Compuf. & Opr Rtr. Vol. 8. pp. 3-l8 Pmgamm Pms Ltd.. l!%I. Pnnwd in Gear ham

MODELINGANDCOMPUTERSIMULATIONSOF ANAEROBICDIGESTIONPROCESSBYDYNAMO

B. N. LOHANI*

Environmental Engineering Division. Asian Institute of Technology, Bangkok, Thailand

Scope and purpose-Modeling and computer simulations are widely used in investigating the system dynamics of biological process, Modeling and simulation of anaerobic digestion process in particular *as

initiated by Andrews(l] who used a digital simulation program called PACTOLUS to validate his model.

This paper presents a DYNAMO model for batch and continuous anaerobic digestion process, which hai

been tested and found to match closely with the results generated by Andrew’s method. The purpose of the research is justified as the digital analog simulator program and Continuous System Modeling Program are

not always available and this is true especially in most of the Asian countries. In such cases, Dynamo

programming system presented in this paper may be utilized.

Abstract-Modeling and computer simulations are found useful in studying the systems dynamics of

biological processes. DYNAMO programming system is used in place of digital analog simulator program

calles PACTOLUS to simulate a dynamic model of anaerobic digestion process. The use of dynamo is found satisfactory for this type of simulation scheme.

INTRODUCTION

Extensive modeling and computer simulations of the dynamic behavior of biological process have become common in today’s field of advanced scientific researches. Edwards[4] in his investigation of the influence of high substrate concentration on microbial kinetics employed the dialect of the FORTRAN IV computer language with an IBM 360/65 series computer. In testing the growth models of culture with two liquid phases, Erickson et a/.[51 conducted the simulation of the model using IBM S/360 Continuous System Modeling Program (CSMP). Murphy et al. [ll], in their study of the dynamic nature of nitrifying biological suspended growth systems made use of mathematical building and simulation, and time series analysis to assess the behavior of the process, Gujer[9] employed Runge-Kutta-Merson method for all numerical integrations to obtain valid computer inputs for a dynamic simulation aimed to facilitate the design of a nitrifying activated sludge process. DiToro et a!.[31 utilized a digital computer and continuous simulation language (CSMP/Il’JO) to solve the differential equations governing the kinetics of a dynamic model of phytoplankton population. Young III et a/.(13) used both analog and digital computers in a dynamic analysis of a microbial growth. Computer modeling and simulations thus have proved to be potent tools in investigative work relating to the behavior and stability of biological metabolism. The use of these techniques has apparently

gained wide recognition during the last decade. The mere utilization of the above modeling and simulation programs, however, does not

guarantee a ready-made solution to the differential equations derived from the model. Almost always, successful operation entails a great deal of mathematical manipulations and trans- formations before the programmed solutions can be run smoothly on a computer. The rigorous mathematical jargons associated with the solution can be minimized and the model can be easily visualized and understood if a straightforward approach is adapted using DYNAMO, DYNAMO as defined by Pugh 111[12] is a programming system translating and running continuous, models which are described by a set of differential equations. Developed and popularized by the industrial dynamics group at M.I.T. headed by Forrester[6], DYNAMO makes available, easy to use computing facilities so that the user can focus his attention on building a useful model undistracted by complex computer requirements. Readers who are not familiar with DYNAMO language and the steps involved such as making causal loop diagram and flow diagrams are referred to the well adopted textbook by Forrester[6].

*Dr B. N. Lohani is Associate Professor of Environmental Engineering at Asian institute of Technology (AIT), Bangkok. His major field of interest is Environmental Systems Engineering. He is technical advisor to Environmental

Information Sanitation Centre at AIT and contributing Editor to the Journal of Environmental Science. U.S.A. for obtaining and reviewing papers by Asian Researchers. He has published over seventy articles in international journals. conferences and as research reports.

39

10

Biological process

B. N. LOHANI

Modeling of any biological waste treatment system is based on an understanding of the basic principles that govern the bacterial growth. For the benefits of some readers who may not be familiar with the principles of bacterial growth in biological waste treatment process, a brief description is included from Metcalf and Eddy [IO], The general growth pattern of bacteria based on the number of cells has four or less distinct phases and they are:

(a) The lag phase. This period represents the time required for the organisms to acclimate the environment;

(b) The growth phase. During this period the cells divide at a rate determined by their generation time and their ability to process food;

(c) The stationary phase. During this period the population remains stationary. Reasons advanced for this phenomenon are (a) that the cells have exhausted the substrate (organic matters serving as food to microorganisms and non-specific measures such as biochemical oxygen demand, chemical oxygen demand, volatile acids are preferably used to determine the concentration subtrate) or nuirients necessary for growth, and (b) that the growth of new cells is offset by the death of old cells.

(d) The log death phase. During this phase the bacterial death rate exceeds the production of new cells. The death rate is usually a function of the viable population and environmental characteristics. In some cases, the log death phase is the inverse of the log growth phase.

The growth pattern can also be discussed in terms of the variation of microorganisms with time. This growth pattern consists of the following three phases:

(a) The log growth phase. There is always an excess amount of food surrounding the microorganisms, and the rate of metabolism (process by which food is built up into living matter or by which living matter is broken down into simple substances) and growth is only a function of the ability of the microorganism to process the substrate.

(bf ~e~~~njng growth phase. The rate of growth and hence the mass of bacteria decreases because of limitations in the food supply.

(c) Endogenous respiration phase. The microorganisms are forced to metabolize their own protoplasm without replacement since the concentration of available food is at a minimum. During this phase, a phenomenon known as lysis can occur in which nutrients remaining in the dead cells diffuse out to furnish the remaining cells with food (known as cryptic growths.

To the santiary engineer, there are two biological cycles involving the growth and decay of organic matter that are of interest and they are the aerobic and the anaerobic cycles. In the aerobic cycle oxygen is used for the decay of the organic matter whereas in anaerobic cycle oxygen is not used. In the aerobic systems, the final products of degradation are more fully oxidized and hence at a lower energy level than the final products of the anaerobic system. This accounts for the fact that much more energy is released in aerobic than in anaerobic degradation. As a result, anaerobic degradation is a much slower process. This paper exclusively deals with the anaerobic systems.

Stute~ent of the problem

This paper is to demonstrate DYNAMO’s potential in dynamic analysis of biological systems, the dynamic model of the anaerobic digestion process reported by Andrew[l] has been purposely chosen for testing. The performance of DYNAMO is then compared with that of PACTOLUS, a digital analog simulator program employed by Andrews[l] to validate his modeI. The corresponding computer output obtained from DYNAMO is found to match closely with the results generated by PACTOLUS. Hence, DYNAMO can soive the more complex differential equations proposed by Graef et al.@] and Carr ef al.[2] for the same previous model. Further, DYNAMO’s simple notational scheme patterned after actual computational method can be very valuable in providing solution to the most complex dynamic model.

Therefore, given Andrew’s [ I] dynamic model, the problem boils down to the formulation of a set of dynamo equations that will give similar computer results. This, of course, necessitates the development of casual-loop and DYNAMO flow diagrams. ConsequentIy, basic relation- ships between varibfes have to be defined in system equations and the appropriate values for the constants have to be supplied.

Modeling and computer simulations

DEVELOPMENT OF z4 DYNAMO MODEL

II

In developing a DYNAMO mode& Goodman[7] suggested the initial construction of causal-loop diagrams to identify the principal feedback loops. Causal-loop diagrams from which the DYNAMO flow diagrams evolve, can serve as preliminary sketches of causal hypothesis and in most cases, can simplify the illustration of a model. Once a DYNAMO flow diagram is completed, DYNAMO equations will readily follow provided the relationships between variables are well defined by specific mathematical equations. Users of DYNAMO must be able to recognize which variables comprise the rates, levels, and auxiliary elements and must be sure that every loop contains at least one rate and one level variable. A DYNAMO model is presented separately for batch and continuous culture.

General assumptions for the model To keep the model as simple as possible, Andrews[l] adapted the following assumptions: (I) The rate of bacterial growth and the rate of substrate utilization are directly proport~onai

to the concentration of microorganisms. (2) The unionized fraction of the volatile acids is the limiting substrate. This unionized

portion (fraction of substrate, as volatile acids, that does not ionize in the liquid phase) is a function of both pH and total substrate concentration.

(3) There is no lag phase, organism death, endogeneous repiration (substrate used for maintan~e energy) or inhibition (adverse effect on growth) by products.

Batch culture DYNAMO model From the general assumptions, the corresponding causal-loop diagram for batch culture is

sketched in Fig. 1. The mathematicat model for a batch reactor can be obtained easily from the continuous flow

reactor model by setting the liquid flow rate equal to zero wherever it appears in the material balances. The basic differential equations can be derived by simple substrate and organism balances and by the use of modified Monod function. The general balance equation in a reactor is defined by:

Ac~umuiation = Input - Output f Reaction.

Organism Balance

dX F -=- dt Vxo

Concentmtion ,X

Substrate

Concentra?ion,S

\ J Unionized Substrate Concentrat0n, HS

KA

Fig. I. Casual loop diagram for batch culturz.

B. N. LOHAN

__.-. _______.-.-.---.-.___._._._b_~_e_~

KA -._._. -.-___ -c

r

X tr, '\ -\

l -\ .x' '\

'\ .' .R

\ ,.' ;_/

.' *'\ I , i ! '1, .*'

'\ .' .'

i i

Ii '\ .,.' \ ,' '\ /

.I i

i \ '\ / ', \ -

i -, i '1 \ \ i ; S '$ '\ '1 i i i i '\ '. i '\ '\ dS i \ '\ '\

.$ '\ _.. *T

‘1 '1 ‘I ./-*

i \ '\ ‘\ \

Fig. 2. Flow diagram for batch culture (a DYNAMO model)

For a batch reactor, F = 0, thus the above equation reduces to the form

Substrate Balance

Again when F = 0, the above equation becomes

dS CL -=_ dt Yx’ (2)

where p = specific growth rate, time-‘, defined by

1

P= K” HS' 1+&+K,

modified Monod function given by Andrew[l]. F = liquid flow rate to reactor, volume/day, V = reactor liquid volume, volume, X = organism concentration in the reactor, mass/volume, S = limiting substrate concentration, mass/volume, fi = maximum specific gro~fh rate in the absence of inhibition (adverse effects on microbial growth), K, = saturation constant, numerically equals the lowest concentration of substrate at which the specific growth rate is equal to one-half the maximum specific growth rate in the absence of inhibition, mass/volume, Ki = inhibition constant, numerically equals the highest substrate concentration at which the specific growth rate is equal to one-half the maximum specific growth rate in the absence of inhibition, mass/volume, t = time, days, Y = yield coefficient, mass organism produced/mass

Modcling and computer simulations

Flow Rate of Organwns in the Influent, DXO

Concerrtration

Flow Rate of Organism in the Eff bent

Fig. 3. Casual loop diagram for continuous culture.

substrate utilized, HS = unionized substrate concentration in the reactor, mass/volume, defined

by

ffS=‘H+l’S1 at pH>6, K,

K, = ionization constant, 10-4.5 for acetic acid at 38°C and an ionic strength of 0.02. Given the above information, the DYNAMO flow diagram is constructed as shown in Fig. 3.

The symbols and notations used are given in Appendices 1 and 2 and the corresponding equations are presented in Appendix 3.

Continuous culture DYNAMO model Again, using the same general assumptions previously outlined and noting that the system is

a continuous culture, the causal-loop diagram is drawn below as in Fig. 4. The mathematical model for continuous flow culture can be derived from substrate and

organism balances. The general equation given in the batch culture also holds true for continuous culture DYNAMO model.

Organism Balance

X,

but 0 = (V/F); therefore

dX X,, X -_L=--J+ dt 8 8

(3)

44 B.N. LOHANI

.Y-'

_,_____.___._._.-.-.-.-.- - -.__

H ___.-.-.__ _,_.C.-.-__, =.*_ \

1 su I i* . .

; , - ._.___._. -. i I i i

Substrate Balance

Again 0 = (V/F); hence

.-_ 5.

-. -. _.__ -.-._._._._ _. -.-.

Fig. 4. Flow diagram for continuous culture (a DYNAMO model).

dS, _ S,, SI 1 -_----- dt e 6 Y (4)

where ~1, fi, F, V, K,, K, and Ki are as defined previously, 0 = mean residence time in the reactor, time-‘, X0= organism concentration in the influent to the reactor, mass/volume, X, = organism concentration in the effluent from the reactor, mass/volume, So = total substrate concentration in the influent to the reactor, mass/volume, S, = total substrate concentration in the e@uent from the reactor, mass/volume, HS, = unionized substrate concentration in the effluent from the reactor, mass/volume.

Based on the preceding set of information the DYNAMO flow diagram is developed as illustrated in Fig. 4 and the equations formed correspondingly are listed in Appendix 4.

The programs for PACTOLLJS system however, is not included here and the interested readers are referred to the work of Andrews [ 11.

RESULTS AND DISCUSSION

To verify the validity of the DYNAMO models the programs developed based on Andrews[l] assumptions and data for batch and continuous cultures were run in IBM 370/145 computer employing a DYNAMO II complier. The effect of Ki on organism growth in batch culture was simulated by initializing Ki with three values, 0.02 g/l, 0.04 g/l and 0.2 g/l respectively. The curves generated by the computer compared well with Andrew’s[l] findings.

Modeling and computer simulations

0.4 - i E

- DYNAMO Ic -- PACTOLUS

Fig. 5. Effect of Ki on organism in batch culture.

Figure 5 lends support to this observation. For continuous culture, the result of the computer simulation for unstable nature of a second “steady state” after increasing the influent substrate concentration level from 10.0 g/t (first steady-state condition) to 10.5 g/I is shown in Fig. 6 in which a striking congruity in DYNAMO II and PACTOLUS outputs are evident. A job CPU time of 8 seconds and a 74 K(V)--storage were used for this simulation process.

CONCLUSION

ModeIing and computer simulations have been found successful in predicting the dynamic behavior of biological processes in both transient and steady state operations. This is made possible by the consistent responses of biological systems to changes in environmental conditions and the availability of established mathematical functions that described these

JO / ,I0

5-

4

3

2- - 0.2

I -

Fig. 6. Unstable nature of the second “Steady State” for a continuous culture.

CAOR Vol. 8. No. I-D

46 B. N. LOHAHI

behaviors. In the absence of, or in case of difficulty encountered in the use of the more popular powerful digital analog simulator program and Continuous System Modeling Program (CSMP). DYNA,MO programming system may be utilized to perform similar dynamic analysis.

4.

5.

6. 7. 8.

9.

IO. II.

12. 13.

REFERENCES

I. F. Andrews. A dynamic model of the anaerobic digestion process. A Publicution of the Enrironmental Sysrems Engineering Department. Clemson University. Clemson (1!%8). A. D. Carr and R. C. O’Donnell, The dynamic behavior of an anaerobic digester. Progess in Water Technology Vol. 9. pp. 727-738. Pergamon Press, New York (1977). D. M. Di Toro, D. J. O’Connor and R. V. Thomann. A dynamic model of the phytoplankton population in the Sacramento - San Joaquin Delta. Nonequilibrium Systems in Natural Water Chemisfry. American Chemical Society (1971). V. H. Edwards, The influence of high substrate concentration on microbial kinetics. Biotechnolog! and Bioengineering XII, 679-712 (1970). L. E. Erickson. L. T. Fan, P. S. Shah and M. S. K. Chen, Growth models of cultures with two liquid phases: IV.. Cell adsorption, drop size distribution, and batch growth. Biofechnology and Bioengineering XII, 713-746 (1970).

J. W. Forrester. Principles of Systems. Wright-Allen Press New York (1968) M. R. Goodman, Study Notes in System Dynnmics. Wright-Allen Press. New York (1974).

S. P. Craef and J. F. Andrews. Stability and control of anaerobic digestion. Wafer Pollution Contru/ Federation J. 46(4)

(1974).

W. Gujer. Design of a nitrifying activated process with the aid of dynamic simulation. Progress in Waler Technology Vol. 9. pp. 323-336. Pergamon Press, New York (1971).

Metcalf and Eddy, Inc., Wastewaler Engineering. McGraw-Hill, New York (1972). K. L. Murphy, P. M. Sutton and B. E. Jank. Dynamic nature of nitrifying biological suspended growth systems. Progress in Wrier Technology Vol. 9, pp. 279-290. Pergamon Press, New York (1977). A. L. Pugh 111, DYNAMO II User’s Munuol. M.I.T. Press Cambridge, Massachusetts, (1970).

T. B. Young III and H. R. Bungay III, Dynamic analysis of a microbial process: A systems engineering approach. Biotechnology ond Bioengineering XV. (1973).

Nofation

APPENDIX I

The following symbols are used in this paper:

p = specific growth rate, time-’

F = liquid flow rate to reactor, volume/time

X = organism concentration in the reactor, mass/volume

S = limiting substrate concentration, mass/volume b = maximum specific growth rate, time-’

K, = saturation constant, mass/volume

K, = inhibition constant, mass/volume

f = time

Y = yield coefficient, mass organism produced/mass substrate utilized

HS = unionized substrate concentration, mass/volume KA = K,, = ionization constant for acetic acid

TETA = 0 = mean residence time in the reactor, time-’ DXDT = DIDT = &I( I + K,IHS + ffslK,)lX; organism growth rate DSDT = DSDT I = ( - I/Y)lli(l + K,lHS + HSIK,)lX; rate of substrate utilization

DXO =X0/0; influent dow’rate of organisms

DfDO =X,/S; effluent flow rate of organisms

DSO = So/f?; influent flow rate of substrate DSIDT = S,/e; effluent flow rate of substrate

MIli = b

pH = H’ = hydrogen ion concentration

KONSI = b/(1 + K,/HS, + HSJK,); specific growth rate in continuous culture Subscripts

0 = influent quantity I = continuous Row culture

APPENDIX 2

DYNAMO/low diogram symbols and notations

-1 I_+ rTL;vyr a Level Equation

I

N = Level initialization

Symbol for a Rate Equation

R = Rate

Modeling and computer simulations

DYNAMO computer prwgrum for botch culture DYNAMO EQUATIONS FOR BATCH CULTURE X.K = X.1 + (DT)(DXDT.JK) ;.

Y c R A

A C c c A

c c L N C R &

PLOT PRINT

SPEC

X=XN XN = 0.0% DXDT.KL = ~~1~U~X.K)~~KONS.K~ KONSK = I + (KSIHS.K) f fHS.K/KI) HS.K = &K)IPH.K)/KA MIU = 0.4 KI = 0.02 KS = 0.001 PH.K = PHI PH f = 3. I&E-I KA = 3.16~~5 S.K = S.J +(RT)(DSDT.JK) S=SN SN = IO.0 DSDT.KI, = - (I/Y)(MIU)(X.K)/(KONS.K) Y = 0.05 X = X(0*.5? X~S~HS~~X~T~DSDT DT = OS/LENGTH = GO!PRTPER = ZiPLTPER = 1

\L/ p, = hixiliary

I--4 p_-* SymWs tar Information TWa-off

~

C = Canttrrnt

Symbda for Sources and Sinks

0 nk

b Material flow Line

e---P

-L Symbol for c0nstont

APPENDIX 3

RUN BASIC c Kf = 0.04

RUN TWO c KI = 0.2 RUN THREE I*

APPENDIX 4

DYNrlMO compuler program for continuous crrltun * DYNAMO ~Q~AT~O~~ FOR CONTINUOUS CULTURE t XLK = X1.J +(D=Q(DIDT^IK + ~XD.JK-DRUNK) N XI=XLN C XlN=0.3XI

48 8. N. LOHAHI

R DIDT.KL=(KONSI.K)(XI.K) A KONSLK =(MIUI)/(I+(KSI/HSl.K)+(HSI.K/KI1)) A HSl.K=(SI.K)(PHl)/KAl c MIUI=O.1 C KII=0.@4 C Kr\i= 3.16E-5 C KSI=0.002 C PHI=I.OE-6 R DXO.KL=DXOC.K A DXOC.K=XO/TETA c x0=0.0 C TETA = 10.0 R DIDO.KL=XI.K/TETA L SI.K=SI.J+(DT)(DSDTI.JK+DSO.JK-DSIDT.JK) N Sl=SIN C SIN=3.78 R DSDTI.KL=(DIDT.JK)(-l/Y) c Y=O.O5 R DSO.KL=SO/TETA c so = IO.5 R DSIDT.KL=SI.K/TETA

PLOT Xl = X(O,l)/Sl = S(O.10) PRINT XI/Sl/DIDT SPEC DT=.S/LENGTH=4O/PRTPER=Z/PLTPER=I

RUN BASIC I*