Effect of the Feed Substrate Concentration on the Dynamic

Article pubs.acs.org/EF

Effect of the Feed Substrate Concentration on the Dynamic Performance of the Bioethanol Fermentation Process Using Zymomonas mobiliz Ibrahim Hassan Mustafa,† Hedia Fgaier,‡ Ali Elkamel,*,† Ali Lohi,§ Gamal Ibrahim,∥ and Said Salah Eldin Hamed Elnashaie⊥ †

Department of Chemical Engineering, University of Waterloo, Waterloo, Ontario N2L 3G1, Canada Department of Mathematics and Statistics, University of Guelph, Guelph, Ontario N1G 2W1, Canada § Department of Chemical Engineering, Ryerson University, Toronto, Ontario M5B 2K3, Canada ∥ Department of Basic Engineering Sciences, Menoufia University, Shebin El-Kom, Egypt ⊥ Department of Chemical and Environmental Engineering, University of Putra Malaysia (UPM), Serdang 43400, Malaysia ‡

ABSTRACT: In this work, a structured verified nonlinear mathematical model for a single continuous fermenter of bioethanol production using Zymomonas mobiliz is developed to investigate the effect of the feed substrate concentration, which is taken as the main bifurcation parameter for performing a bifurcation analysis. The chaotic, oscillatory, and steady-state behaviors of the fermentation system are investigated. It is found that the system is dominated by oscillatory behavior in the medium gravity (MG) region (80.76 < Cso < 213.6), while it is dominated by stable steady state outside this range; however, at high Cso and low dilution rate, the chaotic behavior dominates the system. It is found that the fermentation system accomplishes the maximum averages of substrate conversion and bioethanol yield in the range of 80.76 < Cso < 97 g/L. These results are in agreement with the experimental results in the literature.

1. INTRODUCTION World consumption of oil increases dramatically nowadays, and it is expected to jump to be 100 million barrels per day by 2016. Around two-thirds of the oil consumption is by daily uses of vehicles [U.S. Energy Information Administration (EIA)]. In addition, world oil prices have increased dramatically since the 1970s.1−3 These considerations support that the search and use of sustainable and renewable sources of biofuels are becoming more and more important nowadays. Bioethanol is one of the most prominent renewable and alternative liquid biofuels because it is produced from renewable multiple resources of lignocellulose. The most important features of bioethanol production are that it can save traditional fossil sources, such as oil and natural gas, for the next generation and it is environmentally friendly because of the low emissions of gases; it can also be produced from renewable raw materials (RRMs), contributing to sustainable development (SD). There are multiple microorganisms used for bioethanol production.4−6 The bacterium Zymomonas mobiliz and the yeast Saccharomyces cerevisiae are the most widely used microorganisms for bioethanol production on a large scale.7 It is found that there are many problems when Z. mobiliz ferments sucrose because some byproducts are formed from sucrose and fructose/glucose mixtures, such as fructose polymer levan.14,15 In addition, sorbitol and fructo-oligosaccharide resulted because of the accumulation of glucose and fructose during the hydrolysis of sucrose in the presence of Z. mobiliz. These byproducts lead to lower efficiency of bioethanol production and bioethanol yield from sucrose and fructose/glucose mixtures compared to that from glucose.8,9 The theoretical yield of bioethanol is 0.51 g/g of glucose consumed, and 0.49 g of CO2 is produced. Ingledew32 showed © 2014 American Chemical Society

that only 90−93% of the theoretical yield is produced because a portion of glucose is consumed for the growth of cells and production of undesired byproducts. In addition, toxic bacteria, such as lactic acid bacteria, can contaminate, grow, compete for nutrients, and form byproducts, which inhibit the growth of microorganisms and bioethanol fermentation, thereby reducing the yield of bioethanol.10 Z. mobiliz depends upon sucrose and/or glucose and/or fructose as carbon sources.11,12 A high glucose concentration can be used for Z. mobiliz to produce a greater bioethanol yield; however, cells might be exposed to inhibition because of a high substrate concentration.13 Thatipamala et al.14 showed that the bioethanol yield decreases because of substrate inhibition, where many significant enzymes, which are important for the proper performance of cells, are affected negatively by substrate inhibition, leading to reduction of cell growth and fermentation efficiency.12,14,15 Z. mobiliz kinetics was investigated through a mathematical model obtained by Huang and Chen16 at a high glucose concentration, investigating the effect of substrate and product inhibitions on the performance of the ethanol fermentation system. Z. mobiliz is characterized by many advantages over the yeasts: it has a higher bioethanol yield, greater specific substrate conversion, greater bioethanol tolerance, and higher volumetric substrate uptake rate than the yeasts.17 In addition, Z. mobiliz does not need a controlled supply of oxygen through the fermentation process.18 Only a few substrates, such as fructose, Received: October 9, 2013 Revised: July 16, 2014 Published: July 28, 2014 5543

dx.doi.org/10.1021/ef501225c | Energy Fuels 2014, 28, 5543−5556

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glucose, and sucrose, can be fermented by Z. mobiliz. However, most of the substrates are converted to bioethanol and CO2, and less is converted to biomass using Z. mobiliz.19 Theoretically, bioethanol yields around 1.9 mol of bioethanol/mol of glucose or 0.49 g of bioethanol/g of glucose consumed are obtained.20 There are many challenges for the production of bioethanol from agricultural residues, such as pretreatment and enzymatic hydrolysis of lignocelluslic biomass to simpler sugars, generation of fermentation inhibitors during pretreatment, and fermentation of mixed sugars, such as hexoses and pentoses, to bioethanol. One of the main disadvantages accompanied by bioethanol fermentation by Z. mobiliz is that, at specific operating conditions, oscillatory behavior of the concentration of substrates, bioethanol, and cells takes place, leading to instability of the fermentation system.9 The oscillatory behavior can be defined as the occurrence of some periods where there are higher concentrations of residual substrates and lower product rates, leading to high substrate losses.21 Continuous fermentation systems are a good environment for the oscillatory nonlinear phenomena. Elnashaie et al.21 showed that non-monotonic kinetics and the reaction rates based on temperature variance may lead to the oscillatory and chaotic behaviors of the fermentation systems. In addition, the natural and forced disturbances of the substrate concentration may give a good opportunity for the oscillatory phenomena. Many researchers indicated that the bioethanol fermentation systems are characterized by periodic behaviors in bioethanol fermentation systems by either Z. mobiliz bacterium or S. cerevisiae yeast, where each bioethanol, substrate, microorganism, and other material oscillates periodically at specific operating conditions.22−24 Bai et al.925 showed that each substrate, product, and cell oscillated at a very high glucose concentration using S. cerevisiae and indicated that the oscillations lasted long periods. In addition, Lee et al.26 investigated the periodic behaviors of the continuous fermentation systems using Z. mobiliz and found that each residual substrate, bioethanol, and cell concentration oscillated significantly at a medium concentration of substrates, but these oscillations disappeared when they lowered the feed substrate concentrations. Furthermore, Ghommidh et al.27 observed the periodic phenomena in the fermentation systems with Z. mobiliz. Jobses et al.28,29 indicated that the main reason for the incidence of periodic phenomena appearing in Z. mobiliz fermentation systems is the late response of the microorganisms on bioethanol inhibition. Bai et al.9 showed that the growth of the microorganisms is affected immediately by a high bioethanol concentration as an inhibitor. Li et al.30 proposed a dynamic model to predict the periodic phenomena involved in Z. mobiliz. They found that the rate of the bioethanol concentration changed significantly, affecting the fermentation efficiency and leading to the onset of the oscillatory behavior. Daugulis et al.15 and McLellan et al.31 investigated models to predict the dynamic behavior of bioethanol fermentation systems using Z. mobiliz and found that the history of the bioethanol concentration may play a role for the oscillatory behavior. The mechanisms explaining the oscillatory phenomena are not clearly understood. These phenomena cause a lot of problems for the economics of the bioethanol fuel industry because they lead to the loss of bioethanol and increase of the residual sugar concentration in the fermentation systems, thereby increasing raw material consumption and making fermentation system control difficult. In our previous work, we presented a structured mathematical model and used it to analyze the periodic and chaotic phenomena,

where substrate, cells, and bioethanol oscillate significantly through certain regions of the dilution rates.44 In this structured model, microorganisms were divided into three categories: viable, non-viable, and dead cells. The viable cells have the capability to proliferate and produce bioethanol immediately. Proliferation of cells allows for new generations of cells to arise and helps to maintain bioethanol production. Non-viable cells are unable to grow or multiply but are able to ferment glucose to produce bioethanol. The dead cells have neither the capability to proliferate nor the capability to produce ethanol. Indgledew32 and Bai et al.9 indicated that the capability of nonviable cells for ethanol production is much lower than that of viable cells. This is because ATP accumulated significantly in the non-viable cells, affecting the glycolytic pathways to produce bioethanol. In this work, the effect of the substrate concentration on the oscillatory behaviors of the bioethanol fermentation systems using bifurcation analysis is investigated. Our previous model44 is used to predict the oscillatory behavior and determine the ranges of the substrate concentration, giving different solutions, such as periodic attractors and steady-state attractors. It is expected that changing the substrate concentration might affect the metabolic pathways of the microorganisms, causing a significant influence of the performance, substrate conversion, and bioethanol productivity. This study could give a reasonable explanation for oscillatory phenomena and help improve the bioethanol yield, productivity, and substrate conversion of the whole process. In this paper, the effect of the feed substrate as an external effect and controlling parameter on the oscillatory behavior of the continuous fermentation system is investigated. The inlet substrate concentration is taken as the main bifurcation parameter to identify the ranges at which different solutions, such as steadystate, periodic, and chaotic attractors, are obtained. The impacts of the feed substrate as a possible mechanism explaining the oscillatory behavior of continuous bioethanol fermentation by Z. mobiliz on the productivity and yield of bioethanol and the substrate conversion efficiency and concentrations of viable and non-viable cells are studied with varying feed substrate concentrations.

2. STRUCTURED MATHEMATICAL MODEL In this section, we review the model that we have developed earlier and that will be employed to study the dynamic phenomena as a result of the initial feed substrate concentration. The model assumes that the living cells can be converted to non-viable cells, which are the second phase, where non-viable cells are unable to proliferate but able to produce bioethanol,33 as indicated in the kinetic mechanism proposed by Mustafa et al.44 (Figure 1). In this model, a certain percentage of the

Figure 1. Proposed kinetic mechanism of ethanol production from different categories of microorganisms.44 living cells is assumed to die to give the last type of cells, which is the dead cells, which are neither able to proliferate nor produce bioethanol. In the specific growth rates, the inhibitory effect of both glucose and bioethanol were taken in consideration, as shown in Table 1. The dynamics of levels of produced bioethanol, residual substrate, and viable and non-viable cells are investigated with varying Cso. The regions of Cso causing oscillations, including high periodicity in the form of period 5544


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Table 1. Ordinary Differential Equations of the Bioreactor item

differential equation

parameter

1

viable cells (Cxv)

dCxv = d(Cxvo − Cxv) + Cxv(μv − μnv ) dt

2

non-viable cells (Cxnv)

dCxnv = d(Cxnvo − Cxnv) + Cxv(μnv ) − Cxnvμd dt

3

dead cells (Cxd)

dCxd = d(Cxdo − Cxd) + Cxnv(μd ) dt

4

bioethanol (Cp)

dCp dt

5

substrate (Cs)

= d(Cpo − Cp) + Cxv

μv Yxp

specific growth rate of the viable cells (μv)

+ mpCxnv

definition

μv =

α ⎛ Cp ⎞ 1 − ⎟ ⎜ C2 Pc ⎠ K s + Cs + Ks ⎝

μmax Cs

ss

7

8

specific growth rate of the non-viable cells (μnv)

specific growth rate of the dead cells (μd)

μnv =

μmaxd Cs K sp + Cs +

Cs 2 K ssp

β ⎛ Cp ⎞ ⎟ − μv ⎜1 − Pcd ⎠ ⎝

μd = ϕμnv

doubling (PD), and steady states (point attractors) are determined through bifurcation analysis. A comparison between the numerical results and the experimental results is made to test the model and validate it. The specific growth rates of viable and non-viable cells indicated in Table 1 take into consideration the influence of glucose and bioethanol inhibitions, where their high concentrations lead to breaking the cell membranes, leading to the death of cells.34−36 Considering these categories of cells, mass balance differential equations accounting for the continuous production of bioethanol in the continuous stirred tank reactor (CSTR) in the form of nonlinear ordinary differential equations are indicated in Table 1. The values of the kinetic parameters are given in Table 2.

Pp = C pd

reference 10 10 4 and 9 4 and 9 4, 9, and 40 44 44 4, 9, and 40 4, 9, and 40 40 44 42 42 40 40 43 12 43 Garhyan and Elnashaie (2004)41 Garhyan and Elnashaie (2005)42 Jarzebski, A.B., (1992)12 Jarzebski, A.B., (1992)12 Generated from the simulation Generated from the simulation Generated from the simulation Generated from the simulation Generated from the simulation

(2)

The bioethanol yield (Yp) is calculated as the ratio between the bioethanol productivity and the rate of substrate input. It is indicated as below

The static and dynamic bifurcation analyses are obtained using the software package of XPPAUT 5 and AUTO 2000, where the periodic branches and steady states are indicated through the software. Matlab is used for obtaining time traces and phase planes at certain parameter values determined previously from AUTO 2000.37 Chaotic behavior and high periodicity, such as PD, are investigated using additional techniques, such as a Poincaré map, where discrete points of intersections between trajectories and a hyperplan surface plot the Poincaré map. At a certain value existing inside a limit cycle for one of the variables, the hyperplan surface is chosen to help formulate the Poincaré map via the intersection points between state variables and the hyperplan.38 FORTRAN is used to build the Poincaré map using the ISML library to account for both chaotic and periodic behaviors accurately. The oscillatory and chaotic behaviors of the Z. mobiliz fermentation system occur because of the complex interaction between growth of viable and non-viable cells with substrate and product concentrations.39 Bifurcation analysis is used for studying the dynamics of fermentation systems more efficiently than traditional simulation tools. Dynamic, static, and chaotic behaviors of the fermentation system could be investigated using bifurcation analysis. Equations 1, 2, and 3 illustrate the mathematical expressions for the substrate conversion (Xs), bioethanol production rate per unit volume of the reactor (Pp), and bioethanol yield (Yp) as shown below

Cso − Cs Cso

unit

The bioethanol production rate (Pp) from the fermenter is calculated as below

3. NUMERICAL TECHNIQUES AND COMPUTATIONS

Xs =

value

0.235 h−1 0.22 h−1 1.74 g/L 2.5 g/L 250 g/L 20 g/L 0.2 150 g/L 350 g/L 200 g/L 9.5 g/L 1.9 g g−1 h−1 3.5 g g−1 h−1 250 g/L 350 g/L 0.022 g/g 0.335 g/g 0.05 h−1 Feed Concentrations Cpo 0 g/l Cxvo 0 g/l Cxnvo 0 g/l Cxdo 0 g/l Steady State Concentrations CS 33.47 g/l Cp 54.15 g/l Cxv 6.6 g/l 1.51 g/l Cxnv Cxd 0.0723 g/l μmax μmaxd α β Ks Ki Φ Kss Ksp Kssp Kip mp ms Pc Pcd Yxs Yxp d

⎞ ⎛ μ dCs = d(Cso − Cs) − ⎜Cxv v + msCxnv ⎟ dt ⎠ ⎝ Yxs

item 6

Table 2. Kinetic Parameter Values

Yp =

Cp Cso

(3)

4. RESULTS AND DISCUSSION In the first section, bifurcation analysis is performed on the basis of the inlet glucose concentration as the bifurcation parameter at a certain value of the dilution rate. In the second section, dynamic behavior of the state variables is performed at different values of the feed glucose concentration representing high gravity (HG), medium gravity (MG), and low gravity (LG). In the third section, chaotic behavior using the Poincaré map is investigated, as discussed below. 4.1. Bifurcation Analysis Using the Feed Substrate Concentration (Cso) as the Bifurcation Parameter at d = 0.05 h−1. In this section, the effect of the feed substrate concentration (Cso) on the fermentation system behavior is investigated at d = 0.05 h−1, and the rest of parameters are shown in Table 2. Cso is taken as the main bifurcation parameter because it is considered an external control parameter, which is expected to influence all state variables. Figures 2−4 show the bifurcation diagrams of Cso and are explained in detail as follows. 4.1.1. First Region: Feed Substrate Concentration at Cso < 80.76 g/L. This region is characterized by a unique stable steady-state

(1) 5545


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Figure 2. Bifurcation diagrams with the feed substrate concentration (Cso) as the bifurcation parameter at the values of kinetic parameters shown in Table 2: (●) stable periodic branch, (○) unstable periodic branch, () stable steady-state branch, and (- - -) unstable steady-state branch. (a) Bifurcation diagram for viable cell concentration (Cxv), (b) bifurcation diagram for residual substrate concentration (Cs), (c) bifurcation diagram for bioethanol concentration (Cp), and (d) enlargement of the box in panel c.

Figure 3. Bifurcation diagrams with the feed substrate concentration (Cso) as the bifurcation parameter at the values of kinetic parameters shown in Table 2: (●) stable periodic branch, (○) unstable periodic branch, () stable steady-state branch, and (- - -) unstable steady-state branch. (a) Bifurcation diagram for substrate conversion (Xs), (b) bifurcation diagram for bioethanol yield (Yp), (c) bifurcation diagram for bioethanol production rate (Pp), and (d) enlargement of the box in panel a.

concentration of Cso until Cxv = 1.73 g/L at the point HB2, which occurs at Cso = 80.76 g/L. Figure 2b illustrates that the concentration of residual substrates (Cs) increases with the increase of Cso until Cs = 15.31 g/L at HB2, while Cp increases significantly with the increase of Cso until Cp = 36.14 g/L at HB2.

attractor (point attractor). The concentration of the feed substrate in this region is referred as a LG. This region is characterized by continuous growth of Z. mobiliz and production of bioethanol, where Figure 2a shows that the concentration of viable cells (Cxv) increases from a very low level at a low 5546


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Figure 4. Bifurcation diagrams with the feed substrate concentration (Cso) as the bifurcation parameter at the values of kinetic parameters shown in Table 2: (●) stable periodic branch, (○) unstable periodic branch, () stable steady-state branch, and (- - -) unstable steady-state branch. (a) Bifurcation diagram for non-viable cell concentration (Cxnv) and (b) bifurcation diagram for dead cell concentration (Cxd).

Figure 5. Dynamic characteristics at d = 0.05, Cso = 200 g/L, and the rest of the system parameters, as shown Table 2. (a) Time traces of viable cell concentration (Cxv), (b) time traces of bioethanol concentration (Cp), (c) time traces of residual substrate concentration (Cs), and (d) phase plane for Cxv versus Cp.

1.8 at HB2, as shown in Figure 3c. Figure 3d illustrates an enlargement for the zoom that appeared in Figure 3a, where it is clear that the stable steady-state attractor started from HB2 at Cso = 80.76 g/L. Figure 4a shows that Cxnv increases from 0.45 g/L at Cso = 50 g/L to 0.75 g/L at HB2, while Cxd increases from 0.045 g/L at Cso = 50 g/L to 0.072 g/L in the same region of Cso, as shown in Figure 4b.

Figure 3a illustrates the increase of the substrate conversion (Xs) from 0.725 at Cso = 50 g/L until Xs = 0.79 at HB2. In this region, the cells of Z. mobiliz use most of the available feed substrate, leading to 79% substrate conversion, and it is clarified from the increase of the bioethanol concentration (Cp) in the fermentation system. Yp increases from 0.41 at Cso = 50 g/L to 0.44 at HB2, as shown in Figure 3b, and Pp increases from 1 at Cso = 50 g/L to 5547


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Figure 6. Dynamic characteristics at d = 0.05, Cso = 2000 g/L, and the rest of the system parameters, as shown Table 2. (a) Time traces of substrate conversion (Xs), (b) time traces of bioethanol production rate (Pp), (c) time traces of bioethanol yield (Yp), and (d) phase plane for Cxv versus Cs.

Figure 7. Dynamic characteristics at d = 0.05, Cso = 200 g/L, and the rest of the system parameters, as shown Table 2. (a) Time traces of non-viable cell concentration (Cxnv) and (b) time traces of dead cell concentration (Cxd).

4.1.2. Second Region: Feed Substrate Concentration at 80.76 < Cso < 213.6 g/L. From the bifurcation figures, the feed glucose concentration range of 80.76−213.6 g/L is dominated by the oscillatory behavior. The concentration of the feed substrate in this region is referred as MG and HG, where HG exists in the range of 135 < Cso < 200 g/L and MG exists in the range of 80 < Cso < 135 g/L. In this region, the oscillations dominate the fermentation system, where only a periodic attractor exists with the increase of amplitudes as Cso increases, as shown in Figures 2−4.

The oscillation averages are calculated to give more indication about the system behavior and to compare them to unsteadystate values. The periodic attractors appeared in this region exist between two Hopf bifurcation points (HB1 and HB2). HB1 occurs at Cso = 213.6 g/L and Cxv = 8.459 g/L, as shown in Figure 2a. When Cso increases from HB2 to close to HB1, it gives a wide amplitude for each of state variables. The amplitudes become narrower as Cso decreases, until they disappear at HB2. 5548


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of Cs increases from the lowest level at HB2, which is at Cs = 36.14 g/L and Cso = 80.76 g/L, until the amplitude becomes the largest at Cso = 197 g/L, where Cs oscillates in the range of 15− 75 g/L, and then it decreases again until it reaches HB1 at Cso = 213.6 g/L and Cs = 47.61 g/L. Furthermore, the average of Cs in this region increases slightly to be larger than the unstable steady state in the region of Cso = 80.76−150 g/L, and then the average of Cs increases significantly in the range of Cso = 150−213.6 g/L,

It is observed that the range of Cxv increases through the periodic range of Cso from 1.995 at HB2 to the biggest range of 10.5−3.75 at Cso = 200 g/L, as shown in Figure 2a. This continuous increase of Cxv reflects the rapid and continuous growth of viable cells based on the corresponding concentration of the inlet substrate in this region. Figure 2b shows that the residual substrate concentration (Cs) increases with the increase of Cso. It is observed that, through this region of Cso, the amplitude 5549


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Figure 10. Dynamic characteristics at d = 0.05, Cso = 120 g/L, and the rest of the system parameters, as shown Table 2. (a) Time traces of non-viable cell concentration (Cxnv) and (b) time traces of dead cell concentration (Cxd),.

Figure 11. Dynamic characteristics at d = 0.05, Cso = 165 g/L, and the rest of the system parameters as shown Table 2. (a) Time traces of viable cell concentration (Cxv), (b) time traces of bioethanol concentration (Cp), (c) time traces of residual substrate concentration (Cs), and (d) phase plane for Cxv versus Cp.

concentrations of glucose as well as ethanol inhibitory action. Figure 2c shows that, like the amplitude of Cs, the amplitude of Cp increases from HB2, which is at Cp = 36.14 g/L, then increases to reach the maximum range at Cso = 197.5 g/L, where Cp exists in the range of 62−100 g/L, and then decreases again until HB1, where Cp = 92.53 g/L. From regions 1 and 2, it is illustrated that the feed substrate is consumed efficiently for cell growth and bioethanol production, as shown in panels a and c of Figure 2, respectively. Figure 2d represents an enlargement

where it can be explained by the rate of substrate consumption for production of bioethanol being less than the rate of substrate adding to the fermentation system, which is why Cs increases continuously with Cso. In addition, adding more substrate to the fermentation system in this region gives the opportunity to cause substrate inhibition and reduction in the fermentation performance and lowering in microorganism efficiency. Mouline et al.45 showed that, in fermentation systems, sugars have a strong inhibitory effect on cell growth rates in the high 5550


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Figure 13. Dynamic characteristics at d = 0.05, Cso = 65 g/L, and the rest of the system parameters, as shown Table 2. (a) Time traces of non-viable cell concentration (Cxnv) and (b) time traces of dead cell concentration (Cxd).

with an average of 0.765. Figure 3b shows that the amplitude of Yp occurs in the maximum range of 0.35−0.53 at Cso = 200 g/L. As Cso increases from HB2 at Cso = 80.76 g/L and Yp = 0.441 to the point HB1 at Cso = 213.6 g/L and Yp = 0.42, the average of Yp follows the same behavior as Xs, where it reaches the maximum value of Yp = 0.452 at Cso = 130 g/L and then decreases to reach the lowest value of Yp = 0.4367 at Cso = 200 g/L. Figure 3c shows that the average of Pp increases continuously as Cso increases through the periodic region from HB2 at Pp = 1.75 to HB1 at Pp = 4.5.

of the zoom shown in Figure 2c. It is clear that the average increases until it hits Cp at HB1. Figure 3a shows that the average of substrate conversion increases from a lower value at HB2 at Xs = 0.79 and Cso = 80.76 g/L and then increases to Xs = 0.82 at Cso = 130 g/L. As Cso increases, the average decreases again to reach Xs = 0.75 at Cso = 200 g/L. Finally, the average increases as Cso increases until it reaches HB1 at Xs = 0.755 and Cso = 213.6. The greatest periodic amplitude of Xs occurs in the range of 0.62−0.95 attained at Cso = 197 g/L, 5551


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Yp, and Pp are higher than those of the unsteady-state values, reflecting the advantage of the oscillatory behavior in the lower range of the feed substrate concentration over the steady-state behavior. These results are compatible with the experimental and theoretical results obtained by Garhyan and Elnashaie,41 who showed the improving of the average of the bioethanol concentration, glucose conversion, and bioethanol yield at the feed substrate concentration of 130 g/L.42 However, as the inlet substrate concentration increases in the range of 130 < Cso < 213.6 g/L, the averages of the previous state variables are lower than that of the unsteady-state values, showing the disadvantage of the oscillatory behavior of the fermentation system at HG. The latter observations are compatible with the experimental results obtained by Bai et al.,4 who confirmed the loss of substrate and reduction of the bioethanol yield and ethanol production rate at HG fermentation systems.4 4.1.3. Third Region: Feed Substrate Concentration at Cso > 213.6 g/L. The concentration of feed substrate in this region is referred as very high gravity (VHG). In this range of feed substrate, there is only a unique stable steady state, where there is no periodic attractor. Figure 2a shows that Cxv increases from 11.83 at HB1 (Cso = 213 g/L) to 13.5 g/L at Cso = 245 g/L and then decreases to 6.5 g/L as Cso increases to 300 g/L. The latter reduction of the viable cell concentration can be explained by Cso increasing to the limit that inhibits the cell growth; therefore, the viable cells have not been produced as Cso increases. The substrate inhibition phenomenon also appears in Figure 2c, where Cp increases from Cp = 92.53 g/L at HB1 to 95 g/L at Cso = 245 g/L and then decreases fast to 75 g/L at Cso = 300 g/L. The fermentation system reaches the substrate inhibition limit; therefore, the cells have become unable to produce bioethanol. Because of substrate inhibition, the unconsumed substrate concentration (Cs) increases continuously through this region, as shown in Figure 2c. Figure 2d is an enlargement of the zoom that appeared in Figure 2c, where it shows that HB1 is a triple point representing a meeting of three different solutions, which are stable steady-state (where Cs > HB1), unstable steady-state, and periodic solutions (where Cs < HB1). Panels a and b of Figure 3 show that any increase in Cso that is higher than the first Hopf bifurcation point (where Cs > HB1) leads to the reduction of both substrate conversion (Xs) and bioethanol yield (Yp), where Xs decreases from 0.755 at HB1 to 0.55 at Cso = 275 g/L, while Yp is lowered from 0.441 at HB1 to 0.25 at Cso = 300 g/L, as shown in Figure 3b. Figure 3c shows that the bioethanol production rate (Pp) is affected by substrate inhibition, where Pp decreases as Cso increases further than 245 to be 3.6 at Cso = 300 g/L. Figure 4a shows that non-viable cells (Cxnv) take a steady-state behavior through the region of Cs > HB1, where it decreases as Cso increases from 1.71 g/L at Cso = 213.6 g/L to 1.25 g/L at Cso = 300 g/L. The dead cells (Cxd) through this region, as shown in Figure 4b, take the steady-state behavior also, where it decreases from 0.0171 g/L at Cso = 213.6 g/L to 0.035 g/L at Cso = 300 g/L. At Cso = 140 g/L, the periodic average of Yp = 0.44 and the average of bioethanol conversion Xs = 0.83. These results are compatible with that obtained by Mahecha-Botero et al., who showed that, at Cso =140 g/L, the average of Yp = 0.431 and the average of bioethanol conversion is 0.89.39 In addition, the results are compatible with the experimental results obtained at HG by Garhyan and Elnashaie,40 who showed that, at very high inlet substrate concentrations, all periodic attractors cease and there is only stable steady-state solution (point attractor).40 4.2. Dynamic Modeling Analysis for the Impact of Feed Substrate Concentrations. Figures 5−7 show the time course

Figure 14. (a) Poincaré bifurcation diagram (Poincaré plane is located at Cso = 245 g/L, Cp = 127 g/L, and the rest of the parameters, as shown Table 2, at the corresponding initial conditions). (b) Poincaré bifurcation diagram (Poincaré plane is located at Cso = 245 g/L, Cp = 127 g/L, and the rest of the parameters, as shown Table 2, at the corresponding initial conditions).

The maximum amplitude of Pp is in the range of 3.25−5.257 at Cso = 200 g/L. The increase of Pp in this region of Cso reflects a positive effect on both viable and non-viable cells for bioethanol production. Figure 3d is an enlargement for the box appearing in Figure 3a close to HB2, reflecting the rapid increase of Xs at a lower concentration of feed substrate. Because of the consideration of the structure of cells in the model and given the practical values to all growth rates, the concentrations of each viable, non-viable, and dead cell can be predicted and compared to experimental values. Figure 4a shows that Cxnv oscillates between HB2 (at Cxnv = 0.7 g/L) and HB1 (at Cxnv = 1.6 g/L), where the maximum amplitude of Cxnv is in the range of 055−2.45 g/L at Cso = 200 g/L, while Cxd (Figure 4b) arrives at the maximum amplitude of 0.045−0.125 g/L at the latter value of Cso. HB2 is attained at Cxd = 0.074 83 g/L, while HB1 is at Cxd = 0.017 179 g/L. Furthermore, the results are compatible with both of the theoretical and experimental results obtained by Garhyan and Elnashaie.41,42 At Cso = 140 g/L, our model gives the average bioethanol concentration of 62 g/L, but the model by Garhyan and Elnashaie gave 65.3 g/L and their experiment gave 63.89 g/L. In addition, it is observed from bifurcation diagrams shown in Figures 2 and 3 that, in the lower range of the oscillatory behavior in LG and MG (80.76 < Cso < 130 g/L), the averages of all Cp, Xs, 5552


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0.048−0.121 g/L, with an average of 0.0845 g/L. At Cso = 200 g/L, our results are different from the experimental results obtained by Garhyan and Elnashaie,42 where their Cp oscillated slightly in the range of 83−94 g/L, leading to a stable steady-state attractor around 93.4 g/L and not a periodic attractor, while our results show that Cp oscillates as a periodic attractor in the range of 70− 103 g/L. This discrepancy might be due to our considering of the structure of the cells to viable, non-viable, and dead cells. Figures 8−10 represent the system dynamics at MG, where Cso = 120 g/L. Cxv oscillates in the range of 2.1−4.75 g/L, with an average of 3.425 g/L, which is lower than the average mentioned previously at HG, as shown in Figure 8a. Figure 8b indicates that Cp changes in the range 47−62 g/L, with an average of 54.5 g/L, which is much less than that at HG. Figure 8c illustrates that Cs oscillates in the range of 7−34, with an average of 20.5 g/L, which is less than that at HG. The closed orbits that appeared in Figure 8d show that Cxv and Cp oscillate in ranges less wide than those at HG, confirming the oscillatory behavior of the system. Figure 9a shows that Xs oscillates in the range of 0.72−0.945, with an average of 0.8325, which is higher than that at HG. It is clear that, at MG, the microorganisms can consume most of the substrate, leading to the increase of conversion. Pp in Figure 9b oscillates in the range of 2.4−3.15, with an average of 2.775 g/L, which is much lower than that at HG. Yp oscillates in the range of 0.4−0.52, with an average of 0.46, which is higher than that at HG and compatible with the bifurcation results indicated in Figure 3b. The limit cycles shown in Figure 9d confirm that the oscillatory behavior appeared in panels a and c of Figure 8 at MG, where Cso = 120 g/L.

at Cso = 200 g/L, which represents a HG system. It is clear that the system is dominated by periodic attractors, where Cxv oscillates in the range of 4.5−11 g/L, with an average of 7.75 g/L, where the system at HG requires a high concentration of viable cells, as shown in Figure 5a. Figure 5b illustrates time traces of Cp, which oscillates through a wide range from 70 to 103 g/L, with an average of 86.5 g/L. This latter concentration of bioethanol at HG is expected to be a high value because the feed substrate concentration is still less than the inhibition limit, which will be shown below. The average Cp of 86.5 g/L is close to the simulated and experimental values of the bioethanol concentration obtained by Garhyan and Elnashaie41,42 at Cso = 200 g/L, where they obtained the experimental value of Cp of 93.4 g/L and their simulated value of Cp of 98 g/L. The concentration of residual substrate is expected to be in a high range, where Cs is in the range of 15−75 g/L, with an average of 45 g/L, as shown in Figure 5c. Figure 5d shows that the phase plane between Cp and Cxv is a closed limit cycle, confirming the oscillatory behavior at HG conditions. Figure 6a shows that time trace of Xs, which fluctuates in a wide range of 0.63−0.92, with an average of 0.775. Figure 6b shows that Pp oscillates in the range of 3.5−5.1 g L−1 h−1 of bioethanol, with an average of 4.3 g L−1 h−1, while Yp oscillates in the range of 0.35−0.51, with an average of 0.43. The closed orbit shown in Figure 6d between Cs and Cxv shows complete limit cycles and confirms the oscillatory behavior at HG. Panels a and b of Figure 7 show time traces of Cxnv and Cxd, respectively, where Cxnv oscillates in the range of 1.2−2.4 g/L, with an average of 1.8 g/L, and Cxd oscillates in the range of 5553


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Figure 16. Dynamic characteristics at d = 0.012 35, Cso = 245 g/L, and the rest of the system parameters, as shown Table 2. (a) Time traces of viable cell concentration (Cxv), (b) time traces of bioethanol concentration (Cp), (c) time traces of residual substrate concentration (Cs), and (d) phase plane for Cxv versus Cp.

dilution rate (d) is taken as the bifurcation parameter at Cso = 245 g/L at the corresponding initial conditions. The cutting Poincaré surface, which is assumed to cross the trajectory in the state space is obtained at Cp = 127 g/L. The Poincaré map describes only the intersections of the plane with the trajectories. It appears that the behavior develops via PD, where period one attractor appears in the range of d > 0.026 h−1, period two appears in the range of 0.018 < d < 0.026 h−1, and then period four appears in the range of 0.014 < d < 0.018 h−1; however, period two returns again as d decreases in the range of 0.005 < d < 0.014 h−1, and then period one appears in the range of d < 0.005 h−1. It is observed that the chaotic behavior is not obtained at the relevant initial conditions, as shown from time traces and phase planes obtained in Figure 15. It is observed that, from Figure 15 at d = 0.02 h−1, Cso = 127 g/L, and the other parameters shown in Table 2, every peak of all state variables repeats twice in the form of PD because this value of d exists in the range of PDs. Figure 14b shows the Poincaré map at the same kinetic parameter values of Figure 14a and the same cutting surface, however, at different initial conditions. Period one, period two, and period four appear in the same range of d that appeared in Figure 14a. Furthermore, fully chaotic behavior appears in the range of 0.008 < d < 0.016 h−1, where there is a wide range of Cxv at each value of d. Figure 16 shows time traces and phase planes of state variables at d = 0.012 35 h−1 and Cso = 127 g/L, and the initial conditions shown in Figure 14b. Panels a−d of Figure 16 illustrate the chaotic behavior of the fermentation system, where

Panels a and b of Figure 10 show time traces of Cxnv and Cxd at Cso = 120 g/L or MG. It is clear that Cxnv oscillates in the range of 0.82−1.5, with an average of 1.16 g/L, which is lower than that at HG, while Cxd oscillates in the range of 0.065−0.105, with an average of 0.085 g/L. Figures 11−13 show the dynamics of the fermentation system at a low feed substrate concentration (Cso = 65 g/L), which represents a very low gravity (VLG). It is clear that the solution at the conditions of VLG, where Cso = 65 g/L, is a point attractor. Figure 11a shows that Cxv approaches 1.263 g/L, while Cp settles down around 28.02 g/L, as shown in Figure 11b. Time traces of the residual substrate concentration in Figure 11c show that Cs settles down finally at 14.18 g/L. The phase plane in Figure 11d shows that the system behavior of Cp and Cxv is settled down to a point attractor. Figure 12a shows time traces of substrate conversion Xs, which settles down to a stable steady state, which is 0.7817. Pp as shown in Figure 12b settles down to 1.402, which is less than at HG and MG. Figure 12c shows that Yp settles down to 0.431, and the phase plane between Cs and Cxv shown in Figure 12d confirms the steady-state (point attractor) behavior, which is compatible with the bifurcation diagrams that appeared in Figure 3. Figure 13a shows the stable steady state of the non-viable cells, where Cxnv approaches 0.601 g/L, which is lower than that at MG. The dead cell concentration (Cxd) shows that 0.0632 g/L is the final concentration. 4.3. Chaotic Behavior for the Impact of Feed Substrate Concentrations. Figure 14a shows the Poincaré map, where the 5554


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chaotic solutions settle into irregular oscillations that persist as time approaches infinity, confirming the chaotic behavior of the Poincaré map shown in Figure 14b.

ACKNOWLEDGMENTS

This work is partially supported by MITACS Elevate, Ontario, Canada.

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5. SUMMARY AND CONCLUSION A structured mathematical model verified experimentally for fermentation of bioethanol using Z. mobiliz is developed to investigate the bifurcation analysis and dynamic behavior, including the chaotic behavior of the system. The physiological structures of cells into viable, non-viable, and dead cells are considered. The specific growth rates of both viable and non-viable cells are chosen from the experimental literature to allow for more practical results. The feed substrate concentration (Cso) is selected to be the bifurcation parameter because it can be manipulated to control the fermentation system. From the bifurcation analysis, the effect of Cso as a bifurcation parameter shows that there are two HB points at Cso = 213 and 80.76 g/L. The system is dominated by stable steady-state behavior when Cso > 213.6 (this region is related to HG) and Cso < 80.76 g/L (this region is related to LG). In addition, the system is dominated by oscillatory behavior in the MG region (80.76 < Cso < 213.6), as shown in Figures 2−4. These results are compatible with the experimental results obtained by Garhyan and Elnashaie40 and Mahecha-Botero et al.,39 where, as shown in Figures 5−9, there are different solutions: steady-state and oscillatory solutions obtained with the different ranges of glucose input concentrations in the fermentation system. Furthermore, the system exhibited PD cascade in the form of PD bifurcation, as shown in the Poincaré map, shown in panels a and b of Figure 14, and the dynamics in Figure 15. It is found that the fermentation system accomplishes the maximum averages of substrate conversion and bioethanol yield, which equal the unsteady-state value in the range of 80.76 < Cso < 97 g/L, and then these averages decrease and become lower than that of the unsteady steady as Cso increases until HB2, as shown in panels a and b of Figure 3. The average of the bioethanol production rate increases gradually with the increase of Cso in the oscillatory region, but it is slightly lower than the unsteady state. In the VHG region, where Cso > 250 g/L, the substrate inhibits cell growth and bioethanol production; therefore, the substrate conversion, bioethanol yield, bioethanol production rate, viable cell concentration, and even the concentration of bioethanol decrease with any further increase of the feed substrate concentrations. This analysis could be a good guide for further research for developing the bioethanol production process and improving the efficiency of the fermentation system using cell recycle, bioethanol removal tool, and multiple reactor fermentation cascade, comparing between them considering mass-transfer limitations, and comparing between different microorganisms for optimizing the efficiency of bioethanol production processes. Our structured model is able to predict the fermentation system behavior through a wide range of the feed substrate concentrations, considering the physiological aspects of Z. mobiliz. This work could present strategies for treating oscillations and chaotic behavior in the fermentation system by controlling the main parameters, such as the dilution rate and feed substrate concentration.

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NOMENCLATURE Cs = concentration of residual substrate (g/L or kg/m3) Cp = concentration of bioethanol (g/L or kg/m3) Cxv = concentration of viable cells (g/L or kg/m3) Cxnv = concentration of non-viable cells (g/L or kg/m3) Cxd = concentration of dead cells (g/L or kg/m3) Cso = feed concentration of residual substrate (g/L or kg/m3) Cpo = feed concentration of bioethanol (g/L or kg/m3) Cxvo = feed concentration of viable cells (g/L or kg/m3) Cxnvo = feed concentration of non-viable cells (g/L or kg/m3) Cxdo = feed concentration of dead cells (g/L or kg/m3) d = dilution rate (h−1) Xs = substrate conversion Yp = bioethanol yield Pp = bioethanol production rate (g/h) ms = maintenance constant based on substrate (g g−1 h−1) mp = maintenance constant based on product (g/g) Yxs = yield constant based on substrate (g/g) Yxp = yield constant based on product (g/g) μmax = maximum specific growth rate of viable cells (h−1) Ks = saturation growth constant (g/L) Kss = saturation growth inhibition constant (g/L) Pc = ethanol inhibition term for cell growth (g/L) Pcd = bioethanol inhibition term for cell growth (g/L) μmaxd = maximum specific growth rate of dead cells (h−1) μv = growth rate of viable cells (h−1) μnv = growth rate of non-viable cells (h−1) μd = growth rate of dead cells (h−1) T = time (h) Ksp = saturation constant for bioethanol (g/L) Kssp = saturation constant for bioethanol (g/L) α = inhibition index for cell growth β = inhibition index for bioethanol production Φ = growth rate ratio of dead cells/viable cells

Abbreviations

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CSTR = continuous stirred tank reactor HB = Hopf bifurcation SB = static bifurcation PD = period doubling HG = high gravity MG = medium gravity LG = low gravity

REFERENCES

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Notes

The authors declare no competing financial interest. 5555


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Effect of the Feed Substrate Concentration on the Dynamic

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