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Structured Mathematical Modeling, Bifurcation, and Simulation for the Bioethanol Fermentation Process Using Zymomonas mobilis Ibrahim Hassan Mustafa,† Ali Elkamel,† Ali Lohi,‡ Gamal Ibrahim,§ and Said Salah Eldin Hamed Elnashaie∥ †

Department of Chemical Engineering, University of Waterloo, Waterloo, Ontario N2L3G1, Canada Department of Chemical Engineering, Ryerson University, Toronto, Ontario M5B2K3, Canada § Basic Engineering Sciences Deptartment, Menoufia University, Egypt ∥ Chemical and Environmental Engineering Department, University of Putra Malaysia (UPM), Serdang 43400, Malaysia ‡

ABSTRACT: A structured nonlinear dynamic model for a single continuous fermentor of ethanol production using Zymomonas mobiliz is built and validated to investigate the effect of dilution rate. A detailed bifurcation analysis is performed at different feed substrate concentrations. It is found that oscillatory behavior dominates the system for both feed substrate concentrations at 140 and 200 g/L; however, the high periodicity in the form of a period-doubling cascade appeared at the higher value of feed substrate concentrations. The model gives reasonable results compared with experiments. (1994)4 indicated that the variations in ethanol concentration could be the main reason for the oscillatory behavior. This is in contradiction to the views thought by Li et al. (1995),5 who showed that there is no influence of the ethanol concentration history on the biological efficacy of Z. mobiliz; however, the high change rate of ethanol concentration could affect Z. mobiliz greatly.2,5 Daugulis and colleagues 3,6 investigated the oscillatory behavior showing that the rate change history of ethanol concentration is the reason for the periodicity of the system. To deal with this debate, physiological models considering the different phases of viability such as viable, nonviable, and dead biomass are needed to explain the oscillatory phenomena in the fermentor. In the oscillatory behavior of the continuous fermentation of Z. mobiliz, all of the ethanol, substrate, and biomass concentrations oscillate through specific operation conditions.7 Li et al. (1995)5 proposed the concept of a dynamic specific growth rate. Daugulis et al. (1997)3 used this concept considering external inhibitory effects to describe the growth rate of biomass which is not greater than a growth rate at real fermentation culture conditions and indicated how the biomass responds to any operating conditions such as the upward increase of the ethanol concentrations.3 They found that the delay of the biomass response to the external effects is the main reason leading to the appearance of periodic phenomena in the fermentation systems.3 In the Z. mobiliz fermentation system it is important to take into consideration the quantity of dead cells in consideration, particularly in structured models as this affects the final concentrations of substrates and products as well. Z. mobiliz cells can be divided into three categories: viable, nonviable, and

1. INTRODUCTION Ethanol is a liquid biofuel that has been used extensively for the past few years due to the harmful environmental effects of traditional fossil fuels such as carbon dioxide emissions. In addition, ethanol is a biodegradable fuel and does not produce unhealthy emissions like gasoline and other classical fuels and can be produced from renewable raw materials (RRMs) such as lignoceluolisic biomass. For example, carbon monoxide, which is more harmful than carbon dioxide, emissions from combustion of ethanol are reduced by nearly 30% as the U.S. EPA showed due to the large oxygen content of ethanol.1 Recently, many countries such as Brazil, Germany, the United States, and France have been producing and using ethanol as a fuel or blending it with gasoline at around 21% concentration. In Brazil, for instance, ethanol is used as a complete fuel in about one-half of all cars and as an additive with about 20% to gasoline.1 One of the main microorganisms used for production of bioethanol more efficiently than yeast is Zymomonas mobiliz. It is characterized by a higher ethanol production rate and yield and an ability to resist the high substrate and ethanol concentrations. However, Li (2009)2 showed that the main disadvantage accompanied by production of ethanol using Z. mobiliz is the periodic behavior where all system variables including the substrate, ethanol, and Z. mobiliz biomass oscillate at specific operating conditions. Through the oscillatory periods the yield of ethanol production, substrate conversion rates, and ethanol concentrations decrease with time leading to a high economic loss appearing in the form of high loss of substrate and reduction of ethanol productivity.3 The mechanism explaining the oscillatory behavior is not well understood. There are many mathematical models on the oscillatory behavior of Z. mobiliz. Li (2009)2 indicated that the oscillatory behavior of ethanol production is not attributed to ethanol concentration. In addition, he claimed that both the rate of ethanol change and the ethanol concentration history may lead to the oscillatory behavior. Furthermore, Hobley and Pamment © XXXX American Chemical Society

Received: July 24, 2013 Revised: January 14, 2014 Accepted: February 28, 2014

A

dx.doi.org/10.1021/ie402361b | Ind. Eng. Chem. Res. XXXX, XXX, XXX−XXX

Industrial & Engineering Chemistry Research

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concentration of viable and nonviable cells, and their relation with the oscillatory phenomena of continuous ethanol fermentation by Z. mobiliz are investigated. The regions of d causing oscillations (periodic attractors) including high periodicity in the form of period doubling and steady states (point attractors) are determined through bifurcation analysis. Comparison of the numerical results to experimental results is made to test the model and validate it. The kinetic parameters of the growth rates of the three classes of cells are proposed to test the model and validate the proposed model as well as using literature experimental examples of oscillations in the continuous fermentation of ethanol. In addition, cell growth and inhibition of substrate and product formation are considered together to be able to investigate the oscillatory phenomena of the fermentor. In the structured model, Z. mobiliz cells populations inside the fermentor are assumed to be the same

dead cells. Viable cells are cells which are producing cells and able to grow with enough nutrients.8−10 These living cells can be converted into nonviable cells which have the capability to produce ethanol but are unable to grow. Viable cell cultures have the capability to produce various metabolic levels and vital stages to assess their capabilities for growth and proliferation. However, dead cells are unable to grow or proliferate where there is no vital function. The life cycle of microorganisms is composed of four phases: lag, exponential, stationery, and death. On the basis of operating conditions such as temperatures, pH, nutrients, and oxygen, the length of each phase is different.9,10 In the stationery phase, there is a balance between cell death and cell multiplication. This means that cultures do not grow despite their metabolic activity. Therefore, the disability of cells to divide or grow does not mean death of the cell. The latter type of cell is called nonviable cells.9−11 Nonviable cells can die due to lack of nutrients or accumulation of toxics or other metabolites. In fermentation systems dead cells are broken down, leading to reduction of viable and nonviable cells counts. The rate of cell death is very important as it has a significant influence on the outcome of biological systems. However, in kinetic studies of cultured cell proliferation, the rate of cell death is not easy to be calculated.12 The portion of the viable cells dying varies based on various conditions such as pH, temperature, substrate, and product inhibition. The portion of the dead cells to the viable cells lies in a range from 10% to 30% based on operating conditions.13 Therefore, μd is often assumed at an average value to be 20% of the specific growth rate of viable cells μv.9 The oscillatory behavior of Z. mobiliz has been investigated by Jobses et al. (1985−1986)14−16 through their two-compartment model. Ghommidh (1989)17 proposed that the viability of biomass plays an important role in the oscillations of substrate, biomass, and ethanol because of significant oscillations of cell viability observed in Z. mobiliz cultures. Therefore, structuring of the cell populations into viable, dead, and nonviable cells has an effective role for describing the oscillatory phenomena of Z. mobiliz fermentation cultures in the form of mathematical models. The structured mathematical models considering these physiological phenomena give more realistic results than nonstructured models and can describe reasonably well the conversion of stability to instability. Ghommidh et al. (1989)17 built a mathematical model for describing the oscillatory behavior in continuous culture of Z. mobiliz; however, this model could not express the stable behavior. The models proposed by Ghommidh (1989)17 and Jobses et al. (1985− 1987)14−16 were not able to predict the system behavior in a wide range of feed substrate concentrations and the dilution rates. In addition, the values of the biomass concentrations were not accurately predicted. With the concept of structuring biomass into viable, nonviable, and dead cells, our model is formulated to describe the oscillatory, stable, and unstable behaviors of continuous ethanol fermentation by Z. mobiliz. In this work, the dilution rate is chosen to be the bifurcation parameter as it is considered as an external effect and a controlling parameter for the continuous fermentation system. Therefore, the effect of the dilution rate as a possible mechanism explaining the oscillatory behavior of continuous ethanol fermentation by Z. mobiliz is investigated. Furthermore, it is important to investigate how the ethanol production rate, substrate conversion, and ethanol yield depend upon the dilution rate. The effects of d on the concentration of produced ethanol, concentration of viable cells, concentration of residual substrate,

2. MODEL DEVELOPMENT Continuous culture fermenters allow multiplying the ethanol productivity many times in comparison to other types of reactors, like batch reactors or centrifuges.18 In addition, it is observed from the literature that viability oscillates significantly, and the measured average fermentation rates were higher than expected.17 This means there is a portion of the active population that was not observed. On the basis of the data from the literature, the following assumptions are considered. (1) The substrate is converted into ethanol and cell growth by viable and nonviable biomass. (2) Viable cells can be converted to nonviable cells which are unable to grow but able to produce ethanol. These inactive cells can die through the high substrate and product concentrations leading to loss of a significant portion of the living cells according to the following mechanism μv

X v → 2X v μnv

X v → X nv μd

X nv → Xd (X v + X nv )

S ⎯⎯⎯⎯⎯⎯⎯⎯⎯→ P where the subscripts terms n, nv, and d refer to the viable, nonviable (inactive), and dead fractions of cells. (3) The fermentation continuous system is considered a homogeneous, mixed, isothermal, and continuous tank reactor (CSTR) as shown in Figure 1. The fermentation kinetics made by Jobses et al. (1985 and 1986)14−16 and Phisalaphong et al. (2005)19 depend on developing the rate equations to be compatible with the experimental values. All fermentation cultures are assumed to operate isothermally at 30 °C via cooling the CSTR to be able to work at the maximum rate of the biomass. The rate of substrate consumption based on the maintenance model1 is given below μ rs = Cxv v + msCxnv Yxx (1)

The first term represents the growth rate of the viable microorganisms, while the second term represents the maintenance of the nonviable cells. From experimental observations of the production of ethanol using Z. mobiliz,17,20 B



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Figure 1. Schematic of the CSTR fermentor.

a mathematical model can be proposed where substrate is consumed in terms of ethanol as a product and energy by living cells at the rate of μv. The high substrate ethanol concentrations lead to death of biomass and loss of viable cells. Considering these categories of biomass, mass balance differential equations accounting for continuous production of ethanol in the CSTR can be written as follows. (1) Viable cells (CXV) dCxv = d(Cxvo − Cxv) + C xv(μv − μnv ) dt

If the inhibition of ethanol as a product is considered where ethanol can alter the composition of the cell membranes of the biomass, and if substrate inhibition is considered by adding an inhibition constant into the kinetic equation, eq 7, of the specific growth rate of the viable cells developed by Bai et al. (2004 and 2009)21,22 is modified to express the effect of ethanol (Cp) in a nonlinear form as follows α ⎛ Cp ⎞ μv = ⎜1 − ⎟ C2 Pc ⎠ K s + Cs + Ks ⎝

μmax Cs

(2)

ss

The specific growth rate of the nonviable cells considering both substrate and product inhibition developed by Watt et al. (2010)1 can be expressed in the same way to obtain the rate of nonviable biomass as follows

(2) Nonviable cells (Cxnv) dCxnv = d(Cxnvo − Cxnv) + C xv(μnv ) − Cxnvμd dt

(3)

μnv =

(3) Dead cells (Cxd) dCxd = d(Cxdo − Cxd) + C xnv(μd ) dt

(5)

(6)

All μv, μnv, and μd are controlled by a substrate-limiting effect and inhibition effects of the substrate and ethanol. The Monod model can be used to get the relationship between the growth rates and the limiting substrate concentrations Cs μv = K s + Cs

K sp + Cs +

CS 2 K ssp

β ⎛ Cp ⎞ ⎜1 − ⎟ − μv Pcd ⎠ ⎝

(9)

(10)

The growth rates given by eqs 8, 9, and 10 describe product inhibition and substrate limitation and are similar to those given by Ghommidh et al. (1989)17 and Jarzbeski (1992).23 It is observed that these differential equations are in a highly nonlinear form, and numerical solution is the only way to solve the system using the kinetic parameters shown in Table 1. The terms ((1 − Cp/Pcd)βCS/(Ksp + CS + CS2/KSSp)) and ((1 − Cp/ Pc)αCS/(Ks + CS + CS2/KSS)) indicate the impact of the complex ethanol inhibition effect. The parameter set values are taken from refs 6−10 and 13−15 and are shown in Table 1. It is clear that even the specific growth rates are not a function of the dilution rate but affected by the concentrations of substrates and ethanol which are affected directly by the dilution rates.

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

μmax d Cs

and cell death can be described by μd = φμnv

(4)

(4) Ethanol (Cp) μ dC P = d(Cpo − Cp) + Cxv v + mpCxnv dt Yxp

(8)

3. NUMERICAL TECHNIQUES AND COMPUTATIONS XPPAUT 5 and the bifurcation program AUTO 2000 are used in performing the bifurcation analysis to determine the entire

(7) C



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The fermentation culture system is assessed via calculating the substrate conversion (Xs), the ethanol production rate per unit volume of the reactor (Pp), and ethanol yield (Yp). Xs is defined as the ratio between the rate of substrate utilized by biomass and the rate of feed sugar. The overall variable Xs is calculated as follows

Table 1. Kinetic Parameters Value and Feed and Steady-State Concentrations parameter μmax μmaxd α β Ks Φ Kss Ksp Kssp mp ms Pc Pcd Yxs Yxp d feed concentrations Cso Cpo

value

units

0.23 0.22 1.74 2.5 20 0.2 150 9.5 200 1.9 3.5 250 350 0. 03 0.375 0.06

h−1 h−1 g/L g/L g/L

140 0 0 0 0

g/L g/L g/L g/L g/L

Cxnvo Cxdo steady-state concentrations Cs 33.47 Cp 54.15 Cxv 6.6 Cxnv 1.51 Cxd 0.0723

Xs =

Cso − Cs Cso

(11)

Pp is defined as the productivity exiting from the CSTR and calculated as follows

g/L g/L g/L

Pp = Cpd

(12)

Yp is defined as the ratio between the ethanol production rate and the rate of sugar feed and is calculated as follows

g/L g/L g/g g/g h−1

Yp =

Cp Cso

(13)

5. RESULTS AND DISCUSSION In the first section we consider the inlet substrate concentration to be 140 g/L. In the second section a higher inlet substrate concentration is considered at 200 g/L. 5.1. Dilution Rate (d) as the Bifurcation Parameter at Cso = 140 g/L. Dilution rate is a function of the feed flow rate. It is used as a bifurcation parameter because it can be manipulated and controlled. Thus, it is important to study the effect of the dilution rate on the fermentation culture behavior. Optimization of the dilution rate is critical as it is the reciprocal of the transient time. The dilution rate is expected to affect the substrate conversion, ethanol yield, and rate of ethanol production.20 In addition, it is important to investigate the effect of the dilution rate on both viable and nonviable cells which are responsible for ethanol production. The concentration of the inlet substrate of 140 g/L refers to a medium gravity (MG) of the fermentation system.21 This concentration of Cso was used experimentally by Garhyan et al. (2004)20 and Chen et al. (2005).11 By considering the structure of biomass in the model and giving practical values to all growth rates the concentrations of each viable, nonviable, and dead cell can be predicted. 5.1.1. Region 1: Dilution Rate in the Range 0.1 > d > 0.0728 h−1. This region is characterized by a unique stable steady state (point attractor) at high values of the dilution rate. As d decreases a Hopf bifurcation (HB) arises at d = 0.0728. In this region the ethanol concentration Cp increases from 25 g/L at d = 0.1−52 g/L at HB as shown in Figure 2c. The residual substrate concentration Cs increases to be higher than 36.41 g/L as shown in Figure 2b; however, the viable biomass concentration Cxv increases to be in the range 7.811−9.4 g/L in the range of d = 0.0728−0.087 h−1 and then decreases as d is greater than 0.087 to reach 4.5 g/L at d = 0.1 h−1 as shown in Figure 2a. Although Cxv increases in the range of d = 0.0728−0.087, the biomass is not able to consume all of the substrate, leading to a reduction of Cp in the same region of d as shown in Figure 2c and an increase of Cs (Figure 2b). Comparing our results to the theoretical results obtained by Watt et al. (2010)1 it is found that our viable cell concentrations Cxv are in agreement with their viable cell concentrations which were in the range of 2.5−10 g/L over a wide range of the transient time or the dilution rate, while the residual substrate concentration differs from those of Watt et al. (2010)1 substrate which were in the range of 0.1−100 g/L and are higher than our range. However, their ethanol concentrations were in the range

g/L g/L g/L g/L g/L

periodic branches as either stable or unstable. Matlab 7 is used for getting the initial conditions at which the system achieves a stable steady state. These initial conditions are used by AUTO 200024 to get the static bifurcation and then from the Hopf bifurcation (HB) point the dynamic bifurcation showing that periodic solutions can be obtained. Matlab 7 is used also for performing simulations and getting phase planes and time traces in oscillatory regions of bifurcation parameters and stable steady states as well. For investigating the chaotic behavior and high levels of oscillations, additional techniques are used where discrete points of intersections between trajectories and a hyperplan surface to plot the Poincare map. This hyperplan surface is selected at a certain point of one of the state variables. Discrete points of intersections should be considered where the trajectories intersect the hyperplane transversally and cross it in the same direction. The Poincare map is obtained via a FORTRAN program with the subroutine DIVPAG with IMSL library and automatic step size to stiff differential equations with high accuracy for periodic or chaotic attractors.25

4. BIFURCATION ANALYSIS Bifurcation analysis helps to investigate the behavior of the Z. mobiliz fermentation system better than traditional simulation. Using bifurcation analysis, the model can predict both static and dynamic bifurcation behavior. The oscillatory behavior dominates the system over a wide range of bifurcation parameters. The complex interaction between ethanol production from viable and nonviable biomass, substrate consumption, growth of viable cells, and death of nonviable biomass can cause the periodic behavior of the system.13 D



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Figure 2. Bifurcation diagrams with dilution rate d as the bifurcation parameter at Cso = 140 g/L and the rest of the parameters as shown in Table 1: stable periodic branch (●), unstable periodic branch (○), stable steady-state branch (), unstable steady-state branch (---). (a) Bifurcation diagram for viable cells concentration (Cxv), (b) bifurcation diagram for residual substrate concentration (Cs), (c) bifurcation diagram for ethanol concentration (Cp), and (d) enlargement of the box in c.

Figure 3a illustrates that Xs decreases to be in the range 74− 25%. Figure 3b shows that Yp decreases in this region to be in the range 0.37−0.12 (kg ethanol produced/kg feed substrate). However, Pp (g ethanol produced/l h) increases to reach 3.77−3.81 kg/h in the range of d = 0.0728−0.087 h−1, and then Pp decreases to reach 1.7 kg/h at d = 0.1 h−1 as shown in Figure 3c. Figure 3d shows an enlargement of the zoom area in Figure 3a around HB. Figure 4a and 4b shows the effect of d as a bifurcation parameter on the

of 0−45 g/L, which are lower than ours. The discrepancies of the results of both residual substrate concentrations and ethanol can be attributed to the growth rates of viable and nonviable cells adopted by Watt et al. (2010)1 which were higher than ours, and their rates cause a large difference from the experimental results obtained by Garhyan et al. (2004).20 In addition, the reduction of the growth rates obtained by Watt et al. (2010)1 leads to the oscillatory behavior and makes the system unpredictable. E



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Figure 3. Bifurcation diagrams with dilution rate d as the bifurcation parameter at Cso = 140 g/L and the rest of the parameters as shown in Table 1: stable periodic branch (●), unstable periodic branch (○), stable steady-state branch (), unstable steady-state branch (---). (a) Bifurcation diagram for substrate conversion (Xs), (b) bifurcation diagram for ethanol yield (Yp), (c) bifurcation diagram for ethanol production rate (Pp), and (d) enlargement of the box in a.

Figure 4. Bifurcation diagrams with dilution rate d as the bifurcation parameter at Cso = 140 g/L and the rest of the parameters as shown in Table 1: stable periodic branch (●), unstable periodic branch (○), stable steady-state branch (), unstable steady-state branch (---). (a) Bifurcation diagram for nonviable cells concentration (Cxnv) and (b) bifurcation diagram for dead cells concentration (Cxd).

concentration of nonviable cells (Cxnv) and dead cells (Cd), respectively. It is clear that in the corresponding region that Cxnv

behaves as a point attractor and decreases from 1.5 to 0.5 g/L as shown in Figure 4a, while Cxd decreases from 0.06 to 0.01 g/L. F



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Figure 5. Dynamic characteristics at d = 0.06, Cso = 140 g/L, and the rest of the system parameters as shown in Table 1. (a) Time traces of viable cells concentration (Cxv). (b) Time traces of ethanol concentration (Cp). (c) Time traces of residual substrate concentration (Cs). (d) Phase plane for Cxv vs Cp.

Figure 6. Dynamic characteristics at d = 0.06, Cso = 140 g/L, and the rest of the system parameters as shown in Table 1. (a) Time traces of substrate conversion (Xs). (b) Time traces of ethanol production rate (Pp). (c) Time traces of ethanol yield (Yp). (d) Phase plane for Cxv vs Cs.

5.1.2. Region 2: Dilution Rate in the Range d < 0.0728 h−1. This region is characterized by oscillations where only periodic attractors dominate and surround the unique unstable steady state. It is observed that the amplitudes of oscillations increase with the increase of d as shown in Figures 2, 3, and 4. Figure 2a and 2b

shows that the averages of oscillations of viable biomass and residual substrate increase continuously with the increase of d. However, these averages of oscillations of ethanol decrease with the increase of d as shown in Figure 2c. It is observed that when d decreases to a very small value approaching zero the system produces the G



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Figure 7. Dynamic characteristics at d = 0.06, Cso = 140 g/L, and the rest of the system parameters as shown in Table 1. (a) Time traces of nonviable cells concentration (Cxnv). (b) Time traces of dead cells concentration (Cxd).

the ethanol concentrations Cp oscillates in the range of 42−70 g/L with an average of 56 g/L, which is compatible with the experimental results made by Garhyan et al. (2004),20 who measured the concentration of ethanol at d = 0.06 and found Cp = 57.3 g/L, which is also close to the theoretical value of Cp = 57.9 g/L. In addition, Figure 5c shows that Cs oscillates in the range of 5−58 g/L with an average of 31.5 g/L and the active cells Cxv oscillate in the range of 3−10 g/L with an average of 6.5 g/L, which is close to the experimental values obtained by Jarzebski (1992),23 who ran the experiments at Cso = 138 g/L and d = 0.08 and got periodic behavior qualitatively similar but quite different quantitatively to our results where ethanol oscillated in 20−60 g/L and substrate oscillated in the range 40−100 g/L while viable cells oscillated in the range of 1−5.5 g/L and nonviable cells in the range 0.5−1.8 g/L. The main reason of the discrepancy is Jarzebski (1992),23 who considered d = 0.05 and assumed that viable cell can die directly without passing the transient stage of nonviable cells. Figure 5d shows the phase plane between Cxv and Cp. It is clear that the phase plane is a closed orbit, confirming the oscillatory behavior of the fermentation system where Cp oscillates in the range 44−74 g/L while Cxv oscillates in the range 3.5−10 g/L. Time traces of the substrate conversion and ethanol production rate and yield and the phase plane between Cxv and Cs are indicated in Figure 6. Figure 6a shows that Xs oscillates in the range of 0.6−0.97 with an average of 0.785. However, Pp oscillates in the range of 2.26−4.2 g/L h to give an average of 3.23 g/L as shown in Figure 6b. Furthermore, the ethanol yield in Figure 6c oscillates in the range of 0.3−0.5 g/L with an average of 0.4 g ethanol/g substrate. The phase plane between Cxv and Cs in Figure 6d shows the closed orbit where Cxv changes in the range 3.5−10 while Cs changes in the range 5−58 g/L, confirming the periodic attractor of the system. Figure 7a and 7b illustrates time traces of both Cxnv and Cxd, respectively. It is observed that Cxnv

maximum ethanol concentration 78 g/L as shown in Figure 2c and minimum residual substrate concentration 3 g/L as shown in Figure 2b. This can be explained by the fact that at very low values of d the transient time τ of the biological reactions extends too long, where τ is the reciprocal of d allowing the microorganisms to consume most of the substrates achieving the highest conversion (almost 100%) as shown in Figure 3a and the highest ethanol yield Yp as shown in Figure 3b. On the other hand, at very low values of d the ethanol production rate (Pp) is lowered as well as the concentration of viable cells as shown in Figures 2 a and 3c, respectively. It is observed from Figure 2b and 2c that both the unstable steady-state branch and the average concentrations branch of residual substrate and ethanol are equal as d increases from d = 0 to 0.045 h−1; then as d increases to be greater than 0.045 h−1, the average concentration of C s becomes higher than the corresponding steady state while the average concentration of Cp becomes lower that that of the unstable steady state as shown in Figure 2 b and 2 c. The same phenomenon about Cp is observed with Xs, Yp, and Pp, where their averages are lower than that of the corresponding steady states when d increases. This reflects the negative effect of increasing d on the operation of the fermentation system. Figure 4a and 4 b illustrates the oscillatory behavior of nonviable cells and dead cells, respectively. It is observed that the averages of Cxnv and Cxd increase with the increase of d. This is attributed to the fact that an increase of d leads to an increase of the influent flow rate causing deactivation of the biomass and increases the average of Cxnv as shown in Figure 4b. Figures 5, 6, and 7 show the dynamics of the fermentation system at d = 0.06 and Cso = 140 kg/m3 to be able to compare the results with the experimental results made by Garhyan et al. (2004).20 Figure 5a, 5b, and 5c shows time traces of viable cells, ethanol, and residual substrate, respectively. Figure 5b shows that H



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indicated that Cxnv = 1 g/L. Figure 7b shows that Cxd oscillates in the range of 0.042−0.117 g/L with an average of 0.0795 g/L.

oscillates in the range 0.8−2.2 g/L with an average of 1.5 g/L, which is close to that obtained by Watt et al. (2010), who I



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oscillates in the range of 3−32 g/L while Cxv oscillates in the range of 2−5 g/L. Figure 10 a and 10 b shows time traces of Cxnv and Cxd, respectively, at d = 0.04. Cxnv oscillates in the range of 0.98−1.7 g/L with an average of 1.34 g/L, which is lower than that at d = 0.06 h−1. Figure 10b shows that Cxd oscillations in the range of 0.0875− 0.118 g/L with an average of 0.10275 g/L, which is higher than that at d = 0.06 h−1. 5.2. Dilution Rate (d) as the Bifurcation Parameter at Cso = 200 g/L. To investigate the effect of changing the feed substrate on the fermentor behavior with d as the bifurcation parameter, the influence of d is studied at Cso = 200 g/L, which refers to a high gravity (HG) that was experimentally studied by Garhyan et al. (2004)20 and Bai et al. (2009).21 As shown in Figures 11, 12, and 13, the effect of d can be divided into three regions as follows. 5.2.1. Region 1: d > 0.06297 h−1. In this region, the fermentation system behavior at Cso = 200 g/L is characterized by stable steady-state solution which looks qualitatively like the behavior at Cso = 140 g/L, but both are different quantitatively. Figure 11 shows that HB appears at d = 0.06297 h−1. From Figure 11a it is observed that Cxv increases from 11.51 g/L at d = 0.06297 h−1 (which is HB) to 13.04 g/L at d = 0.073 h−1; then as d increases Cxv decreases to be almost zero at d ≥ 0.086 h−1. It is clear that the increase of d in this region leads to a rapid reduction of the active cells. Therefore, d should be in a certain range to keep Z. mobiliz cells viable and a high concentration of ethanol. The concentration of Cxv in this region of d and Cso = 200 g/L is higher than that at Cso = 140 g/L, reflecting the need to high active cells at high concentrations of feed substrate. Figure 11b shows that the residual concentration of substrate Cs increases from 51.99 g/L to 200 as d increases from 0.06297 h−1 at HB to 0.086 h−1. On the other hand, Cp decreases from 74.32 g/L at d = 0.06297 h−1 to almost zero at d = 0.086 h−1 as shown in Figure 11c. It is clear that

Figures 8, 9, and 10 show the dynamics of the fermentation system at d = 0.04 h−1 and Cso = 140 kg/m3 to be able to compare our results with the experimental results obtained by Garhyan et al. (2004).20 Figure 8a shows that Cxv oscillates in the range 2−5 g/L with an average of 3.5 g/L while Cp oscillates in the range of 55.8−71.8 g/L with an average of 63.8 g/L, which is in agreement with the experimental results obtained by Garhyan (2004),20 who conducted experiments at the same conditions of d = 0.04 h−1 and Cso = 140 g/L. At these conditions the experimental value of Cp = 61.93 g/L while the simulated value obtained by Ghayran et al. (2004)20 was 63.4 g/L. Figure 8 c shows that Cs oscillates in the range of 3−32 g/L with an average of 17.5 g/L. The phase plane between Cp and Cxv (Figure 9d) shows a complete closed orbit, confirming the periodic attractor dominating the fermentor at the corresponding conditions. Figure 9 illustrates the time traces of substrate conversion, ethanol production rate, and ethanol yield and the phase plane between viable cells and residual substrate at d = 0.04 h−1 and Cso = 140 g/L. Figure 9a shows that Xs oscillates in the range of 0.77−0.99 with an average of 0.88, which is higher than that of substrate conversion at d = 0.06 h−1 illustrated previously in Figure 6a. Figure 9b illustrates that Pp disturbs in the range of 2.28−2.9 g/L with an average of 2.59 g/L, which is lower than that at d = 0.06 h−1. The latter result of Pp is compatible with the bifurcation diagram shown in Figure 3c and reasonable because the increase of d reflects the increase of the volumetric flow rate and reduction of the transient time required for completing the fermentation reaction by Z. mobiliz. Figure 9c shows that Yp oscillates in the range of 0.407−0.52 with an average of 0.4635. It is observed that the latter average is greater than that at d = 0.06 h−1 as Yp decreases with the increase of d. The phase plane between Cxv and Cs shown in Figure 9d gives a closed orbit, confirming the periodic behavior of the system where Cs J



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Figure 11. Bifurcation diagrams with dilution rate d as the bifurcation parameter at Cso =200 g/L and the rest of the parameters as shown in Table 1: stable periodic branch (●), unstable periodic branch (○), stable steady-state branch (), unstable steady-state branch (---). (a) Bifurcation diagram for viable cells concentration (Cxv). (b) Bifurcation diagram for residual substrate concentration (Cs). (c) Bifurcation diagram for ethanol concentration (Cp). (d) Enlargement of zoom 1 in c. (e) Enlargement of zoom 2 in c.

5.2.2. Region 2: 0.03518 < d < 0.06297. This region is characterized by a periodic attractor solution. Figure 11a shows that the average of Cxv increases with an increase of d, where Cxv increases from 6.679 g/L at d = 0.03518 to 11.51 g/L at HB. The amplitude of the oscillatory behavior increases as d increases where Cxv reaches the maximum level which is 16 g/L at d = 0.057. Compared to the oscillatory behavior of Cxv through the same region of d at Cso = 140 g/L, the average of Cxv at Cso = 200 g/L is higher than that at Cso = 140 g/L. Cs in Figure 11 b changes periodically as well. The average of Cs increases with the increase of d. However, the average of Cp decreases in this region of d from 101.6 g/L at d = 0.035186 to 74.32 g/L at HB as shown in Figure 11c. The average of Cp is higher than the average of Cp at

the concentration of Cp at HG (Cso = 200 g/L) is higher than that at MG (Cso = 140 g/L). Figure 11d shows an enlargement of the region around HB in Figure 11c. It is clear that HB arises at d = 0.06297 and Cp = 74.32 g/L, which is higher than the concentration of ethanol at Cso = 140 g/L where HB took place at Cp = 52.02 g/L and d = 0.0728 h−1. Figure 12a shows that Xs decreases through this region from 0.73 at HB to be almost 0.4 at d = 0.082. Yp decreases from 0.37 at HB to 0.2 at d = 0.08 (Figure 12b). Figure 12 c shows that Pp decreases from 0.45 at HB to 0 at d = 0.086. The maximum range of Pp is around 4.8 as shown previously in Figure 12 c and is higher than that at Cso =140 g/L as shown in Figure 3 c. K



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Figure 12. Bifurcation diagrams with dilution rate d as the bifurcation parameter at Cso= 200 g/L and the rest of the parameters as shown in Table 1: stable periodic branch (●), unstable periodic branch (○), stable steady-state branch (), unstable steady-state branch (---). (a) Bifurcation diagram for substrate conversion (Xs). (b) Bifurcation diagram for ethanol yield (Yp). (c) Bifurcation diagram for ethanol production rate (Pp). (d) Enlargement of the box in a.

Cso = 140 g/L (as shown in Figure 2c). Figure 11 d shows an enlargement of the zoom region in Figure 11 c. HB is a triple point where three different solutions intercept. These solutions are the stable steady states from the right and other both unstable steady state and an unstable periodic attractor from the left. Figure 12a shows that the average of Xs decreases from 0.85 at d = 0.03518 to 0.73 at d = 0.06297. It is observed also that the average of Xs is higher than that at Cso = 140 g/L. Figure 12b shows that the Yp average decreases also with the increase of d. However, the average of Pp increases through this region of d. Figure 12d is an enlargement of the zoom region appearing in Figure 12a and shows that the largest amplitude occurs at d = 0.058 where Xs changes in the range 0.5−0.97. 5.2.3. Region 3: 0 < d < 0.01121 h−1. This region is similar to region 2 where there is only a periodic attractor. This region is characterized by the lowest concentration of Cxv and Cs as shown in Figure 11a and11b, respectively. However, Cp in this region is the highest concentration where it reaches 107 g/L as shown in Figure 11c. Figure 12a shows that Xs is almost 99%, which means that most of the sugar is converted to ethanol and reflects the behavior of Yp as shown in Figure 12b where Yp reaches the maximum values where it changes to around 0.5 g ethanol/g substrate. In addition, Pp changes in the lowest range, where it oscillates in the range 0−1 g/L in very small amplitude as shown in Figure 12c. 5.2.4. Region 4: 0.01121 < d < 0.03518. This region is characterized by an unstable periodic attractor shown in open circles in Figures 11 and Figure 12. The periodic solutions become unstable via period doublings where there are two period doublings: the first PD1 appears at a dilution rate d = 0.035186 h−1, and PD2 appears at d = 0.01121 h−1. Then the fermentation

system comes back to the original stable periodic solution. The period doubling is one of the main ways leading to chaotic behavior or a high periodicity. To investigate the complexity behavior with period doublings, the Poincaré map for one of the state variables is required25 to investigate as will be illustrated later in Figure 19. Figure 13 shows the fermentation dynamic behavior at Cso = 200 g/L and d = 0.06 h−1, and the rest of the parameters are shown in Table 1 to be able to compare with the results obtained at Cso= 140 g/L. Figure 13a shows that Cxv oscillates in the range of 6−15 g/L with an average of 10.5 g/L, which is higher than that at Cso = 140 g/L as shown previously in Figure 5a. Cp in Figure 13b oscillates in the range of 55−93 g/L with an average of 74 g/L, which is higher than that at Cso = 140 g/L. Figure 13c illustrates that Cs oscillates in the range of 20−95 g/L with an average of 57.5 g/L, which is higher than that shown in Figure 5c at MG where Cso = 140 g/L. Figure 13d confirms the oscillatory behavior of the fermentation system through the closed orbits between Cxv and Cp. Figure 14a shows that Xs oscillates in the range of 0.53−0.91 with an average of 0.72, which is lower than that at Cso = 140 g/L and d = 0.06. Pp oscillates in the range of 3.25−5.52 (g ethanol produced/l h) with an average of 4.385 g ethanol produced/l h (Figure 14b), which is higher than that at Cso =140 g/L and the same value of d. Figure 14c illustrates the time traces of Yp where it oscillates in the range of 0.27−0.46 with an average of 0.365, which is lower than that at MG and the same value of d. Figure 14d shows closed oscillatory orbits between Cxv and Cs in the ranges mentioned above. Figure 15a shows that Cxnv oscillates in the range of 1.25−2.5 g/L with an average of 1.875 g/L, which is higher than that at MG, while Cxd oscillates in L



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Figure 14. Dynamic characteristics at d = 0.06, Cso = 200 g/L, and the rest of the system parameters as shown in Table 1. (a) Time traces of substrate conversion (Xs). (b) Time traces of ethanol production rate (Pp). (c) Time traces of ethanol yield (Yp). (d) Phase plane for Cxv vs Cs. M



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the range of 0.075−0.122 g/L with an average of 0.099 g/L, which is higher than that of MG.

Figures 16, 17, and 18 show the dynamics of the fermentation system at d = 0.04 and HG where Cso = 200 g/L. Figure 16a N



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shows that Cxv oscillates in the range of 3.5−10 g/L with an average of 6.75 g/L, which is higher than that at MG and the same value of d as mentioned in Figure 8a. Figure 16b shows that Cp oscillates in the range of 75−105 g/L with an average of 90 g/L, which is higher than that at MG and d = 0.04. Figure 16c shows that Cs oscillates in the range of 0.2−55 g/L with an average of 27.3 g/L. The closed orbits between Cp and Cxv shown in Figure 16d confirm the oscillatory behavior of the system at HG and d = 0.04 and compatibility with the bifurcation diagrams shown in Figure 11b. Figure 17a shows that Xs oscillates in the range of 0.73−0.999 with an average of 0.86 while Pp oscillates in the range of 3−4.15 with an average of 3.6 (Figure 17b). Yp in Figure 17c oscillates in the range of 0.37−0.52 with an average of 0.445, which is lower than that at MG and d = 0.04. Figure 17d shows the closed orbits between Cxv and Cs, confirming the oscillatory behavior. Figure 18a shows time traces for Cxnv, where it oscillates in the range of 1.25−2.5, while Cxd oscillates in the range 0.075−0.122 with an average of 0.099. To investigate the period doublings appearing in the bifurcation diagrams in Figures 11 and 12, a one-dimensional Poincaré map is studied. Poincaré map is obtained by plotting the intersections between the trajectories in one direction and a hypothetical hyperplan surface taken at a specific value of one of the state variables.25 The state variable value taken into consideration is Cp = 102 g/L. One of the main advantages of the Poincaré map is that it simplifies the problem of closed orbits to a problem of points which is more convenient to deal with than the close orbits problem (Mustafa et al. (2012)). Figure 19a illustrates the Poincaré bifurcation map considered at Cp = 102 g/L, Cso = 200 g/L, and the rest of the parameters as shown in Table 1. The dilution rate (d) is the main bifurcation parameter taken in the rage of 0.01 < d < 0.04. Figure 19a shows that PD is the route leading to a high periodicity where the evolution


is described by PD sequence and can be summarized as follows. Windows of period one attractor in the range 0 < d < 0.01121; then windows of period two in the range 0.01121 < d < 0.02; O



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Figure 19. (a) Poincaré bifurcation diagram (Poincaré plane is located at Cp = 102 g/L and the rest of the parameters as shown in Table 1). (b) Enlargement for the zoom area in a.

Figure 20. Dynamic characteristics at Cs = 200 g/L, Cp =102, d = 0.024 h−1, and the rest of the system parameters as shown in Table 1. (a) Time traces of viable cells concentration (Cxv). (b) Time traces of ethanol concentration (Cp). (c) Time traces of residual substrate concentration (Cs). (d) Phase plane for Cxv vs Cp. P



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Figure 21. Dynamic characteristics at Cs = 200 g/L, Cp =102, d = 0.024 h−1, and the rest of the system parameters as shown in Table 1. (a) Time traces of substrate conversion (Xs). (b) Time traces of ethanol production rate (Pp). (c) Time traces of ethanol yield (Yp). (d) Phase plane for Cxv vs Cs.

Figure 22. Dynamic characteristics at Cs = 200 g/L, Cp = 102, d = 0.024 h−1, and the rest of the system parameters as shown in Table 1. (a) Time traces of nonviable cells concentration (Cxnv). (b) Time traces of dead cells concentration (Cxd).

then windows of period four in the range 0.02 < d < 0.0265; then windows of period two in the range 0.02 < d < 0.0365. Figure 19b is an enlargement for the zoom region of period four shown in Figure 19b. Figure 20a−d shows the time traces and phase planes of period four windows at d = 0.024 and Cso = 200 g/L and the

rest of the parameters as shown in Table 1. Figure 20 shows that the period repeats itself every four peaks/bases. Cxv in Figure 20a oscillates in the range of 1.5−4.5 g/L, while Cp oscillates in the range of 90−107 g/L (Figure 20b); Cs oscillates in the range of 0−33 g/L as shown in Figure 20c, while the closed orbits Q



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between Cp and Cxv confirm the high periodicity shown in windows of period four. Figure 21a shows that Xs is repeated in the range of 0.83−0.99 where it occurs at a small value of d = 0.024. Figure 21 b illustrates that Pp oscillates in the range of 2.1− 2.58, which is repeated as shown in windows period four, while Yp oscillates in the range 0.44−0.52 as shown in Figure 21c. The phase plane between Cs and Cxnv confirms the oscillations occurring in period four as shown in Figure 21d. Figure 22a and 22b shows that Cxnv and Cxd oscillate in the range 0.88−1.6 and 1−1.6 g/L, respectively. Although the periodicity is developed to period four, it is not able to develop to fully chaotic behavior as shown in the Poincaré map (Figure 19).

ACKNOWLEDGMENTS

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

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ABBREVIATIONS CSTR continuous stirred tank reactor HB Hopf bifurcation SB static bifurcation PLP periodic limit point PD period doubling HG high gravity MG medium gravity LG low gravity Cs concentration of residual substrate (g/L or kg/m3) Cp concentration of ethanol (g/L or kg/m3) Cxv concentration of viable cells (g/L or kg/m3) Cxnv concentration of nonviable 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 ethanol (g/L or kg/m3) Cxvo feed concentration of viable cells (g/L or kg/m3) Cxnvo feed concentration of nonviable 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 ethanol yield Pp ethanol production rate (g/h) ms maintenance constant based on substrate (g/g h) mp maintenance constant based on product (g/g) Ysx yield constant based on substrate (g/g) Ypx 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 ethanol 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 nonviable cells h−1 μd growth rate of dead cells h−1 T time (h) Ksp saturation constant for ethanol (g/L) Kssp saturation constant for ethanol (g/L) α inhibition index for cell growth β inhibition index for ethanol production Φ growth rates ratios of dead cells to viable cells

6. SUMMARY AND CONCLUSIONS A structured mathematical model for fermentation of ethanol using Z. mobiliz in a single-well mixed reactor is developed and validated to investigate the bifurcation analysis and oscillatory behavior of the fermentation system including the chaotic behavior of the system. The physiological structures of biomass into viable, nonviable, and dead biomass are considered. The specific growth rates of both viable and nonviable cells are chosen from the experimental literature to give more realistic results. The model gives a whole portrait for the complex effect of a wide range of the dilution rate d on the fermentation system behavior. The dilution rate is selected to be the bifurcation parameter because it can be controlled and handled. The effects of this parameter on the performance of a fermentation continuous perfect mixed reactor are investigated. It is found that there is only one HB point at d = 0.0728 h−1 on the dynamic bifurcation diagram. The simulation results at d = 0.06 and 0.04 h−1 are compatible with the experimental results obtained by Garhyan et al. (2004)20 as shown in Figures 5−9. Steady-state and oscillatory solutions have been observed in the fermentation system. The system exhibited period doubling cascade in the form of perioddoubling bifurcations as shown in Figures 11, 12, and 13 and the Poincaré maps shown in Figure 19. It is found that the maximum substrate conversion, ethanol concentrations, and ethanol yield are achieved when the fermentation system operates at very low dilution rates (around 0.01) when the system operated in the oscillatory conditions as shown in Figures 11 and 12. However, the maximum ethanol production rate was achieved in the periodic behavior when the dilution rate was in the range 0.06−0.07. The mechanism of structuring Z. mobiliz cells into viable, nonviable, and dead cells and the relation between high periodicity (period four) with increase in feed substrate concentration seems to give a reasonable explanation of the periodic behavior observed in continuous ethanol fermentation systems. The high concentration of biomass accompanied with a high concentration of ethanol can be explained by the mechanism at various concentrations of feed substrates. This analysis could be a good guide for further research for developing the bioethanol production process and improving the performance of the fermentation system using cell recycle, ethanol removal tool, and multiple reactor fermentation cascade and comparing them considering mass transfer limitations and utilizing between different microorganisms for optimizing the efficiency of ethanol production processes.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Notes

The authors declare no competing financial interest. R



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