ANALYSIS OF AN ANAEROBIC DIGESTION PROCESS USING OPTIMIZATION TOOLS

Angélica Maria Alzate I1, Luis Gerónimo Matallana P2, Juan Bernardo Restrepo B3

1Universidad Nacional de Colombia - Sede Manizales, Manizales, Colombia, amalzatei@unal.edu.co2Universidad Nacional de Colombia - Sede Manizales, Manizales, Colombia, lgmatallanap@unal.edu.co3Universidad Nacional de Colombia - Sede Manizales, Manizales, Colombia, jbrestrepob@unal.edu.co

keywords: Applications of Dynamical Systems in En-gineering; Bifurcation theory and analysis; Dynamics ofBiosystems.

1. INTRODUCTION

Dynamic analysis is an important mathematical tool inthe study of chemical engineering systems. It provides a sys-tematic framework in which to study the system behavior andthe sensitivity at variations in the parameters. However, somenumerical methods for the detection, computation, and con-tinuation of equilibrium and bifurcation points of dynamicalsystems, such as manifolds and numerical continuation [1],by themselves are sometimes not be the best strategy in orderto obtain the optimal dynamic analysis of the system.Due to the fact that most of chemical engineering processeshave highly non-linear dynamics, and constraints are alsofrequently present on both the state and the control variables[2], it is necessary to solve the mathematical model of thesesystems as optimization problems.

2. OPTIMIZATION PROBLEM

Optimization is becoming more widely used in many ap-plication areas, most of the optimization models that are NLP(Non Linear Programming) in chemical engineering, usu-ally have multiple local solutions. The use of global algo-rithms to find stationary points of differentiable functionsbased on convexification strategies result a good proposal tosolve such problems [3], compared with other iterative meth-ods for finding roots of equations such as Newton’s method,which depend on the starting points and are designed for lin-ear systems.The algorithm for NLP solution requires smooth functions

Figure 1 – Algorithm structure

and the resolution is based on different methods but it is al-most impossible to predict how it will behave given NLPsolver with a NLP model. In this case we use Baron NLPsolver of the GAMS Software. The modeling languagesGAMS have had such a capability to use specific globaloptimization algorithms, perform automatic derivatives,andhave been used in many practical large scale nonlinear appli-cations.The formulation of the optimization with GAMS (GeneralAlgebraic Modeling System) for solve the dynamical systemcorrespond to:

minx,s

s

s.t. h (x)− s ≤ 0

−h (x)− s ≤ 0s ≥ 0

xl ≤ x ≤ xu

(1)

Wheres is slack variable,x represents the state variablesvector andh(x) are the differential equations equaled to zero.Figures 1 shows the structure of the algorithm using for thebifurcation diagrams in dynamical analysis.

3. DYNAMIC ANALYSIS OF ANAEROBIC DIGES-TION PROCESS

We considered the mathematical model of the anaerobicprocess. The anaerobic digestion is a very complex processin which a great number of bacterial populations intervene.We assume that the dynamics of the system present two mainstages, acidogenesis and methanogenesis stage [4]. The dy-namic model that describes the behavior is:

X1 = µ1X1 − αDX1 (2)

X2 = µ2X2 − αDX2 (3)

S1 = D (S1in − S1)− k1µ1X1 (4)

S2 = D (S2in − S2) + k2µ1X1 − k3µ2X2 (5)

C = D (Cin − C)− qC + k4µ1X1 + k5µ2X2 (6)

Z = D (Zin − Z) (7)

The kinetic variablesµ1 andµ2 correspond Monod model foracidogenesis and Haldane model for methanization, respec-tively. The model of the anaerobic digestion process initiallyhas been analized by [7] and [8]. In this work, we proposean analysis of the system based on the reformulation and so-lution of the mathematical model as optimization problem.Figure 2 presents the bifurcation diagram ofX1, X2, S1 andS2 variation of one parameter,D. The graphic presents twolimit points atD = 1.07 and three bifurcation points. Thecurves represent two saddle-node bifurcation curves and twotranscritical bifurcation curves.The first and second equilibrium branch point correspondsto the washout ofX2 andX1, atD = 1.65 andD = 0.97.The third branch point corresponds to the trivial equilibriumto the washout ofX2 but not ofX1 at D = 0.91. The firstlimit point corresponds to the washout ofX1 but not ofX2

at D = 1.07 in X2 = 0.48, the acidogenesis stage is notpresented therefore cannot perform the anaerobic digestionin the bioreactor. The second limit point corresponds to thetrivial equilibrium point atD = 1.07 with X1 = 0.47 andX2 = 0.68.

0.8 0.9 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7−0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

X1 (

g/L)

D (day−1)

UnstableStable

0.8 0.9 1 1.1 1.2 1.3 1.4 1.5 1.6 1.72

4

6

8

10

12

14

16

18

S 1 (g/

L)

D (day−1)

Unstable

Stable

0.8 0.9 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

1.2

X2 (

g/L)

D (day−1)

UnstableStable

0.8 0.9 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7

20

40

60

80

100

120

140

160

180

200

S 2 (m

mol

/L)

D (day−1)

UnstableStable

Figure 2 – Bifurcation diagram obtained varying D

4. CONCLUSION

We presented a dynamic analysis of the anaerobic modelproposed by [4], solved as optimization problem. The opti-mization tools are useful for nonlinear models, have the abil-ity to obtain global solutions.The bifurcation set obtained illustrates the complexity ofdynamical behavior. The bifurcation diagram presents bi-furcations curves not previously reported in the literature.The dynamic analysis of the anaerobic digestion mathemat-ical model through the proposed algorithm allows obtainedglobal solutions that with other methods such as Newton’smethod cannot be found.The dynamic analysis showed that the trivial solution cor-responds to the washout and it takes place in two stages.The first stage corresponds to the washout condition ofmethanogenic bacterias but not of the acidogenic bacteriasand occurs after a combination of fold and transcritical bifur-cations and the washout of the reactor occurs after a trans-

critical bifurcation. The last stage correspond to the washoutcondition of acidogenic and methanogenic bacterias occursafter a fold bifurcation.

Parameter MeaningC,Cin Total inorganic carbon concentration(mmol/L)D dilution rate(d−1)k1 yield for substrate degradation(mmol/g)k2 yield for VFA production(mmol/g)k3 yield for VFA consumption(mmol/g)k4 yield forCO2 production(mmol/g)k5 yield forCO2 production(mmol/g)qC carbon dioxide flow rate(mmmol/Lperday)S1, S1in organic substrate concentration(g/L)S2, S2in volatile fatty acids concentration(mmol/L)X1 concentration of acidogenic bacteria(g/L)X2 concentration of methanogenic bacteria(g/L)

References

[1] W. J. F. Govaerts, "Numerical Methods for the Bifur-cation of Dynamical Equilibrium", SIAM, 2000.

[2] E. Balsa-Canto, J. R. Banga, A. A. Alonso and V. S.Vassiliadis, "Dynamic optimization of chemical andbiochemical processes using restricted second orderinformation", Computers and Chemical Engineering,Vol. 25, pp. 539-546, 2001.

[3] L. G. Matallana, Curso: "Modelado, simulación yoptimización en GAMS", Universidad Nacional deColombia, 30 de Enero al 3 de Febrero de 2012.

[4] O. Bernard, Z. Hadj-Sadok, D. Dochain, A. Genovesiand J. Steyer, "Dynamical model development andParameter identification for an anaerobic wastewatertreatment process", Biotechnology and bioengineer-ing, Vol. 75, No. 4: pp. 143-438, November 2001.

[5] S. Ghosh and F.G. Pohland, "Kinetics of substrate as-similation and product formation in anaerobic diges-tion", Journal Water Pollution Control Federation, Vol.46, No. 4, pp. 748-759, Abril 1974.

[6] C. D. Maranas and C. F. Floudas, "Finding All Solu-tions of Nonlinearly Constrained", Journal of GlobalOptimization, Vol. 7: pp. 143-182, May 1995.

[7] J. Hess and O. Bernard, Advanced dynamical riskanalysis for monitoring anaerobic digestion process,AIChE, Vol. 25, No. 3, pp. 643-653, 2009.

[8] A. Rincon, F. Angulo and G. Olivar, "‘Control of ananaerobic digester through normal form of fold bifur-cation", Journal of Process Control, Vol. 19, pp. 1355-1367, 2009.