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Mod el ing of Ethano l Fermentat ion at High

Yeast Concentrations

A.

6 Jarzebsk i and J . J . Ma l i now sk i

Polish Academy of Sciences Institute of Chemical Engineering

44-100

Gliwice ul. Bahycka 5,Poland

G.

G oma

Departement de Genie Biochimique et Alimentaire lnstitut National des

Sciences Appliquees UA-CNRS-No544 Toulouse Cedex France

Accepted for publication February 1,

1989

Three models of ethanol fermentation at high yeast con-

centrations were developed and verified by comparing

them with experimental data. The conventional ap-

proach, neglecting l oss of cell viability, proved to be the

least accurate. The models, allowi ng both for t he

growth of viable cells and accumulation of the inactive

yeast, satisfactorily portray the fermentation process at

very high

cel l

concentration and may be used to facili-

t a te process optimization. Analysis of results shows that

both intrinsic and nonintrinsic models provide similar

results for ethanol fermentation.

INTRODUCTION

The growing prospects of ethanol as a fuel and future

chemical feedstock have prompted considerable efforts to

increase effectiveness of the fermentation process. Con-

tinuous fermentation markedly reduces costs but low yeast

concentration in the fermentor and the low dilution rates

usually applied seriously limit process productivity. Cy-

sewski and Wilke'.' first proposed yeast cell recycling, and

this increased productivity four times that of a conven-

tional process. Initially, a settling tank or a centrifuge was

used to recycle cells. However, the most recent reports

advocate continuous concentration by means of cross-

flow filtration both in ultra- and microfiltration

The latter technique gives cell concentrations of up to

200

kg m-3 dry weight3 and even more,537while the de-

duced maximum cell density of the yeast cultures is ca.

300

kg m-3.739310

At very high yeast cell concentrations, conditions for

growth and metabolism are less favorable due to hindered

access to nutrients, space limitations, and cell interaction.

These facts together with prolonged residence time of cells

lead to kinetics and compositions different from those usu-

ally encountered. An adequate mathem atical model should

certainly take these factors into account.

In spite of widespread interest in experimental investiga-

tion of the process, mathematical modeling of fermentation

at high cell concentrations has attracted relatively little at-

tention. Le e, Polard, and Coulman developed a model

for glucose-to-ethanol fermentation at high cell concentra-

tions to evaluate the feasibility of improving fermenter

productivity by using a multiple-stage reactor. This model

uses nonintrinsic concentrations, and the growth rate of

cells is expressed by Monod type kinetics extended to al-

low both for product and cell inhibition. The rate of

ethanol formation is assumed to be proportional to the rate

of cell growth while the

loss

of cell viability, which may

be an important factor at high cell

concentration^,^ ^^^

is to-

tally neglected. A comparison of theoretical predictions

with several experimental points does not provide suffi-

cient evidence to draw any overall conclusions as to the

accuracy

of

the developed model. Hence, for computer

simulations relevant parameters were assumed from the

range of feasible values rather than those assigned from

experimental data.

More recently, M onb ~uq uet t e '~mphasised the need to

use the intrinsic concentration approach to describe high

cell density bioreactor operation and indicated that the

nonintr ins ic approach is sui table when the at ta inable

biomass volume fraction is less than ca.

0.1.

Comparing

corresponding predictions from intrinsic and nonintrinsic

models, Monbouquette found that nonintrinsic models can

lead to gross errors in calculated substrate and product

concentrations, substrate conversions, and volumetric pro-

ductivity. Standard Monod-type kinetics were extended to

take into account product inhibition but neglected the cell

inhibition effect and possible death phase, similarly in an-

other report.

To summarize, the available literature does not provide

a verified model for fermentation at high cell concentra-

tions. The aim of this communication is to develop and

investigate several approaches for this process by com par-

ing theoretical predictions with recently published experi-

mental data on glucose-to-ethanol fermentation at high

yeast concentration^.^

Biotechnology and Bioengineering,

Vol. 34, Pp.

1225-1230 1989)

0 1989

John W i ley

&

Sons, Inc.

CCC 0006-3592/89/0901225-06 04.00



FERMENTATION MODEL S

A Conventional Approach

Conventional models for fermentation precesses neglect

loss of cell viability and the appearance

of

inactive cells.

Since both models developed1 re ~ en tl y ~elong to this cate-

gory the questions of whether and to what extent such an

approach can successfully portray the process of fermenta-

tion at high cell concentrations is of practical interest.

The central point in each fermentation model is the ki-

netics. Analysis of the concentration profiles' indicated

that chief inhibiting factors are product and cell concentra-

tions. Adopting Monod-type kinetics and extending them

in the simplest possible form to allow for inhibition effect,

the rate

of

growth is expressed as

i.e ., growth ceases if either product concentration

P

or to-

tal cell concentrations

X ,

reach certain maximum values

P ,

or X, , respectively. Exponents A , and A, take into account

nonlinearity of inhibition effects.

The rate of substrate consumption is1'

s

= -rx/yx,s

m sx ,

2)

and rate of ethanol formation is assumed to have the form

of

the Luedeking-Piret equation

(3)

The typical system of fermentation with mem brane cell re-

cycle module is shown schematically in Figure 1. The fil-

tering device is assumed perfect and the feed, bleed, and

filtrate streams are denoted. by

F, B ,

and L , respectively.

In the conventional nonintrinsic approach, the final set

of mass balance equations describing operation

of

the fer-

mentation system is

r p

= r* /Yxp+ m p x t

4)

L

P, s

x = o

8

*

Figure 1

Fermentation sysiem with membrane filtering module.

5 )

The system

of

eqs. (1)-(6) supplem ented with a set

of

val-

ues

of

operational and adjustable parameters constitute the

model of the fermentation at high yeast cell concentrations.

A Modified Approach

Several authorsL2213ave already noted the loss of cell

viability at high cell concentrations. Analysis of concentra-

tion profiles reported recently by Lafforgue and co-work-

e r ~ *ndicate that inactive cells can constitute as much as

one-quarter of the total dry biomass and hence their ap-

pearance cannot be neglected. M ore thorough investigation

of this question reveals that the rate of death depends on

the rate of growth of the viable phase and also on its con-

centration. Taking all this into account, it is assumed that

the total biomass co mprises a viable (active) phase

X ,

and

an inactive (dead) phase

X,.

Consequently, the growth rate

of the viable phase is

where

X = X , + X,.

Similarly, the rate of death may be expressed as

rd = ( k , w + k2)X

(8)

where

p

is the specific growth rate of the viable phase and

the rate of substrate consumption is of the form

(9)s

=

- r*/yx, s msxv

Since specific ethanol productivity decreases monotoni-

cally with cell concentration, the rate of ethanol production

can be described, as done by Mota et al.,I6 by the expo-

nential function

(10)

p

=

a x ,

exp(

-bX,

.

In the intrinsic approach, the final set

of

equations is

X S S

+ r,V

(13)

1226

BIOTECHNOLOGY AN D BIOENGINEERING, VOL.

34,

NOVEMBER 1989



while

in

the traditional nonintrinsic approach, substrate

and product concentrations are given by eqs. (5) and

(6)

and equations for active (viable) and inactive phases are as

above . The systems of eqs. (7)-(14) or eqs. (5)-(12) con-

stitute two versions of the m odified model of the fermenta-

tion at high yeast cell concentrations.

RESULTS AND D ISCUSSION

The exp erimental observations of L afforgue, Malinow ski,

and Goma' obtained in a fermentation system (fermenter

and cell separator) with total working volume 2.4

X

m3,

S =

150 kg m-3, and dilution rates D

=

0.5 and

B = 0

are compared in Figure 2 with predictions of the

conventional model (solid lines). Model parameters esti-

mated numerical ly took the values : p = 0 . 2 4 h - I ;

K , = 0.5; A , =

0.85; A,

=

1.1;

Y,,, =

0.13;

m, =

0.22;

Y,,, = 0.4; and

m

= 0.13. From results reported previ-

~ u s l y , ' ~

,,,

was assumed to be equal to

90

kg m-3; from

data col la ted by L af fo rgue, Mal inowsk i, and G ~ m a , ~t

was taken that

X , =

320 kg m-3.

It is clear from Figure

2

that the conventional model

fails to predict ethanol concentration and predicts too rapid

a rise of yeast concentration in the exponential growth

phase.

To

overcome the first drawback, eq. (3) was re-

placed by the expression

r, = d

xp(-bX,) (15)

and predictions obtained for

a =

0.43, b

=

0.0048,

Y , , =

0.15, and

m

= 0.21, with other parameters as before, are

shown in Figure 2 (dashed lines). It can be seen that the

modification of the expression for ethanol production rate

markedly improved ethanol concentration profile but had

little, and in fact adverse, effect

on

the predictions of yeast

concentration. On the whole, the conventional modelling

a p p r o a c h a p p e a r s i n a d e q u a t e t o p o r t r a y a c c u r a t e l y

the fermentation process at high yeast concentrations.

The same experimental observations of Lafforgue and

co-workers5 are com pared in Figure

3

with predictions of

the modified approach. Model parameters estimated nu-

merically to provide the best fit of the curves took the val-

ues:

p =

0 . 24 ;

A , =

0 . 85 ;

A , =

1 . 1 ;

K , =

0.5;

a

=

0 .55 ; b

= 0 . 0 0 6 1 ;

Y, , ,

= 0 . 1 2 ; k , = 0 . 2 0 8 ; k , =

2.08

X

and m, = 0.27. Similarly as before

P,,,

was

assumed to be equal to 90 kg m-3 and X ,

=

320 kg m-3.

It is noteworthy that estimated values are in fair agreement

with those reported by previous and exam -

ined in the most recent report of Daugulis and Swaine.I9

From Figure 3 it may be seen that agreement between pre-

dict ions and exper iment is good for a l l prof i les and

throughout the whole time range. Moreover, the two modi-

fied models provide similar results, in particular for yeast

concentration. The largest discrepancy was found in sub-

strate concentration but this was never larger than about

twenty percent.

A s

this agreement remains fair also for

> 0

both models may be used for simulation purposes

and further analysis of the process.

From eqs.

6)

and (10) and also from eqs. (10) and (14)

(after minor rearrangements), it may be seen that ethanol

productivity, equal to the product of dilution rate and etha-

nol abiotic phase conc entration, under steady (o r quasi-

steady) operation may be expressed as

t

( h l

Figure

2.

Comparison

of

experiment with theoretical predictions

of

the conventional approach

COMMUNICATIONS TO THE EDITOR

1227



s

0 20 60 100 140 180

50

20

-

m

E

9 0

-

60

z-

30

0

Figure 3.

and

(-)

nonintrinsic model.

Comparison of experim ent with predictions of the modified models: (---) intrinsic model

DP = d

xp(-bX,)

(16)

It is evident that productivity initially rises with increase in

X,, attains a maxim um value, a nd subseque ntly falls expo-

nentially but specific ethanol productivity falls monotoni-

cally with increase in X,; this agrees with the most recent

observations of R ichter and M eyer.20

The viable cell concentration corresponding to maxi-

mum productivity is

X

= l / b

maxDP

and for the case considered this value was ca.

164

kg m-3.

The corresponding maximum ethanol productivity was

found to be equal to 33.2 kg m-3 h-I and ethanol exit con-

centration of ca.

66.4 kg

m-3. It may be noted that these

computed and experim ental values correspond well at

maximum product concentration (see Fig. 3) . Certainly,

optimum conditions for maximum ethanol production can

be achieved for incomplete cell recycle, i.e. when bleed

stream

B

0.

A

bleed is also necessary to maintain a cell

concentration suitable for undisturbed operation since build

up of cells continues until the fermenter becomes inoper-

able due to high viscosity.2' The most suitable value of ra-

tio B / F , from the ethanol productivity standpoint, together

with the actual values of

;U,

and S can easily be computed

by resolving the system of eqs . (11)-(

13)

or eqs.

5), 1

) ,

and 12), respectively, taking derivatives equal to zero. In

the case analyzed maximum productivity occurred for

B / F

of ca. 0.03. Figure

4

portrays predictions of concentrations

X , , P ,

and

S

vs. time for

B / F

ratios

0.01, 0.03,

and 0.05

and Figure 5 depicts dependence of ethanol productivity

and substrate conversion on the bleeding, obtained from

the nonintrinsic approach. It is notewo rthy that the conven-

tional productivity expression

DP

and the more rigorous

form given by Monbouquette [eq.

15)]14

provide similar

results. The range

B / F 0.03

appears to offer the most

favorable conditions for practical applications.

CONCLUSIONS

The models, allowing both for the growth of the viable

cells and the appearance and accumulation of the inactive

yeast, can be used to portray fermentation at high cell con-

centration. The growth of yeast can be adequately modeled

by conventional Monod kinetics supplemented with terms

taking into account cell and product inhibitions which

prove not to depart markedly from a linear relationship.

Recycling of cells inevitably leads to the appearance of an

inactive (dead) cell phase, which increases in proportion to

the rate of growth of viable cells and cell concentration.

Cell inhibiting effects depend on the total concentration of

viable plus dead phases. With an increase in viable cell

concentration, productivity of ethanol initially rises, attains

a maximum value, and subsequently falls while specific

productivity decreases m onotonically. For ethanol fermen-

tation, both intrinsic and nonintrinsic approaches give

similar predictions and hen ce are equally suitable for simu-

lation purposes.

NOMENCLATURE

A , , A Z

power indices, eqs.

1 )

and

7)

a

parameter in eqs.

10)

and

15) (kg

kg- h-')

1228

BIOTECHNOLOGY AND BIOENGINEERING, VOL.

34,

NOVEMBER 1989



32

280 - / F = O . O l

24

-

m

-

60

120

80

40

0

-

_ _ _ -- - - - - - -

€

_ , _ , _ _ _ . _ . . - . - . - . - . - . -

I-

X

I - . - . I . . -. -,-.

- - - - - _ _ _ _ _ --

- - - ___

0 2

6

100 140 180

t

( h l

Figure

4.

Effect of bleedstream ratio

on

total biomass, product and substrate concentrations

parameter in eqs. 10) and (15) (kg- m3)

bleedstream flow rate (m' h -l)

dilution rate (h-')

feedstream flow rate ( m3 h-')

substrate saturation constant (kg m-3)

parameter in eq. 8)

parameter in eq.

. 8)

(h-I)

filtrate stream flow rate (m 3 h-I)

constant in eq.

(3)

(kg kg- h-')

maintenance coefficient

(kg kg-

h-I)

35

31

.-

Ic

T

27

E

l

x

-

2 23

19

15

product concentration

(kg

m-')

rate of ethanol production (kg m-3 h-')

rate of substrate consumption (kg K -')

rate of yeast cell production (kg m-l h-I)

substrate concentration (kg m-3)

yield of cells based on substrate consumed

(kg

kg- )

yield of cells based on product formed (kg

kg-l)

yeast cell concentration

(kg

IT-')

volume (m )

specific growth rate (h-l)

I

I

I

I

I

0

0 0 0.0

8

0 12 0 1 0.2 0

B / F - I

Figure

5. Effect of bleeding on ethanol productivity and substrate conversion.

1.0

0.9

-

I

0 8

cn

0.7

7

j:

0 6

0.5

COMMUNICATIONS

TO THE

EDITOR 1229



Subscripts

d

inactive (dead) phase

m

maximum value

v viable (active) phase

t total concentration

References

1 .

2.

3.

4.

5

6.

7 .

8

G . R. Cysewski and C . R. Wilke, Biotechnol . B i o e n g . ,

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1230 BIOTECHNOLOG Y AN D BIOENGINEERING, VOL.

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1989

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