the effect of fatty acid diffusion in leachate on the propagation of concentration waves in the...

0097-8078/01/2806- $25.00 © 2001

MAIK “Nauka

/Interperiodica”0691

Water Resources, Vol. 28, No. 6, 2001, pp. 691–697. Translated from Vodnye Resursy, Vol. 28, No. 6, 2001, pp. 756–762.Original Russian Text Copyright © 2001 by Vavilin, Shchelkanov, Lokshina.

INTRODUCTION

Notwithstanding the considerable progress achievedin recent decades in the creation and development ofmunicipal solid waste landfills, many of landfills, eitherclosed or in use, are a substantial hazard as a source ofwater and air pollution. Poor engineering solutionsresult in surface and subsurface water pollution byleachate as well as in substantial methane emission intothe atmosphere. As a source of methane, the landfillsare believed to contribute to global warming.

Municipal solid waste landfills are the place wherea batch process of solid material decomposition canoccur. The anaerobic decomposition of organic matterproceeds in moist media and comprises several stages(hydrolysis, acidogenesis, acetogenesis, and methano-genesis). The rates of decomposition of suspended anddissolved organic material are commonly controlled bythe stages of hydrolysis and methane production,respectively [11]. Of critical importance in the anaero-bic decomposition of municipal solid wastes is the ratioof the rates of polymer hydrolysis and methanogenesis[2]. According to experimental studies [3] the produc-tion of methane to be limited first by the utilization ofvolatile fatty acids and next by the hydrolysis of poly-mers. A high level of volatile fatty acids results in lowpH, which can result in complete inhibiting bot meth-ane production and hydrolysis. Moisture content, thedimensions and geometry of solid waste particles aswell as the leachate density and the rates of circulationand neutralization along with the rate of fluid move-ment are among the most important characteristics tocontrol the rate of solid waste decomposition. The land-fills widely differ in their composition [10], and organicmatter decomposition can proceed with a considerablerate only in specific biochemically active zones. Poormass transfer along with acid inhibition can be fatal forthe waste decomposition process. For example, matter

almost without decomposition can be found in someplaces in old landfills [9].

In recent years, special bioreactors [4] have comeinto practice to replace solid waste disposal in landfills.In such reactors, the process of solid waste decomposi-tion can be made many times more rapid, the concen-tration of methanogens being initially high. A typicalscheme is the bioreactor proposed in the Netherlandsand called BIOCELL [13]. Keeping the production ofvolatile fatty acids in the course of acidogenesis andacetogenesis (a relatively fast process) in balance withthe destruction of volatile fatty acids in the course ofmethanogenesis (relatively slow process) requires ahigh rate of recirculation of partially decomposedorganic matter containing methanogenic microorgan-isms. When this balance is disturbed and pH becomeslow, the decomposition of solid waste can cease com-pletely.

The processes taking place in landfills are very com-plex [17]; therefore, the development of environmen-tally safe practice of waste disposal requires jointefforts of experts in many fields (microbiologists,chemists, engineers, and mathematicians). The avail-ability of experimental data will allow the use of math-ematical modeling. The overall rate of municipal solidwaste decomposition is commonly described by akinetic equation of the first order with respect toorganic matter, however, this description can be inade-quate. Only a few models are known to simulate biogasproduction in a landfill and incorporate hydrologicaland biological variables (model [5] is an example).These models still use a first-order kinetic equation todescribe the hydrolysis and make no allowance for theinhibition of individual processes.

A so-called two-particle model is proposed in [7].This model considers two types of particles (one typeexhibit high methanogenic activity and low organic

The Effect of Fatty Acid Diffusion in Leachate on the Propagation of Concentration Waves

in the Process of Municipal Solid Waste Decomposition

V. A. Vavilin, M. Yu. Shchelkanov, and L. Ya. Lokshina

Water Problems Institute, Russian Academy of Sciences, ul. Gubkina 3, GSP-1, Moscow, 119991 Russia

Received February 8, 2001

Abstract

—A distributed model of municipal solid waste decomposition is proposed. The main model param-eters are concentrations of municipal solid waste, volatile fatty acids, and the biomass of methane-producingmicroorganisms. Numerical analysis of the model is made. Diffusion of fatty acids in leachate, which facilitatesthe formation of anaerobic conditions, is shown to cause the formation of concentration chemical waves prop-agating in the space. The area of initiation methane production is found to expand.

WATER QUALITY AND PROTECTION: ENVIRONMENTAL ASPECTS

692

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No. 6

2001

VAVILIN

et al.

matter content, while the other has low methanogenicactivity and high organic matter concentration) uni-formly distributed over the reactor volume. The masstransfer between the particles is of diffusion type. Thismodel, however, fails to take into account the nonuni-formity in the distribution of solid wastes and microor-ganisms over the reactor volume. The nonuniform dis-tribution results in essentially different rates of metha-nogenesis in different parts of the reactor. Diffusion ofvolatile fatty acids from solid waste particles can be akey mechanism at the microscale of conventional parti-cles in the model [7]. However, the key mechanism atthe macroscale of a real reactor can be methanogenesis,which spreads over the reaction volume from some“initiation” spots. Veeken and Hamelers [16] made anattempt to apply the two-particle model to account forthe effect of leachate recirculation in a laboratory reac-tor containing spatially distributed acidogenic andmethanogenic layers (new and old solid domesticwastes). Using a mathematical model, they showed thatthe leachate recirculation, which facilitates the trans-port of volatile fatty acids from acidogenic into metha-nogenic zones, along with repeated batch implementa-tion of the process enhance the reactor efficiency.

The objective of this study is to develop and analyzea distributed one-dimensional model of activated bio-chemical medium, which forms in the course of munic-ipal solid waste decomposition.

MODEL DESCRIPTION AND ANALYSIS

The model is based on the following assumptions:Hydrolysis of polymers and methanogenesis are

limiting stages of the anaerobic decomposition of solidwaste. For the sake of simplicity, only these two stagesare accounted for in the model. The concentrations ofsolid waste, volatile fatty acids, and the biomass ofmethanogenic microorganisms are model variables;

Hydrolysis and methanogenesis are inhibited byhigh concentrations of volatile fatty acids;

The hydrolysis and acetoclastic methanogenesis aredescribed by a first-order equation and the Monodmodel

Diffusion of solid domestic waste and methano-genic microorganisms is negligible as compared withvolatile fatty acids diffusion, which is assumed constantthroughout the reactor volume.

Now we have the system of partial differential equa-tions:

(1)

∂W∂T-------- kWf S( ),–=

∂S∂T------ D

∂2S

∂X2--------- χkWf S( ) ρmg S( ) SB

KS S+---------------,–+=

∂B∂T------ Yρmg S( ) SB

KS S+---------------,=

(2)

(3)

(4)

(5)

where

W

≡

W

(

X

,

T

)

,

S

≡

S

(

X

,

T

)

, and

B

≡

B

(

X

,

T

)

are theconcentrations of solid waste, volatile fatty acids, andmethanogenic organisms, respectively;

0

≤

X

≤

L

isspace coordinate along the one-dimensional reactor;

0

≤

T

< +

∞

is time;

D

is coefficient of volatile fatty acidsdiffusion;

k

is the constant of first-order hydrolysis;

χ

isa stoichiometric coefficient;

K

s

is half-saturation con-stant;

Y

is an yield coefficient. Initial conditions (2)–(4)allow a nonuniform distribution of the respective vari-ables along the reactor length

L

. Boundary conditions(5) imply zero flux of volatile fatty acids through theboundaries

1-

D

of the reactor. Dimensionless functions

f

(

S

)

and

g

(

S

)

account for the inhibition of hydrolysisand methanogenesis by volatile fatty acids, respec-tively:

(6)

Let us introduce the characteristic value of the concen-tration of volatile fatty acids

S

1/2

, defined by the equa-tion

(7)

The somewhat unusual form of (7) can be attributed tothe fact that both the hydrolysis and methanogenesisare inhibited by the concentrations of volatile fattyacids, and sometimes it is impossible to say which inhi-bition is stronger.

Dividing (1.1), (1.2), and (5) by

S

1/2

D

/

L

2

; (1.3.), by

K

s

D

/

L

2

; (2) and (3), by

S

1/2

; and (4), by

K

s

yields the fol-lowing normalized system of equations:

(8)

(9)

(10)

W X 0,( ) σ̃ X( ),=

S X 0,( ) ϕ̃ X( ),=

B X 0,( ) ψ̃ X( ),=

∂S 0 T,( )∂T

--------------------- ∂S L T,( )∂T

--------------------- 0,= =

f 0( ) g 0( ) 1,= =

dfdS------ 0, if S 0,≥≤

dgdS------ 0, if S 0,≥≤

f S( )S +∞→lim g S( )

S +∞→lim 0.= =

g S1/2( ) f S1/2( ) 14---.=

∂w∂t------- αwf s( ),–=

∂s∂t----- ∂2s

∂x2-------- χα wf s( ) βg s( ) sb

δ s+-----------,–+=

∂b∂t------ Yβg s( ) sb

δ s+-----------,=

w x 0,( ) σ x( ),=

s x 0,( ) ϕ x( ),=

WATER RESOURCES Vol. 28 No. 6 2001

THE EFFECT OF FATTY ACID DIFFUSION IN LEACHATE 693

(11)

(12)

Equations (8)–(12) contain only dimensionless vari-ables.

Analysis of the linearized system of ordinary differ-ential equations following from (1) at D = 0 showed thesystem to have two stationary states (a node and afocus). In this case

w1 = 0, s1 = 0, b1 = YI (stable node), (13)

w2 = 0, s2 = I, b2 = 0 (saddle), (14)

,

where I is integral of lumped model.System (8), which includes equations of parabolic

type, was solved using Galerkin technique, in which therange of the space variable 0 ≤ x ≤ 1 is divided into(N – 1) intervals with N boundary nodes, and the gen-eral solution of the problem amounts to solving Cauchyproblem for ordinary differential equations. The calcu-lations were made for the normalized system (8),whereas the results of calculations are presented inabsolute volumetric form to make them more clear. Sta-ble solution can be obtained at N about 200. The modeldomain in this case is essentially a thin tube (a three-dimensional reactor where significant diffusion takesplace only along the x coordinate). For the volume ofmethane released, it was assumed that

where A is the transformation coefficient of volatilefatty acids into methane.

The inhibition functions were taken in the form

(13)

b x 0,( ) ψ s( ),=

∂s 0 t,( )∂t

------------------ ∂s 1 t,( )∂t

------------------ 0,= =

w w x t,( )≡ WS1/2--------, s s x t,( )≡ S

S1/2--------,= =

b b x t,( )≡ BKs

------, 0 x≤ XL---- 1,≤= =

0 t≤ DT

L2-------- +∞, α< L2k

D--------, β

L2ρm

D------------,= = =

δKs

S1/2--------, σ x( ) σ̃ x( )

S1/2-----------, ϕ x( ) ϕ̃ x( )

S1/2-----------,= = =

ψ x( ) ψ̃ x( )Ks

------------.=

I χ σ x( ) xd

0

1

∫ 1Y--- ψ x( ) xd

0

1

∫ ϕ x( ) xd

0

1

∫+ +=

∂M∂T-------- A

1Y--- 1–

∂B∂T------,=

f S( ) 1/ 1S

K f

------ m f

+ ,=

(14)

where Kf ≥ 0, Kg ≥ 0, mf ≥ 1, and mg ≥ 1 are constants.Functions (13) and (14) describe both weak and stronginhibition and meet requirement (6). Note that at mf =mg = 1, functions (13), (14) are commonly used todescribe inhibition processes [11].

The initial distributions of solid waste, volatile fattyacids, and biomass were taken using the Gauss functionin the form

where g1, …, g9, a1, a2, a3 are coefficients of the distri-butions. Function ϕ(x) approximately satisfies boundarycondition (12) and at a2 = 0.5 describes the case with theinitial zone of methanogenesis located in the centerof 1D-reactor.

RESULTS AND DISCUSSIONHomogeneous (Lumped) Reactor

The coefficients of the distributed model (1) werechosen in the course of precalibration of the lumpedmodel (D = 0) against the experimental data in [8, 12].Visual calibration [14] was made by trying model coef-ficients with the ranges of most important coefficientstaken from literature [11]. Clearly the obtained set ofcoefficients is not unique since the model coefficientsare numerous and the experimental data are scanty.

Pilot acidogenic and methanogenic reactors forsolid municipal waste processing were studied byLagerkvist and Chen [8] using leachate recirculation. Itis obvious that, when the rate of recirculation is high,the time of variables averaging over the reactor volumewill be much less than the characteristic time of varia-tion in the dynamical variables. In such ideal homoge-neous reactor, the reaction proceeds synchronouslythroughout its volume. The concentrations of volatilefatty acids higher than 35 g/l (pH 5.2) were found tostop both the methanogenesis and hydrolysis, whereasat volatile fatty acids about 25 g/l (pH 5.8) both the pro-cesses still proceed. The overall bacteria population inthe acidogenic and methanogenic reactors numbered105 and 1011 cell/ml, respectively. As seen from the sys-tem dynamics shown in Fig. 1, both hydrolysis andmethanogenesis exhibit a characteristic lag phase stem-ming from the high initial concentration of volatilefatty acids. Increasing concentration of methanogenic

g S( ) 1/ 1S

Kg

------ mg

+ ,=

σ x( ) g1 1 g2 0.5x a1–

g3-------------

2

–exp–

,=

ϕ x( ) g4 1 g5 0.5x a2–

g6-------------

2

–exp–

,=

ψ x( ) g7 1 g8 0.5x a3–

g9-------------

2

–exp+

,=

694


VAVILIN et al.

microorganisms, which consume volatile fatty acids,facilitates hydrolysis. The model parameters obtainedfrom its calibration are given in the table.

Figure 1b gives the dynamics of a laboratory reactor[13], which processes preshredded sorted domestic

wastes. The initial concentration of volatile fatty acidsin this reactor was relatively low. Notwithstanding theabsence of leachate recirculation, the lumped modelwas still used since the size of the laboratory reactorwas far smaller than in the previous case. The model

Fig. 1. Dynamics of solid waste decomposition at (a) high and (b) low initial concentration of volatile fatty acids. Symbols and dotsare experimental data in [8] and [12], (a) and (b) respectively; curves are solutions of model (1) at D = 0; Kf = 4.5 g/l, Kg = 9.9 g/l (a),Kf = 12 g/l, Kg = 15 g/l (b)

Kinetic coefficients used in model (1) and obtained through calibration against experimental data [8, 12]*

Data source k, day–1 χ ρm, day–1 Ks, g/l Y mf = mg Kf, g/l Kg, g/l W0, g/l S0, g/l B0, g/l M0, g/l A, l2/g

[8] 0.16 0.80 0.50 4.50 0.30 3 4.50 9.90 15 25 2 0 7.4[12] 0.45 0.80 0.90 4.50 0.18 100 12 15 23 0.5 0.1 0 1.1

* W0, S0, B0, M0 are the initial concentrations of solid waste, volatile fatty acids, methanogenic biomass, and methane.

15

10

5

0

12

8

4

0 50 100 150

Solid

was

te, g

/lB

, g/l

20

10

0

200

100

0 50 100 150

Vol

atile

fat

ty a

cids

, g/l

CH

4, l

Time, day

(a)

20

10

0

20

10

0 20 40 60

Solid

was

te, g

/lB

, g/l

20

10

0

Vol

atile

fat

ty a

cids

, g/l

CH

4, l

20 40 60

20

10

0

Time, day

(b)



parameters are also given in the table. As can be seenfrom Fig. 2, hydrolysis has a two-stage character andbreaks at high concentrations of volatile fatty acids.This pattern of anaerobic decomposition of solidorganic matter was mentioned in [1, 15]. It is worthmentioning, however, that in previous studies [17], thedynamics [12] was described using “Methane” simula-tion model. This model is fairly complicated and, in

particular, can take into account different types of vol-atile fatty acids, hydrogenotrophic and acetotrophicmetanogenesis, and some other features, but fails toallow for diffusion. Model (1) is simpler than “Meth-ane” in terms of both the number of variables and therelationships between them. Therefore, the results ofsimulation using “Methane” model fits better the exper-imental data. However, Fig. 1b shows that model (1)

Fig. 2. Series of simulated spatial distributions of solid waste, volatile fatty acids, and biomass in one-dimensional reactor at a cer-tain time along with (dots) overall methane production at g3 = g6 = g9 = 0.05, D = 10–5 L2/day; (a) (0–3) 0, 30, 60, and 90 days,

respectively, W0 = 15 g/l, S0 = 25 g/l, M0 = 0, A = 20 l2/g, (b) (0–4) 0, 10, 20, 30, and 40 days, respectively, W0 = 23 g/l, S0 = 0.5 g/l,

B0 = 0.1 g/l, M0 = 0, A = 3 l2/g.

20

10

0

0

1

2

Solid

was

te, g

/l

10

0

0 0.4 0.8

1 2

30

Vol

atile

fat

ty a

cids

, g/l

x-Coordinate

x-Coordinate

4

2

0

0 2

3

4

B, g

/l

0.4 0.8

50

40

30

20

10

0 40 80

CH

4, l

Time, day

(b)

.. 4410

0

Solid

was

te, g

/l

20

10

0

0 0.4 0.8

02

3

Vol

atile

fat

ty a

cids

, g/l

x-Coordinate

14

10

6

20 0.4 0.8

0

B, g

/l

x-Coordinate

400

200

0100 200

CH

4, l

Time, day

(a)

0 3

2

32

1

1

1

1

696


VAVILIN et al.

has reflected the most important feature of the processdynamics, that is, the two-stage pattern of hydrolysisand sigma-like pattern of methane accumulation in thesystem.

Distributed Reactor

Figures 2a and 2b give the dynamics of a distributedsystem according to model (1) with the initiation meth-anogenic zone concentrated in the center of the one-dimensional reactor and the kinetic coefficients givenin Table 1. Cases with high and low background con-centrations of volatile fatty acids are shown in Figs. 2aand 2b. Clearly volatile fatty acids are both a substrateand inhibitor of methanogenesis. Activated methano-genesis takes place in the area with a relatively low(non-inhibiting) concentration of volatile fatty acids butceases when concentration of volatile fatty acidsbecomes too low. The supply of volatile fatty acids tothe area of activated methanogenesis is due to their dif-fusion from the reactor zones with high concentrationsof solid waste and volatile fatty acids. On the whole, aconcentration chemical wave propagates in the reactorand the methanogenic zone spreads over the entire reac-tor volume.

It is common knowledge that the motion of liquid inlandfills can stimulate methane production [3]. For anydomain Θ within the reactor, complete transformationof volatile fatty acids into methane is possible onlywhen there exists such t0 ≥ 0 that

(15)

where RH is the rate of volatile fatty acids productionthrough hydrolysis; RM is the rate of volatile fatty acidsutilization by methanogens; JΘ is the influx of volatilefatty acids through the boundary of domain Θ owing todiffusion; t0 is the initial time moment. Otherwise,active methanogenesis will be suppressed by excessiveinput of volatile fatty acids into this zone (Fig. 3). Therates RM and RH are controlled by the respective kineticcoefficients, such as the maximum specific rate of utili-zation of volatile fatty acids(ρm) and the constant of thefirst-order rate of hydrolysis k (see (1)), along with theinitial concentrations of methanogenic biomass, vola-tile fatty acids, and solid municipal waste. The less theinitial biomass concentration, the greater is the proba-bility of activated methanogenesis suppression owingto the diffusion of volatile fatty acids. Solid wastealmost without decomposition can be found in someplaces in old landfills, and we can suppose that the acti-vated chemical wave does not reach these zones.

Separation of Methanogenesis and Acidogenesis Zones in Space

The zones of activated methanogenesis and acido-genesis (hydrolysis) can be separated in space [9]. Pre-liminary pressing of solid waste, which drives the wasteparticles and methanogenic microorganisms closetogether, causes an increase in the rate of methanogen-esis provided that moisture is available. Activated

JΘ RH x t,( ) xd

Θ∫ RM x t,( ) x, t t0≥d

Θ∫<+

Fig. 3. Series of simulated spatial distributions of solid waste, volatile fatty acids, and biomass in one-dimensional reactor at a cer-tain time along with (dots) overall methane production in the case of active methane generation suppressed because of intense dif-fusion of volatile fatty acids. (0–4) 0, 200, 400, 600, 800 days, respectively. Kinetic coefficients based on the data published in [8]are given in the table. Kf = 9.9 g/l, Kg = 4.5 g/l, D = 10–3 L2/day, A = 20 l2/g, g3 = g6 = g9 = 0.025.

15

10

5

0

–5

0

12

4Solid

was

te, g

/l

30

20

10

0 0.4 0.8

10

Vol

atile

fat

ty a

cids

, g/l

x-Coordinate

x-Coordinate

2

1

0 0.4 0.80

4

B, g

/l

3

0 400 800

CH

4

Time, day

2

1

4



methanogenesis can proceed in the margins of the aci-dogenic zone, where the concentration of volatile fattyacids is low enough to considerably reduce the metha-nogenesis inhibition, although it is high enough not tolimit the methanogenesis. Again, the background con-centration of methanogenic organisms should be suffi-ciently high. It is worth noting that in the model consid-ered it was assumed that the solid waste particles do nothamper the diffusion of volatile fatty acids. An increasein the rate of diffusion along with condition (15) willextend the zone of activated methanogenesis, thusincreasing the rate of methane production.

CONCLUSION

The decomposition of solid waste can be accompa-nied by propagation of concentration chemical wavesand extension of the zone of activated methanogenesis.Depending on the conditions, diffusion of volatile fattyacids can either facilitate or inhibit this process.

ACKNOWLEDGMENTS

This study was supported by the Copernicus-2project (ICN-2CT-2001-10001).

REFERENCES

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3. Barlaz, M.A., Schaefer, D.M., and Ham, R.K., Appl.Environ. Microbiol., 1989, vol. 55, p. 55.

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6. Oscillations and Traveling Waves in Chemical Systems,Field, R.J. and Burger, M., Eds., New York: Wiley, 1985,p. 720.

7. Kalyuzhnyi, S., Veeken, A., and Hamelers, B., Wat. Sci.Technol., 2000, vol. 41, no. 3, p. 181.

8. Lagerkvist, A. and Chen, H., Proc. Int. Symposium onAnaerobic Digestion Solid Waste, Venice: Stamperia diVenezia, 1992, p. 75.

9. Martin, D., Trans. Inst. Chem. Eng., 2001, vol. 79,part B, p. 29.

10. Nozhevnikova, A.N., Nekrasova, V.K., Lebedev, V.S.,and Lifshits, A.B., Wat. Sci. Technol., 1993, vol. 27,no. 2, p. 243.

11. Pavlostathis, S.G. and Giraldo-Gomez, E., Crit. Rev.Environ. Control, 1991, vol. 21, p. 411.

12. Salminen, E., Ph.D. Thesis, Finland: Univ. Jyavaskyla,2000.

13. Ten Brummeler, E., Ph.D. Thesis, The Netherlands:Wageningen Agricultural Univ., 1993.

14. Vavilin, V.A., Lokshina, L.Ya., and Rytov, S.V., Vestn.Mosk. Univ., Khim., 2000, vol. 41, no. 6, p. 22.

15. Vavilin, V.A., Rytov, S.V., Lokshina, L.Ya., and Rin-tala, J.A., Proc. II Int. Symposium on Anaerobic Diges-tion Solid Waste, Barcelona, 1999, vol. 2, p. 1.

16. Veeken, A.H.M. and Hamelers, B.V.M., Wat. Sci. Tech-nol., 2000, vol. 41, no. 3, p. 256.

17. Young, A., Microbiology of Landfill Sites, Senior, E.,Ed., Boca Raton: Lewes Publ., 1995, p. 61.

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