The dynamics of coupled nonisothermal continuous stirred tank reactors

Emergent Spaces for Coupled Oscillators

Systems of coupled dynamical units (e.g., oscillators or neurons) are known to exhibit complex, emergent behaviors that may be simplified through coarse-graining: a process in which one discovers coarse variables and derives equations for their evolution. Such coarse-graining procedures often require extensive experience and/or a deep understanding of the system dynamics. In this paper we present a systematic, data-driven approach to discovering “bespoke” coarse variables based on manifold learning algorithms. We illustrate this methodology with the classic Kuramoto phase oscillator model, and demonstrate how our manifold learning technique can successfully identify a coarse variable that is one-to-one with the established Kuramoto order parameter. We then introduce an extension of our coarse-graining methodology which enables us to learn evolution equations for the discovered coarse variables via an artificial neural network architecture templated on numerical time integrators (initial value solvers). This approach allows us to learn accurate approximations of time derivatives of state variables from sparse flow data, and hence discover useful approximate differential equation descriptions of their dynamic behavior. We demonstrate this capability by learning ODEs that agree with the known analytical expression for the Kuramoto order parameter dynamics at the continuum limit. We then show how this approach can also be used to learn the dynamics of coarse variables discovered through our manifold learning methodology. In both of these examples, we compare the results of our neural network based method to typical finite differences complemented with geometric harmonics. Finally, we present a series of computational examples illustrating how a variation of our manifold learning methodology can be used to discover sets of “effective” parameters, reduced parameter combinations, for multi-parameter models with complex coupling. We conclude with a discussion of possible extensions of this approach, including the possibility of obtaining data-driven effective partial differential equations for coarse-grained neuronal network behavior, as illustrated by the synchronization dynamics of Hodgkin–Huxley type neurons with a Chung-Lu network. Thus, we build an integrated suite of tools for obtaining data-driven coarse variables, data-driven effective parameters, and data-driven coarse-grained equations from detailed observations of networks of oscillators.

Mechanism and Phenomenology of an Oscillating Chemical Reaction

Chemical reactions, which are far from equilibrium, are capable of displaying oscillations in species concentrations and hence in colour, electrode potential, pH and/or temperature. The oscillations arise from the interplay between positive and negative kinetic feedback. Mechanisms for such reactions are presented, along with the rich phenomenology that these systems exhibit, from complex oscillations and chemical waves, to stationary concentration patterns. This review will focus on the Belousov-Zhabotinksy reaction but reference to other reactions will be made where appropriate.

Yield Effects in Single and Coupled Nonlinear Thermokinetic Reaction Systems

The influence of nonlinear dynamic effects on yield is considered for consecutive-parallel reaction schemes in a single phase CSTR. In the first part, the behavior of a single uncoupled reactor is investigated. Emphasis is on the reaction dynamics and the yield of the intermediate ethanal using continuation and optimization techniques. We characterize regions of periodic, complex periodic, and chaotic oscillations. The chaotic region is reached by type III intermittency. In all cases, the global optimum yield of the intermediate is a steady state, which is unstable or stable due to the location of the stability boundary. It is found that autonomous periodic operation is only locally better than steady state operation. In the second part, different types of mass and energy coupling between two reactors are studied. The results obtained for a simple consecutive parallel reaction scheme are validated by a more detailed model of the ethanol oxidation by hydrogen peroxide under iron(III) catalysis, which can be described by an extended consecutive-parallel reaction scheme. Analogies are found with respect to the dynamics and the yield of the intermediate.