Dynamic modelling of a single-stage high-rate anaerobic reactor—I. Model derivation

Wat. Res. Vol. 25, No. 7, pp. 847-858, 1991 Printed in Great Britain. All rights reserved

0043-1354/91 $3.00 + 0.00 Copyright © 1991 Pergamon Press pie

DYNAMIC MODELLING OF A SINGLE-STAGE HIGH-RATE ANAEROBIC REACTOR--I. MODEL DERIVATION D. J. COSTELLO, P. F. GREENFIELD O a n d P. L. LEE* Department of Chemical Engineering, University o f Queensland, Australia

(First received March 1990; accepted in revised form January 1991) Abstract--A mathematical model o f a high-rate anaerobic treatment system was developed by defining the biological make-up of the anaerobic ecosystem, the physico-chemical system, and the reactor process. The biological model was extended from current theory to include the accumulation of lactic acid under specific process conditions. Product inhibition and pH inhibition of each group o f bacteria were represented in the model so as to utilize all the available information about the anaerobic process. A rigorous approach was applied to the development o f the physico-chemical model which was used in the determination o f the reactor pH and ionic species concentrations. The overall reactor equations were based on continuous well-mixed processes, with no effective solids residence time parameter introduced to account for bacterial retention in the process. Parameters were taken from models developed by previous researchers, and from mixed and pure culture studies of the anaerobic bacteria.

Key words--anaerobic, modelling, lactate, hydrogen, pH, inhibition, industrial treatment, high-rate, UASB

NOMENCLATURE [acid-] = the concentration of a dissociated organic acid (mmol/1) ATP = adenosine 5'-triphosphate b = the first-order bacterial decay coefficient (day -I ) [CO2] = the concentration of carbon dioxide in the liquid phase (mmol/l) H = Henry's coefficient for carbon dioxide (arm/retool) [H + ] = the concentration of hydrogen ions (retool/l) [HCO~-] = the concentration of bicarbonate ions (retool/l) [I] = the inhibitor concentration (retool/l) [IC] = the total concentration of inorganic carbon (retool/l) K l = product inhibition parameter (mmol/l) KLa = the mass transfer coefficient for carbon dioxide (day- t) K s = the half-velocity constant (retool/l) k = the maximum specific substrate uptake rate (retool/rag/day) k w = the dissociation constant of water (retool/l) k~ = the dissociation constant o f dissolved carbon dioxide (retool/l) [Na ÷] = the net concentration of positive ions (retool/l) Pco2 = the partial pressure of carbon dioxide (atm) P~ = the partial pressure of a gas phase component (atm) QI = the liquid or solid phase flowrate for the each bacterial group (I/day) QL = the liquid flowrate through the reactor (l/day) Qs = the pseudo bacterial solids flowrate (1/day) Qv = the total gas phase fiowrate (l/day) R = the Ideal Gas Law constant rCH, = the methane gas evolution rate from the liquid phase (retool/I/day)

*Author to whom all correspondence should be addressed. 847

rco:T = the rate o f transfer o f carbon dioxide from the gas to the liquid phase (retool/I/day) rHz = the hydrogen gas evolution rate from the liquid phase (retool/I/day) r i = the overall rate o f generation of the component Yi [soluble components (retool/I/day); bacterial components (rag/I/day); or the rate o f evolution of gas from the liquid phase (mmol/l/day)] rp,i = the production rate for product i (rag/retool) r s = the substrate uptake rate (mmol/1/day) r x = the growth rate of the bacteria (retool/I/day) [S] = the substrate concentration (mmol/1) T = the process temperature (K) Vo -- the gas phase volume (litres) VL = the liquid volume (litres) [X] = the bacterial concentration (rag/l) X L = the concentration o f bacteria in the liquid phase (mg/1) X s = the concentration of bacteria in the bacteria solids phase (rag/l) Y = the bacterial yield coefficient for the carbon substrate (mg/mmol) Yi= the bacterial yield coefficient for product i (mg/mmol) y~ = a general reactor concentration [soluble components (retool/l); bacterial components (rag/l)] zn = 2000 x partial pressure o f hydrogen gas at 35°C

INTRODUCTION T h e d y n a m i c m a t h e m a t i c a l m o d e l o f the h i g h - r a t e a n a e r o b i c t r e a t m e n t p r o c e s s d e s c r i b e d in this p a p e r is b a s e d o n t h e w o r k o f M o s e y (1983) a n d i n c o r p o r a t e s t h e m o s t r e c e n t a d v a n c e s m a d e in t h e m o d e l l i n g a n d s t u d y o f t h e general a n a e r o b i c d e g r a d a t i o n process. A f e a t u r e o f a n a e r o b i c d e g r a d a t i o n m o d e l s is t h e c o n s i d e r a t i o n o f different generic g r o u p s o f bacterial

848

D.J. COSTELLOet al.

species used to describe the conversion of raw substrate to the final products of methane and carbon dioxide. Early models (Andrews, 1969; Graef and Andrews, 1974) considered degradation of acetic acid by the acetoclastic bacteria, assuming that this step was rate limiting, and therefore the most important step in the degradation of the influent waste. As the need to model more complex aspects of the anaerobic degradation process grew, models with additional trophic groups of bacteria were developed. Hill and Barth (1977) included a group of hydrolytic bacteria to account for the degradation of a soluble substrate into acetic acid. More complex models (Heyes and Hall, 1981; Mosey, 1983) have since been developed to account for the dynamics of hydrogen gas and the higher volatile acids (propionate and butyrate) produced in an anaerobic reactor undergoing loading changes. The most common intermediate organic acids observed in anaerobic degradation are the volatile acids, acetic, propionic and butyric acid. It has been hypothesized that lactic acid may also be an important intermediate in the anaerobic degradation process (Stronach et aL, 1986). This is supported by the experimental evidence of Pipyn and Verstraete (1981) and Eng et al. (1986) who have shown that lactic acid accumulates after a sudden increase in the loading of a readily degradable substrate to an anaerobic reactor. With the development of more complex models for anaerobic ecosystems, a technique to account for the varying generation rates of the different volatile acids was needed. Research had shown that the uptake rates and product distribution of some species of bacteria were regulated by hydrogen gas (Iannotti et al., 1973; Kaspar and Wuhrmann, 1978). Mosey (1983) defined and incorporated the regulation and inhibition of the hydrolytic bacteria, and the propionic and butyric acid acetogenic bacteria onto the anaerobic degradation model. This was done by using the Nernst equation to link the levels of hydrogen in the reactor to the redox reactions of the pyridine nucleotides (characterized by the nicotinamide adenine dinucleotide redox couple NAD + ,--,NADH). Hydrogen in the gas phase of the reactor determined the ratio of oxidized to reduced N A D within the bacteria, which in turn regulated and inhibited the metabolic reactions within the bacteria that were coupled to the NADS+~--~NADH redox reaction. Other more general modes of inhibition are a feature of most anaerobic degradation models. These fall into the categories of either voltatile acid or pH inhibition. A characteristic of early models was to combine the effects of voltatile acids and pH inhibition with the most commonly used inhibition model being developed by Andrews (1969). The substrate uptake equation for the acetoclastic bacteria was modified by using Haldane kinetics to account for the apparent effects of substrate inhibition by un-ionized

volatile acids. Evidence has shown that this mechanism of inhibition is likely to be caused by salt toxicity (McCarty and McKinney, 1961). It has also been shown that the acetogenic bacteria, rather than the acetoclastic bacteria, are inhibited by acetic acid (Kaspar and Wuhrmann, 1978; Denac, 1986). Reinforcing these results Gizard et al. (1986) showed that utilization of propionate and butyrate substrates followed a competitive inhibition model, with acetic acid acting as the inhibiting agent. Apart from the work of Mosey (1983), much of the recent knowledge of anaerobic ecosystems has not been incorporated into a useful high-rate anaerobic treatment model. The aims of this work are to develop a general model of the anaerobic degradation process incorporating the essential features of the ecosystem, now known to be applicable, particularly in terms of inhibition by building on the work of Mosey (1983). A further paper (Costello et al., 1990) describes the experimental validation of the model using both small-scale and pilot-scale data, and the flexibility and robustness of the model in its application to different high-rate anaerobic process. MODEL DESCRIPTION Development of the anaerobic process model has been divided into three major sections. The first section briefly describes the overall reactor equations for a single-stage treatment process. The second section describes the biological rate equations, and the inhibitions and interactions within the anaerobic ecosystem. The last section discusses the physicochemical reaction system used to determine the ionic concentrations of soluble components in the reactor. The anaerobic process model simulates the degradation of soluble organic substrates. Since most wastes are characterized in terms of their carbon content, glucose is a suitable molecule to represent this waste. By using glucose as the standard substrate, stoichiometric relationships could be developed for each bacterial reaction. Hence, an accurate carbon balance could be maintained around the process. This was very useful in checking and de-bugging the model. The general reaction network by which the influent waste is degraded by the different groups of anaerobic bacteria is shown in Fig. 1. Each group of bacteria is subject to various forms of inhibition. The metabolic reactions of the acidforming and the lactic acid bacteria are inhibited and regulated by hydrogen through the redox reactions of nicotinamide adenine dinucleotide (NAD). The groups of butyric acid and propionic acid bacteria are also subject to hydrogen inhibition. Individual pH inhibition functions are applied to each group of bacteria used in the mathematical model. Product inhibition was included in the acid-forming, propionic, and butyric acid bacteria by modifying their substrate uptake equations; noncompetitive and competitive inhibition terms were used.

Model derivation of an anaerobic reactor

]

849

LIOUID PHASE

Soluble glucose

Solubleglucose add-forming bacteria

I

I

H2

acetic

butyric

lactic

acid

acid

acid

..

lactic acid b a c t e r i a

I

acetic acid

i

butyric bacteria

acetic acid

..

propionic acid

propionic bacteria

f'~!

g~

i i

acetic acid

1 I acetic acid

acetoclastic bacteria

H2 CO2

v

c%

[ ,II-] ~

/

CO2

H2 CO2

H2

~

H2

i

C O 2 ,.__

H 2 C%

CH4

hydrogen- utilising bacteria

c%

h.

..

Fig. 1. A schematic of the relationships between each group of bacteria in the anaerobic ecosystem model.

There are three major gaseous components modelled in the biogas phase; these are hydrogen, methane and carbon dioxide. The gas phase above the liquid is modelled as a constant pressure vapour space, with a total pressure of one atmosphere minus the vapour pressure of water at the specified reactor temperature. Carbon dioxide is the only gas assumed to have any significant solubility in the liquid phase. The rate of transfer of carbon dioxide into the gas phase is controlled by a nonequilibrium driving force between the liquid and gas phases (Andrews and Graef, 1971).

A physico-chemical model was used to determine the pH and the concentrations of ions and undissociated molecules in the liquid phase. The model consists of acid-base equilibrium relationships for each of the organic acids, the dissolved carbon dioxide, and an algebraic mass balance equation for each acid-base pair. The remaining equations necessary to define the system were an overall charge balance equation and the dynamic mass balance equations for the base component (in this system it is caustic soda, represented in the model as sodium ions) and each acid-base pair.

850

D.J. COSTELLOet al.

The anaerobic ecosystem and physico-chemical models were incorporated into the process equations for a single-stage reactor which was modelled using the dynamic rate equations for a continuous stirred tank reactor (CSTR). The flowrate of biological solids through the reactor was considered to be independent of the flowrate of the bulk liquid phase. This was to account for the retention of solids by high-rate industrial anaerobic processes such as upflow anaerobic sludge blanket (UASB) reactors, upflow filter, and fluidized bed reactors. The liquid phase reactor equations

A dynamic mass balance equation is required for each biological component present in the liquid phase of the reactor. The general dynamic reactor equation used to describe the behaviour of each substrate and biomass component is described below: dyi Q1 dt - -~t (Yi.in

-

-

Yl) + ri

(1)

where y, represents the concentration of each process variable. The process variables include the raw substrate concentration (glucose), the volatile acid concentrations, and the individual biomass concentrations. The rate term r i represents the overall rates of production of each of these components. For reactor variables other than carbon dioxide, the rate term represents the net biological production rate of the component. For carbon dioxide two rate terms are required: the first term represents the net biological production rate, and the second term represents the amount of carbon dioxide that evolves into the gas phase. The evolution of carbon dioxide is described by equation (4) in the next section. To simulate solids retention in the reactor, the biomass rate equations use a lower flowrate than that used for the substrate (liquid phase) equations. Hence, the concentrations given by the rate equations for each bacterial species are calculated relative to a pseudo solids flowrate, chosen to give the desired retention of solids in the reactor. To calculate the concentration of bacteria leaving the reactor relative to the liquid flowrate, the following equation is used:

as

XL = QLL × Xs'

(2)

The pseudo bacterial (solids) flowrate is represented by the variable Qs, while QL represents the liquid phase flowrate through the reactor. The solids flowrate was estimated from the solids retention time of the particular anaerobic system being modelled. All reactions in the single-stage process are assumed to be effectively rate controlled, i.e. the effects of diffusional limitations of flocs or agglomerations are constant and incorporated into the kinetic term. This was shown by Denac et al. (1988) to be a reasonable assumption given that diffusional

gradients were not important in the calculation of the bulk concentrations of reactor components. By definition, this model assumes that all bacterial solids present in the reactor are active. Hence, any relationship between the simulated concentration of bacterial solids and reported concentrations of volatile suspended solids (VSS) would have to account for the levels of biological inactivity in an operating system. This model also assumes that the concentration of bacterial solids retained in the reactor by wall growth, adhesion to support media, and agglomeration can be effectively accounted for by choosing an appropriate solids retention time using the pseudo solids flowrate Qs. Hence, the consideration of the dynamics of aggregation and dispersion of biomass from support media or flocculating particles is beyond the scope of the current model. The gas phase reactor equations

In the gas phase of operational anaerobic reactors measurable levels of ammonia, nitrogen, or sulphur dioxide may be present. But this model considers only the four most important gas phase constituents; these are water vapour, carbon dioxide, methane and hydrogen. The partial pressure of water vapour in the gas phase has not been considered as a variable quantity in the model and so remains at the saturation conditions of the specified process temperature. Methane, carbon dioxide, and hydrogen are generated from the growth of the different anaerobic bacteria. Carbon dioxide is also produced from the chemical conversion of bicarbonate to carbon dioxide. Methane and hydrogen are assumed to be insoluble in the liquid phase, so the bacterial generation of these gases will be equivalent to their rate of evolution into the gas phase. The transfer of carbon dioxide into the gas phase cannot be calculated in this way because of its significant solubility in the liquid phase. Hence a rate equation specifying the rate of transfer of carbon dioxide between the gas and liquid phases must also be included in the model. All gases are assumed to obey the Ideal Gas Law and exist at a temperature equivalent to the liquid phase temperature in a constant volume, completely mixed, constant pressure headspace. Using these assumptions the general dynamic mass balance equation used for each gas phase variable can be written as: dP__.2= -- Ov Pi + ~VL R T r , dt VG G _

_

(3)

The mass balance equations for hydrogen and methane use their respective bacterial production rates for the general gas evolution rate term r i. The carbon dioxide mass balance equation, however, used a gas evolution rate term determined from the general theory of two-film mass transfer. This equation, commonly used in anaerobic reactor

Model derivation of an anaerobic reactor models (Andrews and Graef, 1971; Graef and Andrews, 1974; Hill and Barth, 1977; Heyes, 1984) is:

rco2T= KLaIff--~

-- [CO2]].

(4)

To integrate the pressure balance equations the gas flowrate (Qv) from the liquid phase must be known. This is calculated by assuming that the total pressure in the reactor remains constant. At 1 atm this becomes: PH2 q- Pco2 + PCH4+ PH~O= 1.

(5)

By differentiating this equation and substituting for the rate terms using the dynamic pressure balance equations, the equation to calculate the gas flowrate can be determined as Qv =

RT (1 --

Pn2o)

VL(rrl2 + rcn, -- rCO2T).

(6)

The total gas flowrate (Qv) was calculated using equation (6), and substituted along with the bacterial gas evolution rates and equation (4), into the pressure balance equations [represented by equation (3)]. The pressure balance equations were then integrated to calculate the partial pressures of the gas phase components in the reactor.

The physico-chemicalreaction system A physico-chemical reaction system was included in the model of the anaerobic system so as to be able to calculate the pH at any time, given the concentrations of the acidic and basic species in the reactor. To specify completely a physico-chemical system the overall charge balance equation is required along with the mass balance equations and equilibrium relationships for each component species. Previous to the development of this technique, solutions to the physico-chemical equations could only be obtained by using the assumptions proposed by Graef and Andrews (1974) namely: (1) that the pH remained in an interval from 6 to 8 so that the concentrations of hydrogen and hydroxide ions were negligibly small compared to the other ionic species defined in the charge balance; (2) the concentrations of un-ionized volatile acids is small compared with the ionized concentrations; and (3) the concentration of carbonate ions in the solution is negligible. These assumptions simplified the physico-chemical equations allowing the pH, bicarbonate ions, and un-ionized concentration of the volatile acids to be determined at any time from the total component concentrations in the reactor. The physico-chemical model used in this work combines the overall liquid phase dynamic mass balance equations for each reacting species with the equilibrium relationship for the corresponding

851

ionic and molecular concentrations. The equilibrium relationships describe the dissociation of carbon dioxide and each individual organic acid. The central equation of the physico-chemical model is the overall steady-state charge balance. By using the equilibrium equation for the dissociation of water, the hydroxide ion concentration was eliminated from the charge balance equation such that: kw [H + ] + [ N a + ] = [HCOf] + ~[acid-] + ~ - ~

(7)

where Y.[acid-] represents the sum of the concentrations of the dissociated component of each organic acid and [Na + ] the net concentration of neutral positive ions, i.e. ions that are neither acidic nor basic. In the model the only neutral positive ions enter with caustic soda in the influent stream. To solve the charge balance equation for the hydrogen ion concentration the values of all ionic terms were required. For each physico-chemical component, the equilibrium relationship and the steady-state mass balance equation were used to calculate the ionic concentration of the component species. For the carbon dioxide pair (note that similar equations can be written for each organic acid) these equations are: In +] x [ncof ] k~ = (8) [CO2] [IC] = [CO2] + [HCO3 ].

(9)

These equations were then rearranged, as follows, to calculate the bicarbonate ion concentration in the reactor: kl × [IC] [HCOT] = k~ [ n++ ~ "

(10)

The total component concentrations for each dissociating reaction (in the above example this is [IC]), and the sodium ion concentration [Na + ] remained as the only terms to be calculated before the charge balance equation could be defined in terms of the hydrogen ion concentration. The component concentrations were determined from the dynamic mass balances developed from the anaerobic process model. These equations were already used to determine the dynamic process behaviour of each organic acid in the reactor. An extra dynamic equation is necessary to calculate the values for the concentration of inorganic carbon [IC] (dissolved carbon dioxide and bicarbonate ions) in the reactor. For inorganic carbon the rate equation is:

d[IC] dt

QL = ~LL([CO2]in "4-[HCOf ]i. - [CO2] - [HCOf]) + r c o 2. (11)

The concentration of sodium ions [Na + ] in the charge balance equation represents the net positive

852

D.J. COSTELLOet al.

concentration of neutral ionic species in the liquid phase. This variable is representative of the concentration of base added to the reactor since any changes in the net positive ion concentration [Na + ] must be due to the addition of a positive ion and corresponding hydroxide ions, or hydrogen ions and corresponding negative ions. Note that the addition of neutral salts would have no effect on the concentration of net positive ions. The rate equation for the sodium ions used in the single-stage reactor model is: d[Na+ ] dt

QL ([Na+ ]in - [Na+ ]). VL

(12)

Rather than calculate the pH for a given caustic concentration, some simulations may assume a constant reactor pH. Hence the [Na +] concentration could be immediately determined from the charge balance equation making equation (12) redundant. Therefore, by removing the need to solve one differential equation and making the solution of the charge balance equation redundant, the solution of the model equations was simplified. The physico-chemical model assumes that only the modelled acids and bases have any significant effect on the system, and that there is no buffering caused by the adsorption of ions onto biological solids (or other solid matter) present in the influent stream. The effect of ionic adsorption onto solids has been found to be negligible in sewage sludge in a pH range of 6-8 (Graef, 1972). In addition, the model described in this paper is concerned primarily with the degradation of soluble wastes which do not contain significant proportions of solids. Hence ion adsorption effects are unlikely to be significant. An important parameter in the model is the sodium ion concentration. It is used to specify the influent concentration of sodium hydroxide entering the reactor. For model simulations when sodium bicarbonate was a component of the influent stream, equivalent

influent concentrations of sodium and bicarbonate ions were specified in the model. Theoretically, if enough sodium bicarbonate is added to the reactor, a negative influent concentration of sodium ions would be required to maintain the requested reactor pH. This would represent the addition of an acid to the process. The addition of acidic components was not considered in this model, but if it is found to be necessary, an equation for the negative ionic species should be used rather than allowing the sodium ion concentration to become negative. The bacterial reaction system The bacterial ecosystem. Six groups of anaerobic bacteria are used in the biological model to degrade sequentially the raw glucose substrate into methane, carbon dioxide, and hydrogen. The interactions between each group of bacteria are shown schematically in Fig. 1, while the substrate and biomass equations for each group of bacteria are described in Table 1. The biomass equations are used to fix the amount of carbon assimilated into the bacteria to enable a complete carbon balance to be obtained around the reactor. The equations for the calculation of biomass have to be derived from an average molecular formula. The formula used in this model is CsHgOaN (Endo et al., 1986; Mosey, 1983). The biomass rate is calculated by multiplying the substrate uptake rate (or the product rates in the case of the mixed-product bacteria) by the biomass yield coefficient for each substrate reaction. These coefficients were calculated from the theoretical ATP yields of each substrate reaction by assuming that approx. 10 g of biomass are produced per mole of ATP generated (Bauchop and Elsden, 1960). Consideration of variations in this relationship due to uncoupling or increased maintenance requirements are presently beyond the scope of the model. The ATP yields for each substrate reaction used on the model are included in the substrate equations of Table 1.

Table 1. The substrate and biomass equations used in the anaerobic degradation model

Mixed-Product Soluble Glucose Bacteria C6HI206 -* CH3CH2CH2COOH + 2CO 2 + 2H 2 C6H1206 + 2H20 ~ 2CH3COOH + 2CO2 + 4H2 C6H1206 --* 2CH3CHOHCOOH 5C6HI206 + 6NH 3 ~ 6CsHgO3N + 12H20

+ 3 ATP + 4 ATP + 2 ATP

Mixed-Product Lactic Acid Bacteria C H 3 C H O H C O O H + H 2 - , CH3CH2COOH + H20 C H 3 C H O H C O O H + H20 ---, CH3COOH + CO 3 + 2H, 5CH3CHOHCOOH + 3NH~ ~ 3CsHgO3N + 6H20

+ 0,5 ATP + 1 ATP

Acetogenie Propionic Acid Bacteria CH3CH2COOH + 2H20 --* CH3COOH + CO 2 + 3H 2 3CH3CH2COOH + CO 2 + 2NH 3 --* 2CsHgOjN + 2H20 + H 2

+ 0.5 ATP

Acetogenic Butyric Acid Bacteria CH3CH2CH2COOH + 2H20 --, 2CH3COOH + 2H2 CH3CH2CH2COOH + CO 2 + NH 3 --* CsHgO3N + H20

+ 0,75 ATP

Aeetoelastic Methane Bacteria CH3COOH -* CH4 + CO2 5CH3COOH + 2NH~ ~ 2C~HgO3N + 4H20

+ 0.25 ATP

Hydrogen-Utilizing Methane Bacteria 4H 2 + CO 2 - , CH 4 + H20 5CO 2 + 10H 2 + NH 3 ~ C~HgO3N + 7H20

+ 1 ATP

Model derivation of an anaerobic reactor One generic bacterial species represents the degradation of the influent glucose substrate into acetic, butyric, or lactic acid through the EMP pathway (Fig. 2). Another group of acid-forming bacteria (Fig. 3) degrades lactic acid produced by the hydrolytic bacteria into either propionic or acetic acid. Both the hydrolytic bacteria and the acid-forming lactic acid bacteria are subject to hydrogen regulation and inhibition. Two groups of acetogenic bacteria are considered in the model. One group degrades the butyric acid produced by the hydrolytic bacteria into acetic acid, while the other group degrades propionic acid into acetic acid. The substrate uptake rates of both groups of bacteria are inhibited by hydrogen. Methane is generated from either the acetoclastic or the hydrogen-utilizing bacteria. The acetoclastic bacteria produce methane and carbon dioxide from acetic acid, while the hydrogen-utilizing bacteria reduce carbon dioxide with hydrogen to produce methane gas. T h e b a c t e r i a l r a t e e q u a t i o n s . The Monod relationship was used to describe the uninhibited rate of substrate utilization for each group of anaerobic bacteria: k[X][S] rs = Ks + [S-----~'

(13)

The growth rate of the bacteria was related to either the individual production rates rp.~ of acids by

853

the mixed-product bacteria, or the substrate uptake rates rs in the acetogenic and methanogenic bacteria. The production rates are used to calculate the growth rates for the mixed-product bacteria because each product is associated with a different ATP yield, and the production of each acid product varies throughout a simulation. To account for bacterial decay a constant first-order death coefficient term was included in all growth rate equations. The general bacterial growth rate equation for the mixed-product glucose and lactic acid bacteria can be expressed as: rx = ~ { Yi" rp.,} -

b[X].

For the acetogenic and methane bacteria there is only one major carbon product, therefore the yield coefficient Y is related directly to the substrate uptake rate: r x = Y r s -- b [ X ] .

~P

H2

o'. No?

2[acet ,I-CoA]

H2

I~I2

2[Lactic acid]

(15)

The rates of production and consumption for each reactor component were calculated from the stoichiometry of the substrate and biomass equations of each group of bacteria. By summing the associated biological rates of consumption and production for a particular component (such as hydrogen) the overall rates of production for each reactor variable rx,~were calculated. This model has assumed that changes in the growth and substrate utilization of the anaerobic bacteria can be represented by modifying the uninhibited

Glucose

2[Pyruvc acid1---

(14)

i

Butyric acid

2[Acetic acid]

Fig. 2. The reaction network with hydrogen regulation sites used to model the degradation pathway of the acid-forming glucose bacteria.

854

D . J . COSTELLO e t al.

3[Lactic acid] ~ r

1 [acetic acid]

3[Pyruvtc acid]

tt 2

2[Propionic acid] Fig. 3. The reaction network with hydrogen regulation sites used to model the degradation p a t h w a y of the lactic acid bacteria.

substrate utilization equations [as represented by equation (13)] to account for hydrogen, pH, and volatile acid inhibition. For hydrogen and pH inhibition, the substrate rate equations are multipled by inhibition factors. With volatile acid inhibition the substrate rate equations are themselves modified by the inclusion of competitive or noncompetitive inhibition terms. Regulation and inhibition by hydrogen. An important feature of this model is the regulation and inhibition of the acid-producing groups of bacteria by the partial pressure of hydrogen gas. This model uses the same technique as was developed by Mosey (1983) to model hydrogen inhibition and regulation in an anaerobic ecosystem.

A major change to Mosey's model was the modification of the anaerobic ecosystem to include lactic acid bacteria (Fig. 1). The proposed metabolic pathway of the glucose bacteria was modified to produce lactic acid rather than propionic acid, while the lactic acid bacteria produced either acetic or propionic acid according to the level of hydrogen in the biogas. The generation rates of the metabolic products for these bacteria were controlled by the partial pressure of hydrogen in the gas phase via the NAD regulation sites shown in Figs 2 and 3. The hydrogen inhibition and regulation functions used by the model for each bacteria are summarized in Table 2. The functions in Table 2 were used to regulate the substrate and product rate equations,

Table 2. Hydrogen inhibition functions for the modification of the maximum uptake rates in the glucose, lacetic acid, and acetogenic bacteria (zn = 2000 x partial pressure of hydrogen in atmospheres) Product regulation factors Bacteria types Mixed-product glucose Mixed-product lactic acid

Inhibition factor 1 1 + zn 1 1 +zn 1

Acetogenic (propionic)

1 + zn

Acetogenic (butyric)

1 1 + zn

Propionic acid

Buytyric acid

Not applicable

zn

I

zn

(I + z n ) 2

(1 + z n ) 2

(I + z n )

Not applicable

I 1 + zn

Not applicable

zn

(1 + z n )

Acetic acid

Lactic acid

Model derivation of an anaerobic reactor by being multiplied with the unregulated substrate uptake rate equations [as represented by equation (13)]. Consider the glucose bacteria; the uninhibited substrate uptake rate of glucose was modified by the hydrogen inhibition parameter specified in the Table 2. The substrate uptake rate term then becomes: rs[modified] = ~ 1

(k(X][S]~ \ ~ / .

(16)

The production rates for the different acids of the glucose bacteria are calculated in much the same way. The modified substrate uptake rate [equation (16)] is multiplied by the stoichiometric coefficients determined from the substrate equations in Table 1. This term is then regulated by the hydrogen inhibition parameters given in Table 2. For example, the production rates for each acidic product of the glucose bacteria are: ro[acetic acid] = 2 × ~

1

rs[modified]

(17)

rp[butyric acid] = 1 x (1 + zn) 2 r~[modified]

(18)

~l - r z n )

zn

rp[lactic acid] = 2 x ~

zn

rs[modified] .

(19)

Note that the hydrogen inhibition terms of equations (17)-(19) regulate the relative proportions of endproducts generated rather than inhibit the uptake of substrate (and therefore growth) as occurs in equation (16). Thus, the stoichiometry of the bacterial reactions is maintained at all times. p H inhibition of the anaerobic bacteria The pH inhibition function for each group of bacteria is represented by a general linear polynomial equation with a value of one at the upper pH limit (PHuL) and a value of zero at the lower pH limit (pHLL) of the range of inhibition. The shape of the inhibition curve can be modified by manipulating the inhibition coefficient m. A value of m greater than one will induce more severe inhibition at the upper pH limit, while a value of m less than one reduces the severity of inhibition at the upper limit and increases the severity of inhibition as the lower pH limit is approached. The general form of the pH inhibition function is as follows: pH inhibition factor = r p H LpHuL

- P H L L It4 - PHLL-] "

(20)

Four different sets of pH inhibition parameters were assigned to the following groups of bacteria. These are: the mixed-product glucose and lactic acid bacteria; the propionic, and butyric acid acetogenic bacteria; the acetoclastic bacteria; and the hydrogenutilizing bacteria. To obtain the pH inhibited substrate uptake rate for a particular group of bacteria

855

the calculated substrate uptake rate was multiplied by the pH inhibition factor as shown below: rs[pH inhibited] = pH inhibition factor × rs.

(21)

An empirical pH inhibition equation was chosen to model the behaviour of the anaerobic bacteria because there is no research available that has isolated the effects of pH inhibition and incorporated them into a structured model. The only pH inhibition model that was based on some theoretical considerations was the model developed by Andrews (1969) which incorporated the effects of falling pH indirectly into the substrate rate equation. Andrew's model was not used because pH inhibition effects had been considered in that model to be directly coupled to substrate inhibition of the bacteria, and so could not be separately examined from substrate inhibition effects (which have since been shown not to be significant in anaerobic reactors). Also, Andrew's inhibition model involves the modification of the substrate model to account for the uptake of the unionized portion of the substrate. This would make the choice of parameters for each group of bacteria quite difficult since most data has been determined in terms of the total concentration of the substrate. The only information available to set the parameters of the pH inhibition model was taken from the general information available on the range of growth rates for particular groups of bacteria. Experimental studies of the acetoclastic bacteria by Clark and Speece (1971) and Duarte and Anderson (1982) show that methane production is maximized in a pH range of 7.0-7.2 and that methane production falls off on either side of this optimum range to a 50% level at plus or minus approximately one pH unit for Duarte and Anderson (1982) and 1.5 pH units for Clark and Speece (1971). Pure culture studies of acetoclastic bacteria have been undertaken by Zinder and Mah (1979) and Huser et aL (1982). Zinder and Mah (1979) showed that optimum methane production occurred between 6.0 and 7.0, with the methane production rate falling to 50% at pH values of 5.5 and 8.0. However, Huser et al. (1982) found the optimum methane production rate to be at about 7.5, with cessation of methane production occurring at pH values below 6.8 and above 8.2. A pure culture study by Zehender and Wuhrmann (1977) of a bacterium utilizing hydrogen and carbon dioxide showed optimal growth rates at a pH of 7.0 and an overall growth range of approx. 6.0-8.5. Some studies are available on the pH inhibition of the acid-forming bacteria which consume glucose. Russell et al. (1979) examined the pH inhibition of five different types of anaerobic bacteria found in the tureen. Three types of bacteria ceased to grow at a pH of about 5.1, with optimum growth occurring at or above a pH of 6.5. The other two groups of bacteria were considerably more sensitive to changes in the pH. They ceased to grow below a pH of

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approx. 5.5, with the observed growth rates improving from 6.0 up to a pH of 6.8. Graham and Lund (1983) examined the growth of a strain of Streptococcus isolated from alkaline potato-processing wastes. This bacterium produced lactate, formate, acetate, and ethanol and was found to have an optimum pH growth range from 6.0 to 9.0. Nanda et al. (1983) found that Propionibacterium shermanii growth on glucose had an optimum pH range of 5.0-7.0, with significant decreases in the specific growth rate outside this range. A study by Zoetemeyer et al. (1982a) showed that in mixed culture systems the optimum growth of a population of glucose consuming bacteria occurred in a pH range of 5.8-6.2. There was a 50% decrease in the growth rate at a pH of 5.0, while for pH levels greater than 6.0 there was a gradual decrease in biological activity down to 25% at a pH of 8.0. A similar optimum pH range was found for an acidogenic population degrading a complex waste (Dinopoulou et al., 1988). The optimum pH was found to be at 7.0 with significant acidification of the wastewater occurring within the range of 5.0-8.0. While information on pH inhibition is available for the acidogenic and methanogenic bacteria, there are no available data on the pH inhibition of the acetogenic bacteria which consume propionic and butyric acids (Heyes, 1984). Yet it is possible to deduce the overall inhibition characteristics of the acetogens from anaerobic treatment systems. Two-stage reactors show optimal growth conditions for the hydrolytic acidogens in the first stage at a pH of approx. 6.0, while the second stage provides for optimal growth for the methanogens at 7.0. The acetogenic bacteria are more likely to grow in the second stage of the reactor because of the high levels of hydrogen generated in the acidogenic phase and the slow growth rates of these bacteria. If as expected natural selection occurs within this group, the optimal pH growth conditions for the acetogenic bacteria would occur at approximately the same pH levels as the methanogens. It could also be argued that the symbiotic relationship of the acetogenic bacteria with the hydrogen-utilizing bacteria would also reflect in similar pH inhibition functions. Product inhibition

The most important role of product inhibition in the biological model was to account for the accumulation of organic acids at a neutral pH (already accounted for to some extent by the hydrogen regulation functions on the acid-forming and acetogenic bacteria). Product inhibition was included in the model by modifying the substrate utilization rate equations for the glucose and lactic acid bacteria, and the propionic and butyric acid bacteria. Consider the two groups of acetogenic bacteria. A competitive inhibition model is used to account for product inhibition by acetic acid. The concentration of the inhibitor [I] is the millimolar concentration

of acetic acid in the reactor. It is assumed that the inhibition of the bacteria by the volatile acid substrates or lactic acid is insignificant. The general equation for the competitive inhibition of the acetogenic bacteria can be written as follows: r~ =

k[X][S]

(22)

Kj] Equation (22) replaces the uninhibited substrate uptake term as expressed by equation (13). For the two groups of acid-forming bacteria, a noncompetitive model for substrate uptake was used to model product inhibition, that is: k[XI[S]

(K~ + [S]) 1 +

Kj]

The inhibitor concentration [I] fbr the acid-forming bacteria was set as the total millimolar concentrations of the respective biological products of each acidforming group. Such specific associations must be considered carefully since product inhibition is generally considered on the basis of total volatile acids. Another consideration which makes the choice of the parameter K~ difficult is that results are commonly reported with the inhibitor concentration in units of weight rather than molar concentrations. Product inhibition has been discussed in several papers (Kaspar and Wuhrmann, 1978; Zoetemeyer et al., 1982b; Denac, 1986). Product inhibition can be characterized as an inhibitory response to the accumulation of products excreted by the bacteria, which would depend upon the concentration of product relative to the amount of biomass present. In some cases a lowering of the free energy of the substrate reaction by the accumulating product is the major cause of product inhibition. In this case the ratio of product to substrate would be more important than the absolute concentration of the product, Free-energy-mitigated product inhibition is likely to occur in bacteria that utilize substrate reactions with a small physiological free energy. In an anaerobic ecosystem these bacteria would be the propionic and butyric acetogens, and the acetoclastic methanogens. The acid-forming glucose and hydrogen-utilizing methanogens have large negative free energies for their substrate reactions and, consequently, any apparent product inhibition could only be caused by a general toxic or inhibitory effect of the product on the growth of these bacteria. The experimental work of Zoetemeyer et ul. (1982b) found that 4580 rag/1 (52 mmol/l) of butyrate reduced the acidogenesis of glucose by 30%, at a pH of 5.5. A change in product formation towards lactate, with a decrease in propionate and butyrate was also noted. This indicated that the product distribution of the glucose bacteria changed in an attempt to reduce the levels of butyric acid in the reactor.

Model derivation of an anaerobic reactor The product inhibition of the propionic bacteria by acetic acid was considered by Kaspar and Wuhrmann (1978). They found that 80mmol/l of acetic acid (4800 mg/l) completely inhibited the uptake of propionic acid, while no inhibition was observed at half this concentration. Furthermore, calculations of the free energy of the degradation of propionate over a range of hydrogen partial pressures and acetate concentrations found that free-energy considerations were important in the degradation of propionic to acetic acid. Using the results of batch experiments, Denac (1986) found that acetic acid inhibition of propionic and butyric acid acetogenic bacteria could be approximated by competitive inhibition functions. At different levels of acetic acid, different inhibition parameter values were used to fit the experimental data. It was found that significant inhibition of the propionic bacteria occurred at 10 mmol/1 of acetic acid, whilst butyric acid bacteria were inhibited at 26.7 mmol/1 of acetic acid. These results show that product inhibiting mechanisms may operate in the acetogenic bacteria as free-energy considerations become significant. The inhibiting concentrations of acetic acid noted in Denac's experiments (Denac, 1986) is much lower than for the experiments of Kaspar and Wuhrmann (1978), but this may be attributed to differences in experimental conditions. IMPLEMENTATION OF T H E M O D E L

Solution of the model equations required approx. 500 lines of F O R T R A N 77 code. A library routine from the Department of Chemical Engineering, University of Queensland, Australia was used to integrate the combined set of differential and algebraic model equations. This routine uses a diagonally implicit Runge-Kutta (2nd order) technique described by Cameron (1983). Solution of a typical problem took approx. 15 min on an IBM XT clone. DISCUSSION

The major difference between this model and those of earlier researchers (Andrews, 1969; Graef and Andrews, 1974; Hill and Barth, 1977) is in the complexity of the anaerobic ecosystem. Initially anaerobic biosystem models considered either a single-group ecosystem (Andrews, 1969; Graef and Andrews, 1974) or a two-group ecosystem (e.g. Hill and Barth, 1977). If some of the more general characteristics of anaerobic reactors are of interest these models may be sufficiently accurate to reflect such behaviour. However, such models cannot be used if effects such as hydrogen inhibition, product inhibition, or the distribution of the different organic acids in the anaerobic reactor are of interest. Consequently more complete descriptions of the anaerobic ecosystem are needed. This has been provided by the

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model developed in this paper. Hydrogen inhibition and regulation of the anaerobic ecosystem (first developed by Mosey, 1983) were included in an attempt to predict the accumulation of the different organic acids in an anaerobic reactor without having to artifically set the relative production rates of each product. The advantage of taking a structured approach to modelling such a complex process is evident when applying the model to reactors operating under different process conditions. With simpler models specific experimental data are required to estimate lumped model parameters, while a structured model will, ideally, simulate any number of different experimental systems (within the constraints of the model assumptions). For example, the model developed in this paper will simulate the relative accumulation of the major organic acids in an anaerobic reactor, while a model without a hydrogen inhibition and regulation model would have to include estimates of the relative production rates of each acidic product, not only at steady state but also for any type (e.g. hydraulic or organic) and range of possible disturbances to the process. An important feature of the anaerobic ecosystem model is its ability to account for the accumulation of lactic acid in the reactor, possibly an important intermediate in the anaerobic degradation process (Stronach et aL, 1986). Product inhibition functions were also included in the model, but were based only on experimental evidence presented in the literature. Competitive inhibition models were used to model the substrate uptake equations for the acetogenic bacteria, while noncompetitive models were applied to the substrate uptake equations of the acid-forming bacteria. The experimental evidence supporting these models is not substantial, and the importance of such models would be subject to serious consideration when modelling specific anaerobic processes. The pH inhibition models used for each group of bacteria were empirical functions based on the range of inhibition of each particular trophic group. Previous mathematical models have either incorporated pH inhibition with so-called "substrate inhibition" (Graef and Andrews, 1974; Hill and Barth, 1977) or assumed some empirical-type function (Mosey, 1983). There appears to be little research investigating the mechanisms behind pH inhibition in anaerobic ecosystems. The adaptability of bacteria to low pH levels is one area that is beyond the scope of this model, but is an area that requires attention if the model is to be improved. A physico-chemical model allows the calculation of the pH and other ionic species in the reactor. Previous models (Graef, 1972; Graef and Andrews, 1974) have used simplifying assumptions that are only valid in a narrow pH range (6-8). The physico-chemical model developed in this paper is valid over the full pH range, and its generalized structure allows it to be easily

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e x p a n d e d to account for other acidic a n d basic c o m p o n e n t s t h a t m a y be f o u n d to be i m p o r t a n t , e.g. in the modelling o f a n a n a e r o b i c t r e a t m e n t process t h a t degrades protein-type wastes. In this case the buffering capacity o f a m m o n i a could be expected to have a significant effect on the process, a n d so should be considered in the physico-chemical model. To be able to simulate dynamically a single-stage high-rate anaerobic reactor the anaerobic degradation model a n d the physico-chemical model were i n c o r p o r a t e d into the rate equations for a two-phase c o n t i n u o u s stirred tank reactor with a c o n s t a n t pressure headspace. Verification a n d testing of this model is examined in a further p a p e r (Costello et al., 1991) which describes the experimental validation of the model against d y n a m i c experimental data from different reactor systems. Acknowledgements--The financial assistance of the University of Queensland through the Alfred and Olivea Wynne Postgraduate Research Scholarship is gratefully acknowledged. Special thanks to the Department of Chemical Engineering for their continual financial support of this work.

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