMENT
The steady-state differential equation describing reaction with diffusion in a biofilm is given by equation (1) (Rittmann and McCarty, 1981; Suidan, 1986),
d%_ kSrXr Dr dz 2 - K~ + Sf
terface and ensure that no substrate is lost into the biofilm attachment surface:
(l)
in which S t = Substrate concentration at a point within the biofilm (MsL -3) Xr= Density of active biomass in the biofilm (Mx L-3) k = Maximum specific substrate utilization rate coefficient (M,M~-IT-l) Ks = half-maximum-rate concentration (M,L -3) Dr= Molecular diffusion coefficient for the substrate in the biofilm (L2T -l) z = Distance dimension normal to the surface of the biofiim (L). The boundary conditions for equation (l) specify the substrate concentration at the liquid-biofilm in-
J = D,. Sb-~L Ss
(4)
in which J = Substrate flux into the biofilm (MsL-2T -~) D,. = Substrate's molecular diffusion coefficient in water (L -2 T -l) Sb = Substrate concentration in the bulk liquid (MsL -3) Ss = Substrate concentration at the liquid-biofilm surface (M~L -3) L = Thickness of the liquid diffusion layer (L). The number of constants in the biofilm mathemetical model given in equations (1)-(4) can be reduced by introducing dimensionless parameters:
sT= E
ST
(5)
z* = z
(6)
\DrKJ
493
Criteria establishing biofilm.kinetic types
L / kXf \m
t,
= t
(s)
J J * = (K, kX:Oz)m.
(9)
Substituting the definitions from equations (5)-(9) into equations (1)-(4) yields
(lO)
d2 S? = s? dz*2 1 + S? j . = S~' - S* L* S~=S~*
at
dz,=0
at
(S*~/I + S*~) Q = (S*/l + S*)"
(11) z*=0
(12)
z*=L~.
(13)
and
dS7
ation type kinetics, such as the Monod reaction rate, are considered. The failure of a concentration ratio to describe film penetration is illustrated when both S* and S t are much larger than one. In this instance, the rate of bioconversion is essentially constant throughout the film, while, depending upon film depth, the values of the two limiting concentrations may be quite different. To remedy this inadequacy in describing biofilm penetration, a dimensionless rate modulus, Q, was defined as the ratio of the specific reaction rate at the biofilm attachment surface to the specific reaction rate that is attainable at the bulk substrate concentration: thus,
Substituting S* from equation (11) and S,* from equation (16) into equation (15) yields an expression for J* as a function of Q, S*, and L*.
J"[
-~--= S t - J * L * - ( I ANALYTICAL RELATIONSHIPS BETWEEN
FLUX AND S,* Equation (10) may be converted to an equivalent form by defining v as equal to dS7/dz*. dv S7 v dS---~= 1 + $ 7 .
(14)
Equation (14) was integrated by Suidan and Wang (1985) to yield a relationship for the dimensionless flux, J* = v at z* = O, as a function of S* and the substrate concentration at the biofilm attachment surface ( z * = LT), S~*. The relationship is given by: j,2
q - - - (s: - s:)
• [-1 + S*-I
(15)
When the biofilm is of sufficient depth that S* is driven to zero, equation (15) results in a deepfilm expression similar to that derived earlier by Suidan and Wang (1985).
(16)
_est
_] + St_QSt).j
[
+ I n .1-1 I + ~ - Q S * 1
'
(17)
Examination of equation (17) reveals that the dimensionless flux of substrate into a biofilm is uniquely determined by specific values for L*, S~', and Q. Equation (17) is highly non-linear and possesses no explicit form for J*. However, analysis of the information contained by this equation can be facilitated by a graphical technique of plotting the ratio of J*/S* vs S*. The J*/S* vs S* form of data presentation permits rapid analysis of the dependency of the overall flux on the bulk substrate concentration. For example, a constant value of J*/S* (i.e. a horizontal line) indicates a first-order dependency of the flux on the bulk substrate concentration. On the other hand, positive or negative slopes of the J*/S* curve indicate that the overall reaction order is greater than or less than first order, respectively.
DIMENSIONLESS RATE MODULUS, Q
Heterogeneous processes which combine chemical or biochemical transformations and diffusive mass transport in a fixed microbial-reaction phase have substrate-concentration gradients which reduce reaction rates from those rates that would be attainable if all the microbial phase were exposed to the bulk phase substrate concentration. The penetration of substrate utilization is defined here as the ratio of actual reaction rate to the reaction rate for the bulk concentration, St. When first-order reactions are considered, such as the case when S* is much smaller than one, the ratio of S,* to S* can be used as a measure of the penetration of the biofilm system. This ratio fails to satisfactorily describe film penetration when satur-
LIMITING CASES
The purpose of this section is to derive simplified expressions for J*/S t that apply to limiting cases of biofilm penetration. There are four limiting cases to consider: (1) the biofilm is deep, and S t ,~ 1; (2) the biofilm is deep, and S t "> 1; (3) the hiofilm is fully penetrated, and S t ,~ 1; and (4) the biofilm is fully penetrated, and S~' >3, 1. Each case has characteristic expressions, which are presented below. The actual J*/St values, from equation (17), can be compared to the characteristics of the limiting cases to determine for what conditions the simpler, limiting cases hold. The value of J*/S*, are derived first and are appropriate for J*/S* when L~ = 0. Subsequently, general J*/S* characteristics are given.
494
MAKRAM T. SUIDAN et al.
and
The biofilm is deep, and S* ~ 1 By definition, a deep film entails a value for the dimensionless rate modulus, Q, of zero since S~ goes to zero at or before the attachment surface. Equation (17) then becomes: j,2 --
2
= S* - ln(1 + S*).
(18)
A two-term Taylor series expansion of the logarithmic term [i.e. ln(l + S * ) = S * - S'2/2 for S,* ~ 1.0], reduces equation (18) to: J* = S*.
(19)
Dividing equation (19) by S* gives the characteristic expressions for a deep biofilm with S* <~ 1. j* --=
s,*
1.
(20)
Equation (20) demonstrates that, for a deep biofilm, the substrate flux is independent of biofilm thickness. Furthermore, the flux of substrate is directly proportional to the substrate concentration at the liquid-biofilm interface.
J* L~' S-~ = S-~"
(26)
Table 1 summarizes the characteristics (J*/S*) for each of the limiting cases. It also lists the corresponding (J*/S~) characteristics which were obtained by substituting S* = S* - J ' L * and solving for J*/S*. METHODS
Equation (17) is implicit with respect to all three variables--J*, S~', and Q - - a n d , consequently, had to be solved numerically to yield relationships between the variables for the general case. Equation (17) was solved for the values of J* using the iterative Newton-Raphson procedure (Kreyszig, 1972) when Q was fixed at a value from 0.01 to 0.99999999, L* was fixed at a value from 0 to 100, and S~ ranged from 10-5 to 105. The dimensionless biofilm thickness, LT, was computed as a function of J*, L*, and S* by using an explicit expression for L~" that was developed by Suidan and Wang (1985).
The biofilm is deep, and S* >>1 For values of S* >> 1, the logarithmic term in equation (18) becomes a small fraction which is negligible compared to the other terms of that equation, and the characteristic expression for the dimensionless flux becomes:
~
= , f 5 (s,*) -,/2.
(21)
Equation (21) clearly demonstrates that when S* >> 1, the dimensionless substrate flux for a deep biofilm varies with the square root of S*. This behavior is typically observed for diffusion limited zero-order reactions (Harremo~s, 1978).
The biofilm is fully penetrated, and S* ~ 1 A fully penetrated biofilm has no reduction of reaction rate with depth in the biofilm. By definition, Q = 1. Substitution of Q = 1 in equation (17) does not yield a defined solution. However, a very useful case of full penetration is a dispersed-growth system having concentration S* for all biomass. The dimensionless flux is given by J*=
-s*, -
I~£; dz * =
! + s,* J0
s*,'
1 +s,*
L*r.
(22)
When S* ,~ 1, equation (22) reduces to
J* = L? S,*
(23)
and
J*/S* = L'~.
(24)
The biofilm is fully penetrated, and S* >>1 When S* >> i, equation (22) becomes
j * = z,?
(25)
RESULTS
Figure 2 presents the J*/S* and L 7 results, which show when the deep and fully penetrated kinetics are good representations of the actual kinetics. For example, curves h and i (Q = 0.1 and 0.01) had J*/S* values equal to those given by the charcteristic equations for a deep biofilm for the whole range of S*. When S* ~< 10 -L~, the characteristic equation for a first-order reaction (i.e. J*/S* = I) was an accurate representation of the flux. When S* >~ 102, the deep, zero-order characteristic (i.e. J*/S*=x//2(S*) -1/2 held. Thus, when Q ~ 0.99. On the other hand, when S* >> 1, the curves corresponded to the deep, zero-order characteristic (i.e. J*/S* = x//2(S*) -~'2 ~ L~/S*), even though Q approached 1.0. Higher values of Q, up to 0.9999999 (not shown on the figure), gave the same result for very high values of S*. The implication of the last result is that dispersed-growth, zero-order kinetics were never found for high S*. Thus, if the reaction is zero-order in the film, the deep model should be used when S* >> 1. Figure 3 presents J*/S~ for cases of small and large external mass transport control--L* = 0.1 and 10.0, respectively--while Fig. 4 presents the L~ values for L * = 10.0. Six conclusions are obtainable from Figs 3 and 4. First, comparison of the results in the figures with the characteristic expressions for
Criteria establishing biofilm-kin©tic types
I
495
/--~rocteriMic_for Deep,
ff [g/-h,i
i~S~<<
I : J"/SZ-I
-0"4t c
-Qs - h } J'/S;= "7 ~== oJ / Chorocteristic_forFully " "'- [ J Penetroted, Sf<< I
. -2.4-
J'/s; =
-.-
-2.8-
-5
-3
-I
I
3
5
I
3
5
3
2-
If
g h i
~
-I- b o -2
-5
-3
-I
•
s ,•
Fig. 2. J*/S* and L~ vs S*. Q is equal to (a) 0.999, (b) 0.99, (c) 0.9, (d) 0.8, (e) 0.6, (f) 0.4, (g) 0.2, (h) 0.1, and (i) ~<0.01.
S~ >> 1 shows that for S* >~ 104 for L* = 10 and 102 for L * = 0.1, the kinetics collapse to a form of the zero-order, deep case in which L * = 0 ; in other words, J*/S* = x/2(S*) -~/2 for all L*. This means that external mass transport resistance is not important, and the characteristic behaves as if L* = 0, if S~ is sufficiently great. Analysis of the results in Figs 2 and 3 suggest that the simple characteristic is accurate when S*/> (10 + L*2)2. Second, for S* < 10 -1'5 and Q i> 0.99 (curves a and b), the fully penetrated, first-order characteristic with L* = 0 holds for any value of L*. Thus, if S* is low enough, external mass-transport resistance does not affect whether or not a biofilm is fully penetrated. W.R. 21 ~4~H
However, increased external mass transport decreases
J*/S* and L~ significantly, because S* < S~ when L*>O. Third, for very low S~ and Q ~<0.1, the curves approach the deep, first-order characteristic which includes L*, J*/S~ = 1/(1 + L*). Thus, for Q ~<0.1, the biofilm is deep, but external mass transport affects the value of J*/S~. Very low S~' can be defined as S~ ~<(10 -1"5 + L*). Fourth, for the same values of S~' and Q, J*/S* are lowered by having L* become larger in all regions, except for very large S~'. The greatest decreases in J*/S~ occur for fully penetrated biofilms having S* < 1. Thus, fully penetrated biofilms are most
496
MAKt~M T. SUIDANet al.
0 °
~i g
'
-(!.4-
-0~/ llLI~
8' .-I
-I.2-
= LT
-I.6-
-20-
.__/
J
-2.4-2.8i
I
I
I
i
-I
-3
-5
I
I
J
l
3
5
-I.0.,f,,
•
-I.5-2.0m~ Or)
=-.
"3
2
-2.5" -50-
g J
-3,5-4£)" -4.5L u = I0.0
-5.0-5
-3
-I
I
3
5
Fig. 3. J * / S ~ and L~ vs S* for L* = 0.1 and 10. Q values the same as in Fig. 2.
affected by external mass transport resistance, because they have no internal mass transport resistance. Deep biofilm are less affected, since internal resistance already is reducing the substrate concentration to the interior of the biofilm. When S~ >> 1, external mass-transport resistance plays no role at all, because the reactions are controlled completely by internal mass transport resistance. Fifth, some curves in Fig. 3 have " h u m p s " for positive values of S~'. The humps occur with the transition from fully penetrated, first-order kinetics,
which are most highly affected by external mass transport resistance, to zero-order, deep kinetics, which are unaffected. As external mass transport resistance becomes less important, J * / S * rises toward its value for L * = 0.0. Sixth, Fig. 4 shows that, for very large S*, all L~ curves converge. However for low S*, the curves find separate, unique values. A horizontal line at a given value of L~ shows how the biofilm becomes more fully penetrated with higher S~. For example, when L~ = 10 -j, Q is 0.5 for S~ < 10 -~, increases to 0.8 at
Table I. Summarycharacteristicsexpressions Characteristic equations In terms of S* In terms of S~' J*lSt = I/(l + L*) J*/S* = 1
Special cases l ~ p biofilm,S* ~ I Deep biofllm,St ~, I Fully penetra~l biofilm,S~' ~ 1 Fully l~-netrat~lbiofilm,S* } I
J * / s * = ~fi(s.*, )- '~
2*/Sf = (2/S, + L * / S t / S D - ,/2 _ L / s t
J*/S* = L7 J*/s7 = L?/S*
L*/$* = L~/St
J * / S * = L~/(I + L * L t)
Criteria establishing biofilm-kinetic types
497
2-
Ioo
-0,1
O-
d
0.2
m~,.,
_1 g,
-I
9,~
-2
09
J
--I
J
-3
0999-o
-4
J L m = I0
-5 -5
'
'
;
'
'
s
Log S~
Fig. 4. A plot of L~ vs S~' for L~ = 10 shows that a given L~ has a minimum Q.
Sb "" 1, and reaches 0.99 at S~' ~ 10. The important feature illustrated by Fig. 4 is that each Lfhas a unique minimum Q. In other words, a biofilm of constant thickness can become deep only to a limited degree. For example, when/.7 = 10-l, the minimum Q is 0.5. To be completely deep (i.e. Q ~<0.1), the L7 must be at least 10°l when L* = 10. Equation (27) presents exact solutions for the/.7 value needed to give a selected Q, when L* # 1. To determine the value of/.7 that corresponds to a value of L* = 1, equation (28) must be used. Equations (27) and (28) represent solutions of the biofilm mathematical model [equations (10)-(13) and (16)] that correspond to values of S* < 1.
L~ = cosh_~IL *~/l ~+( Q2(L *2~-1) -11 - ~ _---
(27)
x I+Q 2 L , = cosh- [ - - ~ ] .
(28)
Substituting Q = 0.1 into equation (27) gives the minimum L 7 needed to have biofilm kinetics behave as deep; substituting Q = 0.99 gives the maximum L7 to have biofilm kinetics behave as fully penetrated. Figure 5 presents the minimum and maximum /.7 values needed to have a deep (i.e. Q < 0.1) or a fully penetrated (i.e. Q i> 0.99) biofilm for a range of L* values. Having more mass transport resistance lowers
I.O , f L~ = 3,0 (15o -0.5-
/--L?-a2
-ID.J
-I~;-
--I
-2D"
IOIL"~ -2,5-3D-3.5-4.0
-3
-!
I LOg L =
Fig. 5. A plot of L~ corresponding to limiting substrate effectiveness conditions when S* < 1.
498
MAKRAMT. SUIDANet al.
S* values and requires a smaller L 7 to have full penetration or deep kinetics. The Lff range over which neither deep nor fully penetrated kinetics apply increases as L* increases. It is 10132 for L* = 0, but increases to 1031~ for L * = 100. Figure 5 also represents the approximate limits on deep and fully penetrated kinetics for S~' ,~ 1 from Rittmann (1979) and Rittmann and McCarty (1981). In the nomenclature of this paper, the biofilm is deemed deep when L?/> 3.0
(29)
and is deemed fully penetrated when L? ~<0.2.
(30)
The figure shows that the approximate limits are quite accurate for low L*, but require too large values of L? for large L*. Since the main goal in defining approximate limits was to preclude assuming a deep biofilm when the biofilm was shallow, they were successful. However, the current limits are more broadly correct. SUMMARY AND CONCLUSIONS The substrate-utilization penetration was defined according to a new rate modulus, Q, which is the ratio of the dimensionless substrate-utilization rate at the attachment surface (S*/[I + S*]) to the dimensionless substrate-utilization rate for the bulk substrate concentration [S*/(I + S*)]. A deep biofilm has Q = 0 , as S* approaches zero, while a fully penetrated biofilm has Q = 1. Equation (17) was derived as an exact solution relating Q, S~, and the dimensionless flux, J*. It was solved numerically by a Newton-Raphson procedure to yield graphs of J * / S * vs S~'. Then, the dimensionless biofilm thickness was calculated from an expression developed by Suidan and Wang (1985). Comparison of the J * / S ~ and L 7 vs S* results to the characteristic expressions for four limiting cases (zero-order kinetics for deep biofilm and dispersed growth, and first-order kinetics for deep biofilms and dispersed growth) provided the followed conclusions: (1) For Sg >1 (10 + L*2)2, the simple characteristic for deep, zero-order biofilm, J * / S * = ~/2(S*)- 1/2, is accurate for all values of Q. For all combinations of S~ and L* in this region, the one-half-order analytical solution is appropriate. (2) For S~' < 10-)5, the kinetics have a first-order, dispersed-growth characteristic, J * / S g = L~, when Q >t 0.99.
(3) For S~' ~<(10-Ls+ L*), the first-order, deep characteristic, J * / S g = 1/(! + L*), is accurate when Q ~<0.1. (4) The minimum L~' to have a deep biofilm (Q ~<0.1) and the maximum L~to have a fully penetrated biofilm (Q/> 0.99) are defined by equation (27) for S* ~ l and exemplified on Fig. 5. For all combinations of L*, L~ and S~' falling into these classes, the simple, first-order analytical solution given on points 2 and 3 are appropriate. (5) External mass transport resistance has the greatest impact on reducing J * / S * when the biofiim is most fully penetrated, because internal mass is unimportant. Acknowledgement--Funding for this work was provided by
the U.S. Environmental Protection Agency under Contract No. EPA-809750-01. This paper has not been subjected to the Agency's required peer and administrative review and therefore does not necessarily reflect the views of the agency, therefore no official endorsement should be inferred.
REFERENCES
Atkinson B. and Daoud I. S. (1968) The analogy between microbiological "reactions" and heterogeneous catalysts. Trans. Instn chem. Engrs 46, 19. Atkinson B. and Davies I. J. (1974) The overall rate of substrate uptake (reaction) by microbial films. Part I. (a). Biological reaction rate equation. Trans. Instn chem. Engrs 52, 248. Harremo~s P. (1978) Biofilm kinetics. In Water Pollution Microbiology (Edited by Mitchell R.). Wiley, New York. Kreyszig E. (1972) Advanced Engineering Mathematics, 3rd edition. Wiley, New York. Rittmann B. E. (1979) The kinetics of trace-organic utilization by biofilms. Ph.D. dissertation, Department of Civil Engineering, Stanford University, Stanford, Calif. Rittmann B. E. and McCarty P. L. (1978) The variableorder model of bacterial film kinetics. J. envir. Engng Div. Am. Soc. cir. Engrs 104, 889. Rittmann B. E. and McCarty P. L. (1980a) Model of steady state-biofilm kinetics. Biotechnol. Bioengng 22, 2343. Rittmann B. E. and McCarty P. L. (1980b) Evaluation of steady-state-biofilm kinetics. Biotechnol. Bioengng 22, 2359. Rittmann B. E. and McCarty P. L. (1981) Substrate flux into biofilms of any thickness. J. envir. Engng Div. Am. Soc. civ. Engrs 107, 831. Suidan M. J. (1986) Performance of deep biofilm reactors. J. envir. Engng Div. Am. Soc. civ. Engrs 112, 78. Suidan M. T. and Wang Y. T. (1985) Unified analysis of biofilm kinetics. J. envir. Engng Div. Am. Soc. cir. Engrs 111, 634. Williamson K. and McCarty P. L. (1976a) A model of substrate utilization by bacterial films. J. Wat. Pollut. Control Fed. 48, 9.
Williamson K. and McCarty P. L. (1976b) Verification studies of the biofilm model for bacterial substrate utilization. J. l'Vat. Pollut. Control Fed. 45, 281.
0043-1354/87 $3.00+0.00 Copyright © 1987PergamonJournals Ltd
CRITERIA ESTABLISHING BIOFILM-KINETIC TYPES MAKRAM T. SUIDAN, BRUCE E. RITTMANNand ULRICH K. TRAEGNER Department of Civil Engineering, University of Illinois at Urbana-Champaign, 208 North Romine, Urbana, IL 61801, U.S.A.
(Received June 1986) Abstraet--A graphical procedure for assessing the nature of substrate penetration within a biofilm is developed. This procedure, which is based on a measure of the effectiveness of substrate utilization by biofilm microorganisms, is used to determine when simple analytical solutions of biofilm models can be used accurately. These analytical solutions correspond to extreme values of bulk substrate concentration and cases where the biofilm is fully penetrated or deep. Minimum values of the effectivenessare related to biofilm thickness.
Key word~--biofilms, attached growth, kinetics, effectiveness factors, penetration, Monod kinetics
NOMENCLATURE
Df= Substrate molecular diffusivity in the biofilm (L2T -|) Dw= Substrate molecular diffusivity in stagnant liquidlayer (L2T -I) J = Flux of substrate into biofilm (ML -2 T -l) J* = Dimensionless substrate flux into biofilm K, -- Monod half-velocity coefficient (ML -3) k = Maximum specific rate of substrate utilization (T -I) L = Thickness of stagnant liquid-layer (L) L*= Dimensionless thickness of stagnant liquid-layer Lf = Thickness of biofilm (L) L7 = Dimensionless thickness of biofilm Q = Dimensionless rate modulus Sb= Bulk liquid phase substrate concentration (ML -3) S* = Dimensionless substrate concentration in bulk liquid S/-- Snbstrate concentration within biofilm (ML -3) S~ -- Dimensionless substrate concentration within biofilm Ss = Substrate concentration at liquid-biofilm interface (ML -3) S*--Dimensionless substrate concentration at liquidbiofilm interface Sw= Substrate concentration at biofilm-solid interface (ML -3) S~*= Dimensionless substrate concentration at biofilmsolid interface X/-- Biolilm microbial density (ML -3) Z = Distance normal to hioiilm surface (L) Z* = Dimensionless distance normal to biofilm surface v = Substrate concentration gradient within biofilm. INTRODUCTION Previous research (e.g. Atkinson and Daoud, 1968; Williamson and McCarty, 1976a, b; Harremo~s, 1978; Rittmann and McCarty, 1980a, b, 1981) has demonstrated clearly that the substrate-utilization kinetics within biofilms can be described accurately by a model that contains reaction with diffusion. Systems of reaction with diffusion create substrateconcentration profiles within the biofilm, as illustrated in Fig. I. Having decreasing concentration
profiles into the biofilm means that the reaction rate per unit biomass declines as the biomass is located farther away from the outer biofilm surface. Figure 1 also illustrates that external mass transport resistance, which can be modeled with a diffusion layer, affects the substrate concentration in the biofilm. Rittmann and McCarty (1981) defined three characteristic substrate-concentration profiles. Figure 1 illustrates that the fully penetrated case has a negligible decrease from the surface concentration (S,), while the deep case has the maximum decrease, because the substrate concentration reaches zero at or before the attaching surface. The shallow case is intermediate, having a concentration at the attaching surface (Sw) between 0 and S~. A priori knowledge that a biofilm's substrateconcentration profile is deep or fully penetrated offers significant advantage, because these regimes are limiting cases for which the mathematics simplify, in comparison to the general case. Although Atkinson and Davies (1974), Rittmann and McCarty (1981), and Suidan and Wang (1985) presented general-case solutions for substrate flux into a biofilm of known thickness, Ly, these solutions are somewhat cumbersome for hand calculation and, when combined into an overall reactor mass balance, do not allow for an analytical calculation of substrate concentration throughout the biofilm reactor. The simpler solutions for fully penetrated and deep biofilms can allow direct analytical calculation of biofilm-reactor performance. Rittmann and McCarty (1978) and Suidan (1986) demonstrated the direct calculations for deep biofilms. If the advantages of the simpler deep and fully penetrated kinetics are to be obtained, criteria that define when a biofilm has deep or fully penetrated kinetics must be clearly established. Rittmann and McCarty (1981) analyzed the effectiveness factor for substrate flux to establish a set of criteria; however, the criteria were only approximations designed to
491
MAKRAMT. SUIDANet al.
492
X
T Bulk Liquid
Diffusion Layer
----Z
Attaching Surface
Biofilm
Fully
enet rated
Sb
.o
~Sw @ O C O rO
L
Lf
Distance Normal to the Surface of the 8iofilm Fig. 1. Characteristic substrate concentration profiles within a biofilm. ensure that flux was not over-predicted. In addition, the phenomena underlying the criteria were not fully explored. The main goal of this paper is to clearly define the criteria for having deep or fully penetrated kinetics. Accomplishing the goal is advanced through the introduction of a new dimensionless rate modulus, Q. MODEL
Sr = S ~
dSr
d-Z=0
at
z=0
(2)
at
z = L r.
(3)
External mass-transport resistance is modeled with an effective diffusion layer (Rittmann and McCarty, 1981),
DEVELOPMENT
The steady-state differential equation describing reaction with diffusion in a biofilm is given by equation (1) (Rittmann and McCarty, 1981; Suidan, 1986),
d%_ kSrXr Dr dz 2 - K~ + Sf
terface and ensure that no substrate is lost into the biofilm attachment surface:
(l)
in which S t = Substrate concentration at a point within the biofilm (MsL -3) Xr= Density of active biomass in the biofilm (Mx L-3) k = Maximum specific substrate utilization rate coefficient (M,M~-IT-l) Ks = half-maximum-rate concentration (M,L -3) Dr= Molecular diffusion coefficient for the substrate in the biofilm (L2T -l) z = Distance dimension normal to the surface of the biofiim (L). The boundary conditions for equation (l) specify the substrate concentration at the liquid-biofilm in-
J = D,. Sb-~L Ss
(4)
in which J = Substrate flux into the biofilm (MsL-2T -~) D,. = Substrate's molecular diffusion coefficient in water (L -2 T -l) Sb = Substrate concentration in the bulk liquid (MsL -3) Ss = Substrate concentration at the liquid-biofilm surface (M~L -3) L = Thickness of the liquid diffusion layer (L). The number of constants in the biofilm mathemetical model given in equations (1)-(4) can be reduced by introducing dimensionless parameters:
sT= E
ST
(5)
z* = z
(6)
\DrKJ
493
Criteria establishing biofilm.kinetic types
L / kXf \m
t,
= t
(s)
J J * = (K, kX:Oz)m.
(9)
Substituting the definitions from equations (5)-(9) into equations (1)-(4) yields
(lO)
d2 S? = s? dz*2 1 + S? j . = S~' - S* L* S~=S~*
at
dz,=0
at
(S*~/I + S*~) Q = (S*/l + S*)"
(11) z*=0
(12)
z*=L~.
(13)
and
dS7
ation type kinetics, such as the Monod reaction rate, are considered. The failure of a concentration ratio to describe film penetration is illustrated when both S* and S t are much larger than one. In this instance, the rate of bioconversion is essentially constant throughout the film, while, depending upon film depth, the values of the two limiting concentrations may be quite different. To remedy this inadequacy in describing biofilm penetration, a dimensionless rate modulus, Q, was defined as the ratio of the specific reaction rate at the biofilm attachment surface to the specific reaction rate that is attainable at the bulk substrate concentration: thus,
Substituting S* from equation (11) and S,* from equation (16) into equation (15) yields an expression for J* as a function of Q, S*, and L*.
J"[
-~--= S t - J * L * - ( I ANALYTICAL RELATIONSHIPS BETWEEN
FLUX AND S,* Equation (10) may be converted to an equivalent form by defining v as equal to dS7/dz*. dv S7 v dS---~= 1 + $ 7 .
(14)
Equation (14) was integrated by Suidan and Wang (1985) to yield a relationship for the dimensionless flux, J* = v at z* = O, as a function of S* and the substrate concentration at the biofilm attachment surface ( z * = LT), S~*. The relationship is given by: j,2
q - - - (s: - s:)
• [-1 + S*-I
(15)
When the biofilm is of sufficient depth that S* is driven to zero, equation (15) results in a deepfilm expression similar to that derived earlier by Suidan and Wang (1985).
(16)
_est
_] + St_QSt).j
[
+ I n .1-1 I + ~ - Q S * 1
'
(17)
Examination of equation (17) reveals that the dimensionless flux of substrate into a biofilm is uniquely determined by specific values for L*, S~', and Q. Equation (17) is highly non-linear and possesses no explicit form for J*. However, analysis of the information contained by this equation can be facilitated by a graphical technique of plotting the ratio of J*/S* vs S*. The J*/S* vs S* form of data presentation permits rapid analysis of the dependency of the overall flux on the bulk substrate concentration. For example, a constant value of J*/S* (i.e. a horizontal line) indicates a first-order dependency of the flux on the bulk substrate concentration. On the other hand, positive or negative slopes of the J*/S* curve indicate that the overall reaction order is greater than or less than first order, respectively.
DIMENSIONLESS RATE MODULUS, Q
Heterogeneous processes which combine chemical or biochemical transformations and diffusive mass transport in a fixed microbial-reaction phase have substrate-concentration gradients which reduce reaction rates from those rates that would be attainable if all the microbial phase were exposed to the bulk phase substrate concentration. The penetration of substrate utilization is defined here as the ratio of actual reaction rate to the reaction rate for the bulk concentration, St. When first-order reactions are considered, such as the case when S* is much smaller than one, the ratio of S,* to S* can be used as a measure of the penetration of the biofilm system. This ratio fails to satisfactorily describe film penetration when satur-
LIMITING CASES
The purpose of this section is to derive simplified expressions for J*/S t that apply to limiting cases of biofilm penetration. There are four limiting cases to consider: (1) the biofilm is deep, and S t ,~ 1; (2) the biofilm is deep, and S t "> 1; (3) the hiofilm is fully penetrated, and S t ,~ 1; and (4) the biofilm is fully penetrated, and S~' >3, 1. Each case has characteristic expressions, which are presented below. The actual J*/St values, from equation (17), can be compared to the characteristics of the limiting cases to determine for what conditions the simpler, limiting cases hold. The value of J*/S*, are derived first and are appropriate for J*/S* when L~ = 0. Subsequently, general J*/S* characteristics are given.
494
MAKRAM T. SUIDAN et al.
and
The biofilm is deep, and S* ~ 1 By definition, a deep film entails a value for the dimensionless rate modulus, Q, of zero since S~ goes to zero at or before the attachment surface. Equation (17) then becomes: j,2 --
2
= S* - ln(1 + S*).
(18)
A two-term Taylor series expansion of the logarithmic term [i.e. ln(l + S * ) = S * - S'2/2 for S,* ~ 1.0], reduces equation (18) to: J* = S*.
(19)
Dividing equation (19) by S* gives the characteristic expressions for a deep biofilm with S* <~ 1. j* --=
s,*
1.
(20)
Equation (20) demonstrates that, for a deep biofilm, the substrate flux is independent of biofilm thickness. Furthermore, the flux of substrate is directly proportional to the substrate concentration at the liquid-biofilm interface.
J* L~' S-~ = S-~"
(26)
Table 1 summarizes the characteristics (J*/S*) for each of the limiting cases. It also lists the corresponding (J*/S~) characteristics which were obtained by substituting S* = S* - J ' L * and solving for J*/S*. METHODS
Equation (17) is implicit with respect to all three variables--J*, S~', and Q - - a n d , consequently, had to be solved numerically to yield relationships between the variables for the general case. Equation (17) was solved for the values of J* using the iterative Newton-Raphson procedure (Kreyszig, 1972) when Q was fixed at a value from 0.01 to 0.99999999, L* was fixed at a value from 0 to 100, and S~ ranged from 10-5 to 105. The dimensionless biofilm thickness, LT, was computed as a function of J*, L*, and S* by using an explicit expression for L~" that was developed by Suidan and Wang (1985).
The biofilm is deep, and S* >>1 For values of S* >> 1, the logarithmic term in equation (18) becomes a small fraction which is negligible compared to the other terms of that equation, and the characteristic expression for the dimensionless flux becomes:
~
= , f 5 (s,*) -,/2.
(21)
Equation (21) clearly demonstrates that when S* >> 1, the dimensionless substrate flux for a deep biofilm varies with the square root of S*. This behavior is typically observed for diffusion limited zero-order reactions (Harremo~s, 1978).
The biofilm is fully penetrated, and S* ~ 1 A fully penetrated biofilm has no reduction of reaction rate with depth in the biofilm. By definition, Q = 1. Substitution of Q = 1 in equation (17) does not yield a defined solution. However, a very useful case of full penetration is a dispersed-growth system having concentration S* for all biomass. The dimensionless flux is given by J*=
-s*, -
I~£; dz * =
! + s,* J0
s*,'
1 +s,*
L*r.
(22)
When S* ,~ 1, equation (22) reduces to
J* = L? S,*
(23)
and
J*/S* = L'~.
(24)
The biofilm is fully penetrated, and S* >>1 When S* >> i, equation (22) becomes
j * = z,?
(25)
RESULTS
Figure 2 presents the J*/S* and L 7 results, which show when the deep and fully penetrated kinetics are good representations of the actual kinetics. For example, curves h and i (Q = 0.1 and 0.01) had J*/S* values equal to those given by the charcteristic equations for a deep biofilm for the whole range of S*. When S* ~< 10 -L~, the characteristic equation for a first-order reaction (i.e. J*/S* = I) was an accurate representation of the flux. When S* >~ 102, the deep, zero-order characteristic (i.e. J*/S*=x//2(S*) -1/2 held. Thus, when Q ~
Criteria establishing biofilm-kin©tic types
I
495
/--~rocteriMic_for Deep,
ff [g/-h,i
i~S~<<
I : J"/SZ-I
-0"4t c
-Qs - h } J'/S;= "7 ~== oJ / Chorocteristic_forFully " "'- [ J Penetroted, Sf<< I
. -2.4-
J'/s; =
-.-
-2.8-
-5
-3
-I
I
3
5
I
3
5
3
2-
If
g h i
~
-I- b o -2
-5
-3
-I
•
s ,•
Fig. 2. J*/S* and L~ vs S*. Q is equal to (a) 0.999, (b) 0.99, (c) 0.9, (d) 0.8, (e) 0.6, (f) 0.4, (g) 0.2, (h) 0.1, and (i) ~<0.01.
S~ >> 1 shows that for S* >~ 104 for L* = 10 and 102 for L * = 0.1, the kinetics collapse to a form of the zero-order, deep case in which L * = 0 ; in other words, J*/S* = x/2(S*) -~/2 for all L*. This means that external mass transport resistance is not important, and the characteristic behaves as if L* = 0, if S~ is sufficiently great. Analysis of the results in Figs 2 and 3 suggest that the simple characteristic is accurate when S*/> (10 + L*2)2. Second, for S* < 10 -1'5 and Q i> 0.99 (curves a and b), the fully penetrated, first-order characteristic with L* = 0 holds for any value of L*. Thus, if S* is low enough, external mass-transport resistance does not affect whether or not a biofilm is fully penetrated. W.R. 21 ~4~H
However, increased external mass transport decreases
J*/S* and L~ significantly, because S* < S~ when L*>O. Third, for very low S~ and Q ~<0.1, the curves approach the deep, first-order characteristic which includes L*, J*/S~ = 1/(1 + L*). Thus, for Q ~<0.1, the biofilm is deep, but external mass transport affects the value of J*/S~. Very low S~' can be defined as S~ ~<(10 -1"5 + L*). Fourth, for the same values of S~' and Q, J*/S* are lowered by having L* become larger in all regions, except for very large S~'. The greatest decreases in J*/S~ occur for fully penetrated biofilms having S* < 1. Thus, fully penetrated biofilms are most
496
MAKt~M T. SUIDANet al.
0 °
~i g
'
-(!.4-
-0~/ llLI~
8' .-I
-I.2-
= LT
-I.6-
-20-
.__/
J
-2.4-2.8i
I
I
I
i
-I
-3
-5
I
I
J
l
3
5
-I.0.,f,,
•
-I.5-2.0m~ Or)
=-.
"3
2
-2.5" -50-
g J
-3,5-4£)" -4.5L u = I0.0
-5.0-5
-3
-I
I
3
5
Fig. 3. J * / S ~ and L~ vs S* for L* = 0.1 and 10. Q values the same as in Fig. 2.
affected by external mass transport resistance, because they have no internal mass transport resistance. Deep biofilm are less affected, since internal resistance already is reducing the substrate concentration to the interior of the biofilm. When S~ >> 1, external mass-transport resistance plays no role at all, because the reactions are controlled completely by internal mass transport resistance. Fifth, some curves in Fig. 3 have " h u m p s " for positive values of S~'. The humps occur with the transition from fully penetrated, first-order kinetics,
which are most highly affected by external mass transport resistance, to zero-order, deep kinetics, which are unaffected. As external mass transport resistance becomes less important, J * / S * rises toward its value for L * = 0.0. Sixth, Fig. 4 shows that, for very large S*, all L~ curves converge. However for low S*, the curves find separate, unique values. A horizontal line at a given value of L~ shows how the biofilm becomes more fully penetrated with higher S~. For example, when L~ = 10 -j, Q is 0.5 for S~ < 10 -~, increases to 0.8 at
Table I. Summarycharacteristicsexpressions Characteristic equations In terms of S* In terms of S~' J*lSt = I/(l + L*) J*/S* = 1
Special cases l ~ p biofilm,S* ~ I Deep biofllm,St ~, I Fully penetra~l biofilm,S~' ~ 1 Fully l~-netrat~lbiofilm,S* } I
J * / s * = ~fi(s.*, )- '~
2*/Sf = (2/S, + L * / S t / S D - ,/2 _ L / s t
J*/S* = L7 J*/s7 = L?/S*
L*/$* = L~/St
J * / S * = L~/(I + L * L t)
Criteria establishing biofilm-kinetic types
497
2-
Ioo
-0,1
O-
d
0.2
m~,.,
_1 g,
-I
9,~
-2
09
J
--I
J
-3
0999-o
-4
J L m = I0
-5 -5
'
'
;
'
'
s
Log S~
Fig. 4. A plot of L~ vs S~' for L~ = 10 shows that a given L~ has a minimum Q.
Sb "" 1, and reaches 0.99 at S~' ~ 10. The important feature illustrated by Fig. 4 is that each Lfhas a unique minimum Q. In other words, a biofilm of constant thickness can become deep only to a limited degree. For example, when/.7 = 10-l, the minimum Q is 0.5. To be completely deep (i.e. Q ~<0.1), the L7 must be at least 10°l when L* = 10. Equation (27) presents exact solutions for the/.7 value needed to give a selected Q, when L* # 1. To determine the value of/.7 that corresponds to a value of L* = 1, equation (28) must be used. Equations (27) and (28) represent solutions of the biofilm mathematical model [equations (10)-(13) and (16)] that correspond to values of S* < 1.
L~ = cosh_~IL *~/l ~+( Q2(L *2~-1) -11 - ~ _---
(27)
x I+Q 2 L , = cosh- [ - - ~ ] .
(28)
Substituting Q = 0.1 into equation (27) gives the minimum L 7 needed to have biofilm kinetics behave as deep; substituting Q = 0.99 gives the maximum L7 to have biofilm kinetics behave as fully penetrated. Figure 5 presents the minimum and maximum /.7 values needed to have a deep (i.e. Q < 0.1) or a fully penetrated (i.e. Q i> 0.99) biofilm for a range of L* values. Having more mass transport resistance lowers
I.O , f L~ = 3,0 (15o -0.5-
/--L?-a2
-ID.J
-I~;-
--I
-2D"
IOIL"~ -2,5-3D-3.5-4.0
-3
-!
I LOg L =
Fig. 5. A plot of L~ corresponding to limiting substrate effectiveness conditions when S* < 1.
498
MAKRAMT. SUIDANet al.
S* values and requires a smaller L 7 to have full penetration or deep kinetics. The Lff range over which neither deep nor fully penetrated kinetics apply increases as L* increases. It is 10132 for L* = 0, but increases to 1031~ for L * = 100. Figure 5 also represents the approximate limits on deep and fully penetrated kinetics for S~' ,~ 1 from Rittmann (1979) and Rittmann and McCarty (1981). In the nomenclature of this paper, the biofilm is deemed deep when L?/> 3.0
(29)
and is deemed fully penetrated when L? ~<0.2.
(30)
The figure shows that the approximate limits are quite accurate for low L*, but require too large values of L? for large L*. Since the main goal in defining approximate limits was to preclude assuming a deep biofilm when the biofilm was shallow, they were successful. However, the current limits are more broadly correct. SUMMARY AND CONCLUSIONS The substrate-utilization penetration was defined according to a new rate modulus, Q, which is the ratio of the dimensionless substrate-utilization rate at the attachment surface (S*/[I + S*]) to the dimensionless substrate-utilization rate for the bulk substrate concentration [S*/(I + S*)]. A deep biofilm has Q = 0 , as S* approaches zero, while a fully penetrated biofilm has Q = 1. Equation (17) was derived as an exact solution relating Q, S~, and the dimensionless flux, J*. It was solved numerically by a Newton-Raphson procedure to yield graphs of J * / S * vs S~'. Then, the dimensionless biofilm thickness was calculated from an expression developed by Suidan and Wang (1985). Comparison of the J * / S ~ and L 7 vs S* results to the characteristic expressions for four limiting cases (zero-order kinetics for deep biofilm and dispersed growth, and first-order kinetics for deep biofilms and dispersed growth) provided the followed conclusions: (1) For Sg >1 (10 + L*2)2, the simple characteristic for deep, zero-order biofilm, J * / S * = ~/2(S*)- 1/2, is accurate for all values of Q. For all combinations of S~ and L* in this region, the one-half-order analytical solution is appropriate. (2) For S~' < 10-)5, the kinetics have a first-order, dispersed-growth characteristic, J * / S g = L~, when Q >t 0.99.
(3) For S~' ~<(10-Ls+ L*), the first-order, deep characteristic, J * / S g = 1/(! + L*), is accurate when Q ~<0.1. (4) The minimum L~' to have a deep biofilm (Q ~<0.1) and the maximum L~to have a fully penetrated biofilm (Q/> 0.99) are defined by equation (27) for S* ~ l and exemplified on Fig. 5. For all combinations of L*, L~ and S~' falling into these classes, the simple, first-order analytical solution given on points 2 and 3 are appropriate. (5) External mass transport resistance has the greatest impact on reducing J * / S * when the biofiim is most fully penetrated, because internal mass is unimportant. Acknowledgement--Funding for this work was provided by
the U.S. Environmental Protection Agency under Contract No. EPA-809750-01. This paper has not been subjected to the Agency's required peer and administrative review and therefore does not necessarily reflect the views of the agency, therefore no official endorsement should be inferred.
REFERENCES
Atkinson B. and Daoud I. S. (1968) The analogy between microbiological "reactions" and heterogeneous catalysts. Trans. Instn chem. Engrs 46, 19. Atkinson B. and Davies I. J. (1974) The overall rate of substrate uptake (reaction) by microbial films. Part I. (a). Biological reaction rate equation. Trans. Instn chem. Engrs 52, 248. Harremo~s P. (1978) Biofilm kinetics. In Water Pollution Microbiology (Edited by Mitchell R.). Wiley, New York. Kreyszig E. (1972) Advanced Engineering Mathematics, 3rd edition. Wiley, New York. Rittmann B. E. (1979) The kinetics of trace-organic utilization by biofilms. Ph.D. dissertation, Department of Civil Engineering, Stanford University, Stanford, Calif. Rittmann B. E. and McCarty P. L. (1978) The variableorder model of bacterial film kinetics. J. envir. Engng Div. Am. Soc. cir. Engrs 104, 889. Rittmann B. E. and McCarty P. L. (1980a) Model of steady state-biofilm kinetics. Biotechnol. Bioengng 22, 2343. Rittmann B. E. and McCarty P. L. (1980b) Evaluation of steady-state-biofilm kinetics. Biotechnol. Bioengng 22, 2359. Rittmann B. E. and McCarty P. L. (1981) Substrate flux into biofilms of any thickness. J. envir. Engng Div. Am. Soc. civ. Engrs 107, 831. Suidan M. J. (1986) Performance of deep biofilm reactors. J. envir. Engng Div. Am. Soc. civ. Engrs 112, 78. Suidan M. T. and Wang Y. T. (1985) Unified analysis of biofilm kinetics. J. envir. Engng Div. Am. Soc. cir. Engrs 111, 634. Williamson K. and McCarty P. L. (1976a) A model of substrate utilization by bacterial films. J. Wat. Pollut. Control Fed. 48, 9.
Williamson K. and McCarty P. L. (1976b) Verification studies of the biofilm model for bacterial substrate utilization. J. l'Vat. Pollut. Control Fed. 45, 281.