Dynamic Bifurcation and Chaotic Behavior of an Ethanol

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Ind. Eng. Chem. Res. 2004, 43, 1260-1273

Static/Dynamic Bifurcation and Chaotic Behavior of an Ethanol Fermentor Parag Garhyan* and S. S. E. H. Elnashaie Chemical Engineering Department, Auburn University, Auburn, Alabama 36849

The problems associated with the efficient production of fuel ethanol from lignocellulosic waste/ materials necessitate a multidisciplinary integrated systems approach, combining various fields in an optimal fashion. These fields/disciplines are genetic engineering, membrane/polymer science, mathematical modeling and process control, and finally nonlinear dynamics and optimization. This paper explores the conditions for increasing the productivity and yield of ethanol by fermentation using mathematical modeling and nonlinear analysis. A possible increase of the sugar conversion and ethanol productivity using periodic and chaotic operation at high sugar concentrations is investigated for continuous stirred tank fermentors with and without ethanol-selective membranes. Continuous ethanol removal using membranes leads to increased productivity and tends to stabilize the system by eliminating the oscillations. Simulations show that, in some pathological cases, increasing the area of the ethanol-removal membranes can lead to a decreased yield and productivity in certain regions. Introduction Both nature and modern industry, as well as its associated trends, generate huge amounts of lignocellulosic materials that can be useful as raw materials for a wide spectrum of products.1,2 The most important among the spectrum of useful products to be produced from this abundant lignocellulosic raw material is ethanol, a clean fuel, an important solvent, and a source of other important chemicals and fine chemicals. The production of ethanol from carbohydrates can be a difficult task, with the level of difficulty depending on the choice of raw materials. The primary step in the production of ethanol from cellulose/hemicellulose is the separation of lignin from lignocellulose. The production of ethanol from cellulosic waste does not require this step. Sugars are produced by the hydrolysis of biomass and other cellulosic sources. The different hydrolysis technologies for different cellulosic feedstocks always produce a mixture of sugars (glucose, xylose, lactose, arabinose, etc.). These two steps of the process have been the subjects of extensive successful earlier research.3-7 In this paper, we focus our attention on the process of bioconversion (i.e., fermentation) with the aim of efficiently fermenting the sugars produced by hydrolysis and obtaining better productivity and yield of ethanol. Physical Significance The presence of sustained oscillations in the Zymomonas mobilis fermentation process has been widely reported in the scientific literature.8-15 In this paper, using an experimentally verified model, oscillations consistent with experimental data are simulated in certain ranges of the model parameters. Further nonlinear analysis indicates that, with a change in the parameter values, these simple oscillations bifurcate * To whom correspondence should be addressed. E-mail: [email protected]. Fax: 1-334-844-2063. Tel.: 1-334844-2033.

into complex phenomena such as fully developed chaos. Regions of multiple steady states (multiplicity) are identified where more than one stable steady states exist. Simulation results show that the average conversion of sugar and the average yield/productivity of ethanol are sometimes higher for periodic and chaotic attractors than for the corresponding steady states despite the fact that, during oscillations, the values of the state variables fall below the average values of the oscillations for some time. Ethanol is known to be an inhibitor to the microorganisms involved in this fermentation; thus, in situ removal of the ethanol produced in the fermentor will lead to an enhanced yield and productivity. Integrating these phenomena of nonlinear dynamics with membrane science (i.e., using a permselective membrane to remove the product ethanol) provides a higher yield and productivity of ethanol. Simulation results also show that the introduction of ethanol-removing membrane stabilizes oscillations in the fermentor (thus acting as a controller to eliminate instabilities). Continuous Ethanol Removal We consider membrane separation of the ethanol produced in the fermentor. This involves the use of a membrane that has some selectivity for a specific product (ethanol in our case) within a reaction environment, with a gas/liquid “sweep stream” on the nonreaction side to remove product from the membrane surface.16,17 This approach is used here to remove the inhibitory product (ethanol) in situ. For the experimental fermentor considered for modeling in this paper, the permselective membrane used by Jeong et al.18 is considered. Oscillatory Behavior The occurrence of oscillations in anaerobic cultures has been well documented experimentally and theoretically.8-15 These oscillations are suitable for in-depth studies of the microorganism physiology.19,20 Many

10.1021/ie030104t CCC: $27.50 © 2004 American Chemical Society Published on Web 02/07/2004

Ind. Eng. Chem. Res., Vol. 43, No. 5, 2004 1261

experimentally verified models have been formulated to simulate the oscillatory behavior of Zymomonas mobilis. Jarzebski8 proposed a three-compartment model that included substrate limitation and product inhibition (with viable, nonviable, and dead cells). This proposed model was an extension of the work of Ghommidh et al.,9 who considered that sustained oscillations were present only at high substrate concentrations. It was concluded that the mechanism of decrease in product inhibition with decreasing substrate concentration was a feasible explanation for the sustained or slowly damped oscillations at substrate saturation and strongly damped oscillations leading to stable steady state at substrate shortage. The proposed model was able to represent the experimental data during the oscillatory behavior of the Zymomonas mobilis culture. Plate count and slide culture estimations provided the experimental support for the presence of viable and nonviable or dead cells. A more phenomenological model in terms of macroscopic variables such as the ethanol, substrate, and biomass concentrations as the only experimental quantities that need to be measured, in contrast to other structured models, was proposed by Daugulis and coworkers.10,11 Their model considers inhibitory culture conditions in the recent past affecting subsequent cell behavior. They proposed the concept of the “dynamic specific growth rate” to explain the inhibition of the instantaneous specific growth rate due to “ethanol concentration change rate history”. This change in the ethanol concentration change rate history is a result of changes in the physiological state of the culture. An unsegregated, structured two-compartment model representation was proposed by Jobses et al.12-14 It considered biomass as being divided into compartments (K compartment and G compartment) containing specific groupings of macromolecules (e.g., the K compartment is identified with RNA, carbohydrates, and monomers of macromolecules, whereas the G compartment is identified with protein, DNA, and lipids). Jobses and co-workers12-14 studied oscillatory behavior utilizing this model in which the synthesis of a cellular component e (which is essential for both growth and product formation) had a nonlinear dependence on the ethanol concentration. Hence, the inhibition by ethanol did not directly influence the specific growth rate of the culture, but its effect was indirect. The biokinetics developed in this two-compartment model is used in the present investigation. Model Development and Discussion An experimentally verified model,12-14 which is an unsegregated, structured two-compartment representation, considers the biomass as being divided into compartments containing specific groupings of macromolecules (e.g., protein, DNA, and lipids). This twocompartment model is modified for use in the present investigation. A relatively simple unsegregated, structured model was suggested on the basis of the introduction of an internal key compound (e) of the biomass. The activity of this compound is expressed in terms of the concentrations of substrate, product, and the compound (e) of the biomass itself. Thus, the rate of formation of the key compound (re) is given by

re ) f(CS) f(CP)Ce

(1)

Figure 1. Schematic diagrams of the fermentor. All flow rates and concentrations are shown.

where the substrate dependence function, f(CS), is given by a Monod-type relation

f(CS) )

CS K S + CS

(2)

The experimental data of Jobses and co-workers12-14 show that the relation between the alcohol concentration, CP, and the alcohol dependence function, f(CP), is a second-order polynomial in CP of the form

f(CP) ) k1 - k2CP + k3CP2

(3)

The model developed by Jobses et al.12-14 is a fourdimensional model with the following components: substrate (S), product (P), microorganism or biomass (X), and key internal component (e). On the basis of the above discussion, the dynamic model for the four components (e, X, S, and P) is given by the following set of ordinary differential equations (eqs 4-7). The value of rate constant p is taken to be equal to 1 h-1.13 Figure 1 shows schematic diagrams of the fermentor and in situ ethanol-removal membrane module setup, including all flow rates and concentrations.

(

)

CSCe dCe ) (k1 - k2CP + k3CP2) + dt KS + CS DinCe0 - DoutCe (4)

(

)

dCX CSCe )p + DinCX0 - DoutCX dt K S + CS

( )( ( )(

)

(5)

CSCe dCS -1 - mSCX + DinCS0- DoutCS )p dt YSX KS + CS (6)

)

dCP CSCe 1 )p + mPCX + DinCP0 - DoutCP dt YPX KS + CS

()

a (C - CPM) (7) VF P

1262 Ind. Eng. Chem. Res., Vol. 43, No. 5, 2004

ethanol yield

Table 1. Base Set of Parameters Used parameter

value

k1 (h-1) k2 (m3/kg‚h) k3 (m6/kg2.h) mS (kg/kg‚h) mP (kg/kg‚h) YSX (kg/kg) YPX (kg/kg) KS (kg/m3) P (m/h) DM in (h-1) CX0 (kg/m3) CP0 (kg/m3) Ce0 (kg/m3) VF (m3) VM (m3) F (kg/m3)

16.0 4.97 × 10-1 3.83 × 10-3 2.16 1.1 2.444 98 × 10-2 5.263 15 × 10-2 0.5 0.1283 4.0 0 0 0 0.003 0.0003 789

YP )

ethanol production rate per unit volume of the fermentor (kg/m3‚h) PP ) CPDout + CPMDMout

( )

dCPM a (C - CPM) + DM inCPM0 - DM outCPM ) dt VM P (8) a(CP - CPM) VM(F)

a(CP - CPM)

Dout ) Din -

VF(F)

a ) AMP

(9)

(10) (11)

In the present investigation, the nonlinear model (consisting of ODEs 4-8 and algebraic eqs 9-11) is used to explore the complex static/dynamic bifurcation behavior of this system for the cases with and without ethanol removal. The system parameters for one of the experimental runs of Jobses et al.12-14 showing oscillatory behavior are used as the base set of parameters in the present investigation and are given in Table 1. The performance of the fermentor is evaluated in terms of the sugar conversion, ethanol yield, and ethanol production rate. The following simple relations were incorporated into Fortran programs to calculate the performance evaluators

substrate (sugar) conversion XS )

CS0Din - CSDout CS0Din

( ) VM VF

For the oscillatory and chaotic cases, the average conversion, X h S; average yield, Y h P; and average production rate, P h P, are computed. These quantities are defined as

Equation 7 contains a term (the last term on the righthand side) for ethanol removal by the membrane (the membrane-side differential equation is given by eq 8 below). In the Jobses et al. works,12-14 there is no membrane, which, in this extended five-dimensional model, corresponds to a permeation area of AM ) 0 and to a ) 0 in eq 7. Jobses et al.12-14 successfully used the above fourdimensional model to simulate the oscillatory behavior of an experimental continuous fermentor without ethanol removal in the region of high feed sugar concentrations. Assuming perfect mixing in the membrane side to simplify this preliminary analysis, the membrane side equations are given by

DM out ) DM in +

CPDoutVF + CPMDMoutVM - CP0DinVF CS0DinVF

X hS )

∫0τXS dt τ

Y hP )

∫0τYP dt τ

P hP )

∫0τPP dt τ

where the τ values represent one period of the oscillations in the periodic cases and are taken long enough to provide a reasonable representation of the “average” behavior of the chaotic attractor in the chaotic cases. Presentation Techniques and Numerical Tools The bifurcation diagrams were obtained using the software package AUTO97.21 This package is able to perform both steady-state and dynamic bifurcation analyses, including the determination of entire periodic solution branches using efficient continuation techniques.22 The DIVPAG subroutine available with IMSL libraries for Fortran with an automatic step size to ensure accuracy for stiff differential equations is used for numerical simulation of periodic as well as chaotic attractors. A Fortran program was written for plotting the Poincare´ plots.23 For the dynamic behavior, classical time traces and phase planes are used; however, for high-periodicity and chaotic attractors, these techniques are not sufficient. Therefore, other presentation techniques are used that are based on the plotting of discrete points of intersection (return points) between the trajectories and a hypersurface23 (Poincare´ surface) chosen at a constant value of a state variable (CX ) 1.55 kg/m3 in the present investigation). These discrete points of intersection are taken such that the trajectories intersect the hyperplane transversely and cross it in the same direction. The return points are used to construct Poincare´ oneparameter bifurcation and return-point diagrams. Results and Discussion The results are discussed in three different sections. The first two sections include the cases of fermentation without ethanol removal, where the bifurcation analysis is carried out for two different bifurcation parameters, Din (dilution rate) and CSO (influent feed substrate concentration) with AM ) 0.0. The third section includes the cases of fermentation with continuous ethanol removal, where the bifurcation parameter used is the area of permeation for ethanol (AM) for a particular set of values of Din and CSO.


Figure 2. Bifurcation diagrams for CSO ) 150.3 kg/m3 and AM ) 0 with Din as the bifurcation parameter. Legend: s, stable steady-state branch; - - -, unstable steady-state branch; bbb, stable periodic branch; OOO, unstable periodic branch; [[[, average of oscillations. Table 2. Conclusion Table for Different Cases Investigated Case A: Dilution Rate (Din) as the Bifurcation Parameter

CS0 (kg/m3) 150.3 200

Din HB (h-1)

Din SLP (h-1)

Din HT (h-1)

type of periodic attractor before homoclinic termination (HT)

5.20 × 10-2 6.20 × 10-2 4.21 × 10-2 period 4 5.40 × 10-2 2.25 4.5835 × 10-2 fully developed chaos

Case B: Feed Sugar Concentration (CS0) as the Bifurcation Parameter Din (h-1)

CS0 HB (kg/m3)

CS0 SLP (kg/m3)

CS0 HT (kg/m3)

0.045

132.0

147.0

165.7

banded chaos

Case C: Area of Permeation (AM) as the Bifurcation Parameter CS0 (kg/m3)

Din (h-1)

AM HB (m2)

AM SLP (m2)

150.3 200.0

0.0422 0.045 84

0.018 79 0.023 468 6

0.001 84 0.045 56

no HT no HT

The three bifurcation parameters used in this investigation (Din, CSO, and AM) are chosen because they can

be easily manipulated during the design and operation of a laboratory- or full-scale fermentor. Table 2 summarizes the locations of the Hopf bifurcation points (HB), homoclinic termination point (HT), and static limit point (SLP) and the type of periodic attractor before the homoclinic termination with respect to the bifurcation parameters. As shown in Table 2, when CSO increases (with Din as the bifurcation parameter), the positions of HB and SLP move to the right (increase in Din), but the speed of movement of SLP is greater than that of HB. This prevents the formation of ordinary fully developed chaos, because the distance between the HB point (from which the periodic branch emanates) and the saddle point (at which the periodic branch terminates) is not sufficient to produce fully developed chaos. The same observation is true when CSO is taken to be the bifurcation parameter. For the cases with AM as the bifurcation parameter, it is seen that no homoclinic termination occurs and a higher value of AM is required to reach the HB point. (A) Bifurcation Analysis Using the Dilution Rate (Din) as the Bifurcation Parameter. Case A-1: Constant CSO ) 150.3 kg/m3, AM ) 0.0 m2. Figure 2 shows the static and dynamic bifurcation diagrams with


Figure 3. One-dimensional Poincare´ bifurcation diagram for CSO ) 150.3 kg/m3 and AM ) 0.

Figure 4. Period change with Din for CSO ) 150.3 kg/m3 and AM ) 0.

the dilution rate Din as the bifurcation parameter for a high sugar feed concentration. It is clear that the bifurcation diagram is an incomplete S-shaped hysteresis type with a static limit point (SLP) at Din SLP ) 0.062 h-1. The dynamic bifurcation shows a Hopf bifurcation (HB) with a periodic branch emanating from it at Din HB ) 0.052 h-1; the amplitudes of the oscillations increase as Din decreases. The Poincare´ bifurcation diagram (Figure 3) shows that the periodicity of the system changes from periodone (P1) to period-two (P2) to period-four (P4) oscillations, in a sequence of incomplete period doubling. Figure 4 also shows the location of the period-doubling points. The first point (PD1) is at Din ) 0.042 36 h-1 and the second (PD2) is at Din ) 0.042 125 h-1. The periodic attractor terminates homoclinically with periodicity four (P4) at Din HT ) 0.042 12 h-1 with infinite period. The complete bifurcation diagram in this case can be divided into five regions (Figure 2A). Region 1 (Din >

Din SLP) has a unique stable point attractor at the lowconversion branch. It can be seen that the sugar concentration, CS, increases very slightly from 30.93 to 30.997 kg/m3 with increasing Din (Figure 2A). Correspondingly, the ethanol concentration, CP, decreases from 57.86 to 57.394 kg/m3 (Figure 2B) with increasing Din in this region. The sugar conversion, XS remains almost constant at 0.793 (Figure 2C) in this region. In contrast, the production rate, PP, increases from 3.63 to 5.165 kg/m3‚h because of the increase in the value of Din (Figure 2D). Region 2 (Din SLP > Din > Din HB) is characterized by the existence of bistability when there is a very highconversion point attractor as well as a low-conversion point attractor. In this region, a saddle-type unstable steady state also exists that begins at Din SLP. A comparison between the values of the low- and highconversion stable static branches at Din ) 0.06 h-1 shows that the high-conversion branch achieves improvement sof 25.416% for XS, 25.636% for YP, and 27.38% for PP. Thus, it is advisable to avoid the lowconversion stable static branch in this region. The physical significance of this region plays a very important role during the start-up of the fermentor. Depending on the start-up conditions, either of the two stable points (one of which is at low conversion, while the other is the desired high-conversion steady state) can be achieved. Region 3 (Din HB > Din > Din PD1) has a very highconversion stable static branch, as well as a stable periodic branch with period one (P1). Region 4 (Din PD1 > Din > Din HT) has a very highconversion stable static branch, as well as an unstable periodic branch. This region is the characteristic region of this case because of the presence of an incomplete period-doubling sequence from P2 to P4 as shown in Figures 3 and 4. This region begins at PD1, where Din ) 0.042 36 h-1, and P2 is sustained until it reaches PD2 at Din ) 0.042 125 h-1, where P2 changes to P4 (Figures 3 and 4). Region 5 (Din < Din HT) has three steady states, two of which are unstable, leaving only the steady state with the highest conversion as stable. This region has the highest conversion (almost 1.0) and the highest ethanol yield (0.51) as compared to the previous four regions. On the other hand, this region has the lowest ethanol production rate because of the low value of the dilution rate Din. The upper steady state gives the highest ethanol concentration and conversion as compared to all of the other steady states (Figure 2B,C). However, it occurs in a very narrow region at very low Din [i.e., very low (qin/VF)], so its production rate PP is extremely low (Figure 2D). In general, there is a tradeoff between concentration and productivity, which requires an economic optimization study to determine the optimum value of Din. However, such an optimization study will have to take into consideration the fact that some periodic attractors have higher ethanol yields and production rates than the corresponding steady states. Case A-2: Constant CSO ) 200 kg/m3, AM ) 0.0 m2. This is a case with a very high feed sugar concentration. Figure 5 shows the static and dynamic bifurcation diagrams with the dilution rate (Din) as the bifurcation parameter and the enlargement of the chaotic region. The highest conversion can be achieved on the upper


Figure 5. Bifurcation diagrams for CSO ) 200 kg/m3 and AM ) 0 with Din as the bifurcation parameter. See Figure 2 for legend.

branch in the range of Din < 2.25 h-1 (almost complete conversion, Figure 5E), and PP increases with increasing Din (Figure 5F). This case is characterized by the existence of fully developed chaos. The transition from a stable period-one periodic attractor to a chaotic

attractor is shown in Figure 6. This figure shows the time trace and corresponding phase portraits for four different dilution rates (Din), each one of which gives period-one (P1), period-two (P2), period-four (P4), and chaotic attractors, respectively. A similar transition is


Figure 6. Phase portraits and time traces showing P1, P2, P4, and chaotic attractors for CSO ) 200 kg/m3 and AM ) 0 at different Din values.

also shown as the one-dimensional Poincare´ diagrams in Figure 7A,B. The complete bifurcation diagram in this case can be divided into five regions (Figure 5A,B), and the regions of interest are as follows: Region 2 (Din SLP > Din > Din HB), where Din HB ) 0.054 h-1, is the multiplicity region. Bistability exists, and there is a very high-conversion stable static branch as well as a low-conversion stable static branch. Also a saddle-type unstable steady-state exists (Figure 5A,C,E,F). A comparison between the values of the lowand high-conversion stable static branches at Din ) 1.5 h-1 shows that the high-conversion branch achieves improvements of 109.99% for XS, 110.96% for YP, and 120.26% for PP. Bistability behavior plays an important role in the start-up policy, as an improper start-up can eventually lead to an unwanted lower-conversion steady state. Region 3 (Din HB > Din > Din PD1), where Din PD1 ) 0.046 04 h-1, has a very high-conversion stable static branch, as well as a stable periodic branch with P1 (Figure 5B,D). A comparison of the average values of the oscillations and corresponding steady state at Din ) 0.045 h-1 shows improvements of the following

order: 12.01% for X h S, 15.434% for Y h P, and 16.277% for P h P. Nevertheless, the average values are still less as compared to the high-conversion stable static branch. Region 4 (Din HT < Din < Din PD1), where Din HT ) 0.045 835 h-1, has a very high-conversion stable static branch, as well as a periodic (chaotic) branch. This region is the characteristic region of this case because of the occurrence of period doubling to banded chaos (two bands); the sequence is P1 f P2 f P4 f P8 f ‚‚‚ f banded chaos, which terminates homoclinically at Din HT. Figure 7A is enlarged in Figure 7B, where the two bands of chaos and the period-doubling sequence are clearly shown. Figure 7C is the return-point histogram for the variable CS at Din ) 0.045 84 h-1. It should be noted that, as the substrate feed concentration is increased beyond 200 kg/m3, there is no change in the shape of the chaos. (B) Bifurcation Analysis Using the Feed Sugar Concentration (CSO) as the Bifurcation Parameter. Case B: Constant Din ) 0.045 h-1, AM ) 0.0 m2. This case has a constant dilution rate of 0.045 h-1 and an area of permeation equal to zero, while the feed sugar concentration (CSO) is varied as the bifurcation parameter. Figure 8 shows the static and dynamic bifurcation


Figure 7. Dynamic characteristics at CSO ) 200 kg/m3, AM ) 0. (A) One-dimensional Poincare´ bifurcation diagram. (B) Enlargement of chaos region in A. (C) Return point histogram at Din ) 0.045 84 h-1.

diagrams with the substrate feed concentration, CSO, as the bifurcation parameter. This case is characterized by

the presence of a period-doubling route to banded chaos and subsequent homoclinic termination of this chaotic attractor. The bifurcation diagram in this case can be divided into four regions (Figure 8A), and the regions of interest are as follows: Region 2 (CSO SLP < CSO < CSO HT), where CSO SLP ) 147 kg/m3, has a periodic attractor, together with a stable static attractor (the highest-conversion branch). The periodic branch in this region changes its periodicity in a period-doubling sequence leading to chaos, and the chaotic attractor terminates homoclinically at CSO HT ) 165.7 kg/m3 as shown in Figures 8 and 9. A comparison of the average values of the oscillations and corresponding steady state at CSO ) 160 kg/m3 shows improvements of the following order: 15.376% for X h S, 15.15% for Y h P, and 15.064% for P h P. Region 4 (CSO < CSOHB) has a unique stable static attractor, where the substrate concentration, CS, initially increases very slightly from 0.027 to 0.3 kg/m3 with increasing CSO (in the range 100 < CSO < 116.778; this is due to the fact that the sugar fed is completely consumed by the microorganisms). After this point, the sugar concentration increases steadily to 12.17 kg/m3 (Figure 8A). Similar behavior is also observed for CP, which increases toward 58.271 from 48.696 kg/m3 (Figure 8B). The conversion and ethanol yield decrease, whereas the production rate increases with increasing CSO. Figure 9A is a one-dimensional Poincare´ bifurcation diagram for the state variable CS with CSO as the bifurcation parameter, which shows the period-doubling route to chaos. A portion of Figure 9A is enlarged in Figure 9B, where the two bands of chaos and the perioddoubling sequence are clearly shown. (C) Bifurcation Analysis Using the Area of Permeation (AM) as the Bifurcation Parameter. To improve the productivity and yield, continuous removal of ethanol is incorporated in the analysis. Bifurcation analysis is carried out for a system having the area of permeation (AM, m2) as the bifurcation parameter. AM is chosen as the bifurcation parameter because the membrane module used for ethanol removal can be easily modified to change the area of permeation, thereby leading to a change in the permeation rate of ethanol across the membrane. Moreover, a change in area helps in visualizing how the multiplicity (and hence the chaotic or complex attractors) give way to a stable unique steady state with a relatively high production rate; thus, the membrane acts as a controller (or stabilizer) that reduces and eventually eliminates the chaotic and oscillatory steady states. Bifurcation analyses are performed for two different cases having fixed values of CSO and Din. The two cases discussed next correspond to the cases A-1 and A-2 that were discussed earlier in this paper. Case C-1: Constant CSO ) 150.3 kg/m3 and Din ) 0.0424 h-1. This is the case that has the feed sugar concentration, CSO, equal to 150.3 kg/m3 and the dilution rate, Din, equal to 0.0422 h-1. This case corresponds to case A-1 discussed earlier. The bifurcation diagrams are shown in Figure 10 with the area of permeation (AM) as the bifurcation parameter. It can be observed that, for AM ) 0 (which corresponds to the case with no continuous ethanol removal), there is one stable periodic attractor surrounding an unstable static steady state. One stable and one unstable static point attractors also


Figure 8. Bifurcation diagrams for Din ) 0.045 h-1 and AM ) 0 with CSO as the bifurcation parameter. See Figure 2 for legend.

exist at this value of AM. It is also seen that the dynamic bifurcation shows a Hopf bifurcation point (HB) at about AM HB ) 0.015 74 m2. The complete bifurcation diagram in this case can be divided into three regions (Figure 10A). Region 1 (AM > AM HB) has only one unique stable steady state, where the value of CS decreases from 14.37 kg/m3 to almost zero with increasing value of the bifurcation parameter; correspondingly, the value of CP remains almost constant at about 58.3 kg/m3, whereas the value of CPM increases from 1.89 to 3.48 kg/m3. The almost constant value of CP and the corresponding increase in CPM are due to the fact that the ethanol produced is continuously permeated across the membrane to be swept away by the sweep liquid. In the same region, with an increase in AM, the conversion, XS, increases from 0.90 to 0.997 (Figure 10D), and the productivity increases from 9.02 to 10.39 kg/m3‚h (Figure 10E). It is observed that increasing the area of permeation beyond 0.030 m2 no longer affects the conversion (which is almost equal to 1.0). Region 2 (AM SLP < AM < AM HB), where AM SLP ) 0.001 84 m2, has a stable periodic attractor (surrounding the unstable steady state) with increasing amplitude of oscillation with decreasing value of AM, as shown in

Figure 10. It can be observed that the unstable steadystate ethanol concentration (CP) remains almost constant at about 58.3 kg/m3, but the membrane-side ethanol concentration (CPM) increases correspondingly. The characteristic feature of this region is that the h P, and average conversion, yield, and productivity (X h S, Y P h P, respectively) are higher than the corresponding values for the unstable steady state. A comparison between the values of the static branch and the average of the periodic branch at AM ) 0.005 m2 shows that the percentage improvements are as follows: 6.46% for X h S, h P. 6.44% for Y h P, and 6.18% for P Figure 10B-E shows the effects of increasing the area of permeation on the ethanol concentration (CP), substrate conversion (XS), and production rate (PP). It is evident that the average values for the oscillatory attractor are greater than the corresponding values attained by the unstable static attractor (as shown by the diamond-shaped points in this figure). Thus, it can be concluded that operating the fermentor at the oscillatory state will eventually give a better ethanol production rate, yield, and conversion. Another important conclusion is that the membrane acts as a controller for the fermentation process. As seen in Figure 10, as the area of permeation increases (leading to an increase in


Figure 9. One-dimensional Poincare´ bifurcation diagram (CS vs CSO) for Din ) 0.045 h-1 and AM ) 0.

the removal rate of ethanol), the amplitude of the periodic attractor decreases. Moreover, further increasing the area of permeation finally leads to complete elimination of oscillations, thus stabilizing the fermentation process. Another important observation is that the values of the sugar conversion, ethanol yield, and productivity decrease for certain values of AM (0.001 84 < AM < 0.0302 for XS, 0.001 84 < AM < 0.011 38 for YP, and 0.001 84 < AM < 0.018 79 for PP) as the value of AM is increased beyond AM SLP (Figure 10D,E). These regions are marked by vertical dotted lines in Figure 10 and are called the “red zone”. Thus, it can be concluded that the introduction of an ethanol-removal membrane can lead to a lower/inferior conversion, yield, and productivity for certain ranges of AM. To be clearer, membrane areas in the range of 0.001 84-0.0302 m2 should be avoided, as this will lead to an inferior conversion as compared to using a membrane area of less than 0.001 84 m2. Similarly, membrane areas in the range of 0.001 84-0.011 38 m2 should be avoided because of inferior yields, and areas in the range of 0.001 840.018 79 m2 should be avoided to avoid an inferior production rate. This behavior is observed because of

the typical shape of the static bifurcation diagrams in this case, where the complete-conversion branch does not extend beyond the static limit point and, beyond the SLP, only a low-conversion/yield/productivity stable static branch exists. Although the value of this lowconversion/yield/productivity stable static branch increases with further increases in the area of permeation, its value reaches a maximum after a certain value of AM outside the red zone. Usually, such complex behavior is associated with some nonmonotonic process such as the rate of reaction, where the rate of reaction increases in some regions but decreases in other regions with increasing concentration.23 The physical significance of this finding is important for the design of an experimental/industrial membrane fermentor, as these regions of inferior conversion and yield/productivity should be avoided. Case C-2: Constant CSO ) 200 kg/m3 and Din ) 0.045 84 h-1. For the investigation of the permeation area as the bifurcation parameter, the value of CSO and Din were taken such that a chaotic attractor exists for the case of AM ) 0.0 m2, corresponding to case A-2 discussed above. There is a static limit point at AM SLP ) 0.045 56 m2 and a Hopf bifurcation point at AM HB ) 0.023 468 6 m2, as seen in Figure 11. As the value of the bifurcation parameter (AM) is increased, the chaotic attractor at AM ) 0.0 m2 follows a period-halving sequence to give a period-one periodic attractor at AM PD ) 0.000 636 m2 (Figure 12). The complete bifurcation diagram in this case can be divided into four regions (Figure 11A), as discussed below. Region 2 (AM HB < AM < AM SLP) is characterized by the presence of multiple steady states (Figure 11). Bistability exists, with a very high-conversion (almost complete conversion) stable steady state coexisting with a lower-conversion stable static attractor. This loweror moderate-conversion stable static steady state has increasing values of conversion, yield, and productivity with increasing AM. In region 4 (AM < AM PD), with increasing AM, the chaotic attractor stabilizes to give a stable periodic attractor of periodicity one (P1), as shown in the onedimensional Poincare´ diagram (Figure 12). It is seen that, at AM ) 0, two-banded chaos arises that loses its chaotic behavior by a period-halving route with increasing AM (Figure 12A). At about AM ) 0.000 636 m2, a period-one stable attractor is obtained (Figure 12B). Again, the averages of the oscillations for the chaotic h P, and and periodic attractors give higher values of X h S, Y P h P than the corresponding steady states. It is observed that, for the region of AM < AM SLP, one high-conversion stable steady state exists, together with an oscillatory state. The conversion, yield, and production rate of the periodic attractor increase with increasing area of permeation (where the average of the oscillatory state is higher than that of the corresponding steady state). The amplitude of the periodic attractor decreases with increasing AM, and eventually, after the HB point, it gives rise to a unique stable steady state with almost complete conversion. For the region with AM > AM SLP, only one complete-conversion stable steady state is present. This reconfirms the conclusion that the membrane (which results in the continuous removal of


Figure 10. Bifurcation diagrams for CSO ) 150.3 kg/m3 and Din ) 0.0424 h-1 with AM as the bifurcation parameter. See Figure 2 for legend.

ethanol from the fermentation broth) acts as a controller for the process. It is seen that the multiplicity and oscillations are reduced and finally eliminated, thus stabilizing the process (Figures 11 and 12).

The red zone (marked by dotted vertical lines) for this case consists of the ranges 0.045 56 < AM < 0.091 for XS, 0.045 56 < AM < 0.066 for YP, and 0.045 56 < AM < 0.075 for PP as the value of AM is increased beyond


Figure 11. Bifurcation diagrams for CSO ) 200 kg/m3 and Din ) 0.045 84 h-1 with AM as the bifurcation parameter. See Figure 2 for legend.

AM SLP ) 0.045 56 m2 (Figure 11D,E). Thus, it can be inferred that increasing the area of permeation (AM) can lead to a lower/inferior conversion, yield, and productivity within certain ranges of AM, which should be avoided during the design of a fermentor.

Conclusions and Recommendations As the need for more environmentally friendly and economically viable technologies/processes is being emphasized in all fields of science and technology, it is


include periodic and chaotic attractors, the operation of the fermentor under periodic/chaotic attractor conditions gives higher substrate conversions, higher product yields, and higher production rates than the corresponding steady states. It is also shown that the inhibitory effect of product ethanol can be overcome by the continuous removal of ethanol from the fermentor. Ethanol removal also causes a change of the static and dynamic characteristics of the system, as well as increasing the yield and productivity of ethanol. The ethanol-removal membranes act as controllers that help to stabilize the fermentation process by removing the multiplicity regions and eliminating the oscillations. This phenomenon can be used to effectively design a fermentor incorporating continuous ethanol removal and eliminating the inherent Z. mobilis oscillations. It is also shown that increasing the area of permeation can lead to regions (red zones) where the substrate conversion and ethanol yield/productivity decrease and then increase gradually. Our future work will include the experimental verification of this phenomenon associated with the pathological behavior of such systems with nonmonotonic characteristics. These observations are made possible by the use of bifurcation analysis, which provides the complete static and dynamic behavior of the process, unlike mathematical simulation, where some process characteristics can be completely missed or neglected, as only a limited number of simulation runs can be performed. The future direction of our work will concentrate on the experimental verification of the findings of this investigation and the economic and environmental evaluation of their impact. Acknowledgment This research was supported by Auburn University though Grant 2-12085. Nomenclature Figure 12. One-dimensional Poincare´ bifurcation diagram (CS vs AM) for CSO ) 200 kg/m3 and Din ) 0.045 84 h-1.

imperative that the process of converting lignocellulosic materials to ethanol also be made more economical and environmentally friendly. As ethanol is an important chemical, to help in environmental protection, any improvement that makes use of lignocellulosic wastes to produce ethanol is indeed contributing greatly toward environmental improvement. To achieve this goal, it is important that different disciplines such as genetic engineering, material science, mathematical modeling and process control, and finally nonlinear dynamics be integrated efficiently. This paper concentrates mainly on the mathematical modeling and nonlinear dynamics parts of the overall interdisciplinary exercise. The investigation reveals the rich static and dynamic bifurcation behavior of the fivedimensional process discussed herein. Efforts are concentrated on the effects of different operating parameters such as the dilution rate, the substrate feed concentration, and the area of permeation on the behavior of the system. The rich static and dynamic behavior of the membrane fermentor has important practical and physical implications. The importance with regard to the start-up policy is evident in the multiplicity regions, where the lower-conversion/yield steady state should be avoided. In the ranges that

Ci ) concentration of component i (kg/m3) D ) dilution rate (h-1) V ) volume (m3) E ) fraction of biomass (kg/kg) k1 ) empirical constant (h-1) k2 ) empirical constant (m3/kg‚h) k3 ) empirical constant (m6/kg2.h) KS ) Monod constant (kg/m3) mS ) maintenance factor based on substrate requirement (kg/kg‚h) mP ) maintenance factor based on product formation (kg/ kg‚h) ri ) production rate of component i (kg/m3‚h) P ) permeability of the membrane (m/h) AM ) area of permeation (m2) PP ) production rate of ethanol (kg/m3‚h or kg/h) P h P ) average production rate of ethanol (kg/m3‚h or kg/h) XS ) substrate conversion X h S ) average substrate conversion YP ) yield of product (ethanol) Y h P ) average yield of product (ethanol) YSX ) yield factor of biomass on substrate (kg/kg) YPX ) yield factor of biomass on product (kg/kg) Greek Symbols µ ) specific growth rate (h-1) τ ) period of oscillation (h) Subscripts of Concentration Ci e ) key component of the biomass e0 ) influent key component of the biomass

Ind. Eng. Chem. Res., Vol. 43, No. 5, 2004 1273 P ) product (ethanol) in the fermentor side PM ) product (ethanol) in the membrane side P0 ) influent product (ethanol) to fermentor S ) substrate (sugar) S0 ) influent substrate (sugar) X ) biomass (microorganisms) X0 ) influent biomass (microorganisms) M ) membrane side F ) fermentor side in ) inlet out ) outlet Abbreviations CSTR ) continuous stirred tank reactor HB ) Hopf bifurcation HT ) homoclinic termination PD ) period doubling PDi ) ith period-doubling point Pi ) periodicity i of the periodic orbit SLP ) static limit point

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Received for review February 6, 2003 Revised manuscript received October 21, 2003 Accepted January 9, 2004 IE030104T

Dynamic Bifurcation and Chaotic Behavior of an Ethanol

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