_{1}

In this paper, a study of the bifurcation analysis of fermentation of sugar to ethanol in presence of
*Saccharomyces cerivisiae* at very high gravity is described. The bifurcation analysis was done for a concentration 280 gl
^{−1} of sugar, and the dilution rate was taken as the parameter of bifurcation. Two Hopf bifurcations (HB) at 280 gl
^{−1} were found. At dilution rate of 0.027 h
^{−1} the system exhibits damped oscillations and not sustained oscillations as previously reported because the system is close to a point of attraction, and we can attenuate these oscillations by the choice of initial conditions. The system exhibits sustained oscillations between the two Hopf Bifurcations, the first at 0.08028 h
^{−1} and the second at 0.04395 h
^{−1}. These oscillations are the consequence of synchrony between the daughter and the mother yeast. Indeed, it is better to take a dilution rate between the two Hopf bifurcations (self sustained oscillations), in order to increase the ethanol productivity.

The increase of CO_{2} concentration in the atmosphere, with the possibility of energy deficit, led to an orientation toward biofuels. Bioethanol, being one of the most promising biofuels, is ecologically clean, and has the potential to greatly reduce the dependence to oil. As reported by [

Because it is simple and efficient, batch fermentation is currently the predominant process for ethanol production. On the other hand, the continuous approaches have some advantages with respect to fermentation. However, for mixtures having a lower ethanol concentration, the product must be distilled [

Very high gravity (VHG) fermentation has been thoroughly studied in [^{−1}, it inhibits the yeast metabolism and when it exceeds 90 gl^{−1} it becomes toxic [

Clearly, these effects should be taken into consideration in the design of efficient processes for bioethanol production because they can act negatively on ethanol production [

The ethanol production by fermentation is realized in the presence of the yeast baker Saccharomyce cerevisiae. For continuous fermentation process, oscillations are observed that are difficult to eliminate. Typically, these appear for dilution rates between 0.09 and 0.25 h^{−1} [

In order to avoid or to eliminate these oscillations [

In this work, the bifurcation analysis of fermentation of sugar to ethanol in presence of Saccharomyces cerivisiae at very high gravity is presented.

The mathematical model for the fermentation in continuous bioreactor and the kinetic of the bioreaction, and all the constants are taken from Fengwu Bai thesis [^{−1}. The dilution rate was taken as the parameter of bifurcation, and the results are present in the form of diagram of bifurcation for the different variables of the system.

In the fermentation process with Saccharomyces cerevisiae two cells, a small one called the daughter and a large one, the mother, are formed at division stage [

Because of their small size, daughter cells cannot enter into the division process after separation, only mother cells are able to bud [

At high feed glucose concentrations, high inhibitory ethanol concentrations are observed only under VHG conditions. Ethanol concentrations over 40 gl^{−1} exert inhibition on yeast metabolism, and turns to toxicity if there is an excess in ethanol concentration up to about a 90 gl^{−1} [

Cells in high ethanol concentration may either die or lose irreversibly their ability to divide. In the latter case, they can still produce ethanol since the fermentation remains active [

The stress due to either yeast growth or ethanol production yields certain metabolites such as Glycerol and trehalose. At the same, time ATP (Adenosine triphosphate) is produced to satisfy the increased energy demands of the cell under stress.

According to the observations of Ding [

The model of Li et al. for the oscillations observed in continuous ethanol fermentation with Z. mobilis is based on the ethanol concentration change rate history [

Since the ethanol concentration history is more significant than the ethanol concentration change rate history, Bai [

The Equations of the proposed model are [

d X d t = X ( Z − D ) (1)

Parameter | Value |
---|---|

D | 0.027 |

Mu_{max} | 0.40 |

K_{s} | 10.8 |

K_{i} | 293.2 |

K_{sp} | 23.0 |

K_{ip} | 1751.7 |

α | 3.86 |

β | 1.66 |

P_{m} | 167 |

μ_{0}_{ } | 0 (if S > K S ) 0.0378 (if S ≤ K S ) |

ν_{0} | 0 (if S > 2 K S P ) 0.246 (if S ≤ 2 K S P ) |

ν_{max} | 1.66335 |

ω | 0.0617 |

Y_{ps} | 0.45 |

d S d t = D ( S 0 − S ) − 1 Y P / S Z p X (2)

d P d t = − D P + Z p X (3)

d Z d t = ω ( V G − Z ) (4)

d V G d t = ω ( µ − V G ) (5)

d Z p d t = ω ( V P − Z p ) (6)

d V P d t = ω ( ν − V P ) (7)

In the kinetic model the specific growth rate is expressed into two terms. One non linear term for the ethanol concentration, and another term for the substrate inhibition with the introduction of substrate constant into the kinetic expressions [_{0} that equals the dilution rate at which the limiting substrate concentration falls below a detection limit.

µ = µ max S K S + S + S 2 / K i ( 1 − P P max ) α + µ 0

Since the growth of cells depends tightly on ethanol production, the following similar model was proposed [

ν = ν max S K S P + S + S 2 / K i P ( 1 − P P max ) β + ν 0

Numerical results were obtained using the software XPPAUT Version 6.10 [^{−1}.

As the dilution rate is decreased, we observed a second region, in which a first supercritical Hopf bifurcation HB1 occurs with D = 0.08028 h^{−1}. This region contains a unique stable periodic branch, represented black bold line in ^{−1}, taken between the two Hopf bifurcations, the system exhibits sustained oscillations as consequence of the synchrony of the cycle, and there is an increase of the productivity over the steady state, as shown in

In the case synchrony of cell growth, modulation of the metabolism appears clearly on the reactor the measured variables.

With the decrease of the dilution rate a second Hopf bifurcation HB2 occurs at D = 0.04395 h^{−1} where the limit cycle disappears, and a stable steady state is created. This is represented by the black bold line in

At dilution rate 0.027 h^{−1} and for initial concentration of sugar 280 gl^{−1}, as shown in

An asynchronous step may be made synchronous in at least two ways [^{−1} all cells start dividing synchronously for several generations until the loss of synchrony.

Shen [^{−1} and run continuously until the steady state was established. The

dilution rate was then switched to 0.027 h^{−1} for another steady state. Oscillations were introduced when the dilution rate was switched back to 0.04 h^{−1} and the fermentation system was maintained at the oscillatory state for five periods, about 4 weeks. This indicates that there are two steady states, one at 0.027 h^{−1}

and another at 0.04 h^{−1}. Therefore, the oscillations are damped, as we can see on

If the initial conditions for the fermentation do not correspond to the steady state, the system oscillates around the steady state and will converge to it over the time. This phenomenon is due to the change in the time required for cell division that causes a decay of cell synchrony as stated before [

When we are far from the steady state, it will take a long time to reach it, and the oscillations seem to be quasi sustained. ^{−1} and for moderate time the oscillations seems to be sustained (quasi limit cycle), but if the time is too long the oscillations are damped (a steady state), this is apparent in

At low dilution rate the oscillations are damped, and with the increase of the dilution rate, the behavior is more and more synchronous and the oscillations tend to be sustained, as shown in

In this paper, the bifurcation analysis of the fermentation of sugar into an ethanol in the presence of yeast Saccharomyces cerivisiae was studied. It was found that the system stability is dependent on dilution rate and on the initial concentration of sugar.

At initial concentration of sugar equal 280 gl^{−1}, the system exhibits two Hopf bifurcations at high dilution rate. The first HB is located at 0.04395 h^{−1} and the second at 0.08028 h^{−1}. The system presents a sustained oscillation between the Hopf bifurcations as shown by the red circle in

At dilution rate 0.027 h^{−1}, there is a steady state where the oscillations are damped, and a not quasi-steady sate, with a sustained oscillations as reported by [

As a consequence, we can attenuate or eliminate the oscillations simply by choosing initial concentrations corresponding to the steady state. The ethanol concentration influences directly the production by influencing the viability of the cells Indeed, the ethanol concentrations over 40 gl^{−1} exert inhibition on yeast metabolism, and turns to toxicity when ethanol concentrations in the fermentation broth exceed 90 gl^{−1} [

The author declares no conflicts of interest regarding the publication of this paper.

Abdelghani, G.-L. (2018) Continuous Ethanol Fermentation at Very High Gravity in the Presence of Saccharomyces cerevisiae: A Bifurcation Analysis. Journal of Sustainable Bioenergy Systems, 8, 116-126. https://doi.org/10.4236/jsbs.2018.84009

D: dilution rate, h^{−1}

K_{S}: intrinsic Monod constant for growth, gl^{−1}

K_{i}: intrinsic substrate inhibition constant for growth, gl^{−1}

K_{sp}: observed Monod constant for growth, gl^{−1}

K_{ip}: observed substrate inhibition constant for growth, gl^{−1}

P: Ethanol concentration, g l^{−1}

P_{max}: maximum ethanol concentration for growth and ethanol formation, gl^{−1}

S: residual glucose concentration, gl^{−1}

S_{0}: Input glucose concentration, gl^{−1}

X: biomass concentration, gl^{−1}

µ: intrinsic specific growth rate, h^{−1}

µ_{0}: specific growth rate at lower dilution rates for lower gravity medium, h^{−1}

µmax: intrinsic maximum specific growth rate, h^{−1}

ν: intrinsic specific ethanol production rate, h^{−1}

ν_{0}: specific ethanol production rate at lower dilution rates for lower gravity medium, h^{−1}

α: ethanol inhibition constant for growth

β: ethanol inhibition for ethanol formation