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Effectiveness Factors in Anaerobic Biofilms A Pseudoanalytical Equation for Parallel and Consecutive Reactions.

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Effectiveness Factors in Anaerobic
Biofilms: A Pseudoanalytical Equation
for Parallel and Consecutive Reactions
C.R. Escalera', A. Maezawa', S. Uchida'. and M. Kuroda3
'Department of Chemical Engineering,
2Graduate of Electronics Science and Technology,
Shizuoka University, 3-5- I Johoku, Hamamatsu, 432, Japan
Department of Civil Engineering,
Gunma University, 1-5- I Tenjincho, Kiryu, Gunma, 376, Japan
The effectiveness factors for an anaerobic biofilm in which a parallel-consecutive
reaction occurs were studied theoretically. A pseudo-analytical equation to represent
the effectiveness factors (Efi)was developed as a function of a dimensionless
parameter called the overall biofilm property value (Moj), and dimensionless
concentrations at the surface of the biofilm (Bj). The parameter MOj is a combination
of the individual biofilm property values (Mj) which represent biofilm characteristics.
The applicability of these approximation equations is demonstrated experimentally
for the degradation of a mixture of volatile fatty acids in which acetic acid is
predominant. As the decompositions of these acids are considered to be rate-limiting
in anaerobic biofilm processes, this approximation equation may be used for the
design and/or performance evaluation of biofilm reactors.
The decomposition of the soluble organic fraction of industrial and domestic
wastewaters can be accomplished successfully by the use of anaerobic biofilm
treatment systems such as packed, expanded and fluidised beds. In these systems,
the immobilisation of purifying bacteria on the surface of solid-supports permits
loading rates that are several times higher than those possible in conventional
suspended systems,' and better stability records are observed in biofilm reactors
when stressed by shock loadings.
In attempts to predict or evaluate the performance of such processes, several
mathematical biofilm models have been proposed. They basically consider
intrabiofilm diffusion and decomposition of substrates, liquid layer resistance to
substrate diffusion, and biofilm growth and decay. Several models24 assumed a
single rate-limiting substrate under steady-state conditions, and pseudo-analytical
overall rate equations were developed in terms of dimensionless concentrations at
the surface of the biofilm, and parameters which were usually some form of a
modified Thiele modulus.
'To whom correspondence should be addressed.
Developments in Chemical Engineering and Mineral Processing, Vol. 1, No 2/3,page 158
Eflectiveness Factors in Anaerobic Biofilms
Recently, biofilm modelling has been improved by considering the existence
of mixed bacterial populations, and two limiting substrates which are degraded
et d 7proposed simplified equations
consecutively within the b i ~ f i l m . ~Kuroda
for the effectiveness factors for two-step consecutive reactions systems in biofilm
reactors operated under a complete-mixed flow regime, and their applicability was
demonstrated for butyrate and propionate degradations. They used a biofilm
model6 which considers the liquid layer resistance to be negligibly small for large
Reynolds numbers. However, differences larger than 20% occurred between the
experimental data and the numerical solution of the basic differential equations,
especially in the case of large bulk concentrations and large values of biomass
concentrations in biofilm. Furthermore, two-step consecutive systems like
propionate and butyrate degradations are only part of the more complex parallel
and consecutive reaction system of the anaerobic decomposition of organic
wastewaters, e.g. those containing carbohydrates.
This paper includes a theoretical study of the effectiveness factors for a
biofilm in which a mixed bacterial population degrades a mixture of rate-limiting
substrates, in a combination of parallel and consecutive reactions. The liquid-layer
diffusional resistance is also taken into consideration. Analytical solutions for the
effectiveness factors for a first-order reaction approximation are presented, and a
pseudo-analytical general equation is developed. This represents approximately
t h e effectiveness factors for one-step reaction, and the combined
parallel-consecutive reaction system for a large range of biofilm surface
Reactor Model
The reactor model (Figure 1) proposed by Sakakibara’ was the conceptual basis
used in this study. The model considers a complete-mixed biofilm reactor with a
planar biofilm of uniform density (p) and uniform thickness (LF), where the
mixture of primary substrates is degraded by the attached micro-organisms
according to the following equation:
B - A - G
where P and B = primary substrates; A = intermediate product; G = final product.
As discussed above, this type of reaction represents the rate-limiting
decomposition steps of the general bio-reaction pathway occurring in the
anaerobic degradation of soluble organics contained in wastewaters, e.g. sugar
It is assumed that both the substrates and the intermediate are transported by
molecular diffusion within the biofilm and the laminar liquid layer, and that their
degradation rates can be represented by Monod-type relations. Furthermore,
considering that biofilm growth and decay are very slow compared with
concentration changes, quasi-steady-state mass balances for the substrates and the
intermediate in the biofilm yield the following equations.
For the substrates B and P:
C.R. Escalera, A. Maezawa, S. Uchida and M.Kuroda
Figure 1 Biofilm Reactor Model as proposed by Sakakib~zra.~
For the intermediate A:
The boundary conditions for the system of equations (2) and (3) are:
EfSectiveness Factors in Anaerobic Bioflms
Mass balances for the substrate and intermediates in the bulk liquid lead to the
following equations:
The mass fluxes for primary substrate and intermediate products are:
In the absence of any diffusional resistance within the biofilm, or when the
diffusion rates are much larger than the decomposition rates, the concentrations of
substrate and intermediates throughout the film are everywhere equal to the
surface concentrations. Therefore, the corresponding fluxes are given by:
The effectiveness factors are then defined as:
Equations (2)-(5) can be represented in terms of dimensionless parameters as
C.R. Escalera, A. Maezawa, S. Uchida and M. Kuroda
From equation (6), in dimensionless variables:
In the foregoing equations, M, values are biofilm property values for the
substrates (B) and the intermediate (A), and represent the corresponding ratios of
utilisation rates to diffusion rates in the biofilm. Parameters Pe, are operational
parameters for substrates (B) and intermediate (A), and represent the
corresponding ratios of loading rates per unit area to diffusion rates. Parameters
Bi, are the corresponding Biot numbers which represent the ratios of diffusion
rates in the liquid layer to those in the biofilm.
Numerical Rates
The effectiveness factors (Efi) were calculated for values of M, varying from 0.1
to 10, and values of the biofilm surface concentrations B, varying from 0.01 to
100. Kuroda el d7calculated the effectiveness factors for two-step consecutive
reaction systems within these ranges. These ranges of the parameter values were
chosen by taking into account typical values of the specific growth rates,
diffusivities, half-velocity coefficients and biofilm thicknesses occurring in
anaerobic processes."
As an example, Figure 2 shows the relation between EfA and BA, for
M B = MA = 4 and M p = 0 . 7 M ~and Bjj = 10, taking the distribution of the
concentrations of A , B and P in the feed as a parameter. The figure shows clearly
that when the concentration of the intermediate (A) in the feed is predominant, the
effectiveness factors are smaller than those corresponding to lower concentrations
of A. This is because the intermediate A produced by the degradation of the
substrates B and P is subsequently rapidly decomposed, and does not accumulate
within the biofilm. However, when larger relative amounts of B or P are present
in the feed, the intermediate A accumulates within the biofilm yielding larger
effectiveness factors.
First Order Approximation
When the concentrations are much larger than the half-velocity coefficients
(E, > l), the reaction rates in equations (2) and (3) can be approximated to
zero-order reaction rate and the effectiveness factor is near to 1 (see Figure 2).
When the concentrations are much smaller than the half-velocity coefficients
(Cj < K,, i.e. E, < l), the rate terms in equations (2) and (3) can be approximated
to first-order reaction rates. The following analytical solutions for the
effectiveness factors can then be obtained:
EfSectiveness Factors in Anaerobic Biofilms
Figure 2 Relation between effectivenessfactor Ef, and dimensionless
concentration at biofilm suvace (B,) for intermediate decomposition.
EJs = tanh M s / M s (S = E , P )
where the parameter Op, is defined as:
When Pe, is much smaller than 1 (small loading rates), the values of the
parameter Op, approximate to Pej For these cases and for M, values larger than
3, the analytical solution of the effectiveness factors for the substrates and the
final intermediate approximate to the following equations:
C.R. Escalera, A. Maezawa, S. Uchida and M. Kuroda
By following the same procedure, it can be shown that the expressions
MA + ~ / M Band
MA + 1 / M p ) are the effectiveness factors for a first order
approximation which correspond to the B-A-G consecutive reaction and P-A-G
consecutive reaction, respectively.
The parameters MOB-A and MOP-A are defined as the pathway biofilm property
values, and correspond to the individual two-step consecutive reactions as defined
by equation (1) (B-A-G and P-A-G reactions). The overall biofilm property
(MoA) is a combination of the biofilm property values corresponding to the
individual consecutive reaction pathways, arranged in the same way as a
parallel-series diffusion resistance configuration. This configuration suggests that
an overall biofilm property value (Mo,)can be defined for any system of
consecutive and/or parallel reactions, as a combination of the individual pathway
biofilm property values.
Figure 3 Relation between parameters a and p and dimensionless concentrations
at bioflm surface Bs and B, for primary substrates and intermediate uptakes.
Eflectiveness Factors in Anaerobic Biofilms
Figure 4 Error obtained by using equations (26) and (27) in calculation of
effectivenessfactors with respect to numerical solutions.
Development of Approximation Equations
Equations (1 8) and (23) can be utilised to represent the effectiveness factors when
the values of the biofilm surface concentrations are very small. However, when
these concentrations become larger (for example B, > l), the Monod-type
relationships in equations (2) and (3) cannot be simplified to first-order rate
equations. Numerical solutions for the system of differential equations are then
needed to obtain the effectiveness factors. In order to avoid the numerical work,
the following development of a general approximation equation was performed.
The effectiveness factor for the primary substrates B and P (or first steps of
the corresponding consecutive reactions) and the intermediate A can be
represented approximately by the empirical function:
Ef,= tanh(aMo,8)
( j = A, B , P)
C.R. Escalera, A. Maezawa, S. Uchida and M.Kuroda
Figure 5 Comparison of experimental and calculated values of effectiveness
factors for propionate and butyrate decompositions in a packed bed reactor.
which can be rearranged in the following form:
Plotting equation (25) by using data from the numerical solution for different
values of the surface concentration (B,), the relationships can be obtained for a
and p. Figure 3 shows the relationship for the primary substrate. These parameters
can be approximated by the following equations as functions of the concentration
+ 0.6573B,+ 0.0228;
- 0.0448, + 0.000828:
a = 1.059
/9 = -1.0
for the range of 0.01 < B, < 100.
The errors resulting from the use of the approximation equations relative to the
numerical solution are presented in Figure 4. The errors ES for the first steps are
consistently less than 10% for all the values of Ms and Bs considered. The errors
&A for the parallel-consecutive reaction system are consistently less than 10% for
Eflectiveness Factors in Anaerobic BioJilms
Figure 6 Comparison of experimental and calculated results of the removal
eficiency and bulk concentrations for decomposition of a mixture of acetic,
propionic and butyric acids (Ace:Pro:Bru = 0.56:0.32:0.12).
C.R. Escalera, A. Maezawa, S. Uchida and M.Kuroda
all the values of M A and BA considered, when the proportion of A in the feed is
larger than one third of the total. It can also be observed that errors less than 5%
occur for larger proportions of A.
Data have been reported11912for anaerobic reactors treating agricultural
wastewaters that are not stressed by shock-loadings, e.g. molasses, beet sugar, and
fruit-processing waste liquors. It was observed that among the acids produced in
the acid-forming decomposition step, the intermediate acetic acid was produced
in large proportions compared to propionic and butyric acids. Considering these
data, equations (24) and (27) can be applied to evaluate or design anaerobic
biofilm reactors that treat such wastewaters, and other wastewaters in which the
decompositions of the mixture of acids become the rate-limiting steps.
Applicability of the Proposed Equations
In order to illustrate the application of the approximation equations, Figure 5
shows a comparison of the effectiveness factors obtained experimentally and the
values calculated by using the pseudo-analytical equation, for the first steps of the
consecutive proprionate and butyrate decompositions.
The experimental procedures and the experimental data are given by
Sakakibara et al. l 3 They used packed-bed anaerobic reactors operated under high
wastewater-recycle conditions. Therefore, the liquid-layer mass transfer resistance
was minimised and represented less than about 20% of the intrabiofilm diffusion
resistance. As shown in Figure 5 , there was good agreement between the
experimental and calculated results obtained for a wide range of biofilm surface
Figures 6(a) and (b) show a comparison of the experimental and calculated
values (obtained by using the pseudo-analytical equations) of the removal
efficiency and effluent concentrations (bulk concentrations) for the anaerobic
decomposition of a mixture of acetic, propionic and butyric acids i n a
laboratory-scale packed bed reactor. Experimental details and data are a ~ a i l a b l e . ~
Good agreement is obtained between the experimental and calculated results.
The effectiveness factors for the decomposition of a mixture of primary substrates
and the final intermediate occurring in an anaerobic biofilm were studied. The
theoretical basis was a biofilm model that considered a system of parallel and
consecutive anaerobic reactions and diffusion rates in the liquid layer and biofilm.
The following results were obtained:
(i) The effectiveness factor of the intermediate depends on the composition of
substrates and intermediate in the feed.
(ii) The effectiveness factors of the primary substrates and the intermediate (Ed)
can be approximately represented by a pseudo-analytical equation expressed
in terms of the overall biofilm properties (Moj). These are defined by
equations (21) and (22) and their corresponding concentrations at the biofilm
surface (Bj).
(iii) These approximation equations can be applied to the design and/or
performance evaluation of anaerobic biofilm reactors, in which the
degradations of a mixture of organic volatile fatty acids (i.e. acetic, propionic
and butyric acids) are the rate-limiting steps.
Effectiveness Factors in Anaerobic Biofilms
specific surface area of biofilm, (In)
concentration of component j in biofilm, (M/L3)
concentration of componentj at biofilm surface, (M/L3)
concentration of componentj in bulk liquid, (ML3)
primary substrate concentration in feed, (M/L3)
overall substrate concentration in feed, (M/L3)
effective diffusivity of component j in biofilm, (L2/r>
removal rate of componentj , (M/(L~T>>
removal rate of component j without biofilm diffusion resistance, (h4/(L2T))
half-velocity coefficient for component j uptake, (M/L3)
mass transfer coefficient of component j in liquid layer, (YT)
biofilm thickness (L)
distance from the solid support surface (L)
Dimensionless variables and parameters
normalised concentration of component j in biofilm, CJCTF
normalised concentration of component j in biofilm surface, CyCTF
normalised concentration of component j in bulk liquid, C’JCTF
normalised concentration of component j in feed, C.,JCTF
dimensionless concentration of component j at biodlm surface, BTFKA//j*
dj Biot number for componentj , L$L/D,
BTF dimensionless substrate feed concentration, CT,JKA
ratio of effective diffusivities, D/DA
effectiveness factor for componentj , J p *
K,,, ratio of half-velocity coefficient, KJKi
M . biofilm property value for componentj , LF(u,ajp/KjDj)ln
d o j overall biofilm property value for component j
Opj operational parameter for component j defined by equation (20)
Pej operational parameter for componentj , L/a8Dj
normalised distance from solid support surface, x/LF
yield coefficient for final intermediate formation from substrate S
YGIA yield coefficient for final product formation
Greek Symbols
parameter defined in equation (26)
mass fraction of component j in biofilm
parameter defined in equation (27)
committed in calculation of effectiveness factor of componentj , (%)
specific decomposition rate of componentj , (1/T)
biofilm density (h4/L3)
hydraulic retention time (T)
intermediate A
substrate B
substrate P
C.R. Escalera, A. Maezawa, S. Uchida and M. Kuroda
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Received: 29 January 1992; Accepted: 20 November 1992
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factors, parallel, pseudoanalytical, reaction, effectiveness, equations, consecutive, biofilm, anaerobic
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