mathematical modeling of non-ideal mixing continuous flow reactors for anaerobic digestion of cattle...

Mathematical modeling of non-ideal mixing continuous flowreactors for anaerobic digestion of cattle manure

A. Keshtkar a,b, B. Meyssami b,*, G. Abolhamd b, H. Ghaforian a,M. Khalagi Asadi a

a Center of Renewable Energies for Research and Application, Atomic Energy Organization of Iran, Tehran, Iranb Faculty of Engineering, Department of Chemical Engineering, Tehran University, P.O. Box 11365-4563, Tehran, Iran

Received 6 August 2001; received in revised form 14 January 2002; accepted 23 April 2002

Abstract

Most conventional digesters used for animal wastewater treatment include continuously stirred-tank reactors. While imperfect

mixing patterns are more common than ideal ones in real reactors, anaerobic digestion models often assume complete mixing

conditions. Therefore, their applicability appears to be limited. In this study, a mathematical model for anaerobic digestion of cattle

manure was developed to describe the dynamic behavior of non-ideal mixing continuous flow reactors. The microbial kinetic model

includes an enzymatic hydrolysis step and four microbial growth steps, together with the effects of substrate inhibition, pH and

thermodynamic considerations. The biokinetic expressions were linked to a simple two-region liquid mixing model, which con-

sidered the reactor volume in two separate sections, the flow-through and the retention regions. Deviations from an ideal completely

mixed regime were represented by changing the relative volume of the flow-through region (a) and the ratio of the internal exchange

flow rate to the feed flow rate (b). The effects of the hydraulic retention time, the composition of feed, the initial conditions of the

reactor and the degree of mixing on process performance can be evaluated by the dynamic model. The simulation results under

different conditions showed that deviations from the ideal mixing regime decreased the methane yield and resulted in a reduced

performance of the anaerobic reactors. The evaluation of the impact of the characteristic mixing parameters (a) and (b) on the

anaerobic digestion of cattle manure showed that both liquid mixing parameters had significant effects on reactor performance.

� 2002 Published by Elsevier Science Ltd.

Keywords: Anaerobic digestion; Mathematical modeling; Continuous-flow reactor; Imperfect mixing; Cattle manure

1. Introduction

Anaerobic digestion is a biodegradation process,

which uses a consortium of natural bacteria to convert a

large portion of the organic solids in the wastewater into

biogas. The biogas is mainly a mixture of methane andcarbon dioxide, and if captured, is a gas fuel used for

heat and/or power generation. Most conventional di-

gesters used for animal wastewater treatment are either

continuously stirred-tank reactors or plug-flow reactors.

Most of the previous research on animal wastewater

treatment was constructed on these two types of the

anaerobic digesters (Varel et al., 1977; Hills and Me-

hlschan, 1984; Angelidaki and Ahring, 1993; Hansenet al., 1999). In these two types of digesters, the HRT

equals the SRT and the active biomass is removed from

the digester in the effluent on a daily basis. The HRT

needs to be long enough to ensure a sufficient SRT in the

digester so that a viable bacterial population necessary

to complete the anaerobic digestion process is main-

tained in the reactor. Typical HRTs of conventionalmesophilic (35 �C) digesters for treating animal wastesare usually controlled at 10–20 days, depending on the

solids content of the wastes. The long retention time

required for animal manure digestion may be attributed

not only to the presence of complex organic compounds,

but also to high concentrations of the ammonia nitro-

gen that affect the anaerobic decomposition process

(Zeeman et al., 1985).The HRT is one of the most important design para-

meters affecting the economics of digesters. For a given

volume of wastewater, a shorter HRT translates into a

smaller digester and therefore more favorable econom-

ics. Digester developers have taken various approaches

Bioresource Technology 87 (2003) 113–124

*Corresponding author. Fax: +98-21-6498982.

E-mail address: [email protected] (B. Meyssami).

0960-8524/03/$ - see front matter � 2002 Published by Elsevier Science Ltd.

PII: S0960-8524 (02 )00104-9

mail to: [email protected]

in the past to reduce HRT, e.g., efficient mixing of the

reactor. Good mixing promotes the efficient transfer of

substrates and heat to the microorganisms, maintains

uniformity in other environmental factors and assureseffective use of the entire reactor volume by preventing

stratification and formation of dead spots and prevents

pockets of the VFA from forming. Scum formation can

also be greatly reduced or even eliminated by suitable

agitation. It is recognized that inhomogeneities in the

medium can have a profound influence, especially on

production of metabolites (Nielsen and Villadesen, 1992).

While imperfect mixing patterns are more commonthan ideal ones in real reactors, anaerobic digestion

models often assume complete mixing conditions.

Therefore, their applicability appears to be limited.

However, by using an appropriate configuration that

corresponds to the hydraulic characteristics of the real

reactor flow pattern, it is possible to simulate non-ideal

reactor performance. For example, by using the proper

liquid mixing models based on the hydrodynamic con-

figuration of the reactor studied, Reinhold et al. (1996)

calculated and predicted the mixing behavior in a biogastower reactor to be scaled-up. One of the classical

mixing models is the two-region model which, despite its

simplicity, is used in chemical engineering for the de-

scription of retention time distribution in real reactors

(Levenspiel, 1972). It has proved to be a useful tool for

the theoretical study of the effects of inhomogeneity in

chemical and biological systems. For example, Bello-

Mendoza and Sharratt (1998) used this mixing modelfor the effect of imperfect mixing on performance of

anaerobic sewage sludge digestion.

The objective of this study was to develop a mathe-

matical model, which combines the two-region mixing

model with a proper structured kinetic model, for the

simulation of anaerobic cattle manure digestion in non-

Nomenclature

a mixing parameter

b mixing parameter

C liquid concentration, g/l

[CO2] free CO2 in liquid concentration, mol/l

f individual bacterial fraction in initial total

biomassfpr mass conversion factor of propionate to ac-

etate ¼ 0.8108

fbut mass conversion factor of butyrate to acetate

¼ 0.6818

Ft biogas transfer rate, mol/d

F(pH) pH function

H Henry’s constant, atm l/mol

HRT hydraulic retention timeSRT sludge retention time

k hydrolysis rate constant, d�1

K0 non-inhibited hydrolysis rate constant, d�1

Ka dissociation constant

kd bacterial decay rate constant, d�1

Ki inhibition constant, g/l

Ks Monod saturation constant, g/l

m feed constant used in Eq. (1)n feed constant used in Eq. (1)

N gas transfer rate, g/d

[NH3] free NH3 in liquid concentration, mol/l

P pressure, atm

pKh constant used in Eq. (16)

pKl constant used in Eq. (16)

Q volumetric flow rate, d�1

rd bacterial decay rate, g/l drh hydrolysis reaction rate, g/l d

rs substrate consumption rate, g/l d

rx bacterial growth rate, g/l d

R gas constant, atm l/molK

t time, d

T temperature, K

Vg gas volume of reactor, l

Vl liquid volume of reactor, l

VFA volatile fatty acidsX microorganisms concentration, g/l

ye yield factor used in Eq. (10)

a flow-through region

b retention region

h HRT, d

l specific growth rate, d�1

lmax maximum specific growth rate, d�1

Subscripts

ac acetate

am ammonia

A acidogenic bacteria

AB butyric degrading acetogenic bacteria

AP propionate degrading acetogenic bacteriabut butyrate

c carbon dioxide

e exchange between zones

f feed

i component i

is insoluble substrate

m methane

M methanogenic bacteriapr propionate

s soluble substrate

t total

w water

114 A. Keshtkar et al. / Bioresource Technology 87 (2003) 113–124

ideal continuous flow reactors. By computer simulation,the effect of mixing can be predicted on the performance

of the process, and the relation between mixing and the

kinetic parameters can also be described.

2. Mathematical model

2.1. Kinetic model

The stoichiometry of anaerobic digestion of cattle

manure has been described by Hill (1982) and developed

by Angelidaki et al. (1993). The kinetic model distin-

guishes five different processes: hydrolysis of particulate

substrate by extracellular enzymes, consumption of

soluble substrates by acidogenic bacteria, consumption

of VFA and formation of acetate by propionate andbutyrate degrading acetogenic bacteria, and finally

consumption of acetate and generation of methane by

methanogenic bacteria. The model includes VFA inhi-

bition of the hydrolysis step, acetate inhibition of

the acetogenic steps, free ammonia inhibition of the

methanogenic step, and pH inhibition of all biological

steps. In the model, the primary substrates in the ma-

nure are represented as soluble (s) and insoluble (is)carbohydrate units, with the basic formula (C6H10O5)sand (C6H10O5 � nNH3)is respectively. Cell mass is repre-sented by the empirical formula. Model statements are

as follows:

ðC6H10O5 � nNH3Þis ! yeðC6H10O5Þsþ ð1� yeÞðC6H10O5 � mNH3Þisþ ðn� ð1� yeÞmÞNH3 ð1Þ

ðC6H10O5Þs þ 0:1115NH3 ! 0:1115C5H7NO2

þ 0:744CH3COOHþ 0:5CH3CH2COOHþ 0:4409CH3ðCH2Þ2COOHþ 0:6909CO2 þ 0:0254H2O

ð2ÞCH3CH2COOH þ 0:06198NH3 þ 0:314H2O

! 0:06198C5H7NO2 þ 0:9345CH3COOHþ 0:6604CH4 þ 0:1607CO2 ð3Þ

CH3ðCH2Þ2COOH þ 0:0653NH3 þ 0:5543CO2þ 0:5543H2O! 0:0653C5H7NO2

þ 1:8909CH3COOH þ 0:4452CH4 ð4Þ

CH3COOHþ 0:022NH3 ! 0:022C5H7NO2

þ 0:945CH4þ 0:945CO2þ 0:066H2O ð5Þ

In Reaction (1), ye is the enzymatic efficiency or yieldfactor and the subscript in represents the non-biode-

gradable inert organic material. The coefficients ye, n,

and m, together with the ratio of the soluble to the in-

soluble substrate depend on the type of manure. The

processes of hydrolysis and biomass decay are described

by the first order reactions shown below:

rh ¼ kCis ð6Þrd ¼ kdX ð7Þwhere k is the hydrolysis rate constant and kd is the

decay rate constant. The hydrolytic reaction rate is as-

sumed to be inhibited by the presence of VFA. The in-hibition function chosen is non-competitive:

k ¼ k0ki;VFAP

VFAþ ki;VFAð8Þ

XVFA ¼ Cac þ fprCpr þ fbutCbut ð9Þ

Consumption of soluble substrates and volatile acids as

well as growth of anaerobic microorganisms, are as-

sumed to obey Monod-type kinetics, as follows:

rs ¼ Ys=xlX ð10Þrx ¼ lX ð11ÞAll the yield coefficients (Ys=x), expressed as gram per

gram of bacteria synthesized, can be directly calculated

from the stoichiometric relationships. Specific growth

rates (l) with non-competitive inhibition effects and pHmodulation for the biological steps can be written as:

lA ¼ lmax ACs

Kss þ Csð12Þ

lAP ¼ lmax APCpr

Kspr þ Cpr

Ki prKi pr þ Cac

FAPðpHÞ ð13Þ

lAB ¼ lmax ABCbut

Ks but þ Cbut

Ki butKi but þ Cac

FABðpHÞ ð14Þ

lM ¼ lmax MCac

Ksac þ Cac

Ki amKi am þ Cam

FMðpHÞ ð15Þ

where F ðpHÞ is the pH modulation function and is de-scribed by a Michaelis pH function normalized to give a

value of 1.0 as the center value (Angelidaki et al., 1993):

F ðpHÞ ¼ 1þ 2� 100:5ðpKl�pKhÞ1þ 10ðpH�pKhÞ þ 10ðpKl�pHÞ ð16Þ

The parameters pKl and pKh, denote the lower and theupper pH drop off values, respectively, where growth

rates are approximately 50% of the uninhibited rate. Ingeneral, the coefficients pKl and pKh are different forvarious microbial groups.

2.2. Two-region liquid mixing model

In the two-region mixing model, it is assumed that the

reactor volume is split into two sections: the flow-through

A. Keshtkar et al. / Bioresource Technology 87 (2003) 113–124 115

(a) region and the retention (b) region. Both regions areassumed to be perfectly mixed but the transfer of mate-

rials between the zones is limited. The retention regionhas the features of the behavior shown by a stag-

nant zone. Different levels of mixing are accomplished by

adjusting the relative volume of the flow-through region

(a) and the ratio of the exchange flow rate between re-

gions to the feed flow rate (b). A conceptual represen-

tation of the two-region mixing model is illustrated in

Fig. 1.

2.3. Model development

The generic model consists of a set of ordinary dif-

ferential equations, which represent mass balances using

different variables (see Appendix A). Differential vari-

ables include total concentrations of substrates, different

intermediate products and bacterial groups. The model

is based on the following assumptions and consider-ations:

1. The gas phase and the two liquid phases, a and b,were each assumed to be uniform.

2. All reactions are effectively rate controlled, i.e., the

effects of diffusional limitations in the biomass aggre-

gates are constant and incorporated into the kinetic

terms.3. The non-competitive type inhibition was considered

in all microbial steps as described in the previous sec-

tion.

4. First order reaction rates were applied for the bacte-

rial decay and the enzymatic hydrolytic steps.

5. Decay rate constants of the different bacterial groups

were assumed to be 5% of their maximum growth

rate (Angelidaki et al., 1993).6. Mass transfer to the gas phase only occurs in liquid

phase a.7. The influent and effluent streams are located in the

flow-through region.

8. The b liquid phase exchanges materials only with thea liquid phase.

9. The system pressure and retention volume are con-

stant.

10. Energetic effects are not considered with temperature

perfectly controlled.

11. At the operational temperature and pressure, biogas

is considered to be an ideal gas.

12. The biogas consists of methane, CO2 and water.13. The water vapor in the biogas stream is at the satu-

ration state.

14. The CO2 present in the a liquid phase is at thermody-namic equilibrium with the CO2 in the gas phase and

it obeys Henry’s law.

15. The concentration of methane in the a liquid phaseis assumed to be negligible, i.e., it is immediately

transferred to the gas phase because of its low solu-bility.

16. In the ionic charge balance (Eq. (A.32)), the algebraic

sum of the concentrations of other ionic compounds

in the process, [A�Cþ], are assumed to be constant

during the anaerobic digestion process and are calcu-

lated from the initial pH of the system for both a andb liquid phases.

With these assumptions and considerations the resulting

material balances the gas phase and a and b liquidphases and the ionic charge balance equations for the

two liquid phases are summarized and presented in

Appendix A.

The ionic charge balance equations should be itera-

tively solved for the pH calculation since the concen-

trations of the ionic compounds, in turn, are functionsof the pH according to Eqs. (A.25)–(A.31). Of course,

we need to use an additional iterative procedure for the

pH calculation of the a liquid phase, because accordingto Eq. (A.24), the total CO2 in the a liquid phase is afunction of the pH of this phase and the partial pressure

of the CO2 in the gas phase. Therefore, the following

steps should be done for the calculation of pH and

different component concentrations of the a liquidphase:

• Guess a pH for the time step of t (the pH of the pre-

vious time step is a good initial guess for the pH of

the next time step).

• Solve the set of ordinary differential equations for the

a liquid phase together with Eq. (A.24).• Solve the ionic charge balance (Eq. (A.32)) by atrial and error procedure for the calculation of

pH.

• Compare the calculated pH with the guessed pH.

• If jpHcalc � pHguessj > 10�3, consider the calculatedpH as the new guess for pH and go to step 2 to repeat

the above steps. Otherwise, save the final results for

time step of t.

Fig. 1. Two-region mixing model.


2.4. Computer simulations

For the dynamic prediction of anaerobic digestion of

cattle manure in non-ideal continuous flow reactors at

different operating conditions and also for evaluating

effects of the characteristic mixing parameters (a and b)

on reactor performance, the model described in this

paper was programmed in a generalized form in For-

tran, where a variable number of steps, feed composi-tion, initial conditions of the a and b liquid and gasphases, and the operating conditions could be specified

through an input file. The simulations were performed

by numeric first order integration of the relevant equa-

tions with a fixed time step by a computer program

based on Euler’s method. The program created an out-

put data file in a format suitable for graphic processing.

Kinetic model parameters were taken directly fromthe literature and are given in Table 1. Physico-chemical

model parameters at 35 �C are given in Table 2. Valuesof the mixing parameters were selected on the basis of

information found in the literature. Tracer studies con-

ducted in full-scale anaerobic digesters have revealed

well-mixed portions of digester volumes ranging widely

from 23% to 88%, while the balances have been in

stagnant volumes (Montieth and Stephenson, 1981).There is less evidence regarding the average interchange

rates between mixed and stagnant volumes in anaerobic

digesters. Smith et al. (1993) applied the same tracer

technique used by Montieth and Stephenson (1981) to

evaluate liquid mixing characteristics of a pilot scale

contact process anaerobic digester. In one of these

studies, they found the mixed and the stagnant volumes

to be 49% and 51%, respectively, whereas the ratio of the

flow rate of the internal exchange to the feed flow rate

(b) was found to be equal to 0.333.

3. Results and discussion

The manure composition used in the model simula-tions is given in Table 3 and was based on the cattle

manure used in the experiments of Angelidaki and

Ahring (1993). The simulations were performed with the

following conditions: 35 �C, 15 days HRT, atmosphericgas pressure, and a gas to liquid volume ratio of 0.1. On

the basis of the two-region mixing model described

above, the liquid composition of the stream flowing out

of the reactor has the same composition as the flow-through zone content. By definition, for a relative vol-

ume in the flow-through region (a) close to unity and,

for any value of ‘a’, with an interchange rate of the

material between regions to feed flow rate ratio (b) ap-

proaching infinity, the dynamic model produces results

closely approaching those of a completely mixed reac-

tor. Otherwise, for any ‘a’ with ‘b’ close to zero (i.e. no

interchange of material between regions) the systemconsists of a reactor with a completely dead zone of

volume (1� a)Vl. For values of the mixing parametersother than those mentioned above, the mathematical

Table 1

Kinetic parameters used in the model (Angelidaki et al., 1993)

Parameter Value

Kss (g/l) 0.5

Ks pr (g/l) 0.259

Ks but (g/l) 0.176

Ksac (g/l) 0.12

Ki VFA (g/l) 0.33

Ki pr (g/l) 0.96

Ki but (g/l) 0.72

Ki am (g/l) 0.26

K0 (d�1) 1.0

lmax A (d�1) 5.0

lmax AP (d�1) 0.54

lmax AB (d�1) 0.68

lmax M (d�1) 0.6

ye 0.55

n 0.454

m 0.34

pKhAP 8.5

pKl AP 6.0

pKhAB 8.5

pKl AB 6.0

pKhM 8.5

pK1M 6.0

Table 2

Physico-chemical parameters at 35 �C (Dean, 1992)

Parameter Value

Kw (M) 2:065� 10�14Ka1 (M) 4:909� 10�7Ka2 (M) 5:623� 10�11Ka3 (M) 1:730� 10�5Ka4 (M) 1:445� 10�5Ka5 (M) 1:445� 10�5Ka6 (M) 1:567� 10�9Hc (atm l/mol) 37.67a

aArcher (1983).

Table 3

Characteristics of the feed

Characteristic Value

Insoluble substrate 30.4 (g/l)

Soluble substrate 5.4 (g/l)

Total acetate 4.5 (g/l)

Total propionate 2.3 (g/l)

Total butyrate 0.2 (g/l)

Total ammonia 3.0357 (gNH3/l)

Total carbon dioxide 0.0 (g/l)

Total microbial biomass 0.2 (g/l)

Fraction of acidogens 0.65

Fraction of propionate acetogens 0.025

Fraction of butyrate acetogens 0.025

Fraction of methanogens 0.3

pH 8.0


model simulates the performance of an imperfectlymixed digester.

Simulations of the anaerobic digestion process ap-

plied to cattle manure with different degrees of mixing

consisting of ða; bÞ equal to ð0:9; 10Þ, ð0:6; 0:5Þ andð0:2; 0:2Þ are shown in Figs. 2–5. The three degrees ofmixing considered were chosen to simulate reactor be-

havior approaching a completely mixed reactor, an im-

perfectly mixed reactor and an incompletely mixedreactor, respectively. The dynamic results of the insol-

uble substrate, total acetate, total propionate and total

ammonia concentrations are illustrated in these figures

for both regions, respectively. Significant differences

between the concentration patterns shown by these

systems arose due to the different degrees of mixing

considered. As can be seen from the figures, the medium

concentrations in both zones for the well-mixed reactorare the same throughout the duration of the simulation,

because the materials quickly distribute from the flow-through region to the retention region due to the good

degree of mixing applied. In contrast, for the poorly

mixed groups the dynamic simulation results of the

medium concentrations show non-homogeneous distri-

butions of components in the reactor and less volume

available for the active digestion due to the limited

interchange between zones. The resulting homoge-

neous and non-homogeneous medium concentrationsthroughout the volume of the reactor due to the high

and low interchange rates used show the ability of the

two-region model to simulate anaerobic reactors with

either ideal or non-ideal mixing. It also shows the effect

of mixing parameters on the residence time distribu-

tion pattern as well as the distribution of components

in the reactor. Therefore, mixing influences rates in the

anaerobic digestion process. This influence on reac-tion rates is a result of the number of non-linear rate

Fig. 2. Dynamic simulation of anaerobic digestion of cattle manure in a continuous flow reactor under HRT¼ 15 days and different degrees ofmixing for prediction of the insoluble substrate concentration in flow-through and retention regions.

Fig. 3. Dynamic simulation of anaerobic digestion of cattle manure in a continuous flow reactor under HRT¼ 15 days and different degrees ofmixing for prediction of the total acetate concentration in the flow-through and the retention regions.


expressions and the substrate-dependent Monod rela-

tionship by which anaerobic digestion is represented.

Dynamic simulation of the methane yield and the pH

of the liquid stream outflow of the reactor are shown in

Figs. 6 and 7 for the three different degrees of mixingdefined above. As the model results show, the methane

yield of the process depends on the variations of the pH

shown in Figs. 6 and 7 for the start-up period. As shown

in Fig. 6, the methane yield decrease with the degree of

mixing where this rate of decrease depends on how far

the conditions are from ideality. The methane yield in

the reactor with (a ¼ 0:6 and b ¼ 0:5) and in the reactorwith (a ¼ 0:2 and b ¼ 0:2) were respectively 3.2% and85% lower than that of the reactor with (a ¼ 0:9 andb ¼ 10). Conversely as can be seen from Figs. 2–4, theconcentrations of the insoluble substrate, acetate and

propionate in the effluent of the reactor with (a ¼ 0:6and b ¼ 0:5) and the reactor with (a ¼ 0:2 and b ¼ 0:2)

were respectively (1.7; 3.3 times), (1.6; 137.6 times) and

(1.7; 53.4 times) higher than those of the reactor with

(a ¼ 0:9 and b ¼ 10). According to these results, it seemslikely that there is a threshold level of deviation from

ideal mixing where reactor performance declines sub-stantially, however, ideal mixing may not be required to

have nearly ideal performance with regard to methane

yield and VFA concentrations in the effluent.

The effect of HRT on the methane yield under dif-

ferent mixing conditions was also evaluated. The steady-

state results are shown in Fig. 8. Methane yield showed

an increase with retention time and degree of mixing.

For HRT¼ 15 days the methane yield for the poorermixed reactor (a ¼ 0:4 and b ¼ 0:5) was 8% lower thanthat attained by the better mixed reactor (a ¼ 0:4 andb ¼ 2). As shown in Fig. 8, extending the retention timecould improve the methane yield of imperfectly mixed

reactors. This is valid for reactors where the hydraulic

Fig. 5. Dynamic simulation of the anaerobic digestion of cattle manure in a continuous flow reactor under HRT¼ 15 days and different degrees ofmixing for prediction of total ammonia concentration in the flow-through and the retention regions.

Fig. 4. Dynamic simulation of the anaerobic digestion of cattle manure in a continuous flow reactor under HRT¼ 15 days and different degrees ofmixing for prediction of total propionate concentration in the flow-through and the retention regions.


characteristic time is greater than the mixing charac-

teristic time.

The methane yield for anaerobic digestion of cattle

manure as a function of the relative volume of the flow-

through region (a) is shown in Fig. 9 for different values

of the mixing parameter b. As expected, the steady-state

methane yield of the reactor increased with mixing

parameter a. It is obvious from the physical consider-ations that increasing the volume of the flow-through

region results in more volume available for immedi-

ate anaerobic digestion activity and, therefore, highermethane yield. It was also observed that decreasing the

mixing parameter a increased the effect of the mixing

parameter b on methane yield. On the other hand, the

model also showed that by increasing the mixing pa-

rameter a, the effect of mixing parameter b on methane

yield would decrease so that all the curves converge to a

fixed value. The steady-state methane yield as a function

of mixing parameter b is shown in Fig. 10 for differentvalues of the relative volume of the flow-through region.

As expected, with decreasing b values, the steady-state

Fig. 8. Effect of HRT on the methane yield of anaerobic digestion of

cattle manure.Fig. 6. Dynamic simulation of anaerobic digestion of cattle manure in

a continuous flow reactor under HRT¼ 15 days and different degreesof mixing for prediction of methane yield.

Fig. 7. Dynamic simulation of anaerobic digestion of cattle manure in

a continuous flow reactor under HRT¼ 15 days and different degreesof mixing for pH prediction of outlet stream from reactor.

Fig. 9. Effect of the relative volume of the flow-through region (a) on

the methane yield of anaerobic digestion of cattle manure.


methane yield of the reactor would decrease. How-

ever, in the range of values evaluated, mixing parame-

ter b showed a lower impact on the performance of the

steady-state anaerobic digestion than mixing para-

meter a.

3.1. Potential applicability of the two-region model of

liquid mixing

The validity of the kinetic model described was

evaluated by Angelidaki et al. (1993) for thermophilic

conditions and Keshtkar et al. (2001) for mesophilic

conditions. Although, the simulation results obtained

are qualitatively in agreement with what could be ex-

pected theoretically and with the published experimentalobservations (Perot et al., 1988; Lin and Pearce, 1991),

the proposed dynamic model requires experimental

verification in order to assess its applicability. In Figs.

11 and 12, model simulation results were compared to

experimental data given by Dugba and Zhang (1999).

They conducted a series of experiments for anaerobic

digestion of dairy manure in a two-stage anaerobic se-

quencing batch reactor system. The volume and theoperating parameters of their experiments are shown in

Table 4.

To evaluate the applicability of the model, prelimi-

nary simulations were compared to sequencing batch

experimental runs measuring methane yield at various

organic loading rate for an HRT of 3 days to determine

the most appropriate set of mixing model parameters. In

Fig. 11, the best fit curve for the experimental data isshown. The estimated a and HRT=b mixing parametersof the reactor are equal to 0.3 and 4.0, respectively.

Steady-state methane yields for an HRT of 6 days were

then predicted for different organic loading rates using

the mixing parameters estimated. Predicted values are

Fig. 11. Model prediction versus experimental data (Dugba and

Zhang, 1999) of methane yield––organic loading rate for selecting the

most appropriate set of mixing parameters.

Fig. 12. Comparison between experimental data (Dugba and Zhang,

1999) and prediction of methane yield as a function of organic loading

rate at a temperature of 35 �C.

Fig. 10. Effect of ratio of the internal exchange flow rate to the feed

flow rate (b) on the methane yield of anaerobic digestion of cattle

manure.

Table 4

Operating parameters of the reactor

Operational parameters Values

Total volume 15 l

Temperature 35 �CpH Controlled at 6.7–7.3

Mixing of reactor 1 min every hour

VS loading rate for HRT¼ 3 days 2, 3, 4, 6, 8 gVS/l/day

VS loading rate for HRT¼ 6 days 2, 3, 4 gVS/l/day


compared with experimental data in Fig. 12. As seen,since the appropriate adjustments to this model were

made for a sequencing batch reactor, good agreement

was obtained between the predicted values and the ex-

perimental data.

For future application of the model, methods to

measure the characteristic mixing parameters are nee-

ded. For modeling the liquid mixing behavior of a bio-

gas tower reactor Reinhold et al. (1996) applied a simplemixing model with characteristic mixing parameters

similar to those required by the two-region mixing

model. The mixing parameters were calculated from

experimental tracer-response curves. This suggests the

possibility of calculating the parameters a and b using a

similar approach. Simple liquid mixing models, such as

the two-region model described here, appear to be useful

for the simulation of anaerobic reactors under non-idealmixing conditions. Imperfect mixing models may also be

useful tools for reactor scale-up.

4. Conclusions

A kinetic model describing the effects of substrate

inhibition, pH and thermodynamic considerations for

anaerobic digestion of cattle manure was applied with a

two-region liquid mixing model to evaluate performanceof non-ideal continuous flow reactors. The resulting

mathematical model could be used for simulation of

reactors with different degrees of mixing. Simulation

results showed that deviations from ideal mixing regime

result in decreased performance of anaerobic reactors. It

was also shown that methane yield is strongly dependent

on pH of the reactor. In addition, methane yield was

shown to increase with greater HRTs and increaseddegree of mixing in the reactor. Completely mixed re-

actors required a shorter HRT than incompletely mixed

reactors to achieve the same methane yield. On the other

hand, it was seen that whenever the hydraulic charac-

teristic time is significantly greater than the mixing

characteristic time, differences between methane yields

for imperfectly mixed reactors decrease. Evaluation of

the impact of the characteristic mixing parameters a andb on anaerobic digestion of cattle manure showed that

both liquid mixing parameters had significant effects on

the digestion process and that methane yield is a com-

plex function of both parameters.

Acknowledgements

The authors would like to express their appreciationfor the financial support provided by Center of Re-

newable Energies for Research and Application and Dr.

A. Ahmadi, the former head of the center.

Appendix A. Material balances

A.1. Liquid phase

Microbial biomass, Xi, i ¼ A, AP, AB, MdX a

i

dt¼ Xi;f � X a

i

ahþ X b

i � X ai

ah=bþ ðla

i � biÞX ai ðA:1Þ

dX bi

dt¼ X a

i � X bi

ð1� aÞh=bþ ðlbi � biÞX b

i ðA:2Þ

Insoluble substrate, Cis

dCais

dt¼ Cis;f � Ca

is

ahþ Cb

is � Cais

ah=b� kaCa

is ðA:3Þ

dCbis

dt¼ Ca

is � Cbais

ð1� aÞh=b� kbCbis ðA:4Þ

Soluble substrate, Cs

dCas

dt¼ Cs;f � Ca

s

ahþ Cb

s � Cas

ah=bþ 162ye162þ 17n k

aCais

� 12:858laAX

aA ðA:5Þ

dCbs

dt¼ Ca

s � Cbs

ð1� aÞh=bþ162ye

162þ 17n kbCb

is

� 12:858lbAX

bA ðA:6Þ

Total acetate, Cac

dCaac

dt¼ Cac;f � Ca

ac

ahþ Cb

ac � Caac

ah=bþ 3:54la

AXaA

þ 8:006laAPX

aAP þ 15:366la

ABXaAB

� 24:135laMX

aM ðA:7Þ

dCbac

dt¼ Ca

ac � Cbac

ð1� aÞh=bþ 3:54lbAX

bA þ 8:006lb

APXbAP

þ 15:366lbABX

bAB � 24:135l

bMX

bM ðA:8Þ

Total propionate, Cpr

dCapr

dt¼

Cpr;f � Capr

ahþCbpr � Ca

pr

ah=bþ 2:937la

AXaA

� 10:566laAPX

bAP ðA:9Þ

dCbpr

dt¼

Capr � Cb

pr


bA � 10:566lb

APXbAP

ðA:10ÞTotal butyrate, Cbut

dCabut

dt¼ Cbut;f � Ca

but

ahþ Cb

but � Cabut

ah=bþ 3:079la

AXaA

� 11:919laBPX

aAB ðA:11Þ

dCbbut

dt¼ Ca

but � Cbbut

ð1� aÞh=b þ 3:079lbAX

bA � 11:919lb

BPXbAB

ðA:12Þ


Total ammonium, Cam

dCaam

dt¼ Cam;f � Ca

am

ahþ Cb

am � Caam

ab17ðn� mð1� yeÞÞ162þ 17n kaCa

is

� 0:15ðlaAX

aA þ la

APXaAP þ la

ABXaAB þ la

MXaMÞðA:13Þ

dCbam

dt¼ Ca

am � Cbam

ð1� aÞh=bþ17ðn� mð1� yeÞÞ162þ 17n kbCb

is

� 0:15ðlbAX

bA þ lb

APXbAP þ lb

ABXbAB þ lb

MXbMÞðA:14Þ

Total carbon dioxide in the liquid phase, Cc

dCac

dt¼ Cc;f � Ca

c

ah� Cb

c � Cac

ah=bþ 2:413la

AXaA

þ 1:01laAPX

aAP � 3:303la

ABXaAB

þ 16:726laMX

aM � N a

c

aVlðA:15Þ

dCbc

dt¼ þ Ca

c � Cbc


bA

þ 1:01lbAPX

bAP � 3:303l

bABX

bAB

þ 16:726lbMX

bM ðA:16Þ

Methane in the liquid phase, Cm

Cbm

ah=bþ 1:509la

APXaAP þ 0:956la

ABXaAB

þ 6:082laMX

aM � N a

m

aVl¼ 0 ðA:17Þ

dCbm

dt¼ Cb

m

ð1� aÞh=bþ 1:509lbAPX

bAP

þ 0:956lbABX

bAB þ 6:082l

bMX

bM ðA:18Þ

where

h ¼ VlQf

ðA:19Þ

b ¼ QeQf

ðA:20Þ

A.2. Gas phase

Carbon dioxide in the gas phase, Pc

dPcdt

¼ RTVg

N ac

44

�� Pc

PFt

�ðA:21Þ

Methane in the gas phase, Pm

dPmdt

¼ RTVg

N am

16

�� Pm

PFt

�ðA:22Þ

Total material balance in the gas phase, Ft

Ft ¼P

P � Pw

N am

16

�þ N a

c

44

�ðA:23Þ

A.3. Thermodynamic equilibrium

The relation between free CO2 concentration in the aliquid phase and partial pressure of CO2 in the gas

phase, according to Henry’s law

½CO2�a ¼PcHc

ðA:24Þ

A.4. Liquid phase equilibrium chemistry

Ionic dissociation equations

CO2 þH2O$ HCO�3 þHþ ka1 ¼

½HCO�3 �½Hþ�

½CO2�ðA:25Þ

HCO�3 $ CO2�3 þHþ ka2 ¼

½CO2�3 �½Hþ�½HCO�

3 �ðA:26Þ

HAc$ Ac� þHþ ka3 ¼½AC��½Hþ�½HAc� ðA:27Þ

HPr$ Pr� þHþ ka4 ¼½Pr��½Hþ�½HPr� ðA:28Þ

HBut$ But� þHþ ka5 ¼½But��½Hþ�½HBut� ðA:29Þ

NHþ4 $ NH3 þHþ ka6 ¼

½NH3�½Hþ�½NHþ

4 �ðA:30Þ

H2O$ OH� þHþ kw ¼ ½OH��½Hþ� ðA:31ÞIonic balance equations for both a and b liquid

phases

½Hþ� þ ½NHþ4 � ¼ ½OH�� þ ½HCO�

3 � þ 2½CO2�3 �

þ ½Ac�� þ ½Pr�� þ ½But�� þ ½A�Cþ�ðA:32Þ

where

½NHþ4 � ¼

Cam=171þ ka6=½Hþ� ðA:33Þ

½OH�� ¼ kw=½Hþ� ðA:34Þ

½HCO�3 � ¼

Cc=441þ ½Hþ�=ka1 þ ka2=½Hþ� ðA:35Þ

½CO2�2 � ¼ Cc=44

1þ ½Hþ�=ka2 þ ½Hþ�2=ka1ka2ðA:36Þ

½Ac�� ¼ Cac=601þ ½Hþ�=ka3

ðA:37Þ

½Pr�� ¼ Cpr=741þ ½Hþ�=ka4

ðA:38Þ

½But�� ¼ Cbut=881þ ½Hþ�=ka5

ðA:39Þ


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