Modeling the kinetics of growth of acetic acid bacteria to increase vinegar
production : analogy with mechanical modeling
*
Céline Pochat-Bohatier , Claude Bohatier
*
and Charles Ghommidh
*
Laboratoire de Génie Biologique et Sciences des Aliments
Université Montpellier II - Place Eugène Bataillon
CC 023 - 34095 Montpellier cedex 5, France
**
Laboratoire de Mécanique et de Génie Civil
Université Montpellier II - Place Eugène Bataillon
CC 048 - 34095 Montpellier cedex 5, France
bohatier@gbsa.arpb.univ-montp2.fr
Keywords: Fermentation, Acetic Acid, Mechanical Analogy,
Modeling, Optimal Control.
Abstract
Vinegar process is based on the cyclic submerged culture of
acetic acid bacteria, which oxidize ethanol into acetic acid
with the oxygen of the air. A analogy with mechanical
equations has been made to model the culture behavior. A
strengthening and damage formulation has been used to
describe the bacteria growth and the recessive process caused
by the product inhibition. The kinetic model may be used to
define the optimal conditions to activate the end of a cycle,
before a too important inhibition of the bacteria growth by
acetic acid.
1 Introduction
Vinegar is the result of the oxidative fermentation of ethanol
by acetic acid bacteria according to the overall equation [1] :
C2H5 -OH + O2
**
CH3-COOH + H2O (1)
Acetic acid fermentation is certainly one of the oldest
biochemical processes. The history of the vinegar making
shows that development had been brought all along centuries
to accelerate the production [2]. It was first known that air
was necessary for the souring of wine, then in the nineteenth
century, the microbiological action had been identified.
Nowadays, vinegar process is based on the submerged
culture of acetic acid bacteria. Good performances have been
achieved with this process. For instance, in 1997 in France,
the balance between alcohol consumption and acetic acid
production showed that the overall fermentation yield
reached 95% of the maximal theoretical yield, calculated
from reaction stochiometry. The largest percentage of
vinegar production is devoted to alcohol vinegar, also called
white or spirit vinegar. This product contains upwards of 100
g/L acetic acid. Besides the acidic environment, the medium
contains also ethanol, known as an inhibitory substrate. These
unfavorable conditions make difficult the growth of the
bacteria.
In spite of the technology involved, vinegar manufactory is
regarded as a low profit making trade and much effort has
been made to reduce manufactory costs. Nevertheless, further
improvements still remain possible in vinegar production.
Optimal control could be involved to raise productivity and
final acetic acid concentrations.
2 Description of the fermentation kinetics
Vinegar production involves the conversion of ethanol to
acetic acid by microbial cells and strongly depends on the
difficulties to carry on acetic acid bacteria cultures. High acid
concentrations decrease the pH down to 2,2 severely affect
microbial growth and can lead to cell death.
To model acetic acid production, it is therefore important to
begin to describe microbial kinetics. The specific growth
rate, µ (h-1), is defined as the ratio between the biomass
production rate, rX (g/l.h), and the biomass concentration, X
(g/l), equation (2).
µ = rX .
1
X
(2)
When the fermentation is carried out in batch culture, the
biomass balance is expressed as:
µ =
1 dX
.
X dt
(3)
Then, the evolution is defined by the non-linear differential
equation:
The product and substrate balances in batch culture lead to
the expression of the acetic acid production rate, rP, and the
ethanol consumption rate, rS.
dP
rP =
dt
dS
rS = −
dt
F + θ .
θ =
(4)
(5)
dF
du
= η.
dt
dt
η
k
(10)
(11)
with the boundary condition defined by (9).
We consider the analogous variables and coefficients:
P and S represent the product (acetic acid) and the substrate
(ethanol) concentration (g/L).
The specific acetic acid production rate, νP, is defined as the
ratio between rP and X and describes the acetic acid
production rate of one unit of biomass. It is expressed in g
acetic acid per g biomass per hour, and can be calculated by
expression 6.
νP =
1 dP
X dt
(6)
An analytical or numerical solution of the mass balances is
possible once the function µ has been specified.
3 Analogy with mechanical equations
- the bacteria biomass concentration X and the force F;
- the acetic acid concentration P and the displacement u2;
- the ethanol concentration S and the displacement u1;
1
η
k
µ→ −
η
The evolution looks like a relaxation process of Maxwell
where θ is constant.
The variation of µ is related to mechanical damage process
associated to the variation of θ.
It is important to notice here that at the beginning of the
evolution θ < 0 then F is a motive force and after a critical
point θ ≥ 0 and F becomes a dissipative force.
Therefore we can define:
νp →
µ=µmax(1-D)
F
u1
u2
The damage formulation have to satisfy: dD/dt≥0, when
Y=P/Pc, it is possible to propose
u
Figure 1 : Maxwell element.
When we consider a non-linear Maxwell model with one
degree of freedom related to the displacement at one
extremity bar submitted to a traction force -F (Fig. 1).
The displacement u is the sum of the elastic part u1 (7) and
the disipative part u2 (8). After an instantaneous variation of
u1 (or u), the evolution is made as it is shown in (9).
F = + k . u1
F =+ η.
du 2
dt
du du1 du2
=
+
=0
dt
dt
dt
(12)
(7)
(8)
(9)
k is the stiffness and η is the dissipation coefficient.
D=Y
(13)
D=Y e(Y-1)
(14)
or
the second choice has the same value D=0 at Y=0 and D=1 at
Y=1.
4 Materials and methods
4.1 Microorganisms and media
The microorganisms used, in this study, were industrial
Acetobacter bacteria strains provided by a vinegar
manufacture (Vinaigrerie Nîmoise, Lunel, F-34400). In
industry, they were adapted to high acidity, during long-term
submerged fermentation, for the production of alcohol
vinegar. At laboratory, they were maintained by continuous
culture in chemostat at 28°C, in a 4-L stirred reactor. Air
flow rate was held at 0,1 vvm. The culture broth of this
reactor was used to inoculate batch cultures.
The fermentative media were prepared from ethanol (Carlo
Erba), acetic acid (SDS, 99.8%), water and a nutrient mixture
supplied by Frings GmbH & Co KG, as source of glucose,
ammonium, mineral salts and vitamins. The composition of
the media were (in g/L distilled water), for chemostat
cultures : ethanol, 72 ; acetic acid, 20 ; nutrient mixture, 1,7,
and for batch cultures : ethanol, 105 ; acetic acid, 20 ;
nutrient mixture, 1,7.
4.2 Culture conditions
All the fermentations were carried out in an 7-L-reactor
(Sétric Génie Industriel), fitted with conventional control
systems (temperature, dissolved O2, mixing rate, pH), and
specific sensors : a mass flow controller to regulate aeration
(Novatron), a sensor measuring differential pressure between
the top and the middle of the reactor to determine the volume
of the fermentation media (Microswitch), a sensor
determining oxygen volumetric percentage in the gas phase
(Abyss CM12AT) for on-line analysis in exhaust gas. pH and
O2 electrodes (Ingold) were connected to a station Mod7F
SGI. The pH was not controlled and the values were between
2,2 and 2,4 at all times. Therefore, the media were no
sterilised. Temperature was maintained constant at 28°C. Air
flow rate was held at 0,1 vvm. Agitation speed was controlled
to 900 rpm in order to maintain the dissolved oxygen
concentration above 30 % of the air saturation value.
Two solenoid valves were activated to direct fresh mash by a
pump to fill the reactor or to empty a part of the vinegar.
The activation of the end of the repeat-batch cultures was
based on ethanol concentration, determined by on-line
chromatographic analysis. The set-point concentration was
fixed to 2,5 g/L.
When the ethanol concentration was reached, a given portion
of fermentation broth (between 35 and 40%) was removed
and an amount of medium was pumped into the reactor.
Then, a new cycle could start.
The expression of rO2 is derived from a mass balance on
oxygen:
rO2
(χ
MO 2
=
Vmol . Vr
O2 i
. Q airi − χO 2 o . Qair o
χO2 is the oxygen volumetric percentage in the gas phase, Qair
represents the air flow rate, subscripts i and o are used to
design input and output respectively.
A mass balance on nitrogen, between the input and the
output of the reactor, leads to the expression of the air flow
rate (equation 16), considering the following assumptions: (a)
there is neither consumption nor production of nitrogen by
the culture, (b) the bacteria respiration has been considered
negligible.
Qair =
o
1 - χO 2
1- χ O 2
i
. Qair
(16)
i
o
From equations (1) and (2),
rO2
M O2
=
V mol . Vr
. Q airi .
χ O2 i - χ O 2 o
1 − χO 2o
Microbial growth was determined by measuring optical
density (OD) at 540 nm (Séquoia-Turner 340), and then
evaluated on a dry weight basis by a calibration curve of dry
cell weight (DCW) against OD (1 OD unit = 0,31 g DCW/L).
Acetic acid concentrations were determined off-line by
titration with 1 N NaOH.
4.4 Oxygen consumption rate (rO2)
(17)
5 Results and discussion
5.1 Batch cultures
The time courses of acetic acid, ethanol and biomass
concentrations during two repeated batch cultures versus time
are plotted in Fig. 2.
160
0,2
140
120
4.3 Analytical methods
) (15)
0,15
100
80
0,1
60
40
Acetic acid
Ethanol
X
0,05
20
0
0
0
20
40
60
Time (h)
Figure 2 : Ethanol, acetic acid and biomass concentrations
versus time.
The quantity of acetic acid produced (g/L) versus the amount
of oxygen consumed (g/L) (Fig. 3) is used to determine the
acetic acid/oxygen conversion yield (y P/O2). A linear
regression was fitted to the chart with a correlation
coefficient higher than 0,99. Results shown that the practical
yield
∆P
YP =
(18)
∆O 2
2
O
is equal to 97% of the theoretical conversion yield (= 1,875).
In others words, it means that the acetic acid production rate
can be deduced from the oxygen consumption rate, (equation
19). Even if the value of rP is slightly over-evaluated, on-line
oxygen analysis provides a good monitoring along the
fermentation, with better precision than off-line acetic acid
analysis.
r P = Y P . rO2
0,040
0,030
0,020
0,010
0,000
100
(19)
120
140
160
180
200
P (g/L)
O2
Figure 4 : Influence of acetic acid on the specific growth rate
of Acetobacter.
50
45
The acid negative influence on the growth of the bacteria
may be described by equation (20).
40
35
30
P

µ = µmax 1 − 

Pc 
y = 1,8251x
R 2 = 0,9979
25
(20)
where, Pc is the maximum acid level at which growth stops
(Pc = 183,7 g/L) and µmax is the maximum specific growth
rate of the biomass (µmax = 0,055 h-1).
20
15
10
30
5
0
0
10
20
30
25
O2 consumed (g/L)
Figure 3 : Determination of the pratical production yield in
acetic acid fermentation.
Fig. 4 shows the specific growth rate versus acetic acid
concentrations.
20
15
10
5
0
0,006
0,007
0,008
1/P (g/L)
0,009
0,01
-1
Figure 5 : Acetic acid specific production rate versus 1/P.
The determination of νP from oxygen analysis allowed to
draw up the time course of νP along the fermentation.
The influence of acetic acid concentration on the specific
production rate (Fig. 5) can be expressed by :
b
νP = a P
where a and b are constant values.
(21)
This behaviour has already been reported by Czuba (1991),
[3].
0,12
0,10
0,08
X (g/L)
Model
0,06
0,04
5.2 Mathematical models
All the calculations were conducted with D=Y.
Firstly, the assumption that νp was constant versus time has
been used, in order to obtain an analytical solution. This
hypothesis is based on the results obtained in continuous
culture of Acetobacter for lower acetic acid concentrations by
Divies (1973) [4] and Ghommidh [5].
The use of equations (2:4), (6) and (20) leads to the
analytical equations (22) and (23) :
µ
µ
dP
= − max .P 2 + µmax .P + X 0 .ν P + max .P0 2 − µ max .P0
dt
2 Pc
2 Pc
X = X0 −
µ
µmax
µ
µ
.P 2 + ma x .P + max .P0 2 − max .P0
νp
νp
2Pc .ν p
2Pc ν p
Solving equation 22 corresponds to resolve an equation
similar to :
dP
(24)
= - dt
2
(a.P + b.P + c)
a,b and c are constant values.
Regarding the experimental results (Fig. 2), a good
approximation of a linear function of P versus time can be
used :
r P = ˜rP . t + P0
(25)
0,02
0,00
0
10
20
40
Time (h)
Figure 6 : Comparison of model simulation with
experimental results, (νp=24,5 g/g.h, Pc=183,7 g/L, P0=105
˜rP = 2
g/L, µmax=0,055 h-1, X0= 0,067 g/L,
g/L.h).
(15)
The data modelled are in agrement with the experimental
results, although the hypothesis done on the specific acetic
acid production rate.
The model leads to the determination of the biomass
production rate versus time (Fig. 7).
In practise, the fermentation cycle was stopped for t= 23 h, so
rX was nearly of 0,0009 g/L.h. If the fermentation is
prolonged the biomass production slows down too much, the
bacteria loss their ability to multiply and the culture is
damaged by the high acetic acid concentrations. The
productivity of the overall process of vinegar is then
decreasing.
with ˜rP a mean value of the acetic acid production rate.
(17)
Fig. 6 shows X plotted against time, determined from
experimental data and with the modelling equations.
30

P
X t +1 = X t + X t . µ max .1  . dt
P

c
0,0025
0,002
(27)
0,12
0,0015
0,10
0,001
0,08
0,0005
0,06
X (g/L)
Model 2
0,04
0
0
10
20
30
40
0,02
Time (h)
Figure 7 : Modelling of the biomass production rate (rX)
versus time of Acetobacter during acetic acid fermentation.
The resolution of mass balances was then conducted
considering the influence of the acetic acid on the specific
production rate (equation 21).
P versus time was first calculated using the expression (Fig.
8) :
P t+1 = Pt + (a -
b
) . X t . dt
P
0,00
0
10
20
30
40
Time (h)
Figure 9 : Comparison of model simulation with
experimental results, (P0=105 g/L, X0= 0,067 g/L,
µmax=0,055 h-1).
These results show that experimental and modelling data are
in agreement. Once the relationship between νP and P known,
it is not necessary to follow the oxygen consumption rate to
determine the change in biomass and acetic acid
concentrations.
(26)
200
180
160
140
6
120
100
P (g/L)
Model
80
60
40
20
0
0
10
20
30
Time (h)
Figure 8 : Comparison of model simulation
experimental results, (P0=105 g/L, X0= 0,067 g/L,
= 44,075, b = 2372,2).
40
with
a
Then, X has been determined and compared to the
experimental results (Fig. 9).
Conclusions
The use of a kinetic model to describe the bacteria growth
lead to the determination of the acetic acid production rate
during vinegar production, and allowed a better
understanding of the fermentation process. Different damage
formulations can be identified to improve the model and be
used in process control. The submerged acetic acid
fermentation is carried out in repeated batch culture. It is
therefore a cyclic operation. The improvement of the overall
productivity of the fermentation process must take into
account the experimental data of the previous batch to predict
the initial conditions of the next cycle. The study of the
optimal conditions for the end of a cycle, which depend on
the occurrence of the critical point, will help to set up a new
culture mode with continuous feeding of ethanol.
References
[1] Ebner H. and Follmann H. Acetic Acid. Biotechnology.
ed. Rehm H.-J. and Reed G., Vol. 3, pp. 387-407, 1983.
[2] Adams M.R. Vinegar. Microbiology of fermented foods.
ed. Wood. Elsevier. Vol. 1, pp. 1-47, 1985.
[3] Czuba J. Growth of Acetobacter biomass and product
formation during acetic acid fermentation. Acta Alimentaria
Polonica, N°3, pp. 205-211, 1991.
[4] Divies C. Contribution à l’étude de la production d’acide
acétique et du métabolisme de l’éthanol par les bactéries
acétiques. PhD. Paris VII. 1973.
[5] Ghommidh C. Utilisation de microorganismes
immobilisés en réacteur aéré. Oxydation de l’éthanol par
Acetobacter aceti. Thèse de docteur-ingénieur de l’INSA de
Toulouse. France. 1980.