465306 Nonlinearity Analysis of Periodically Forced Bioreactors

Monday, November 14, 2016
Grand Ballroom B (Hilton San Francisco Union Square)
Chi Zhai1, Ahmet Palazoglu2, Wei Sun3 and Zengzhi Du1, (1)College of Chemical Engineering, Beijing University of Chemical Technology, Beijng, China, (2)Department of Chemical Engineering, University of California, Davis, Davis, CA, (3)College of Chemical Engineering, Beijing University of Chemical Technology, Beijing, China

Nonlinearity Analysis of Periodically Forced Bioreactors

Chi Zhaia, Ahmet Palazoglub, Wei Suna*, Zengzhi Dua

a China 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

Nonlinearity is the underlying characteristic in process systems that allows for time average performance enhancement by periodic forcing of external parameters1, 2. The well-known pi-criterion provides a sufficient condition for optimal periodic operation3, and is based on the linearization of the system and second-order truncation of the performance index, which is valid only for weakly nonlinear systems with infinitesimal forcing amplitudes. On the other hand, bioreactors often exhibit highly nonlinear dynamics which poses difficulties on the systematical analysis of periodic operation. Hence, it becomes desirable to explore how inherent nonlinearities would contribute to performance improvement if periodic forcing is applied on the process. In this paper, a two-compartment structured model for ethanol production is studied4. By judiciously setting the scope of the design/operation parameters and implementing bifurcation analysis on the unforced model, Hopf bifurcation points are detected which separate the parameter space into a hyperbolically stable region and a self-oscillatory region.

  For a hyperbolically stable point, analysis of the periodic operation can be accomplished by applying the center manifold theory to the forced nonlinear system. With a multivariable expansion of the center manifold equations, higher-order terms of the nonlinear system are evaluated5. This method is a higher-order correction of the pi-criterion, which will reduce to the pi-criterion by truncation up to 2nd order terms. But this method is only suitable for sinusoidal-like forcing inputs. Typically, when pulsed periodic inputs are applied on the system, Fourier transform of the inputs would complicate this approach significantly6. Previous studies7 used Carleman linearization as another method for pulsed periodic optimization problems. This method is carried out by Carleman linearization on the nonlinear model and solving it analytically, and then the state terms are substituted into the Taylor expansion of the performance index. Since Carleman linearization is cumbersome and suffers from lack of parsimony, a more compact functional expansion method using the Laplace-Borel transform is applied8. This method is found to be comparatively easy for dealing with both pulsed inputs and sinusoidal-like inputs9. These three methods are compared with a case study on the two-compartment bioreactor model, and the superiority of the functional expansion method is addressed.

  This bioreactor may exhibit self-oscillatory behavior by adjusting the parameters properly. We show that for a self-oscillatory point, external periodic forcing may bifurcate the system to more complex dynamic behaviors. Bifurcation analysis can be implemented on the forcing waveform that is continuously differentiable, i.e., when the sinusoidal input is exerted on the self-oscillator, complex dynamics such as invariant torus or chaos would emerge. In this regard, the route to chaos by period-doubling cascades is discussed in this case study. While, intermittency10 may also lead the system to chaos, and one may only concentrate on the onset of chaos in industry as chaotic behavior is viewed as an operational state to be avoided. An algebraic criterion on the onset of chaos is adopted in this case study, which is the Laplace-Borel transformation of the autocorrelation function. Unlike other criterion for the formation of chaos, which is usually simulation-based, this criterion is analysis-based and offers better insight.

Keywords: Carleman linearization; functional expansion; Laplace-Borel transform; central manifold; the onset of chaos.

References:

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9. Harris, K. R., Palazoglu, A., 1997. Studies on the analysis of nonlinear processes via functional expansions-II. Forced dynamic responses.  Chem. Eng. Sci. 52, 3197-3207.

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