Strategies for the Analysis of Continuous Bioethanol Fermentation

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Strategies for the Analysis of Continuous Bioethanol Fermentation under Periodical Forcing Chi Zhai, Ahmet Palazoglu, Shining Wang, and Wei Sun Ind. Eng. Chem. Res., Just Accepted Manuscript • DOI: 10.1021/acs.iecr.6b03930 • Publication Date (Web): 22 Mar 2017 Downloaded from http://pubs.acs.org on March 27, 2017

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Industrial & Engineering Chemistry Research

Strategies for the Analysis of Continuous Bio-ethanol Fermentation under Periodical Forcing Chi Zhaia,b, Ahmet Palazoglub, Shining Wanga, Wei Suna* a

Beijing Key Lab of Membrane Science and Technology, College of Chemical

Engineering, Beijing University of Chemical Technology, 100029 Beijing, China b

Department of Chemical Engineering, University of California, Davis, CA 95616, USA

ABSTRACT: The process of continuous bio-ethanol fermentation often exhibits strong nonlinear dynamics and tends to generate self-oscillatory state trajectories. The aim of this work is to further investigate such oscillators by coupling with outside periodic forcing, so as to improve process performance. To that end, several strategies for analyzing periodically forced self-oscillatory processes are proposed. By periodically and deliberately forcing an operational parameter, the behavior of the forced fermentor is investigated using limit cycle bifurcation analysis. Specifically, periodic doubling and Neimark-Sacker bifurcations are investigated with limit cycle continuation diagrams. The limit cycle bifurcation analysis is compared with the simulation-based methods, such as stroboscopic and maximum bifurcation maps, and it is shown to be an efficient analysis tool for the periodically forced self-oscillatory system. Process performance enhancement by designing the fermentor to be self-oscillatory is also discussed, as well as the possibility of controlling the oscillatory behavior using external periodic forcing.

Keywords: Self-oscillation; limit cycle bifurcation analysis; Poincaré bifurcation diagram; route to chaos.

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Introduction As fossil fuels are facing depletion and the associated environmental issues can no longer be ignored, efforts are being made to acquire energy and commodity chemicals from renewable sources, including the manufacture of industrial products from biomass. Among various bio-products, bio-ethanol is an important renewable and sustainable alternative fuel source, either directly in the form of fuel ethanol or blended with gasoline1. The yeast Saccharomyces cerevisiae or bacterium Zymomonas mobilis are microorganisms that are particularly important in the production of ethanol from continuous fermentation processes2. The growth kinetics of these microorganisms, however, is known to be highly nonlinear3, and continuous ethanol fermentation tends to generate steady-state multiplicity4 or self-sustained oscillations5-7. Since the richness in dynamic behavior is often considered undesirable from the perspectives of process design and operation, a complete knowledge of the static and dynamic behaviors for such processes is necessary8. The studies on the oscillatory fermentation processes can be classified into two distinct areas: one is the elimination of the oscillations and the other is the coupling of external oscillations, as reviewed in Table 1. In this paper, we focus on the operation of continuous bio-ethanol fermentation (CBEF) process where some operational parameters are varied continuously, and explore possible strategies to analyze the dynamic system with periodically varied parameters. Regarding the investigation of periodically forced self-oscillatory systems, most previous analysis methods have been based on simulations. Abashar9-11 investigated the dynamic response of periodically forced CBEF processes by using stroboscopic Poincaré bifurcation maps. Ajbar12, 13 studied types of singularities for unstructured CBEF models and carried out an analysis by periodically adjusting the substrate feed concentration. One more problem is that these original studies only focused on the impact of the forcing amplitude on the dynamic behavior of the oscillatory CBEF processes, while the responses to both the forcing amplitude and the forcing frequency have not been thoroughly explored. 2


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Table 1. Review on study of self-oscillatory processes. Elimination of the oscillations Authors

Model / Merits

Mahecha14, 2005

Structured model; selective membrane; bifurcation analysis

Sridhar15, 2007

Structured model; key components; bifurcation analysis

Zhang et al.16, 2016

Bifurcation control by washout filter; bifurcation analysis

Haelssig17, 2008

Structured model; extractive fermentation; configurations

Skupin et al.18, 2015

Unstructured model; multiple feeding

Cysewski et al.19, 1977

Vacuum fermentation

Coupling of the oscillations Authors

Model / Merits

Yang and Su20, 1993

Oscillator in series configuration; bifurcation analysis

Cinar et al.21, 1987

Process stabilization; vibrational control

Azhar et al.22, 2014

MPC control of periodic forcing

Ajbar13, 2011

Unstructured model; chaos; Lyapunov coefficient

Abashar9,10, 2010; 2011 Structured model; stroboscopic Poincaré map Skupin et al.18, 2015

Unstructured model; multiple feeding;

The unforced CBEF model exhibits oscillations; we can choose the maximum bifurcation map to investigate the impact of forcing frequency as well as the stroboscopic bifurcation map to that of the forcing amplitude. Both methods are simulation-based and the associated inefficiency and significant computational cost 3



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are inescapable. Yet the qualitative change of a dynamic system can be analyzed efficiently based on the bifurcation analysis, which narrows the range of the parameter space, as only one or a very few sets of dynamic behaviors are acceptable for an industrial process. In the present study, the impact of external forcing parameters, amplitude or frequency, can be studied based on the limit cycle bifurcation analysis. The limit cycle bifurcation has codimension one and is analogous to the stationary state bifurcation on dimension one stroboscopic Poincaré maps. It is shown in our study that the limit cycle bifurcation analysis can be directly implemented on periodically forced self-oscillatory CBEF models, and the results are cross validated with those of simulation-based methods.

Analysis Strategies The inspiration for the self-oscillatory CBEF process design stems from the unsteady-state operation where the process may vary with time, either as a result of instability in the process itself or due to external disturbances23, 24. All variations can be considered or approximated by periodical changes or certain combinations thereof according to Fourier theory. Therefore, it is important to study sinusoidally forced self-oscillatory systems to develop proper control strategies or to exploit them for process improvements. For this purpose, we propose the analysis strategy shown in Figure 1. The procedure begins with the mathematical modeling of the process. The unstructured model is employed in this paper, but the analysis strategy developed is not limited to this specific model or process. The procedure is subsequently carried out by designing an unforced system that exhibits self-oscillatory behavior and developing a corresponding forced system, followed by identifying the steady-state solutions of the unforced system to determine whether or not there is multiplicity of solutions. If multiple solutions are detected, a step response analysis of the forced system is needed to evaluate the transient effects due to external disturbances, as shown in the left branch of Figure 1. If the transient effects are negligible, the 4


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autonomous equivalence for the forced system can be used to implement the limit cycle bifurcation analysis, and the results from the limit cycle bifurcation analysis are cross validated with simulation-based methods, i.e., stroboscopic or maximum bifurcation maps, as shown in the right branch of Figure 1. Then, the forcing parameters can be determined to improve the process performance or to avoid catastrophic behavior. Process

Establish dynamic model

Locate oscillatory states on the unforced system Excite with sinusoidal disturbance Analyze periodic forcing parameters

Impact analysis of

N Stop

ϕ

Impact analysis of A

Multiplicity in unforced system

Y

Stroboscopic bifurcation map

Impact analysis of f

Cross validation

Limit cycle bifurcation analysis

Cross validation

Maximum bifurcation map

Assume quasi-steady state operation Simulate dynamic behavior

Conduct sensitivity analysis by step response

Find proper forcing parameters to improve performance/ identify catastrophic forcing

Decide ϕ that leads to catastrophic change by phase space plot

Figure 1. Schematic of the strategy for analyzing the periodically forced process, where φ is the shift angle, A is the forcing amplitude and f is the forcing frequency of a sinusoidal perturbation.

Unforced Process Modeling and Analysis Modeling of an Unforced System. The simple unstructured model for the CBEF system is used to demonstrate the analysis procedure, which is described by the mass balances for the substrate (S), biomass (X) and product (P) as follows, 5



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dX = − DX + µ X dt dS = D( S f − S ) − σ X dt dP = − DP + ε X dt

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(1)

where D is the dilution rate and Sf is the feed substrate concentration. The yield coefficient Y, which describes the relationship between the biomass growth rate σ and the specific growth rate µ, is assumed to have a linear relationship of the form, σ (CS ) = Y = Y0 + Y1 * CS µ (C S )

(2)

For many cases, the product generation rate ε is proportional to µ. Hence the 1st and 3rd equations in Eq. 1 are mathematically identical, which is clearly seen after the Laplace-Borel transform25 of both equations, 1  P LB[ε X ]  1 =  + D  + D  X = LB[ µ X ] = c  x0   x0 c

(3)

where x0 is the transform variable, and c is the constant satisfying ε = cµ. The integrating factor could eliminate P if Eq. 3 is applied. Therefore, the dynamic system can be described without any loss of information by the first two equations in Eq. 1. dX = − DX + µ X dt dS µX = D( S f − S ) − dt Y

(4)

The specific growth rate follows the Haldane relation26 to describe the substrate-inhibited kinetics, µ=

µm S K S + S + S 2 / Ki

(5)

where µm is the maximum specific growth rate, Ks is the saturation constant and ki is the substrate inhibition constant. When Ki takes large values, the Monod type growth is obtained. µ=

µm S KS + S

(6)

Both specific growth rates in Eqs. 5 and 6 are studied in this paper, and the data used in the numerical simulation and analysis are listed in Table 2. The differences in their 6


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state trajectories reveal that different modeling strategies will have a profound influence on decision making at the preliminary stage for process development, and choosing a proper model to characterize a process system is of vital importance.

Table 2. Data used for two types of continuous bio-ethanol fermentor models. Monod kinetics

Haldane kinetics

Parameter

Value

Parameter

Value

µm (h-1)

0.3

µm (h-1)

0.3

Y0 (g/g)

0.01

Y0 (g/g)

0.01

Y1 (l/g)

0.03

Y1 (l/g)

0.03

Ks (g/l)

1.75

Ks (g/l)

7

Ki (g/l)

64

Review of periodic solutions. The existence of periodic solutions can be examined by applying the Bendixson criterion. If periodic solutions exist, limit cycles are obtained for an autonomous system, and the stability property can be assessed by the Floquet theory. However, the graphical nature of the Bendixson criterion and the difficulty in finding the fundamental matrix for the Floquet theory hinders their application to the autonomous system analytically. Hence, the bifurcation theory is adopted. The perturbation of a general dynamic system and a linearized system around the steady-state is obtained:

dx = f ( x, u ), x ∈ R n dt

(7)

d (δ x) = J (u )δ x + F (δ x, u ) dt SS

(8)

where δx is the perturbation of the state vector at the steady state, and in this study, x=[X, S]; u is the bifurcation parameter; J(u) is the Jacobian matrix of the system. Setting the remaining term F(δx, u) = 0, the solution of the linearized system results in 7



the general form, n

δ x SS = ∑ ea t [Ci sin bit + Di cos bit ] i

(9)

i =1

where λi = ai+ibi are the complex eigenvalues of J, and Ci and Di are constants determined by the initial conditions. It is clear that ai < 0 and bi ≠ 0 are critical conditions for the existence of a periodic solution in Eq. 9, and the critical point that causes a pair of eigenvalues of J(u) to cross the imaginary axis is the Hopf bifurcation point, which gives birth to a limit cycle from equilibrium in dynamic systems modeled by ODEs. 50

Input Substrate concentration (g/l)

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45 40

Pd

35 30

H2

H1 S2

25 20 15 10

S1 5 0

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

Dilution rate (1/h) Figure 2. Hopf plot with Monod kinetics.

For the 2-dimensional system, represented by Eq. 4, the test function for the Hopf bifurcation is, det J tr J

SS

SS

=0

0, we obtain the subcritical Hopf bifurcation and an unstable limit cycle is generated. Bifurcation Analysis of the Unforced System. To implement the numerical bifurcation analysis28, two steps are carried out: (1) Draw steady-state curves with variation of some parameters in the process; a numerical continuation algorithm, e.g., arch-length continuation27, is available for solving such a problem. (2) Detect bifurcation points along these curves, where the qualitative or topological structure may change, leading the manifold to also change significantly.

9



45

a Substrate concentration (g/l)

40

BP

35

Stable steady-state

30

Self-sustained oscillation 25

lp=-2.316e-5

15 10

0

Wash-out condition

H2

20

H1 lp=-1.480

5 0

0.05

0.1

0.15

Stable steady-state

0.2

0.25

0.3

0.35

0.4

0.45

Dilution rate (1/h) 45

b 40

Substrate concentration (g/l)

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Wash-out condition BP

35 30 25

P3 Stable steady-state

P2 LP Self-sustained oscillation

20

H2 lp=-8.384e-4

15 10 5 0 0

P1

H1 lp=-6.070e-2 0.05

0.1

0.15

0.2

0.25

0.3

Dilution rate (1/h) Figure 3. Continuation diagram of the unstructured bio-ethanol fermentor of (a) Monod type growth kinetics and (b) Haldane type growth kinetics for Sf = 35 g/l.

Both Eq. 5 and 6 are used to examine the causes of steady-state multiplicity and periodic solutions of the CBEF model, represented by Eq. 4. In Figure 3, D is taken as the bifurcation parameter because it is the reciprocal of the residence time of the 10


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reactor. For the Monod type growth kinetics, Figure 3a displays the steady-state portrait by using the software Matcont29. The horizontal dashed line in Figure 2 is equivalent to Figure 3a, and the two Hopf points DH1=0.095 h-1 and DH2=0.271 h-1 are labeled, which are supercritical. The operation of the process for any D between the two Hopf points will lead to stable oscillations for any initial condition. When D is larger than the Branch point, DBP=0.286 h-1, the operation of the bioreactor leads to wash-out. For the Haldane type growth kinetics, an extra limit point, DLP=0.180g/l, is detected as shown in Figure 3b. Between DH1 = 0.060 h-1 and DH2 = 0.178 h-1, the fermentor will stably oscillate for any initial condition. Points in the branch connected by LP and BP are detected as saddle points, and any D in between will lead the steady-state portrait to bi-stability where two attractors exist. For example, the dashed line D = 0.176 h-1 intersects with the curve at three points, P1, P2 and P3, and the corresponding phase portrait (see Figure 6a) shows P2 as a saddle point. The oscillatory attractor (P1) may drift towards the wash-out attractor (P3) as a result of the fluctuations of the operating parameters. The idea of designing the fermentor at a self-oscillatory state is originally due to Rippin23. Taking Pd in Figure 2 as an example, the time average substrate concentration is 4.2% lower than the corresponding steady-state output of the same design conditions, which could reduce the utility demand for the downstream separations. Numerical Continuation of the Limit Cycle. For the investigation of the self-oscillatory behavior of the CBEF, the inherent period of the unforced system becomes important. There are two methods to calculate the period of a stable limit cycle: the shooting method30, and the period continuation of limit cycle (CLC) method. Since the CLC method can deal with systems with varying parameters easily, and as our subsequent analysis of the forced CBEF is based on the bifurcation of limit cycle, the CLC method is adopted in this paper. Assuming that Eq. 7 has an isolated orbit with period T0, the periodic trajectory would satisfy the boundary condition, 11



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x(t = 0) = x(t = T0 )

(13)

The original problem can be reduced to a periodic boundary-value problem (BVP), and normalized in the range [0, 1]: & − T0 f ( w ) = 0 w   w (0) − w (1) = 0

(14)

where w(τ) = x (T0τ + φ) is a solution to this BVP with T0 for any phase shift φ. Since T0 is unknown, one more constraint is required to obtain a unique solution for Eq. 14. The following equation is often employed, which guarantees the orbit closure:

∫

1

0

& old (τ ) dτ = 0 w (τ ), w

(15)

where w& old (τ ) is the derivative vector of a previously calculated limit cycle and is therefore known: iteration steps are taken to update the vector of w(τ) for some constant u, and the initial vector is taken as the results of last continuation step. However, the precise solution of the limit cycle with u at the beginning bifurcation point is needed, which can be calculated using the shooting method as briefly reviewed in Appendix 2. The symbol〈x, v〉is the inner product of two vectors. Because Eqs. 14 and 15 need to be discretized, the discretization nature demands that & (τ ) in Eq. 15. & old (τ ) is used instead of w w Orthogonal collocation method is applied to convert Eqs. 14 and 15 to a system of nonlinear algebraic equations. The solutions of the shifted Legendre polynomial are set as the collocation points, and the Lagrange polynomial is used to approximate the differentiation and integration terms, as follows,

(

)

(

)

m m  wi , j l i′, j (ς i ,k ) − T0 f ∑ j = 0 wi , j l i , j (ς i , k ), u = 0  ∑ j =0  0,0 N −1, m =0 w − w  N −1 m −1 N ,0 i, j i, j i, j ∑ i = 0 ∑ j = 0 σ i , j w , w& old + σ N ,0 w , w& old = 0 

(16)

We set the updating times (iterations) as i = 0,1,2,…,N-1, and discretization of τ as k = 0,1,2,…,m-1, respectively. The first expression in Eq. 16 consists of Nm equations. The li,j are the Lagrange basis polynomials, ζi,k are roots of the Legendre polynomials relative to the interval [τi, τi+1], and σi,j are the Lagrange quadrature coefficients31. It is 12


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clear that Eq. 16 is sparse and by applying the subroutine SNOPT32, it can be solved efficiently. When w& old (τ ) in Eq, 15, the problem of providing proper initial conditions is solved, which is critical for the numerical solution of the system. 12

H2

10

Biomass concentration (g/l)

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8

Poincare map

6

4

2

0

H1

5

Substrate (g/l)

0

0.1

0.12

0.14

0.16

0.18

0.2

0.22

0.24

0.26

0.28

0.3

Dilution rate (1/h)

Figure 4. Numerical continuation of limit cycles for the unstructured bio-ethanol fermentor of Monod type growth kinetics.

In correspondence with Fig. 3a, a few limit cycles near H1 are displayed in Figure 4, though the continuation of the limit cycle would not stop till D = DH2. Proper projection of the isolated cycles to the Poincaré map can form a curve as the green dots in Figure 4 indicate. The Hopf and limit points on this curve are the period doubling and Neimark-Sacker bifurcation, respectively, which belong to the bifurcation of limit cycle and are discussed next. The period at the design point Pd is equal to T0 = 13.5569505211829 h, and there is a 13



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sharp change of the inherent period around Pd, which challenges the controller design strategy to guarantee the oscillatory fermentor maintain a specific period. Compared with the internal oscillation, an external periodic forcing is easier to control, and the possibility of controlling and managing the oscillatory behavior using external periodic forcing is discussed in the next section.

Discussion of Forced Process Modeling When the self-oscillatory fermentor is coupled with external periodic forcing, complex dynamic trajectories may emerge which can be analyzed by two kinds of bifurcation on the continuation of the limit cycle diagram, namely, the periodic doubling (PD) and Neimark-Sacker (NS) bifurcations. And the analysis method is cross validated with the simulation-based methods. However, the forced CBEF needs to be converted to an equivalent autonomous form to implement the bifurcation analysis. Therefore, the condition for developing the autonomous equivalence of the forced CBEF is discussed before any bifurcation analysis. Following the previous study, Sf is chosen as the perturbation variable. Two important aspects of this scenario are noted. First, the perturbation of upstream units may cause the parameters to be periodic, thus a preliminary analysis would aid decision making on whether to add a surge tank or not, to avoid the generation of complex dynamics; secondly, when there are occasions that multiple feed streams with different concentrations are supplied to the fermentor18, simple control of each flow actuator may lead the substrate concentration Sf to be periodic, which may be offered as an alternative tool for process improvement. In this paper, a sinusoidal function is used for simplicity, S f = S f 0 + A sin(2π tf + ϕ )

(17)

where Sf0 is the nominal substrate concentration, equal to Sf of the unforced system. A is the forcing amplitude, φ is the shift angle, and f is the forcing frequency. The forced fermentor is obtained by substituting Eq. 17 into Eq. 4, which is a non-autonomous system. Eq. 17 can be converted to an autonomous form by adding two dimensions, u 14


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and v.

S f = S f 0 + A⋅u du = u + 2π vf − (u 2 + v 2 )u dt dv = −2π uf + v − (u 2 + v 2 )v dt

(18)

Note that Eqs. 17 and 18 are not equivalent: the former is a cycle, and the latter represents a limit cycle. The equivalency condition is discussed next.

The autonomous equivalence condition. Under extreme conditions, a step response is considered which can be viewed as forcing with an infinitely large period. A step disturbance satisfies the quasi-steady-state (QSS) assumption24, i.e., the forcing frequency is very low in comparison with the inherent frequency. As shown in Figure 5, a step disturbance of Sf ≈ Sf0+Asin(φ) at t =1300 h leads the process to shift from a stable oscillation to a stable point after a time interval. Since substrate inhibition is considered in the Haldane kinetics, a sudden increase of the substrate concentration may shut down the process. 60


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50 40

Apply periodic forcing

30 20 10 0

0

500

1000

1300 1500

2000

2500

3000

time (h)

Figure 5. The dynamic behavior of CBEF with Haldane kinetics, where D = 0.176 h-1, and a step variation, Sf = Sf0+1.75 sin(π/2), is added at t = 1300 h.

The scenario above is undesirable from the process safety viewpoint. Very small fluctuations of about 5% in the dilution rate may push the process from an oscillatory behavior to wash-out conditions. The phase portrait before and after adding a step 15



disturbance is compared in Figures 6a and 6b. The state (S, X) = (25.7, 7.35) is labeled, and it is clear in Figs. 6a and 6b that a step disturbance leads the oscillatory fermentor to wash-out. a

15

10

P1 X (g/l)

X: 25.7 Y: 7.35

P2

5

P3 0 0

5

10

15

20

25

30

35

40

45

S (g/l) 15

b

P1

10

X: 25.7 Y: 7.35

X (g/l)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

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P2 5

P3

0 0

5

10

15

20

25

30

35

40

45

S (g/l)

Figure 6. Phase portrait of the bio-ethanol fermentor model with Haldane type growth kinetics, where Sf = 35 g/l and D = 0.0176 h-1. (a) φ =0 and (b) φ =π/2.

Bi-stability is the fundamental reason that external disturbances may push the 16


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process from oscillatory behavior to wash-out conditions. To carry out a bifurcation analysis on the forced CBEF, Monod kinetics does not exhibit multiplicity, and is used in the subsequent discussion.

Simulation-based analysis of a forced system. The impact of forcing amplitude is investigated using a stroboscopic Poincaré map9-1, 34. Instead of observing the whole trajectories, a discrete sequence of phase points at fixed time intervals is recorded on the Poincaré map. We take Pd (0.099, 35) as the design condition for the unforced system, which generates self-oscillations with a frequency f0 =1/13.5569505211829 h-1. The phase projection of the trajectories is inspected at multiples of the forcing period, i.e., ts = m/f, where m = 1, 2, 3…. Figures 7a and 7b depict the Poincaré maps for the forced system with f/f0 equal to 2 and 3, respectively. The main phenomenon associated with small forcing amplitude is the entrainment of the internal limit cycle. The rational number f/f0 decides the number of branches that emerge on the stroboscopic map, and the limiting case for this region is the unforced oscillatory system (A = 0.0). Taking Figure 7a as an example, where f/f0 equals to 2, two branches occur as frequency locking and the points of intersection at each amplitude forcing are on the surface of a three dimensional torus. Then, the forced system follows PD cascades to chaos. As the forcing amplitude increases, the trajectory loses its stability and a stable trajectory with a doubled period is born. The number of stroboscopic points for PD is 1, 2, 4, 8, etc, and chaotic dynamics may be generated as A increases further. When the maximum Lyapunov exponent is positive, the system is chaotic, and the numerical development of the Lyapunov exponent is given in Appendix 3. For example, when A = 1.25 g/l and f/f0 = 2, the system is chaotic. Chaos occurs in a deterministic system when the system is sensitive to initial conditions, which means that by knowing the state at the current time and the system equations, the state at the next time interval cannot be predicted. Therefore, chaotic behavior in a process system is to be avoided. Similar to the investigation of the forcing amplitude, the impact of forcing frequency is also carried out using a maximum bifurcation map. Likewise, instead of observing the whole trajectories, a discrete sequence of phase points at each local maximum is 17



recorded, and the points are recorded by varying the forcing frequency, and each plot has a fixed forcing amplitude, as shown in Figs. 8a and 8b. 4.5

Sunstrate concentration (g/l)

4

a

3.5

PW5

PW4

Stripe of chaos

3

PW3 Period-halving bifurcations

2.5 2 1.5 1 0.5

Period-doubling bifurcations

0 -0.5

0

2

4

6

PW5 8

10

12

14

16

18

20

18

20

Forcing amplitude(g/l) 1.6

b Frequency locking region

1.4


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

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1.2

Chaos 1

Simple region

0.8

Bubbling region

0.6

Fully entrained

0.4

NS 0.2

0

2

4

PD

PD 6

8

10

12

14

16

Forcing amplitude (g/l) Figure 7. Stroboscopic Poincaré bifurcation diagrams (strobing every one forcing period) for the design bio-ethanol fermentor with Monod kinetics, where (a) and (b) have different ratio of natural and forcing frequency, f/f0 = 2, and 3, respectively. 18


Page 19 of 33

1.265

1.4 1.26

1.3

1.255

1.35

1.28 1.25

1.3

1.245

X (g /l)

X (g /l)

1.26

X (g /l)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


1.24

1.24

1.25

1.235 1.23

1.22

1.2 1.225

1.2

1.22

1.15

1.215

1.18 0.5

S (g/l)

1

1.5

0.6

0.7

0.8

0.9

1

1.1

1.2

S (g/l)

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

1.2

1.3

1.4

S (g/l)

Figure 8. Local maximum bifurcation diagram (recording every peak of the wave) for the design bio-ethanol fermentor with Monod kinetics, where (a) and (b) have different ratio of natural and forcing frequency, A = 3.5g/l and 7g/l, respectively. 19



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As shown in Figure 7, complex dynamic behavior emerges immediately after frequency forcing increases from f = 0 h-1, where the forcing amplitude is fixed at A = 7g/l; then high amplitude resonances are formulated, which may be accompanied by strips of chaos. As f increases, branches may converge to a single line by periodic-halving; the single branch begins with a PD point and ends with a NS point. This single branch will evolve to a long tail of periodic or quasi-periodic invariant as f increases further. It is worth to mention that the conclusion, i.e., chaotic behavior fading away by periodic-halving cascades, cannot be drawn directly in the maximum bifurcation map; neither does the emergence of invariant torus after the simple branch. These conclusions are supported by dynamic simulation results as shown in the sub-plot of Figure 8a while limit cycle bifurcation analysis can provide the information directly and precisely.

Limit Cycle Bifurcation of a Forced Fermentor. The stroboscopic/maximum bifurcation maps focus on how the inherent limit cycle may vary with external periodic disturbances. However, dynamic simulations can be rather inefficient and ineffective for analyzing fermentation models that exhibit complex nonlinear behavior. Instead, a limit cycle bifurcation analysis would prove to be a powerful tool for obtaining more efficient and complete characterization of the system behavior. Due to obvious operational adversities, the presence of complex dynamic behavior is generally avoided in industrial applications, and only simple branches, such as points between PD and NS in Figures. 7 and 8 would be of interest if the goal of period forcing is to improve process performance. Therefore, it is important to detect the bifurcation points where catastrophic dynamic behavior may emerge as parameters vary. Limit cycle bifurcation analysis detects those forcing parameters that may lead simple oscillatory behavior to complex ones. Analogous to the analysis of limit cycles by Hopf bifurcation diagrams, the forced system can also be analyzed by bifurcation of limit cycles, and their similarity is interpreted in the Poincaré map because a limit cycle in the Poincare map is viewed as an “equilibrium point”. Based on the continuation curves, two kinds of bifurcation are discussed in this study. Neimark-Sacker bifurcation (NSB) is analogous to the Hopf 20


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bifurcations in the Poincaré map. Period doubling bifurcation (PDB) of limit cycles takes place when the fixed point in the Poincaré map changes its stability, i.e., the fixed point loses stability and becomes a saddle point. NSB generically corresponds to a bifurcation to an invariant torus, on which the flow contains periodic or quasi-periodic motion, and PD occurs when a new limit cycle emerges from an existing limit cycle, and the period of the new limit cycle is (approximately) twice the old one. 25

PDB curve 20

Simple region

VL2

15

A (g/l)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


VL1

Invariant region

VL3

10

HL2 NSB curve 5

0

HL1

0

0.2

0.4

0.6

0.8

1

1.2

1.4

f (h-1)

Figure 9. Limit cycle continuation of forced bio-ethanol fermentor with Monod kinetics, where the forcing parameters are forcing frequency f, and forcing amplitude A. PDB stands for period doubling bifurcation and NSB stands for Neimark-Sacker bifurcation.

Since Pd point does not exhibit multiplicity characteristics, an autonomous representation by Eq. 18 is used to implement the bifurcation analysis. Taking (f, A) as the parameter space, the limit cycle bifurcation analysis of the periodically forced self-oscillatory fermentor model is shown in Figure 9, and f and A are considered simultaneously in the analysis. The horizontal lines HL1 and HL2 correspond to Figure 8a-b, and each intersects with the PDB curve with a PD point on the left hand 21



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Page 22 of 33

side, which are (0.207, 3.5) and (0.230, 7), respectively; and intersects with the NSB curve with a NS point on the right hand side, which are (0.228, 3.5) and (0.417, 7), respectively. The branches between PD and NS points are detected by the maximum bifurcation map to exhibit simple periodic behaviors. The vertical lines VL1 and VL2 correspond to Figure 9 (a) and (b), respectively. By small amplitude forcing, VL1 exhibits period doubling cascades by entering PD point (0.1475, 0.882), the intersection of NS point (0.1475, 1.837) is found to be the terminator of the stripe of chaos, as is shown in Figure 7a, and a period doubling cascade to chaotic behavior is formed again as A increases further. Figure 7b does not exhibit period doubling cascades, and the closed invariant region vanishes because a NS point (0.221, 3.362) is detected in VL2, then the branch between two PD points, (0.221, 5.233) and (0.221, 11.051), are found to be the bubbling region in Figure 9b, and VL2 enters simple periodic region after A increases further.

Process Enhancement by Periodic Forcing. Not all dynamic behaviors shown in bifurcation maps of Figs. 7 and 8 are supported by an analytical explanation using limit cycle bifurcation analysis, but only the closed region bounded by the PDB and NSB curves is of interest. This is because any point in this region generates simple oscillatory behavior, which is comparatively easy to manage and control. For the oscillatory fermentor, time-average substrate output concentration is obtained as,

∫ S=

t2

t1

S (t ) dt

t 2 − t1

(19)

Equation 19 is integrated numerically using a 4th-order Runge-Kutta algorithm for a time span of many forcing periods. We expect that the periodic operation of the fermentor would yield better performance than the corresponding unforced operation. For example, when the forcing parameters are set as Pf (0.24, 15), time-average substrate consumption rate is 13.1% higher than that of corresponding unforced one.

Oscillation Control by Periodic Forcing. Challenges facing the design and operation of periodic operations include building a proper control system. The conventional PID controllers may lead self-oscillatory system to chaos. Other control configurations, such as repetitive control and relay feedback control, can lead the 22


Page 23 of 33

process to operate periodically, but high accuracy is required for the determination of the inherent period of this unperturbed oscillatory state. 3.5 No forcing, D=0.099/h Periodic forcing, D=0.099/h Periodic forcing, D=0.097/h Periodic forcing, D=1.05/h

3


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


2.5

2

1.5

1

0.5

0

0

10

20

30

40

50

60

70

80

time (h) Figure 10. Oscillatory behavior in time domain, where the black line is for the unforced fermentor at design point Pd, and the red line is for the forced fermentor.

For an oscillatory fermentor, errors in experiments, model inaccuracies and calculation errors are inevitable, which hinders the precision of the inherent period calculation and the proper control of the oscillator. On the other hand, periodical forcing of the process parameters may alter the dynamics of the original process. Similar to the concept of vibrational control21, 33, forced periodic control becomes an open-loop control strategy. Taking the current fermentor design as an example, with A = 15 g/l and f = 0.24 h-1, the output frequency shifts from its natural frequency to the forcing frequency of about 0.24 h-1, hence the properties of the unforced system is subordinate to the forcing properties, as shown in Figure 10. This information is valuable especially when the measurements are costly and cumbersome or the process is subjected to disturbances. For example, one obtains almost the same dynamic response as the red curve in Figure 10 by varying the dilution rate D ∈ (0.097, 0.105 23



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Page 24 of 33

h-1), i.e., the system behavior is well controlled as long as the forcing input is under control, regardless of its internal oscillations.

Conclusions In this work, strategies for analyzing periodically forced self-oscillatory processes are proposed. The CBEF may exhibit oscillatory behavior, and a bifurcation analysis is implemented on an unstructured model. Based on a two-parameter Hopf bifurcation diagram in the (D, Sf) plane, a self-oscillatory fermentor is designed and labeled as Pd. By designing the fermentor at the design point Pd, the substrate consumption rate is recorded as being 4.2% higher than its corresponding nominal steady-state. While designing the process to be self-oscillatory may yield higher overall substrate conversion rates than that in the corresponding steady-state situation, the regulation and management of the fermentor dynamics in the presence of periodic behavior is challenging. CLC algorithm is used to calculate the inherent period of the oscillatory process, and the inherent period is found to be sensitive to D around Pd. By introducing sinusoidal variations to Sf, the bio-ethanol fermentation process becomes a forced system. The effect of the forcing parameters on the oscillatory fermentor is analyzed. The analysis of the forced system is accomplished with a two-parameter limit cycle bifurcation diagram in the (f, A) plane. Two kinds of limit cycle bifurcation are investigated in this study: NS bifurcation leads a periodic orbit to an invariant torus, and PD bifurcation leads a periodic orbit to a new orbit with (approximately) doubled period. The limit cycle bifurcation analysis is presented as the appropriate method for investigating the dynamics of periodically forced self-oscillatory systems. The analysis results are cross validated with the simulation-based methods, i.e., stroboscopic and maximum bifurcation maps. By implementing the limit cycle bifurcation analysis on the forced fermentor, a Simple region is identified in which any parameter pair (f, A) leads to an oscillatory behavior. Taking A = 15 g/l and f = 0.24 h-1 as example, the substrate consumption rate increases by 13.1% compared to that of the unforced one. Moreover, the period of the process is dominated by the 24


Page 25 of 33

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forcing frequency regardless of its internal oscillations. The concept of open-loop control strategy by periodic forcing is also discussed.

Corresponding Author *Tel.: +86 10 64445826. E-mail: [email protected].

Acknowledgements The authors gratefully acknowledge the following institutions for support: the National Natural Science Foundation of China (Grant No.2157015); the National Basic Research Program of China, 973 program (Grant No. 2013CB733600). The authors are also grateful to Professor Ali Cinar and Dr. Mustafa Cagdas Ozturk for providing constructive comments on a draft of the manuscript.

25



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Page 26 of 33

Appendix 1 The normal form of the Hopf bifurcation is a 2-dimensional system, which can be expressed by the central manifold theorem:  dx1   dt   β  =  dx1   1    dt 

 x1 ( x12 + x22 )  −1 + σ   2 2  β   x2 ( x1 + x2 ) 

(A1-1)

where σ = sign (lp) = ±1. If σ = -1, the equilibrium at the origin is stable when β < 0, and unstable when β > 0; a stable limit cycle exists with radius sqrt(abs(β)). This is the supercritical Hopf point. If σ = 1, the equilibrium at the origin is stable when β < 0, and unstable when β > 0; a unstable limit cycle exists with radius sqrt(abs(β)). This is the subcritical Hopf point. A general 2-dimensional system under a Hopf bifurcation is structurally equivalent to Eq. A1-1, and the stability of the limit cycle is determined by σ = lp (0), which is calculated by the Taylor expansion of f in Eq. 2, 1 1 4 f ( x, 0) = A0 x + B( x, x) + C ( x, x, x) + O( x ) 2 6 where n

B j ( x, y ) = ∑ k ,l

C j ( x, y, z ) =

∂ 2 fi (ξ , 0) xk xl ∂ξ k ∂ξl n

∂ 3 fi (ξ , 0)

∑ ∂ξ ∂ξ ∂ξ

k ,l , m

k

l

(A1-2)

xk xl xm

m

Let q be a complex eigenvector satisfying A0 q = i ω q, A0T p = -i ω p, and
= 1. Then lp is given as follows,  p, C ( q, q, q ) − 2 p, B(q, A0T B( q, q )) 1 lp(0) = Re  2ω0  + p, B( q , (2iω0 I n − A0 ) −1 B(q, q )) 

   

(A1-3)

where In is the identity matrix. Appling Eq. A1-3 to Eq. 11 gives the lp as is shown in Eq. 12.

26


Page 27 of 33

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


Appendix 2 Similar to Eq. 15, the extra constraint for the shooting method is an anchor function with which a point on the periodic trajectory coincides after period T0, and represented in the following general form, N (x, T0 ) = 0

(A2-1)

Equation 14 represented in the iterated form, F ( x k , T0 ) − x k = 0

(A2-2)

where F is the integration function of f. Extending the formula x=F(x, T0) with Eq. A2-1, the well-posed system is given as follows,

 F ( x k , T0 )  Gk ( x k , T0 ) =    N ( x k , T0 ) 

(A2-3)

Typically, Newton-like method of rank one correction is carried out to numerically solve this nonlinear set of equations,

x% k +1 = x% k − Ak−1Gk ∆x% k = x% k +1 − x% k ∆Gk = Gk +1 − Gk ∆Ak =

(A2-4)

( ∆Gk − ∆Ak ∆x% k ) ∆x% ∆x% Tk ∆x% k

T k

Ak +1 = Ak + ∆Ak where x% denotes the extended state vector. And the Jacobian in Eq. (16) is given as:

 ∂F  ∂x − I A=  ∂N  ∂x 

∂F  ∂T0   ∂N  ∂T0 

27



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Page 28 of 33

Appendix 3

The Lyapunov exponent of a dynamical system is a quantity that characterizes the rate of separation of infinitesimally close trajectories Z(t),

δ Z (t ) ≈ e λ t δ Z (t = 0)

(A3-1)

where λ is the Lyapunov exponent. There is a spectrum of Lyapunov exponents which are equal in number to the dimensionality of the phase space. From Eq. A3-1, if there is a positive λ, the system will diverge and the predictability of the system is hindered, so we can use the maximal Lyapunov exponent as the criterion for the onset of chaos, which is defined as follows,

1 t

λ = lim lim ln t →∞ δΖ0 → 0

δ Z (t ) δ Z (0)

(A3-2)

δZ(0) approaching zero ensures the validity of the linear approximation at any time, and the numerical computation algorithm by Wolf et al35 is used in this paper.

28


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Page 33 of 33

Table of Contents 4.5 4

Sunstrate concentration (g/l)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


a

3.5

PW5

PW4

Stripe of chaos

3

PW3 Period-halving bifurcations

2.5 2 1.5 1 0.5

Period-doubling bifurcations

0 -0.5

0

2

4

6

PW5 8

10

12

14

16

18

20

Forcing amplitude(g/l) The complex dynamics of the peroidcally forced self-oscillatory fermentor is equally analyzed by the limit cycle bifurcation method.

33


Strategies for the Analysis of Continuous Bioethanol Fermentation

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