Environ. Sci. Technol. 1987, 21, 273-280
Mathematical Modeling of Biofilm on Activated Carbon H. Ted Chang" and Bruce E. Rittmann
Department of Civil Engineering, University of Illinois, Urbana-Champaign, Illinois 6 1801
w A mathematical description for the kinetics of biofilm on activated carbon (BFAC model) is derived. The model incorporates film transfer, biodegradation, and adsorption of a substrate, as well as biofilm growth. The modeling problem is uniquely characterized, because it involves diffusion across a moving boundary, diffusion with nonlinear reaction, and diffusion in layered media. The detailed derivation and solution procedure are developed to account for the unique features of the model. The BFAC model is solved with a global orthogonal collocation method (GOCM), which not only provides an efficient solution for the substrate concentration but also allows easy computation of the other key quantities, such as substrate fluxes.
Introduction Because biofilm is a dominant form of biological activity in an environment where the organic concentration is low and specific surface area is high (1-3), understanding biofilm kinetics can help researchers understand the fate of the organic contaminants in many natural environments. In addition, biofilm processes have been employed in traditional wastewater treatment to remove organic materials, nitrogen species, and other contaminants. In recent years, adsorptive media, such as activated carbon, have been used in anaerobic fluidized-bed biofilm reactors (4), in biologically active filters of granular activated carbon (GAC) used to produce drinking water (5, 6), and to upgrade activated sludge treatment (7,B). Adsorbing media offer at least three potential advantages: (1)adsorption and sequestering of inhibitory materials (9),(2) adsorption and retention of slowly biodegradable compounds (IO),and (3) superior attachment of the microorganisms (11). Advancements in the fundamental understanding of how the microorganisms, substrate, and surface interact have not kept pace with the increased interest in and use of biofilm on activated carbon. The objective of this paper is to derive, explain, and demonstrate a model of biofilm on activated carbon (the BFAC model). The model is based upon the fundamental mechanisms of film transport, biodegradation within the biofilm, adsorption within the GAC, and growth of the biofilm. Combination of all these mechanisms provides a modeling problem having unique characteristics and requiring a specially tailored solution technique. A companion paper (12) presents experimental results that verify that the model accurately describes the activity of biofilm on activated carbon. Theoretical Concepts The physical concept of biofilm on activated carbon (BFAC) is shown in Figure 1. The first major component is the activated carbon. The external surface of activated carbon acts as a supporting medium for the attached growth of bacteria. Activated carbon is highly porous, and ita interior surface has high adsorptive capacity for organic solutes, including many of those found in water sources or industrial and municipal wastewaters (13-15). The second major component is the biofilm. A biofilm is a layer-like aggregation of microorganisms attached to 0013-936X/87/0921-0273$01.50/0
a solid surface (3). Biofilms are idealized as being composed of a homogeneous matrix of bacteria and the extracellular polymers that bind the bacteria together and to the surface (16). Models of biofilm kinetics (1-3,17-20) have been derived and shown to be adequate representations for the simpler case of an inert (i.e., nonadsorbing) surface. Basic Assumptions. In order to write the mathematical expressions describing a biofilm on activated carbon, 12 necessary assumptions must be made: (1)activated carbon is a homogeneous spherical particle (2) no biological reaction occurs inside the carbon particle (3) adsorption of the substrate is completely reversible (4) local equilibrium between the surface concentration and liquid concentration occurs at the biofilm-carbon interface ( 5 ) the biofilm is homogeneous (6) biofilm density is constant (7) biofilm curvature effects are negligible (8) any increase in biofilm thickness is due to the growth of biofilm, because attachment of bacteria from the liquid is negligible (9) the biofilm shear loss rate is first order with respect to biofilm mass (10) a single substrate is both diffusion and reaction limiting in the biofilm (11)Fick's law governs diffusion in the biofilm and activated carbon (12) biofilm growth does not affect the flow pattern of liquid in a reactor because the duration of the experiment was sufficiently short that biofilm growth did not alter the porosity significantly Two implications of the above assumptions are important to note here. First, the assumption of a homogeneous biofilm (number 5 ) implies that the substrate concentration does not vary laterally on the carbon surface and that diffusion is in the radial direction in the biofilm only. Second, the constant biofilm density assumption (number 6 ) implies that the biofilm is a dynamic one. The decaying portion continuouslygives up space to the growing portion in such a manner that the biofilm density remains constant in all parts of the biofilm (2). The shear loss rate of biofilm seems to be a complicated function of factors such as biofilm density, extracellular polymerization, medium shape, and shear stress. The mechanisms are not well understood, and mechanistic models are not available yet. Hence, the first-order approach developed by Rittmann (21) is used here. Variations in the first-order shearing coefficient are used to represent the different interactions between biofilm and activated carbon. Figure l a illustrates typical substrate-concentration profiles in biofilm and activated carbon, while Figure l b defines the coordinate systems used for modeling. The mathematical equations describing substrate transport, utilization, and adsorption, as well as b i o f i i accumulation, can be written, going from inside the carbon particle out, with appropriate boundary conditions imposed. Intraparticle Diffusion. A homogeneous solid diffusion model for activated carbon adsorption is used in this
0 1987 Amerlcan Chemical Society
Environ. Sci. Technol., Vol. 21, No. 3, 1987 273
BC3: qw I= Kq(S,)l'n
(4)
where K = Freundlich isotherm coefficient (M,'-1fnL-3fn/Mq),n = Freundlich isotherm exponent, qw = surface concentration at biofilm/activated carbon interface ( M J M J , and S, = substrate liquid concentration at biofilm/activated carbon interface ( M s / L 3 ) . Diffusion with Reaction i n Biofilm. Biofilm modeling was advanced by Rittmann and McCarty (1-3),who based their models on diffusion (Fick's law) and biological reaction (Monod kinetics) within the biofilm and liquidlayer mass transfer from the bulk liquid. Put in a nonsteady-state form, the differential equation for diffusion with Monod reaction within the biofilm is (2) BULK LIQUID
DlFNSlON LAYER
BlOFlLM
ACTIVATED CARBON
Figure 1. Conceptual basis of biofilm on activated carbon: (a) concentration profiles; (b) coordinate systems.
research. The equation is a typical diffusion equation in spherical coordinates (22):
where q = surface concentration of adsorbed substrate (M,/Mq),D, = surface diffusivity (L2/T),r, = radial coordinate in activated carbon ( L ) ,t = time (T), and R = radius of carbon particle (L). The first boundary condition (BC1) for the above equation is the symmetry of the concentration profile with respect to the center of the carbon particle (15); thus
aq -0 BC1: - r, = 0, t 1 to ar, The other boundary condition (BC2) can be found at the activated carbon/biofilm interface. As long as extracellular enzymes are not actively adsorbed in the micropores, no reaction occurs inside the carbon particle. Hence, the substrate mass entering the activated carbon from the biofilm must be equal to the increase in adsorbed substrate in the activated carbon:
where Sf = rate-limiting substrate concentration in the biofilm (M,/L3),D f = substrate diffusion coefficient in the biofilm ( L 2 / T ) ,rf = radial coordinate in the biofilm ( L ) , and p = apparent particle density of dry activated carbon (Mq/E3).The initial condition depends on the operational condition and will be shown later. Adsorption Isotherm. An adsorption isotherm is used to relate the concentrations in the liquid phase to the adsorbed solid phase. Since a Freundlich isotherm describes the equilibrium data over a wide range (15),it is used: 274
Envlron. Sci. Technol., Vol. 21,
No. 3, 1987
as,
a2sf
ksf
-=Df2X fat drf K, + Sf
0 5 rf 5 L f
(5)
where K = maximum specific rate of substrate utilization [ M s / ( M x T ) K, ] , = half-velocity concentration (M,/L3),X , = cell density of biofilm ( M x / L 3 )and , L f = biofilm thickness (L). Two boundary conditions (BCs) and one initial condition (IC) are required for this equation. Since the biofilm shares one boundary with activated carbon, the boundary condition at the biofilm/activated carbon interface (eq 3) must also be satisfied for eq 5. On the other side of the biofilm, at the liquid/biofilm interface, the flux entering the interface from the liquid film must be equal to the flux leaving the interface into the biofilm:
where S b = substrate concentration in the bulk liquid (M,/L3), S, = substrate concentration at liquid/biofilm interface (M,/L3), and kf = liquid-film mass transfer coefficient (L/T). The mass transfer coefficient is inversely proportional to the thickness of the diffusion layer (3) L1:
L1 = D / k f
t 7)
where D = substrate diffusion coefficient in the bulk liquid (L2/r )* Growth of Biofilm. As the substrate diffuses into and through the biofilm, the biomass utilizes substrate for biosynthesis and respiration. The biomass can increase or decrease with time until the growth rate is just balanced by the biomass decay and shear losses, at which time a steady-state biofilm results (2). Since the density of the biomass in the biofilm is assumed constant, the volume of biofilm, and thus the thickness of biofilm, must increase with time as the biofilm grows. Therefore, the substrate diffuses through a boundary, the liquid/biofilm interface, which can be moving with time. One equation is required to describe the movement of this moving boundary (2):
where Y = true yield of biomass ( M J M , ) , b = biofilm decay coefficient (l/!P), and b, = biofilm shear loss coefficient (l/T) (21). Biofilm density, Xf, drops out when it is assumed constant. Only one initial condition is required for this ordinary differential equation (ODE): an initial biofilm thickness must be known for the biofilm. The above differential equations, boundary conditions, and initial conditions represent the fundamental phenomena of the BFAC model. Features of the Model. Recognizing the critical features of the model is necessary to select an efficient solu-
tion technique. The model has the following special features: (1) Moving Boundary. The biofilm thickness (Lf)and, hence, the boundary through which substrate diffuses can change during the transient state. Examples of other moving-boundary problems can be found in heat transfer with freezing or melting ice (23),in gas absorption with rapid chemical reaction, and in diffusing of oxygen in absorbing tissue. (2) Diffusion with Nonlinear Reaction. The biological reaction is described by the Monod relationship, which is a nonlinear expression. The nonlinear term makes analytical solution impossible, except in certain restrictive cases (24). (3) Diffusion in Layered Media. There are two layers in series (Figure 1)in this process: namely, the biofilm and the activated carbon. Analogous problems can be found in heat transfer in composite material and in groundwater hydrology where groundwater may flow through an aquifer that consists of heterogeneous media (25). Completely Mixed Biofilm Model A simple reactor system in which the BFAC model can be applied is the completely mixed biofilm reactor. All the particles are the same, and all bacteria at the liquid/biofilm interface are exposed to the same substrate concentration. A completely mixed biofilm reactor containing biofilm on activated carbon can be a stirred tank reactor or column reactor having a sufficiently high recycle flow rate. The complete BFAC model for the completely mixed flow reactor is assembled from the component models described in previous sections and the following equations of reactor mass balance and standard initial conditions:
Solution to BFAC Model The above differential equations for the model can be simplified by first defining the following dimensionless variables, which have an asterisk q* = 4 / q 0
(16)
s,* = Sf/&
(17)
sb* = s b / s O
(18)
K,* = K,/So
(19)
Xf* = Xf/SO
(20)
x,* = X,/Xf
(21)
re* = r,/R
(22)
rf* = r f / L f
(23)
where qo = Kq(S0)l/".Then the equations for the model become, in dimensionless form
asf* Df a2sf* -=--at ~ f drf*2 2
-=(
dX,* dt
ksf* K,* Sf*Xf*
0 Irf* I1 (25)
+
YkSf* K,* + Sf*
dt Yksb* K,* sb*
(27)
+
kSb* Xs*Xf* (28) K,* + sb*
0 Ir,* I1, t = to IC2: Sf*= 0 0 Irf* I1, t = to IC3: Lf = Lfo t = to IC1: q* = 0
IC1: q = 0
0 Ir, IR , t = to
(11)
IC2: Sf = 0
0 Irf ILf, t = to
(12)
IC3: Lf = Lfo IC4: X, =X,o
t = to (13) t = to (14) IC5: S b = 0 t = to (15) where So = substrate concentration in the feed (M8/L3), X, = weight of carbon in the reactor (MJ, X, = biomass concentration for suspended growth (Mx/L3),Q = flow rate of the feed solution (L3/T), V = empty bed volume of the reactor (L3),and e = porosity of the reactor. Equation 9 is the mass balance for the suspended biomass, and it is similar to the mass balance for a chemostat reactor, except that an extra term is included for the increase of suspended biomass due to the shear of attached biomass. Equation 10 is a material balance relationship for the substrate in the bulk solution. All the initial substrate concentrations in the reactor are assumed zero for a standard run; however, initial biofilm thickness must be set to a small initial value in order that the biofilm can start to grow. A solution for eq 1-15 consists of the substrate concentrations in the carbon particle, the biofilm, and the bulk liquid, as well as the biofilm thickness, as function of time.
IC4: X,* = Xso* IC5: sb* = 0 BC1: aq*/ar,* = 0
-1
0 8 0 as,* BC2: =p Lf arf* ff.=o pq
t = to t = to rs* = 0, t = to
&-s a
at o
BC3: S,* = (qw*In
1
(29) (30) (31) (32) (33) (34)
q*r,*2 dr,*
(35)
r,* = 1
(36)
+
where b' = b b, (21). Because the above model consists of integral and differential equations with nonlinear reaction and adsorption terms, analytical solution is impossible. Numerical techniques must be employed. Finite difference (FDM) and finite element (FEM) methods have been used extensively to solve diffusion equations since the advent of the high-speed digital computer. The conveniences of the two methods are combined by using orthogonal polynomial expansions, fitted by a collocation technique (26). The orthogonal collocation method (OCM) has been gaining favor in solving the activated carbon adsorption model. It is generally more efficient than FDM with respect to computer time (27-29). Environ. Sci. Technol., Vol. 21, No. 3, 1987
275
However, the choice and application of the OCM to solve the BFAC model require consideration of the three model features. Numerical techniques solving moving boundary problems were discussed extensively by Crank (30). In general, the problem with a moving boundary is that the domain of interest changes constantly and thus needs to be rediscretized at every time step. In one example, he first used a similarity transformation to remove the singularity at the fixed boundary at time zero and then used a second transformation to fix the moving boundary. The fixing of the boundary was nothing more than normalizing the domain from zero to one. Therefore, the moving boundary was fixed at x = 1 in the dimensionless sense, and the diffusion equation was solved in the dimensionless domain. The equation of a moving boundary was then solved to keep track of the movement of the boundary in the physical domain. Fixing the boundary in a dimensionless domain (from x = 0 to x = 1) is an easy and efficient technique to handle the growth of biofilm. The technique also renders the diffusion in the biofilm equation easily solvable by OCM. Notice that eq 23 fixes the moving boundary by normalizing the location in the biofilm, rf, to the biofilm thickness, Lfi With the moving boundary fixed in the dimensionless domain, eq 25 is solved for the domain of zero to one. Equation 25 can be solved easily for the dimensionless concentration profile in the biofilm. The position of the moving boundary (Le., the actual biofilm thickness in physical domain) can, in turn, be found by solving eq 26. Finlayson (31) reported that OCM is particularly effective for nonlinear diffusion problems, such as a temperature-dependent catalytic reaction. The nonlinearity of the biological reaction in the activated carbon/ biofilm model also can be handled easily by this technique. The boundary conditions of the problem of diffusion in layered media generally are continuities of concentrations and fluxes. A solute-transport problem with nonlinear adsorption in layered media was solved with OCM successfully (25). The boundary conditions were handled easily and satisfied exactly in terms of the trial orthogonal functions. The complicated boundary conditions in the activated carbon/biofilm model also can be handled easily and accurately by OCM. On the basis of the above analyses, an OCM is well suited to solve the BFAC model. In particular, a global orthogonal collocation method (GOCM) is used in this research. The prefix global differentiates the GOCM from another method that uses orthogonal collocation in finite elements. Details of the GOCM can be found in Villadsen and Stewart (26) and Finlayson (31). The choice of the trial orthogonal polynomials depends highly on the domain of interest and boundary conditions. Since the concentration profile in the carbon is symmetric with respect to the origin (i.e,, particle center), Legendre polynomials (an even function) in spherical coordinates are used to approximate the exact concentration profile. Shifted Legendre polynomials in planar geometry with no special characteristics are used to approximate the concentration profile in the biofilm. The distribution of collocation points in biofilm and activated carbon is shown in Figure 2. The coordinates of the collocation points are dictated by the roots of the corresponding orthogonal polynomial. Therefore, they need not be equally distributed in the biofilm and activated carbon, nor need they be distributed in the same way in each component, unless the same or276
Envlron. Sci. Technoi., Vol. 21, No. 3, 1987
BIOFILM
LIQUID F'ILM
ACnVAlED CARBON
L
I
I
I
I
I
I
I
I
I
I
I
i
I
I
i
I
.. 3
2
N+2 N + l N .
1 N + l N N-1
2 1 Flgure 2. Distribution of collocation points In biofilm and actlvated carbon domains.
..,.e
represents the center of the activated carbon.
thogonal polynomial is chosen for both components. The collocation point 1 in the carbon is near the center of the adsorbent particle, whereas the point N + 1 is at the biofilm/activated carbon interface. The collocation point 1 in the biofilm is at the biofilm/activated carbon interface, whereas the point N 2 is at the liquid/biofilm interface. Defining the following symbols helps simplify the GOCM equations:
+
(44) (45)
Applying GOCM and the newly defined variables to the dimensionless equations (eq 24-37) yields the following equations for the BFAC model: dq*i dt
N+1-
- D,, j=l C BQijq*j .
i = 1, 2, ...,N
Df x 108N+2XfkSf*i -dSf*i -BFijSf*j dt Lf2 j=1 K,* + Sf*i
i = 1, 2, ..., N
(47)
+ 1 (49)
flux means substrate is desorbed out of activated carbon. The flux across the biofilm/activated carbon interface is computed with I
+1
(53)
...,N + 2
(54)
IC1: q*i = 0
i = 1, 2, ..., N
IC2: Sf*j= 0
i = 1,2,
t = to
IC3: Lf = Lfo IC4: X,* = X,o* IC5:
sb*
t = to
=0
t = to
sf*1 = (q*N+l)n
BC3:
(55) (56) (57) (58)
N+1KfdL+b*
-
j=l
AFN+2JSf*j
(59) AFN+2,N+2 + KfdLf -In the above equations, BF, AF, and are collocation BC4: sf*,+, =
e
matrices to replace the Laplacian, gradient, and integral operators,respectively, in the biofilm domain. Collocation matrices BQ, AQ, and function similarily in the activated carbon domain. The collocation matrices are generated by using the roots of trial polynomials (32)and the procedure outlined by Finlayson (31). Boundary condition 1 (eq 34) is satisfied automatically by Legendre polynomials; thus, it no longer appears explicitly in the model. Boundary condition 2 is contained in eq 48, which is a combination of eq 35 and 47. The unit for biofilm thickness is micrometers (pm), while all other length units are in centimeters (cm). The application of GOCM results in the replacement of spatial derivatives and integrals with matrices, thus reducing the number of independent variables. Elimination of the spatial derivatives in the diffusion equations and the integral in the biofilm growth equation reduces the activated carbon/biofilm model to a system of first-order ODES that is an initial value problem with respect to time. The ODES are then integrated numerically to simulate the transient operation of the BFAC process. Since the time scale for biological growth is muclilarger than for physical diffusion, the above system is relatively stiff. A subroutine called LSODE (33)was chosen to integrate the resulting stiff system of first order ODES.
Jq
= Ppa-
where Jq= flux of substrate across the biofilm/actictated carbon interface [M,/(L2T)]. GOCM solution of the BFAC model provides an easy way to compute the concentration gradients (31)in biofilm and activated carbon. Equation 62 is transformed into the following equation to compute the flux: P$sQoN+l-
Jq = -C AQN+ljQ*j R j=1
(63)
The rate of the substrate utilized by the biofilm can be computed by integrating the substrate utilization across the entire thickness of the biofilm. This value is denoted as rb and is computed with
m,
Additional Computations Some quantities, besides substrate concentrations, are important for understanding the BFAC process. These quantities are easily computed during the GOCM solution. The first of these quantities is the flux of the substrate from the bulk liquid into the biofilm. Since the concentration profile across the stagnant liquid film is linear, the flux is easily computed with
Jb
The amount of substrate adsorbed per unit mass of the activated carbon can be computed by integration of the concentration profile in the carbon particle. This value is denoted as W, and is computed with
The integral is replaced with the quadrature formula, resulting in (67) The mass of the substrate stored in the biofilm can be computed by integrating the concentration profile across the entire thickness of the biofilm. This value is denoted as w b and is computed with
The equation is then solved in GOCM by the quadrature formula as follows:
= kf(sb - sa)
(60) where J b = flux of substrate from bulk liquid into biofilm [Ma/ (L2T)]and other variables were defined previously. The GOCM formulation of eq 60 is Jb
The scheme of computing integration with GOCM is equivalent to using the Gauss-Legendre quadrature formula (31). The equation is then solved by the quadrature formula in GOCM as follows:
= kfSO(Sb* - s f * N + d
(61)
A second important quantity is the flux of substrate across the biofilm/activated carbon interface. The flux is important because it tells whether activated carbon is adsorbing or desorbing substrate. According to the coordinate systems defined in Figure lb, when the value of flux is positive, substrate is diffusing from the biofilm into the activated carbon. On the other hand, a negative value of
In order to compute the concentration of carbon dioxide produced due to bioreaction, a mass balance must be formulated for the utilization of the organic substrate by the biofilm and suspended biomass, as well as the growth and respiration of the biofilm and the suspended biomass:
where C = concentration of carbon dioxide due to the Envlron. Sci. Technol., Vol. 21, No. 3, 1987
277
1
n.
I
NUMBER OF COLLO. POINTS IN ACT. CARBON- 3 NUMBER OF COLLO. POINTS IN ACT. CARBON= 6 NUMBER OF COLLO. POINTS IN ACT. CARBON= 9
1.0
I
-.....-
NUMBER OF COLLO. _ _ _NUMBER OF COLLO.
POINTS IN BIOFILM- 3 POINTS IN BlOFlLM- 6 NUMBER OF COLLO. POINTS IN BIOFILM- 9
"t
2
0.6
0
0
L
/
l
0.0 I 0
I
\
I PO
40
60
Time, h o u r s
80
100
110
Figure 3. Effect of number of internal collocatlon points in activated carbon to computer solution of BFAC model. Note that all three curves coincide with one another.
biodegradation of organic substrate ( M / L 3 ) ,A = total surface area of the attaching medium (L2), a = conversion factor for the utilization of organic substrate to carbon dioxide (=2.81 mg of C02/mg utilized for phenol, which was used in experimental verification), and = conversion factor for the respiration of cells ( 4 . 9 5 mg of C02/mg of cell oxidized). Equation 70 can be solved for quasi steady state, for which carbon dioxide concentration can be computed by
Total carbon in the effluent is the summation of the carbons in the substrate, carbon dioxide, and suspended biomass.
Analysis of Numerical Accuracy As the number of internal collocation points, N , used in the GOCM is increased, the accuracy of the solution improves. An investigation of the effects of the number of internal collocation points on the effluent substrate prediction is shown in Figures 3 and 4. The test data used were those for the BFACl experiment, as listed in the accompanying article (12). Figure 3 shows the effect that the number of internal collocation points in activated carbon had on the computer solution) when the number of internal collocation points in the biofilm was fixed at 6. The results show that using three, six, and nine points in activated carbon gave virtually the same effluent substrate curves. Figure 4 shows that the effluent substrate curves were virtually the same for six and nine points in the biofilm, when the number of internal collocation points in activated carbon was fixed at six. For three points in the biofilm, the substrate curves were the same up to about 50 h. Thereafter)three points gave slightly higher effluent concentrations. As a result of this analysis, the value of N was chosen to be six in the biofilm and the activated carbon. In general, N need not be the same for activated carbon and biofilm. The total number of ODES to be integrated is 2(N + 1) + 1. The time step for the integration was controlled by the error tolerance of the solution. To reach the same time, a stricter error tolerance requires a smaller time step and expends more computer time. The computer solution was checked by calculating mass balances for the substrate accumulated in the biofilm and in the activated carbon. Mass balance error is defined by comparing the total mass input to the biofilm during a time step with the net increase in the accumulated mass in the biofilm and acti278
Environ. Sci. Technol., Vol. 21, No. 3, 1987
0.0
I
0
20
40
60
Time, hours
80
100
110
Figure 4. Effect of number of internal collocation points in biofllm to computer solution of BFAC model. Note that the curves for six and nine points coincide with each other.
vated carbon plus the mass utilized by the biofilm. The mass balance error was always less than 1 % )when the error tolerance was set equal to 10" and the number of the collocation points in the biofilm and the activated carbon was 6. A tolerance of was used for the subwquent simulations. All the programs were coded in FORTRAN language and executed with a Cyber 175 computer (Control Data Corp., Sunnyvale, CA) at the University of Illinois) Urbana-Champaign, IL.
Discussion The BFAC model presented in this paper is an extension of the original biofilm models (1-3) to include the adsorption of substrate by the supporting medium to which the biofilm is attached. All the basic mechanisms of the biofilm kinetics are included. The incorporation of the adsorption effect does not alter the equations of the biofilm model. However, a homogeneous solid diffusion model is included to describe the movement of the substrate in the activated carbon. A new boundary condition for the continuity of the substrate flux must be satisfied at the biofilm/activated carbon interface. The resulting BFAC model has three features: diffusion across a moving boundary) diffusion with nonlinear reaction, and diffusion in layered media. To account for these unique features, the GOCM is employed to provide an easy and accurate solution for the model. Since the solution of the substrate profile is a single piece of continuous curve in the domain of interest (31))the solution also provides easy and accurate computations for the fluxes. The flux across the liquid/ biofilm interface and the surface area determine the rate of substrate removal in a biofilm process. The basic BFAC model is applied to a completely mixed flow reactor in this study. Its application should not be limited to a particular type of reactor. For example, in a fixed-bed, once-throughreactor, the basic BFAC equations (eq 1-8) remain the same, but the reactor model should take into account the variation of concentration along axial positions in the column. Several other workers have presented models that combined concepts of activated carbon adsorption and biofilm biodegradation. DiGiano et al. (34) derived a model of biofilm on granular activated carbon. The model included most of the basic mechanisms of the BFAC model. However, shear loss was not included explicitly as a mechanism of biofilm loss but was modeled as part of the first-order rate of endogenous respiration (b). Recently, Speitel et al. (35) revised the model of DiGiano et al. (34) to include shear loss explicitly. The model differed from the BFAC model in that it was for a plug flow reactor and had dif-
ferent initial conditions. Also, the BFAC model includes the prediction of the suspended biomass, which was neglected in the model of DiGiano and co-workers (34,35). Andrews and Tien's model (36) considered substrate diffusion and utilization in the biafilm, adsorption of the substrate, and the growth and decay of the biofilm. However, many simplifications were made in the model. Film transfer of the substrate, shear loss of the biofilm, and diffusion in the adsorbent were not considered. To facilitate an analytical solution, the utilization of the substrate was assumed to be a first-order rate and solved for a quasi steady state condition. Ying and Weber (37) developed a model that included liquid-film transfer, intraparticle diffusion, substrate utilization by Monod kinetics, and the increase of the biofilm density. The model was derived for a fluidized-bed reactor with plug flow and completely mixed regimes. One drawback of the model was that it did not consider that the biofilm had a physical thickness. Instead, the biofilm density ( X f )increased up to a maximum density with time. As a result, the model neglected the resistance of the substrate diffusing through the biofilm. Irl addition, shear loss was not included as a mechanism for biofilm loss. Omitting diffusion resistance caused the biofilm to grow faster and, hence, made the breakthrough curve reach steady state faster than experimental results. Acknowledgments We thank A. J. Valocchi of the Civil Engineering Department for his valuable discussion on the numerical techniques.
Glossary specific surface area (l/L) collocation matrix for the first derivative in biofilm collocation matrix for the first derivative in activated carbon specific decay coefficient (1/ 7') specific shear loss coefficient (1/ 2) total biofilm loss coefficient (l/T), b' = b + b, collocation matrix for the second derivative in biofilm collocation matrix for the second derivative in activated carbon particle diameter ( L ) molecular diffusivity in bulk liquid (L2 molecular diffusivity within biofilm ( L / T ) surface diffusivity ( L 2 / T ) substrate flux from liquid phase into biofilm [Ma/ (L2T)1 substrate flux from biofilm into activated carbon [Ms/(L2T)1 maximum specific rate of substrate utilization [Ma/(MxT)I liquid-film mass transfer coefficient (L/T ) Freundlich isotherm coefficient half-velocity concentration (Ms/L3) length biofilm thickness ( L ) thickness of diffusion layer\ ( L ) mass, in general mass of substrate (Ma) mass of activated carbon (Mq) mass of bacteria (M,) Freundlich isotherm coefficient number of internal collocation point surface concentration (MJM,) surface concentration in' eGuilibrium with So (MsIMq) surface concentration at biofilm/carbon interface (M.I M J influent flow rate ( L ~ / T )
4Q
substrate utilization rate of the biofilm [Ma/(L3T)] radial distance in biofilm ( L ) radial distance in activated carbon ( L ) particle radius ( L ) substrate concentration in bulk liquid (Ma/L3) substrate concentration in the biofilm (Ms/L3) substrate concentration at liquid/biofilm interface (M,/L3) substrate concentration in the feed (Ms/L3) time (29 superficial flow velocity through reactor ( L I T ) volume of reactor (L3) collocation matrix for the integral in biofilm collocation matrix for the integral in activated carbon biofilm density (M,/L3) * weight of activated carbon (M,) true yield coefficient of biomass (M,/Ms) conversion factor for the utilization of phenol to carbon dioxide conversion factor for the respiration of cells reactor porosity maximum specific growth rate (1/T) apparent particle density of activated carbon (Mq/L3) Literature Cited (1) Rittmann, B. E.; McCarty, P. L. J. Enuiron.Eng. Diu. (Am. SOC.Ciu. Eng.) 1981, 107, 831-849. (2) Rittmann, ,B. E.; McCarty, P.'L.'Biotechnol. Bioeng. 1980, 22, 2343-2357. (3) Rittmann, B. E.; McCarty,P. L. J. Enuiron. Eng. Diu. (Am. SOC.Civ. Eng.) 1978, 104, 889-900. (4) Suidan, M. T.; Cross, W. H.; Fong, M.; Calvert, W. J. Enuiron. Eng. Diu. (Am.Soc. Ciu. Eng.) 1981,107,563-579. (5) Sontheimer,H.; Heilker, E.; Jekel, M. R.; Noh, H.; Vollmer, F. H. J.-Am. Water Works Assoc. 1978, 70, 393-396. (6) Bancroft, K.; Maloney, S. W.; McElhaney, J.; Suffet, I. H.; Pipes, W. 0. Appl. Enuiron. Microbiol. 1983,46,683-688. (7) Flynn, B. P.; Robertaccio, F. L.; Barry, L. T. Proceedings of the 31th Purdue Industrial Waste Conference;Ann
Arbor Science: Ann Arbor, MI, 1976; p 855. (8) Scaramelli,A. B.; DiGiano, F. A. Water Sewage Works 1973, 120, 90. (9) Nayar, S. C.; Sylvestor, N. D. Water Res. 1979, 13, 201. (10) Schultz,J. R.; Keinath, T. M. J.-Water Pollut. Control Fed. 1984, $6, 143. (11) den Blanken, J. G. J. Enuiron. Eng. Diu. (Am. SOC.Ciu. Eng.) 1982,108,405-425. (12) Chang, H. T.; Rittmann, B. E. Enuiron. Sci. Technol., following paper in this issue. (13) Yen, C.-Y.; Singer, P. C. J. Enuiron. Eng. Diu. (Am. SOC. Ciu. Eng.) 1984, 110, 976. (14) Kim, B. R.; Snoeyink, V. L.; Saunders, F. M. J. Enuiron. Eng. Diu. (Am. SOC.Ciu. Eng.) 1976, 102, 55. (15) Weber, W. J., Jr. Physicochemical Process for Water Quality Control;Wiley: New York, 1972; Chapter 5. (16) Characklis, W. G. Water Res. 1973, 7, 1113. (17) Suidan, M. T.; Wang, Y. T. J. Enuiron. Eng. Diu. (Am.SOC. Ciu. Eng.) 1985, 111, 634-646. (18) Williamson,K. J.; McCarty, P. L. J.-Water Pollut. Control Fed. 1976,48, 9-24. (19) Atkinson, B.; Davies, I. J. Trans. Inst. Chen. Eng. 1974, 52, 248-259. (20) LaMotta, E. J. Appl. Enuiron. Microbial. 1976, 31, 286. (21) Rittmann, B. E. Biotechnol. Bioeng. 1982, 24, 501-506. (22) Bird, R. B.; Stewart, W. E.; Lightfoot, E. N. Transport Phenomena; Wiley: New York, 1960; Chapter 18. (23) Danckwerts, P. V. Trans. Faraday Soc. 1950,46,701-712. (24) Wang, S. C. P.; Tien, C. AIChE J. 1984, 30, 786. (25) Lin, S. H. J. Hydraul. Diu., Am. SOC.Ciu. Eng. 1977,103, 951-958. (26) Villadson, J. V.; Stewart, W. E. Chem. Eng. Sci. 1967,22, 1483-1501. (27) Thacker, W. E., Ph.D. Dissertation, University of Illinois, Urbana, IL, 1980. Environ. Sci. Technol., Vol. 21, No. 3, 1987
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Kim, B. R., Ph.D. Dissertation, University of Illinois, Urbana, IL, 1977. Crittenden, J. C.; Wong, B. W. C.; Thacker, W. E.; Snoeyink, V. L.; Hinrichs,R. L. J.-Water Pollut. Control Fed. 1980, 52, 2780.
Crank, J. Q. J. Mech. Appl. Math. 1957,10, 220-231. Finlayson, B. A. The Method of Weighted Residuals and Variational Principles; Academic: New York, 1972; Chapter 5. Stroud, A. H.; Secrest, D. Gaussian QuadratureFormulas; Prentice-Hall: Englewood Cliffs, NJ, 1966; Chapter 3. Hindmarsh, A. C. ACM-SIGNUMNewsletter, 1980, 15, 10-11.
DiGiano, F. A.; Dovantzis, K.; Speitel, G. E., Jr. Environmental Engineering, Proceedings of the 1984 Specialty Conference of ASCE, Los Angeles, CA; Pirbazari, M.; Devinny, J. S., Eds.; American Society of Civil Engineers: New York, 1984.
(35) Speitel, G. E., Jr.; Dovantzis, K.; DiGiano, F. A. J. Environ. Eng. Diu. (Am. SOC. Civ. Eng.), in press. (36) Andrews, G. F.; Tien, C AZChE J . 1981,27, 396-403. (37) Ying, W.; Weber, W. J., Jr. Proceedings of the 33th Purdue Industrial Waste Conference; Ann Arbor Science: Ann Arbor, MI, 1978; pp 128-141.
Received for review February 10, 1986. Accepted October 27, 1986. Although the informationdescribed in this paper has been funded wholly by the U.S. Environmental Protection Agency under Assistance Agreement EPA Cooperative Agreement CR810462 to the Advanced Environmental Control Technology Research Center, it has not been subjected to the Agency's required peer and administrative review and therefore does not necessarily reflect the views of the Agency, and no official endorsement should be inferred. Mention of trade names or commercial products does not constitute endorsement or recommendation for use.
Verification of the Model of Biofilm on Activated Carbon H. Ted Chang" and Bruce E. Rittmann
Department of Civil Engineering, University of Illinois, Urbana-Champaign, Illinois 6 1801
rn Laboratory-scale column reactors were used to verify the model of biofilm on activated carbon (BFAC model). The effluent from the reactor was recycled so that the reactor approached a completely mixed regime. Twelve Coefficientswere measured independently from the column studies, while three parameters had to be assumed so that the model simulated the substrate concentration results very well. The BFAC model was verified with biofilms grown on spherical activated carbon (BACM) packed in the column reactor. The detailed sequence of the bioregeneration, modeled by the BFAC model, showed that the substrate flux in the activated carbon sequentially was positive during adsorption, was negative during bioregeneration, and approached zero toward steady state. Although the BFAC model accurately described the substrate concentration and the sequence of bioregeneration, it predicted lower effluent suspended biomass than the experimental results. Further research is needed to understand the mechanisms of the shearing loss of biofilms.
Introduction Chang and Rittmann (I)presented a model for the kinetics of biofilm on activated carbon (the BFAC model). The model incorporated the mechanisms of film transfer, biodegradation, and adsorption of a substrate, as well as biofilm growth. Model formulation and solution were tailored to the unique features of the BFAC system: namely, diffusion across a moving boundary, diffusion with nonlinear reaction, and diffusion in layered media. Solution was by the global orthogonal collocation method (GOCM) (I), which provides time-dependent values of substrate concentration, biofilm thickness, and substrate fluxes. This paper presents experimental results for the verification of the BFAC model. The verification includes experimental evidence for and mechanistic explanations of an important characteristic of a BFAC system, bioregeneration. Bioregeneration, defined here as the removal of previously adsorbed substrate from the adsorbent surface through biological means, has been demonstrated in previous studies (2-6). Because bioregeneration involves 280
Environ. Sci. Technol., Vol. 21, No. 3, 1987
Table I. Composition of Mineral Medium salt concn, mg/L KH2P04 8.5 KzHPO4 21.75 Na2HP04 17.7 MgS04 11.0
salt
concn, mg/L
FeC1, NaHCO, KNO, CaClz
0.15 1.0 3.215 27.5
Table 11. Composition of Growth Medium salt KHzPO4 FeC1, NaHC03
concn, mg/L
salt
concn, mg/L
155.0 12.0 100.0
KN03 CaC1,
1172.0 200.0
the sequential adsorption of a solute, growth of a biofilm, and desorption of the solute, it gives the BFAC model a rigorous test. An experiment with biofilm grown on nonadsorbing media, glass beads, was used as a control. Readers interest in detailed procedures and results for the determination of coefficients is directed to Chang (7).
Materials and Methods Supporting Media. For the purpose of verifying the mathematical model, bead-shaped activated carbon (BACM, Kureha Chemical Industry Co., Ltd., Tokyo, Japan) was chosen as one supporting medium for the biofilm attachment. The carbon is sturdy, and its bead-shaped configuration matches the assumption of diffusion to and in a spherical particle (I). The activated carbon was sieved with U.S. standard sieves, and particles that passed through No. 30 (0.59 mm) and were retained on No. 40 (0.42 mm) sieves were used. The geometric mean diameter, 0.50 mm, was used in the mathematical modeling. The carbon was washed several times with distilled/deionized water (DIDW) to remove carbon fines and stored in an oven at 110 "C before use. Glass beads with diameters ranging from 0.9 to 1.23 mm (geometric mean 1.05 mm) also were chosen as a supporting medium for biofilm growth. The nonadsorbing characteristics of glass beads provide a control for biofilm activity alone. The glass beads were not sieved but were
0013-936X/87/0921-0280$01.50/0
0 1987 American Chemical Soclety
