World Journal of Microbiology and Biotechnology 8, 439-445

Technical Communicat ion

Use of the Golden Section search method to estimate the parameters of the Monod model employing spread-sheets

C. Rolz* and J. Mata-Alvarez

An alternative procedure to obtain the parameters of Monod's growth model in batch culture is presented. It is based on the integral kinetic analysis methodology, employs a one-dimensional Golden Section search optimization method and is implemented on a spread-sheet programme. The procedure is discussed in detail and is illustrated by analysis of batch substrate consumption data by an aerobic bacterial consortium.

Key words: Growth model, kinetic analysis, parameters (for growth), spread-sheet, wastewater treatment.

Spread-sheets are powerful tools for chemical and biochemical engineers, and have been discussed by Rosen & Adams (1987). In this contribution, the LOTUS 1-2-3 spread-sheet is used to implement the one-dimensional Golden Section search optimization method, by means of a short MACRO procedure. The methodology is used to fit the kinetic constants of the Monod model according to the methodology proposed by Ong (1983), which can be considered as a refinement of the Gates & Marlar (1968) procedure, widely employed in wastewater treatment modelling studies (Sundstrom & Klei 1979).

T h e o r y

The Monod model describes the changes with time of cell biomass concentration (X) and limiting substrate concentra- tion (S):

dX 12IaxSX (1)

dt - K s + S

dS I dX - (2)

dt Y dt

C. Rolz is with the Center for Scientific and Technological Studies, Central American Research Institute for Industry, (ICAITI), P.O. Box 1552, 01901, Guatemala. J. Mata-Alvarez is with the Dpt. Enginyeria Quimica, Uni- versitat de Barcelona, Marti I Franques 1, E-08028 Barcelona, Spain. *Corresponding author.

~) 1992 Rapid Communications of Oxford Ltd

Ong (1983) derived the following integrated equation:

I S (ln[l + a(So - S)]) - l n - - = b - c t So t

where:

(3)

a = u o (4)

b = 1 + (X o + YSo)/YK~ (5)

C = /Umax(X 0 + YSo)/(yKs) (6)

Equation (3) is a linear relationship between (1/t)(ln S/So) vs In [I + a(S o - S)]/t. Plotting experimental values of both groups leads to the estimation of parameters b and c; however a needs to be known. Gates & Marlar (1968) suggested varying a within an interval and choosing the one that originated the best straight line as judged graphically. Ong (1983) obtained the fit by a modification of the Powell optimization algorithm, applied to maximize the coefficient of determination, R 2, for the regression. Here an alternative procedure is suggested. Appropriate units are chosen for X 0 such that parameter a lies between 0 and 1. These two bounds suggest the use of the Golden Section search to carrry out the one-dimensional optimization. Taking into account the flexibility introduced by the use of a spread-sheet, another modification is suggested, consisting in minimizing the difference between experimental and estimated values of the left-hand side of equation (3).

The Golden Section search is among the most efficient region elimination methods to optimize functions of a single

World Journal o~ Microbiology and Biotechnology, Vol 8, 1992 4 ~ 9

C. Rolz and ]. Ma~a-Alvarez

S t e p 1. Setting the two initial inner points

0 L I I I1

1 - - X X

X + ( 1 - -X) X Condition:

x 1 - x

If: x + ( 1 - x ) = l

1 x Then:

x 1 - - x

Or: ,~ - - (1 - - x ) = 0

Solving for x: x = 0.618

Hence: (1 - x) = 0.382

S t e p 2 , S u p p o s e the right segment was removed Jeaving a subinterval of length x

0 k . I I x = 0.618

1 - - x = 0.382

S t e p 3. Location of the third inner point to keep symmetry

x 2

0 I_ I I , x = 0.618

1 - x = 0.382

0.236

F i g u r e 1. The Golden Section one-dimensional optimization search on a bounded interval of length 1.

variable, provided upper and lower bounds are defined (Edgar & Himmelblau 1988). Basically the method consists of performing two initial trials that are located within the initial interval (assumed for convenience to be of length 1), in a way in which the ratio of the whole line Ix + (1 - x)] to the larger segment (x) is the same as the ratio of the larger segment (x) to the smaller (1 -x)- - -see Figure i, step 1. Then, assuming the function to be unimodal, one of the subintervals is eliminated depending on the result of comparison of the two function values evaluated at the two inner points (Figure 1, step 2). The remaining subinterval of length x, has one trial located interior to it at a distance 1 - x from one end. The symmetry of the search pattern must be kept, hence the distance i - x maintains the relationship x 2 = 1 - x (Figure 1, step 3). The new inner point is calculated as: (0.618 - 0) x 0.382 = 0.236. That is, performing one new trial, with the one kepL the reduced interval has two trials located equidistant from the ends. This pattern is continued until the difference of function

values calculated is smaller than a predefined ending criterion. Note that on every step of the iteration only one new function value needs to be calculated.

T h e S p r e a d - s h e e t

The experimental data are placed in columns A to C from rows five to 14. Figure 2 shows the spread-sheet in which data of Stiebitz et al. (1987) have been placed. Column A is time in h, B and C give biomass and substrate concentrations in m g l - k

The following two columns are defined and set as follows:

Column D: Left-hand side of equation (3); cell Dr:((1/A6),@LN(C6/$C$5))

Column E: Right-hand side of equation (3); cell E6: (@LN(I + $C$I*($C$5-C6))/A6)

Note that for simplicity these have been called Yexp and Xe• respectively.

The linear regression commands are then set: /DR; X-Range:E6..EI2; Y-Range: D6 . .D12; Output-Range: A15; G. These commands produce the regression coefficients and list them in the space of the spread-sheet limited by A15 .. D23.

These values are used to calculate Ye• column F, as: ($D$16 + $C$22,E6). Column G contains the square of the difference between Y~p and Y~, called lack of fit, and calculated as (Dr-F6)^2, Cell G15 contains the sum of the squared lack of fit, calculated as @SUM(G6.. G12). For these initial calculations and only for illustration purposes a value of a = 0.1 was assumed, as shown in cell C1.

The use of the spread-sheet to fit the data and obtain the model parameters presents some advantages: (a) it can use the method of Gates & Marlar (1968) to display the data graphically, and also the value of r 2, so that, if desired, a manual optimization can be carried out; (b) it can use an algorithm to optimize r 2 similarly to the Ong method; (c) it can use other optimization criteria, such as the minimization of the lack of fit or sum of squares of the difference between experimental and calculated values.

Here, the optimization algorithm of the Golden Section has been implemented. A straight MACRO procedure, named/G, has been set up (Figure 3). Instructions are placed in a specific section of the spread-sheet. In this case, parameter a is taken as the independent variable, and the squared lack of fit as the objective function. Initial values for a are usually 0 and 1, if a proper selection of X 0 units is carried out, and they should be placed in cells N4 and N5 respectively. Termination criteria are based on a pre-specified tolerance, expressed as % of differences between the two last response evaluations, here set at 0.001 and placed in cell N6.

440 World ]~urnal of Microbiology mTd Biotechnology, Vol 8, 199g

Estimation of model parameters employing spread-sheets

Row Columns

A B C D E F G

4 Time X S Yexp Xexp Yest Lack of fit

square

5 0.00 15.50 151.00 6 0.54 23.00 123.00 - 0.3798 2.4722 0.0248 0.1637 7 0.90 30.00 105.00 - 0.4037 1.9142 -- 0.6466 0.0590 8 1.23 38.80 75.00 -- 0.5689 1.7494 -- 0.8449 0.0761 9 1.58 48.50 43.50 - 0.7877 1.5594 -- 1.0735 0.0817

10 1.95 58.30 14.50 - 1.2016 1.3766 -- 1.2933 0.0084 11 2.33 61.30 3.50 - 1.6157 1.1832 -- 1.5261 0.0080 12 2.70 62.50 0.50 -2.1150 1.0280 -- 1.7127 0.1618 13 14 15 Regression output: Sum of squares: 0.5587 16 Constant: - 2.9496 17 Standard error of Yest: 0.3343 18 R squared: 0.7878 19 No. of observations: 7.0000 20 Degrees of freedom: 5.0000 21 22 x coefficient: 1.2031 23 Standard error 0.2792

of coeff. :

Figure 2. Initial spread-sheet with experimental data shown in the space limited by [A5, C12], the experimental values according to equation (3) in [D6, E12], the l inear regression parameters in [A15, D23] with a = 0.1 (cell C1). Ye,t (column F) and the 'lack of fit ' or difference squared of Yexp - Ye,t (column G). Cell G15 gives the sum of squares.

Row Column

J K L

4 5 6

13 {LET N8,1} 14 /XGJ17- 15 (IF JS>K8}{LET N4,J5} 16 {IF J8(K8}{LET N5,K5} 17 {LET J5,N4 + ( N 5 - N4)*0.382} 18 (LET C1,J5} 19 (CALC}/DRG" 20 {LET J8,G15} 21 {LET K5,N4+ ( N 5 - N4)*0.618} 22 (LET C1,K5} 23 {CALC}/DRG 24 {LET KS,G15} 25 {LET N8,N8+ 1} 26 {IF @ABS((J8- K8)/

K8)(N6}/XGJ28 ~ 27 /XGJ15- 28 {IF JS(K8}{LET NIO,J5} 29 (IF JS>K8}{LET N10,K5} 30 (GOTO}E16 ~ 31 {BEEP} 32 /XQ ~

M N O P C)

Initial value: 0.0000 Initial value: 1.0000 Tolerance: 0.0010

Counter to keep track iterations Branch to cell J17 Drop lower region Drop upper region At cell J5 place Golden Rule point

Let a at C1 equal J5 Calculate and perform regression Place sum of squares in J8 At cell K5 place Golden Rule point Let a at C1 equal K5 Calculate and perform regression Place sum of squares in K8 Iteration counter increased Sum of squares difference/ tolerance Conditional no. branch to J15 Place final a value in NIO Place final a value in N10 Move screen view to cell E16 Sound beep Macro end

Figure 3. Section of the spread-sheet that contains MACRO /G.

World Journal of Microbiology and Biotechnology, Vol 8, I992 4 4 1_.

C. Rolz and ]. Mata-Alvarez

Row Column

A B C D E F G

1 0.0220 2 3 4 5 6 7 8 9

10 11 12 13 14 15 16 17 18 19 20 21 22 23

Lack o f f i t

Time X S Y~xp X~xn Y,s~ square 0.00 15.50 151.00 0.54 23.00 123.00 --0.3798 0.8892 -0.1391 0.0579 0.90 30.00 105.00 --0.4037 0.7771 -0.7510 0.1207 1.23 38.80 75.00 --0.5689 0.7993 -0.6298 0.0037 1.58 48.50 43.50 --0.7877 0.7682 -0.7995 0.0010 1.95 58.30 14.50 --1.2016 0.7115 - 1.1092 0.0085 2.33 61.30 3,50 -1,6157 0.8207 - 1.6053 0,000t 2.70 62.50 0.50 -2,1150 0.5413 -2.0385 0.0059

Regression output: Sum of squares: 0.1969 Constant -4.9943 Constants: Standard error of Ys,t: 0.1985 Y = 0.3412 R squared: 0.9252 Ks = 44.0391 No. of observations: 7.0000 #m,x = 1.1197 Degrees of freedom: 5.0000

X coefficient: 5.4604 Standard error of coeff.: 0.6943

Row Column

J K L M N

4 Initial value: 0.0213 5 0.0217 0.0220 Initial value: 0.0225 6 Tolerance: 0.0010 7 8 0.1970 0.1969 16,000 9

10 0.0220

Figure 4. Final spread-sheet showing results. In 18 iterations (cell N6) an optimum value of a (cells N10 and C1) has been found equal to 0.022. Optimum values for b and c are found in cells C22 and D16 respectively. With these values and equations (4 to 6) the three Monod parameters have been estimated and are shown in cells G17 to G19.

Selection of Units for Xo. The parameter a is defined by equation (4). It changes during batch culture because the overall biomass yield coefficient does so. The yield is always below 1, hence the value of X 0 should be one that secures an interval for a between 0 and 1. In most instances this will be the case, in the example, X 0 = 15.5 g 1 - t and the expected Y is about 0.3. If not, the value of X 0 should be modified either by changing units or by a variable transformation, as illustrated in example 2.11 of Reklaitis el al. (1983).

Results of the calculations in the spread-sheet are shown in Figure 4. The algorithm converges quickly to the optimum solution and parameter values obtained are very similar to those obtained by the Powell search (Table 1).

Table 1. Summary of Monod parameters obtained by different optimization algorithms and those from previous workers.

Methodology Monod parameters

# (h-') K, (mg t-')

Ong integral method Golden Section search 1.12 44.04 Powell algorithm* 1.12 43.98

Monod--as cited by Stiebitz et al. (1987) 1.20 20.00

Levenspiel (1980) 0.72 26.3 Stiebitz et al. (1987) 0.92 25.7

* Implemented as described by Ong (1983)

442 World Journal of Microbiology and Biotechnology, Vol 8, I992

They are slightly different, as expected, from those obtained by previous workers employing other analysis techniques. In Figure 5 the linear regression fit is illustrated. (Note that two experimental Monod points seem to be outliers.)

The regression fit gives the following confidence intervals at c~ = 0.05:

0.81 < ]Xm. • < 1.53

11.70 </(5 < 104.00

which bound practically all data reported in Table 1.

Estimation of model parameters employing spread-sheets

in 12 iterations (cell NS) converges to a = 0.1985 (cells CI and K5), having started again in an interval of unit length. With these values the Monod parameters are: Y = 0.79, /XI, x = 0.75 h -I , and K s = 15.05 mg 1-1 (cells G17, G18, and G19, respectively). These differ from those given by Sundstrom & Klei (1979) but the differences are within the bounds one would expect, as explained previously.

The Monod model with the above quoted parameter values predicts quite nicely the substrate consumption data as shown in Figure 7.

A n A p p l i c a t i o n E x a m p l e

A problem from Sundstrom & Klei (1979) dealing with chemical oxygen demand (COD) consumption in an industrial effluent by an aerobic bacterial consortium in a batch-operated reactor, will be solved. The data for the problem is shown in Figure 6. A modification has been introduced in this case. The only experimental data available are the substrate consumption with time and the initial biomass concentration, cell B5. The Golden Section search

Conclusion

The method of Ong for determining parameters of the Monod equation (Ong 1983) can be easily implemented in a spread-sheet, using the Golden Section search method instead of the Powell algorithm. The whole spread-sheet is very simple and can be set up in 30 rain for a user with previous experience with spread-sheet software. Its application to analyse COD batch consumption data in industrial effluents is recommended.

0.0

-0.5

-I .0

N

~-I .5 O0

r

-2.0

-2.5

-3.0 0.50

. i / J

I I

l l l I l I i I

0.55 0.60 0.65 0.'/0 0.75 0.80 0.85 0.90 InC 1+CSo-S))/t

Figure 5. Linear regression fit with optimum values for a, b and c~equation (3).

! I

0.95 1.00

World Journal of Microbiology and Biotechnology, Vol 8, 1992 443

C. Rolz and ]. Mata-Alvarez

Row Column

A B C D E F G

1 0.1985 2 3 4 5 6 7 8 9

10 11 12 13 14 15 16 17 18 19 20 21 22 23

Lack offit

Time X S Yo,o Xexp Ye,~ square 0.00 4.00 19.00 1.00 16.50 -0.1411 0.4030 --0.1463 0.0000 2.00 13.00 -0.1897 0.3922 -0.1743 0.0002 3.00 9.00 -0.2491 0.3645 -0.2461 0.0000 4.00 5.50 -0.3099 0.3257 -0.3470 0.0014 5.00 2.00 -0.4503 0.2952 - 1.4263 0.0006

Regression output: Sum of squares: 0.0022 Co nstant - 1.1928 Constants: Standard error of Yest: 0.0272 Y = 0.7940 R squared: 0.9615 Ks = 15.0520 No. of observations: 5.0000 #max = 0.7469 Degrees of freedom : 3.0000

X coefficient: 2.5970 Standard error of coeff. : 0.3002

Row Column

J K L M

4 Initial value: 5 0.1966 0.1985 Initial value: 6 Tolerance: 7 8 0.0022 0.0022 9

10

N

0.1935 0.0216 0.0010

12.0000

0.1966

Figure 6. Spread-sheet showing results on the batch chemical oxygen demand consumption by a mixed bacterial consortium.

N o m e n c l a t u r e

a ,b ,c - -parameters defined by equations (4 to O) k--iteration index in Golden Section search K~--Monod's parameter saturation constant L--subinterval in Golden Section search S----substrate concentration at time t So--substrate concentration at time t = 0 t--batch time /~max---maximum specific growth-rate

x--length of interval on Golden Section search X--Biomass concentration at time t X0--Biomass concentration at time t = 0 Xe• side of equation (3) calculated with the experimental data Y--overall biomass yield coefficient Ye~t--left-hand side of equation (3) calculated with the experimental parameters Yexp--left-hand side of equation (3) calculated with the experimental data (1 - x)----length of interval in Golden Section search.

444 Worla ]ournal of Microbiology and Biotechnology, Vol 8, 1992

Estimation of model parameters employing spread-sheets

15

q

,0

8 f.3

5

o

0 a J i , ,I l

0 I 2 3 4 5 6 Tlme [h)

Figure 7. Simulation of chemical oxygen demand (COD) consumption data by Monod's model with the following parameter values:

Y = 0.79, ,umax = 0.75, and K s = 15.05 mg 1 I O--experimental points.

References

Edgar, T.F. & Himmelblau, D.M. 1988 Optimization of Chemical Processes. New York: McGraw-Hill.

Gates, W.E. & Marlar, J.T. 1968 Graphical analysis of batch culture data using the Monod expressions. Journal of the Water Pollution Control Federation 40, R469.

Levenspiel, O. 1980 The Monod equation: A revisit and a generalization to product inhibition situations. Biotechnology and Bioengineering 22, 1671-I687.

Ong, S.L. 1983 Least-squares estimation of batch culture kinetic parameters. Biotechnology and Bioengineering 25, 2347-2358.

Reklaitis, G.W., Ravindran, A. & Ragsdell, K.M. 1983 Engineering Optimization. Methods and Applications. New York: John Wiley.

Rosen, E.M. & Adams, R.N. 1987 A review of spread-sheet usage in chemical engineering calculations. Computers in Chemical Engineering. 11, 723-736.

Stiebitz, O., Wolf, K.H. & Kloden, W. 1987 Auswirkungen der Methode der Linearisierung des /~-C-Verlaufs auf die Parameteranpassung bet dem Monodschen Modell. Acta Biotechnologica 7, 167-172.

Sundstrom, D.W. & Klei, H.E. 1979 Waste Water Treatment. Englewood Cliffs, NJ: Prentice-Hall.

(Received in revised form 9 January I992; accepted 27 January 1992)

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