Dynamics of Continuous Stirred-Tank Biochemical Reactor Utilizing Inhibitory Substrate

W. Sok61 Institute of Technology and Chemical Engineering, Technical and Agricultural Academy, ul. Seminaryjna 3, 85-326 Bydgoszcz, Poland

Accepted for publication January 22, 1987

The model of a continuous-stirred tank biochemical reac- tor was developed in which the instant uptake rate of substrate was used. The solutions of the model found for the oxidation of phenol by Pseudornonas putida fitted the experimental data better than the results obtained from the models cited in the literature. The model en- ables control of the culture parameters so that the un- wanted washout of the biomass from the bioreactor can be avoided. A review of the models cited in the literature is also presented.

INTRODUCTION

A continuous-stirred tank biochemical reactor (CSTBR) usually operates at such parameters that practically all sub- strate introduced into it is oxidized. However, a change of the culture parameters caused by a non-steady-state process before the CSTBR can take place. It is important to be able to predict whether occurrence of the disturbance will cause only a transient increase of substrate concentration in a CSTBR; hence, the oxidation of substrate will take place at a high conversion, as in the steady state prior to disturbance, or the disturbance will cause the washout of biomass from a bioreactor. In the latter case the biomass concentration in the CSTBR will continuously decrease to zero, while the substrate concentration will increase gradually until it reaches the substrate concentration in the feed. The descrip- tion of the dynamics of a CSTBR enables prediction of the conversion of substrate after disturbance and control of the culture parameters so that the unwanted washout of the biomass from the bioreactor can be avoided.

The dynamics of a CSTBR is described in this article by the material balance equations in which the instant uptake rate of substrate was used. The solutions of the model were found for the oxidation of phenol by Pseudomonas putida in the non-steady state caused by the step increase of the dilu- tion rate or the step increase of the phenol concentration in the feed.

Biotechnology and Bioengineering, Vol. 31, Pp. 198-202 (1988) 0 1988 John Wiley & Sons, Inc.

REVIEW OF MODELS CITED IN LITERATURE

When a metabolic activity of microorganisms is deter- mined by a specific growth rate, the material balances on substrate and biomass are described by the equations

ccx = D(S0 - S ) - - dS dt Y -

= -DX + fl a dt -

The specific growth rate p in the non-steady state depends on substrate concentration S and the history of the culture and the way in which S has varied in the past.”’

Powell’ suggested that the values of p in the non-steady state can be described by

p = y m s )

Developing Powell’s theory, Chi and Howell3 described the oxidation of an inhibitory substrate by

(3) dQ - Q ~ m s - YQ ’S dt

- - Ks + S + Sz/Ki nKs + S + nS2/Ki

(4) dS QSW + Xo) dt

= D(S0 - S ) - - nKs + S + nS2/Ki

The solutions of the above equations found by the authors for the oxidation of phenol by P. putida have shown an agreement of calculated and experimental results only for small disturbances.

Chase4 assumed that the oxidation rate of substrate de- pends on an enzyme concentration in the cells that is related to the substrate concentration in the bioreactor and the du- ration of the unsteady state:

_ - dsb - -Axc(S, t ) dt

CCC 0006-3592/88/030198-05$04.00

The enzyme concentration in the cells in the non-steady state was described by the equations

- - dc - k(c,* - C ) for c 5 c,* dt (7)

(8) dc - _ - -kc dt

when c > cp

where

c:s* c p =

KQ + S* Combining a s . (6)-(9), the author obtained

(9)

= o d2Sb 1 dsbdx cis, 1 c: + k- + -kX dt2 X dt dt dt Y KQ + S *

for c 5 c t (10)

and

d2Sb 1 dSb dx dsb + - = 0 whenc > c,* (11) dt2 X dt dt dt

The relation between change in substrate concentration in the bioreactor (dS / d t ) and change in substrate concentration caused by biooxidation only (dsb /d t ) is described by the equation

dsb - dS dSo dt dt dt

To identify this model, experimental data are necessary for various substrates and microorganisms.

Yang and Humphrey’ used the values of p calculated from the following equation in Eqs. (1) and (2):

(13)

The symbol p(0) refers to the specific growth rate just after disturbance, whereas a parameter a is characteristic of the biological model (substrate-microorganism). It can be no- ticed that according to Eq. (13), the values of p are inde- pendent of substrate concentration. It was found, however, that the values of p in the unsteady state depend also on substrate concentration. ‘*‘,’ The model showed the discrep- ancies between calculated and experimental results for larger disturbances.

Lee et a1.* developed a two-state model distinguishing the different growth conditions of cells in suspension and those in aggregates. Assuming that aggregation and decay pro- cesses are reversible, the authors described the dynamics of continuous culture by the equations

p = p(0) + at

-- - -OXs + psXs + KaSXa - KsXs

5

(15)

(16)

dt

= -OXa + paxa + KsXs - KaSXa dt

The usefulness of the model was not evaluated by the authors.

Young and Bungay’ assumed that the values of p depend on substrate concentration S, inside the cells and described the material balance on the reactor by the equations

- -DX + pX dx dt - _

The authors determined the values of parameters Kl , K2, and K y for culturing of Sacharomyces cerevisiae on glucose. The solutions of the model found by the authors to step increase dilution rate and step increase glucose concen- tration in the feed were in an agreement with experimental data only for an increasing substrate concentration in the bioreactor.

As can be seen, there were discrepancies between calcu- lated and experimental results. It can be expected to be excessively difficult to describe exactly the concentrations of substrate and biomass in the non-steady state for a wide range of disturbances. A prediction of the conversion of substrate after the disturbance will, however, enable the con- trol of culture parameters so that the oxidation of the sub- strate will occur at high conversion as prior to disturbance.

MATERIAL BALANCE ON BIOREACTOR

Let the metabolic activity of microorganisms be described by a specific uptake rate of substrate defined as the mass of substrate oxidized by a unit mass of microorganisms in unit of time. Then material balances on substrate and biomass are described by the equations

ds - = D(S0 - S ) - UX dt

- _ - -DX + uyX dt

In order to predict the transient behavior of the bioreactor, the equation governing a specific uptake rate has to be established experimentally for the microorganisms and sub- strate used.

The specific uptake rate of phenol by P . putida grown at various dilution rates has been studied by Sokol and Howell” and Sokol.” The instant values of u at high con- version of phenol were described by the equation“

(22) 2.86 S*S

(2.1 + S * ) (Ks + S + S 2 / K j ) u =

SOK6L: DYNAMICS OF STIRRED-TANK BIOCHEMICAL REACTOR 199

where concentration in the feed are shown in Figs. 1-3. Growth yield Y used in Eq. (21) was calculated from the experi-

(23) mental formula Ks = 0.98S*0.4 - 0.21

U Ki = 7.6S'0.9 + 4.2 (24)

These are valid for 0.46 mg/L 6 S * S 2.1 mg/L. Exam- ples of solutions to Eqs. (20)-(22) found for the step in- crease of the dilution rate and the step increase of the phenol

Y = (25)

It can be seen in Figs. 1-3 that the results obtained from

1 . 7 7 ~ + 0.09

140-

A A n n A r. A

n 1 ----------- _ _ _ _ _ _ _ _ _ _

c A

120..

110-, - d __ \ -- m E - 30- X

20 - .

- - _ _ _ - - - -- - - - ----- , # f - - -

/ /

/

lo- ' / 0 0

c( .-._ f - - " " O 0 0

l o r

O O 012 014 016 O h 1 f4 116 1.b 210 - 0 0 1 0 0

0 0 0 0

..I5

-10 - d

\ Ul E v) u

.- 5

0

Figure 1. Experimental concentrations of substrate S and biomass X calculated from the following models: (- - - - ), ref. 3; (-. - -), ref. 5; (-), Eqs. (20)-(22). (0) experimental S, (A) experimental X. Dilution rate was step increased from D = 0.13 h-' to D = 0.19 h-' at constant phenol concentration in feed, So = 250 mg/L.

Figure 2. Experimental concentrations of substrate S and biomass X calculated from the following models: (- - - -), ref. 3; (---*), ref. 5; (-), Eqs. (20)-(22). (0) experimental S, (A) experimental X. Phenol concentration in feed was step increased from So = 500 mg/L to So = 700 mg/L at constant dilution rate D = 0.19 h-I.

200 BIOTECHNOLOGY AND BIOENGINEERING, VOL. 31, FEBRUARY 1988

600

500

4oc

- < E30C

X Y

20c

1 oc

( 5 t [h l x)

1200

1100

1000

900

800

700

600 2 u

500 * 400

300

200

100

0

Figure 3. Comparison of experimental concentrations of substrate S and biomass X with those calculated from the following models: ---- ,ref. 3; -.-., ref. 5;- , Eqs. (20)-(22). (0) experimental S; (A) experimental X. Washout occurred after step increase of dilution rate fromD = 0. I3 h-' to D = 0.25 h-' at constant phenol concentration in feed, So = loo0 mg/L.

Figure 4. Solutions of Eqs. (20)-(22) for step increase of substrate concentration in feed from So = 250 mg/L to So = 500 mg/L at various dilution rates (h-I): ----, 0.19; ~ , 0.26; - - - -, 0.30.

Eqs. (20)-(22) and experimental data are in agreement. The conversion of phenol predicted by the model was confirmed experimentally for all cases.

The concentrations S and X calculated from the models cited in the literature3s9 are also shown in Figs. 1-3. In

Eqs. (1) and (2) the authors3 have applied the values of p given by the formula

(26) P J = Ks + S + S'/Ki

SOK6L: DYNAMICS OF STIRRED-TANK BIOCHEMICAL REACTOR 201

For phenol oxidation by P . putida they found p , = 0.369 h-', Ks = 5.94 mg/L, and Ki = 227 mg/L. It can be noted in Figs. 1-3 that the solutions of Eqs. (20)-(22) fit experimental data better than the results obtained from Eqs. (I ) , (21, and (26).

Next, the solutions of Eqs. ( l) , (2), and (13) suggest that the occurrence of any disturbance will not cause the wash- out. This is contradictory to experiment^^.'^ that showed that large disturbances, particularly occumng at high dilution rates, caused washout.

As already mentioned, the occurrence of large dis- turbances can lead to washout, thus reducing the oxidation of substrate. In order to estimate the magnitude of the dis- turbances causing the washout, Eqs. (20)-(22) were solved for various disturbances. It can be seen in Fig. 4 that the moderate step increase of So occumng at a small dilution rate will not cause washout. Even a moderate step increase of So occurring at a high dilution rate causes washout. Simi- lar conclusions can be drawn for the step increase of D and the simultaneous step increases of D and So.

Analysis of the highest values of dilution rate D, and phenol concentration in the feed So, after a disturbance whose occurrence did not cause washout has shown that the values of D, and So, calculated from the model were about 15% lower than those found experimentally.

CONCLUSIONS

A review of the models of CSTBRs cited in the literature showed that there were discrepancies between calculated and experimental results. Furthermore, in some cases the models predicted total oxidation of substrate when the ex- perimental data showed that washout had occurred.

In the model of the CSTBR developed in this article the instant uptake rate of substrate was applied. The solutions of the model, found for oxidation of phenol by P . putida in non-steady state caused by the step increase of the dilution rate or the step increase of the phenol concentration in the feed, fitted the experimental data better than the results obtained from the models cited in the literature. The con- version of phenol predicted by the model was confirmed experimentally for all disturbances.

The analysis of the highest dilution rate D, and phenol concentration in the feed So, whose occurrence did not cause washout showed that the values of D, and So, calcu- lated from the model were about 15% lower than those found experimentally. Then the values of D, and So, ob- tained from the model were not substantially underestimated compared to experimental data.

NOMENCLATURE

constant in Eq. (6) enzyme concentration in cells (mg/L) equilibrium enzyme concentration in cells (mg/L) dilution rate (h-') rate constant for enzyme synthesis (h-l) kinetic parameter in Eqs. (15) and (16) (L/mg h) inhibition constant (mg/L) kinetic parameter in Eqs. (15) and (16) (h-I) saturation constant (mg/L) parameter (mg h/mg) kinetic parameter in Eqs. (17) and (19) (L/mg h) kinetic parameter in Eq. (19) (h-') quotient of smallest and highest values of Q (dimensionless) metabolic activity factor substrate concentration in unsteady state (mg/L) substrate concentration in cells (mg/L) substrate concentration in feed (mg/L) substrate concentration in steady state prior to disturbance (mg/L) time (h) specific uptake rate (mg/mg h) biomass concentration in unsteady state (mg/L) biomass concentration due to wall growth (mg/L) growth yield (mg/mg) specific growth rate (h-I)

Subscripts a aggregated b biooxidation m maximum s suspension

References

1. 2.

3. 4. 5.

6.

7.

8.

9.

10. 11. 12.

C. J. Perret, J . Gen. Microbiol., 22, 589 (1960). E. 0. Powell, in Proceedings of the Third International Symposium, E. 0. Powell, C. G. T. Evans, R. E. Strange, and D. W. Tempest, Eds. (Her Majesty's Stationery Office, London, 1967). C. T. Chi and J. A. Howell, Biotechnol. Bioeng., 18, 63 (1976). L. M. Chase, Biotechnol. Bioeng., 19, 1431 (1977). R.D. Yang and A.E. Humphrey, Biotechnol. Bioeng., 17, 1211 (1975). V. H. Edwards, T. E. Kinsella, and D. B. Sholiton, Biotechnol. Bio- eng., 14, 123 (1972). A. F. Gaudy, Jr., and R. Srinivasaraghaven, Biotechnol. Bioeng., 16, 723 (1974). S. S. Lee, A. P. Jackman, and E. D. Schrijder, Wat. Res. 9, 491 (1975). T.B. Young I11 and H.R. Bungay, Biotechnol. Bioeng., 15, 377 ( 1975). W. Sokol and J. A. Howell, Biorechnol. Bioeng., 23, 2039 (1981). W. Sokol, Biotechnol, Bioeng. (in press). I. Malek, K. Beran, 2. Fencl, V. Munk, J. Ritica, and H. SmrCkowa, Eds., Continuous Cultivation of Microorganisms (Academic, New York, 1969).

202 BIOTECHNOLOGY AND BIOENGINEERING, VOL. 31, FEBRUARY 1988