ISSN 0040-5795, Theoretical Foundations of Chemical Engineering, 2006, Vol. 40, No. 5, pp. 496–502. © Pleiades Publishing, Inc., 2006. Original Russian Text © L.L. Min’kov, S.V. Pyl’nik, J.H. Dueck, 2006, published in Teoreticheskie Osnovy Khimicheskoi Tekhnologii, 2006, Vol. 40, No. 5, pp. 533–539.
Steady-State Problem of Substrate Consumption in a Biofilm for a Square Law of Microbial Death Rate L. L. Min’kova, S. V. Pyl’nika, and J. H. Dueckb a
Physicotechnical Faculty, Tomsk State University, pr. Lenina 36, Tomsk, 634050 Russia b Department for Environmental Process, Engineering, and Recycling, University of Erlangen–Nuremberg, Paul-Gordan-Str. 3, 91054 Erlangen, Germany e-mail:
[email protected] Received June 20, 2005
Abstract—The steady-state problem of substrate consumption by a biofilm is solved over a wide range of biochemical parameters. It is shown that, regardless of the biofilm thickness, it can be completely saturated with the substrate. Ranges of applicability of analytical solutions for various problem parameters are presented. DOI: 10.1134/S004057950605006X
Biofilms occur when microorganisms (bacteria, algae, unicellular organisms) adhere to the interfaces between gas and liquid phases (e.g., the water surface), liquid and solid phases (e.g., the stony bottom), or two liquid phases (e.g., an oil drop in water) [1]. The adhesion of microorganisms to the interface can occur by various mechanisms. Of greatest importance are van der Waals forces, electrostatic attraction, or hydrogen bonds. In the course of their life, physiologically active microorganisms not only multiply but also produce a large number of substances, some of which form a biopolymer matrix used by microorganisms for their fixation. On scales much larger than the size of individual cells or organisms, biofilms are water-saturated thin mucous films at the interface. The overwhelming majority of microorganisms in nature live in such an aggregated medium. Their activity can have an adverse effect, e.g., if biofilms damage materials (biocorrosion) [2], or it can also be beneficial, e.g., in water purification in nature and technology [3]. The concentration of physiologically active microorganisms is maintained by continuous reproduction, which is possible if the supply of a substrate (and oxygen if the process is aerobic) is ensured. In the microporous matrix of the biofilm, the main mechanism of transport of reactants is diffusion. The rate of supply of the substrate to microorganisms within the biofilm obviously affects the process macrokinetics. The situation is largely similar to the case of heterogeneous reactions in a porous layer [4, 5]; however, for the biofilm kinetics, there are a number of specific features: the dependence of the biochemical reaction rate on the substrate concentration is generally described by a saturating curve and cannot be characterized by a power function [3, 6]; substrate consumption is possible only at a sufficiently high concentration of physiologically active
microorganisms in the biofilm, which is determined by both the microbial reproduction rate and the microbial death rate; biofilms are rather thin, on the order of 10–3–10–4 m; therefore, a model based on solving the diffusion equation in a semi-infinite region is here of limited use. The purpose of this work is to obtain and analyze approximate analytical solutions of the steady-state problem of substrate consumption in a biofilm for the previously substantiated square law of microbial death rate [7–9]. MATHEMATICAL FORMULATION OF THE PROBLEM The steady-state set of basis equations describing the biofilm kinetics in the one-dimensional approximation includes the following: the equation of balance between supply and consumption of the substrate in the biofilm 2
Sf d Sf = q --------------X D f ---------2 K + Sf f dz
(1)
under the boundary conditions z=0
dS f -------- = 0, dz
z = Lf,
S f = S1
(2)
(here, as usual, it is assumed that the substrate consumption is described by the Michaelis–Menten kinetics, the equations of which are derived on the basis of the theory of enzymatic reactions [3, 5, 6]); the biomass balance equation Sf 2 Yq --------------X = bX f (3) K + Sf f
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STEADY-STATE PROBLEM OF SUBSTRATE CONSUMPTION
(in this case, unlike the usually accepted linear law [3, 10], which does not ensure stability of the solution [7−9], it is assumed that the microbial death rate is proportional to the square of the biomass concentration. The square law is based on the reasoning that the death rate is actually proportional not only to the biomass concentration but also to the concentration of metabolic waste products, which, in turn, is proportional to Xf [8]). In formulation of the problem, the biofilm thickness is supposed to be a given constant value [3]. Actually, it is likely to depend on the properties of the environment of the biofilm, in particular, on the characteristics of water flow past the biofilm [7–9]. From Eq. (3), the concentration of physiologically active biomass can be expressed through the substrate concentration. As a result, Eq. (1) takes the form
SOLUTIONS AT LARGE BIOFILM THICKNESSES Let us seek the substrate flux into the biofilm at the boundary x =1 as δ ∞. Let us introduce the new variable y = x δ . Then, Eq. (5) under boundary conditions (6) can be written as 2
S 2 d S -------2- = ⎛ ------------⎞ , ⎝ 1 + S⎠ dy y = 0,
If the dependence of Xf on Sf is taken into account, the true reaction order varies from 2 at low Sf to 0 at high Sf. The effective reaction rate is determined by the dS substrate flux into the biofilm: J = Df --------f . To find dz z = L
S S = -----f , K
z x = -----, Lf
2
2
Yq L δ = ---------------f , bK D f
S S L = -----1 K
transforms problem (2), (4) to the form 2
S 2 d S --------2 = δ ⎛ ------------⎞ , ⎝ 1 + S⎠ dx x = 0,
dS ------ = 0; dx
x = 1,
(5) S = SL.
(6)
For further estimates, let us take the following values: for the kinetic parameters, K = 0.01 mg/cm3, q = 10 day–1, Y = 0.5, and b = 0.1 cm3/(mg day); for the film parameters, Df = 0.64 cm2/day and Lf = 0.001 cm; and for the substrate concentration in the liquid, S1 = 0.005 mg/cm3 [3]. At these values, δ = 0.8 and SL = 0.5. The biofilm thickness may reach several hundred micrometers. Correspondingly, δ may be as large as 100 or more. Similarly, a change in the substrate concentration outside the film within a typical range from 0.0005 to 0.05 mg/cm3 leads to a change in SL from 0.05 to 5. Thus, of interest are solutions of problem (5), (6) over wide ranges of the parameters δ and SL.
δ,
S = SL.
(8)
2
Ψ = 0;
S = SL,
Ψ = Ψ(
(9) δ)
;
(10)
where S0 is the (generally unknown) substrate concentration at the boundary y = 0 and Ψ ( δ ) is the sought value of the flux at the boundary y = δ . From Eqs. (9) and (10), we obtain the solution SL
Ψ(
f
this rate, it is necessary to solve a boundary-value problem. Introduction of the dimensionless variables and parameters
y =
d Ψ S 2 ------ ⎛ ------⎞ = ⎛ ------------⎞ , ⎝ 1 + S⎠ dS ⎝ 2 ⎠ S = S0 ,
(4)
dS ------ = 0; dy
(7)
dS Denoting Ψ = ------ , we reduce the order of Eq. (7): dy
2
2 d Sf q Y⎛ Sf ⎞2 -------- --------------- . D f ---------= 2 b ⎝ K + S f⎠ dz
497
δ)
dS = -----dy
= δ
y=
S 2 2 ⎛ ------------⎞ dS . ⎝ 1 + S⎠
∫
(11)
S0
b dS - J is the dimensionless = -----= ------------------2 dy KY q D f y= δ concentration flux into the biofilm and, consequently, the effective reaction rate. Obviously, as δ ∞ and at finite values of SL, the concentration S0 tends to zero (an unsaturated biofilm). Then, from Eq. (11), the expression for the substrate flux into the biofilm is obtained: Ψ(
δ)
Ψ(
δ)
=
1 2 ⎛ 1 + S L – -------------- – 2 ln ( 1 + S L )⎞ . ⎝ ⎠ 1 + SL
(12)
As numerical computation shows, Eq. (12) is valid at SL > exp[–1.7(δ – 1)0.25] and δ > 10. At SL 1, Eq. (12) implies 2 3/2 (13) ≈ --- S L , 3 which can be matched with the above applicability range at very large δ (e.g., δ > 150 if the solution is sought at 0.01 < SL < 0.1). At SL 1, the flux at the boundary can be approximately represented as Ψ(
Ψ(
δ)
δ)
≈ 2S L .
(14)
Thus, at δ 1, the effective reaction order varies from 3/2 at low SL to 1/2 at high SL.
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Thus, at sufficiently large δ ≥ 50, the effective reaction order varies from 3/2 at S1 < K to 1/2 at S1 > K and, with a further increase in the substrate concentration in the solution, changes to 0 at SL > δ/2. The maximal Ψ
Ψ(δ0.5) 100 d
value at a given δ is δ . The transition from the unsaturated to the saturated biofilm can also be illustrated by the case of a finite δ value. Equations (5) and (6) imply
c
10 b
δ
1 a
Ψ(
1
δ)
=
∫
S ⎞2 ⎛ ----------- dy . ⎝ 1 + S⎠
(16)
0
Introducing a certain average value of the integrand, we write
2
0.1
3 0.01 0.1
Ψ( 1
10
100
1000 SL
Fig. 1. Substrate flux on the biofilm surface: (1) solution (18) and solutions of problem (7) at δ = (2) 100 and (3) 400. The thin lines are asymptotes. Line a is the asymptote of solution (13); line b, solution (14); and lines c and d, solution (19).
Restrictions on the applicability of solution (12) can be obtained by considering the solution of Eq. (5), the right-hand side of which is unity (since SL 1), under 2 y –δ boundary conditions (6): S(y) = ------------- + SL. 2 This solution implies that the substrate concentraδ tion on the biofilm support is S0 = SL – --- . When SL is 2 sufficiently high, the concentration profiles at a given film thickness are similar and the difference of the concentrations on the outer and inner sides of the film is independent of SL. Once the concentration on the support becomes S0 = 0, we have 1 S L = --- δ. 2
δ)
S 2 δ ⎛ ------------⎞ . ⎝ 1 + S⎠
=
(17)
A concentrated substrate solution supplied by intense diffusion at relatively small biofilm thicknesses has no time to be completely consumed and saturates the biofilm. Thus, one can assume that the average integral substrate concentration in this case can be taken to be SL, which gives the following expression for the concentration flux: SL ⎞ 2 - . Ψ ( δ ) = δ ⎛ ------------(18) ⎝ 1 + S L⎠ Here, with an increase in the substrate concentration, the effective reaction order varies from 2 to 0. At SL 1, Eq. (18) leads to the expression Ψ(
δ)
=
δ,
(19) 2
Yq L or, in dimensional variables, the expression J = ---------------f , b which is independent of Df. Equation (19) describes the substrate flux into the saturated biofilm provided that SL 1. Equating the fluxes from Eqs. (14) and (18), we obtain the condition for the transition from the unsaturated to the saturated biofilm, which coincides with Eq. (15).
(15)
δ Thus, at substrate concentrations SL < --- , the film is 2 δ unsaturated, and at SL > --- , the film is saturated. 2 Figure 1 presents the substrate flux into the biofilm as a function of the substrate concentration on the biofilm surface. Figure 1 shows that, at fixed biofilm thickness, with an increase in the concentration SL, the substrate flux initially varies according to the law describing the unsaturated biofilm (Eq. (12)) and then, once the concentration SL = δ/2 is reached, the flux behaves as dictated by the law describing the saturated biofilm (Eq. (15)).
SOLUTIONS AT RELATIVELY SMALL BIOFILM THICKNESSES Up to now, only asymptotic concepts of the distributions of S across the film have been used. For relatively thin biofilms, of great importance is to calculate the concentration profiles, which can be done using a certain approximate method. Let us seek the solutions at small δ and SL by the method of linearization of the source. Let us consider the approximate equation obtained from Eq. (15) by linearizing its right-hand side: 〈 S〉 d S --------2 = δ ------------------------S, 2 ( 1 + 〈 S〉 ) dx 2
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STEADY-STATE PROBLEM OF SUBSTRATE CONSUMPTION Ψ(δ0.5)
1
∫
where 〈S〉 = S d x .
4
0
The analytical solution of Eq. (20) under boundary conditions (6) for the substrate concentration is obtained in the form SL - cosh ( δ 1 x ), S = ------------------------(21) cosh ( δ 1 ) 〈 S〉 where δ1 = δ ------------------------2 . ( 1 + 〈 S〉 ) Correspondingly, SL - tanh ( δ 1 ). 〈 S〉 = -------δ1
δ)
2 12S L -⎞ . δ ⎛ ---------------------------------⎝ 12 ( S L + 1 ) + δ⎠
=
(24)
(25)
and the flux is written as δ)
=
2S L ⎞ 2 - . δ ⎛ --------------------⎝ δ + 2S ⎠ L
3 4
(22)
after taking δ 1 3, at which tanh δ 1 ≈ 1. In this case, the average integral substrate concentration is represented as
Ψ(
2
3
1
This formula qualitatively describes the behavior of the flux as a function of SL for δ values up to 20, as is seen from Fig. 2. At sufficiently high δ values 10 < δ < 50 and moderate substrate concentrations SL, we use Eq. (22) again
2S 〈 S〉 = --------L δ
1
2
δ Since the maximal δ1 is --- , we will consider δ1 < 1 4 tanh ( δ 1 ) ⎛ δ –1 - ≈ 1 + -----1⎞ . To find and use the expansion -----------------------⎝ 3⎠ δ1 〈S〉, we solve Eq. (22) using the iteration method after substituting 〈S〉 = 1 into the right-hand side. In this approximation, the averaged concentration is represented as 12S L 〈 S〉 = -------------(23) 12 + δ and the expression for the substrate flux based on Eq. (17) takes the form Ψ(
499
(26)
The effective reaction rate as a function of the substrate concentration varies here, as in Eq. (24), from one described by the square law for a dilute solution (an unsaturated biofilm) to one independent of the substrate concentration for a concentrated solution (a saturated biofilm).
0 0.1
1
10
100 SL
Fig. 2. Substrate flux into the biofilm at δ = (1) 1, (2) 4, (3) 10, and (4) 20. The solid lines are numerical solutions. Dashed lines 1 and 2 are described by formula (24), and dashed lines 3 and 4, by formula (26).
It is easy to see that Eqs. (18), (24), and (26) are similar and can be represented in the form 1 ⎞2 Ψ( δ) - , ----------------- = ⎛ ----------------(27) ⎝ 1+αS –1⎠ δ L where α in Eqs. (24) and (26) differently depends on δ. Ψ δ Figure 3 presents ------------ as a function of SL at δ extreme α values bordering the δ range from 1 to 50. Obviously, the transition from the square law to the Ψ δ complete independence of ------------ from SL is gradual. δ One can determine (as Fig. 3 shows) the center SL, t of the transition range as the point of intersection of the S 2 Ψ δ Ψ δ asymptote ------------ = ⎛ ----L-⎞ with the ordinate ------------ = 1; ⎝ ⎠ α δ δ Ψ δ i.e., SL, t = α. At this point, ------------ = 0.25. The asymptote δ reaches this value at SL = 0.5α. This point can be taken to be the beginning of the transition range. Then, the transition range width ∆SL, t = α. In Eq. (24), we have α = 1 + δ/12, whereas, in Eq. (26), we have α = δ /2. Therefore, it can be concluded that, the thicker the biofilm, the higher the substrate concentration at which the biofilm becomes saturated and the longer the transition to the saturated state. The biofilm saturation can be characterized by the ratio of the substrate concentrations on the support and
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MIN’KOV et al. Ψ( δ) ----------------δ 1
S0/SL 1
∆SL, t
0.8 0.1 0.6 1 0.01
0.4 2
1 2 3 4
0.2 0.001 0 0.001 0.0001 0.01
0.1
1
10
100 SL
Ψ δ Fig. 3. ------------ as a function of SL at α = (1) 1.1 and (2) 3.5. δ
at the interface with the aqueous solution. Approximately, this ratio can be obtained from Eq. (21): S 〈 S〉 ----L- = cosh ⎛ δ ------------------------2⎞ . ⎝ S0 ( 1 + 〈 S〉 ) ⎠
(28)
Figure 4 shows the curves calculated by Eq. (28) using Eqs. (23) and (25). Obviously, the thicker the biofilm (the larger δ), the more difficult its saturation by diffusion and the lower the curve in Fig. 4. At low SL, an increase in the substrate concentration in the solution leads to acceleration of the biochemical reaction at the interface between the film and the solution and, consequently, to a decrease in the substrate concentration S0 near the support. At SL > 1, the reaction rate becomes less and less sensitive to an increase in the concentration and the biofilm saturation with the substrate increases. The minimum in the curve for the saturation slightly shifts toward higher SL. At δ on the order of 100 or more, the minimum drops down to almost zero and a noticeable increase in S0 is observed at SL on the order of 1000, which is likely to be beyond the range of realistic values. RESULTS AND DISCUSSION
0.01
0.1
1
10
100
Fig. 4. Biofilm saturation as a function of SL at δ = (1) 1, (2) 4, (3) 10, and (4) 40. The points represent the results of calculations by formula (28); the solid lines are the results of numerical calculations.
The comparison parameter was taken to be the quantity Ψ an ( δ, S L ) -, Φ ( δ, S L ) = 1 – ------------------------------Ψ com ( δ, S L ) where Ψan( δ , SL) and Ψcom( δ , SL) are the results of calculating the concentration flux using one of the analytical formulas and the numerical solution, respectively. For quite a large set of points in the plane ( δ , SL), we sought the minimal value of Φ( δ , SL) by testing all the analytical formulas. The analytical formula for which Φ( δ , SL) was minimal was considered preferable for a given point. Figure 5 shows the boundaries of five regions for each of which a preferable formula can be presented. The boundaries of applicability of the formulas can be described by the following functions: curve 1, SL = exp[±1.7(δ – 1)0.25],
(29)
where the plus and minus signs refer to the upper and lower branches, respectively; curve 2, SL = 0.5δ2/(δ – 10);
(30)
curve 3, SL = 0.0013δ3.8;
(31)
curve 4, δ = 4.5 + 31.7SL – 30.2 S L . 2
The analytical formulas were obtained by various approximate methods, the strongest substantiation of which can be only their comparison with the numerical solution of problem (7), (8).
1000 SL
(32)
On crossing each of the boundaries, there is no continuous transition from one solution to another. The differences between the calculated values of the concen-
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STEADY-STATE PROBLEM OF SUBSTRATE CONSUMPTION
trated flux into the biofilm on opposite sides of a boundary reach 20%. Within each of the regions, the discrepancies between the solution obtained by the preferable formula and the numerical solution are within the range 5–10%; i.e., Φ( δ , SL) < 0.1. The total result of the study is that the biofilm productivity increases with an increase in the biofilm thickness and the substrate concentration in the solution. An analysis of the problem shows that, for each biofilm of fixed thickness, there is a limit of the consumed substrate flux for a highly concentrated solution. This follows from Eq. (18) rewritten in dimensional vari2 Yq Lf ables as Jmax = --------------- , where Jmax is independent of Df. b The limit exists because the reaction rate reaches a maximum. The maximal consumed substrate flux is proportional to the biofilm thickness. The expression for Jmax also contains the maximal biochemical reaction rate q and the rate of death of the active biomass, which naturally affect the biofilm thickness Lf. In this context, it is necessary to further develop the model involving the balance of production of active biomass and biofilm erosion [7–9]. For a dilute solution, the biofilm productivity is lower than the maximum possible and depends on the substrate concentration in the solution. The dependence of the substrate consumption rate can be described by the formula 2
Yq L S 2 J = ---------------f ⎛ -----1⎞ . b ⎝ K⎠ At very large Lf, J depends on SL and the kinetically controlled mode is replaced by the mode controlled simultaneously by diffusion and kinetics. In this case, regardless of Lf, the concentration flux into the biofilm can be represented as 2
S KY q D J = --------------------f f ⎛ -----1⎞ , ⎝ K⎠ b S where f ⎛ -----1⎞ can be expressed as a power-law function ⎝ K⎠ S S n S f ⎛ -----1⎞ = ⎛ -----1⎞ , with n depending on -----1 . ⎝ K⎠ ⎝ K⎠ K S At -----1 < 1, Eq. (13) implies that n ~ 1.5, and in a K S wide range -----1 > 1, n = 0.5. In this range, Eq. (14) is K 2
2S 1 Y q D f preferable, according to which J = ------------------------ is indeb pendent of Lf.
501
SL 100 II 10 1 III
2
1 IV
V
3
0.1
4
I 0.01 0.001 0.1
1
10
100
1000 δ
Fig. 5. Regions of preferable analytical solutions: (I) (18), (II) (19), (III) (24), (IV) (26), and (V) (12). The curves are described by the following equations: (1) (29), (2) (30), (3) (31), and (4) (32).
2bD f ------------ , the lim2 Yq iting value Jmax is reached, at which the reaction order tends to zero. Finally, as Eq. (15) shows, at Lf >
Thus, the region of the parameters SL and δ consists of subregions in each of which there are approximate analytical expressions for the diffusion substrate flux into the biofilm. The productivity of substrate consumption by the biofilm increases with an increase in the biofilm thickness and the substrate concentration in the solution. The biofilm saturation depends nonmonotonically on the substrate concentration in the solution. The saturation is close to unity at low and high substrate concentrations; in the former case, this is caused by a low reaction rate, and in the latter case, this is because of a weak concentration sensitivity of the reaction rate. ACKNOWLEDGMENTS This work was supported by Bayerische Forschungsstiftung (Germany) and also in part by the Ministry of Education and Science of the Russian Federation and the US Civilian Research and Development Foundation within the framework of the program “Basic Research and Higher Education” (project no. 016-02).
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NOTATION b—microbial death constant, cm3/(mg day); Df —diffusion coefficient within the biofilm, cm2/day; J—substrate flux into the biofilm, (mg cm2)/day; K—Michaelis–Menten constant, mg/cm3; Lf —biofilm thickness, cm; q—substrate consumption rate constant, day–1; S—dimensionless substrate concentration in the biofilm; Sf—substrate concentration in the biofilm, mg/cm3; SL—dimensionless substrate concentration outside the biofilm; S1—substrate concentration outside the biofilm, mg/cm3; t—time, days; Xf —concentration of physiologically active microorganisms, mg/cm3; x, y—dimensionless coordinates; Y—biomass yield per unit amount of substrate consumed, mg/mg; z—coordinate, cm; δ—dimensionless biofilm thickness; Ψ ( δ )—concentration flux into the biofilm. SUBSCRIPTS AND SUPERSCRIPTS an—analytical value; com—computational value; f—film; L, l—solution; max—maximal value; t—transition; 0—interface between the biofilm and the support.
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