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Notes: Periodic variations are applied to the influxes of oxygen and methane entering a reaction vessel in which takes place a combustion reaction. We measure the temperature and chemical responses of the system as we change the forcing amplitudes, periods, and equivalence ratios. Using a simple model of a Carnot engine we calculate efficiency changes in an externally varied flux mode (VFM) of operation relative to the constant flux mode (CFM) of operation. We find increases and decreases in the average temperature, efficiency, and unburnt fuel concentrations in the VFM relative to the CFM. For certain constraints we find regions where the average temperature in the VFM is less than that of the CFM and there is an efficiency increase. We find other regions where the entire temperature response in the VFM is greater than that of the CFM and this also can lead to an efficiency increase which is due to changes in extent of combustion and heat losses. The effects of forcing amplitudes, periods, and equivalence ratios on the system are explored, and predictions of numerical calculations agree with much of the data. A simple model of the reaction limited by one reagent, and the absorption of heat by the products and the other reagent predicts for variations of one reagent the number of maxima in the response, and phase relations of external variations and the response.

Notes: As in a prior article (Ref. 1), we consider an oscillatory dissipative system driven by external sinusoidal perturbations of given amplitude Q and frequency ω. The kinetic equations are transformed to normal form and solved for small Q near a Hopf bifurcation to oscillations in the autonomous system. Whereas before we chose irrational ratios of the frequency of the autonomous system ωn to ω, with quasiperiodic response of the system to the perturbation, we now choose rational coprime ratios, with periodic response (entrainment). The dissipative system has either two variables or is adequately described by two variables near the bifurcation. We obtain explicit solutions and develop these in detail for ωn/ω=1; 1:2; 2:1; 1:3; 3:1. We choose a specific dissipative model (Brusselator) and test the theory by comparison with full numerical solutions. The analytic solutions of the theory give an excellent approximation for the autonomous system near the bifurcation. The theoretically predicted and calculated entrainment bands agree very well for small Q in the vicinity of the bifurcation (small μ); deviations increase with increasing Q and μ. The theory is applicable to one or two external periodic perturbations.

Notes: The rate of entropy production due to chemical reaction is calculated for a variety of parameter values in the reversible Oregonator model. The average values over cycles of oscillation are compared to those in the coexisting stationary states. The present work corrects an earlier calculation [A. K. Dutt, J. Chem. Phys. 86, 3959 (1987)]. Contrary to previous impressions and claims, there is no consistent relationship between the magnitudes of the entropy production in coexisting stationary and oscillatory states, in this case.

Notes: We present a thermodynamic analysis of global validity for effectively one-variable, irreversible chemical systems with multiple steady states. A hypothetical reaction chamber is held at constant temperature and volume and is connected by selectively permeable membranes to reservoirs of reactant(s) and product(s), each at constant selected pressures. An appropriate free energy function, which yields criteria of evolution to equilibrium for the composite system of reaction chamber and reservoirs, is a hybrid of Gibbs and Helmholtz free energies. The one variable in the reaction chamber is the pressure of a chemical intermediate which varies in time according to a given reaction mechanism. With the hybrid free energy, the kinetics for a given mechanism, and a concept of instantaneous indistinguishability of systems with different mechanisms, we establish a thermodynamic driving force, or species-specific affinity, for the intermediate. The species-specific affinity vanishes at steady states, and upon its differentiation we obtain necessary and sufficient conditions for the stability of steady states and for critical points. The integral of the species-specific affinity globally provides valid Liapunov functions for the evolution of the intermediate. These results are independent of the number of steady states of the system, and they hold both near to and far from equilibrium. For a large class of mechanisms with a single intermediate, the integral of the species-specific affinity appears in the irreversible partof the time-dependent transition probability of the single-variable Master equation and in its stationary solution. Hence for these mechanisms we obtain a direct interpretation of the stochastic results in terms of thermodynamic quantities. The time rate of change in the pressure of the intermediate multiplied by its species-specific affinity yields a species-specific term in the dissipation. The total system dissipation (or entropy production) is not in general a minimum at a nonequilibrium steady state, but the species-specific term is minimized at every such state. The expression of the stationary solution of the master equation in terms of the species-specific affinity provides a generalization of the Einstein relation for the probability of equilibrium fluctuations to far-from-equilibrium conditions. The functional form of the species-specific term in the dissipation parallels a form that appears in Boltzmann's H theorem for the momentum relaxation of a dilute gas.

Notes: Periodic variations are applied to the influxes of oxygen and methane entering a reaction vessel in which takes place a combustion reaction and, in the absence of these variations, the system is in a stable focus. We measure the reaction of the system to these variations and find a resonant response, and changes of phase relations between the forcing and response of the system, near the autonomous frequency. We calculate the enthalpy content of the gases and using a simple model of a Carnot engine we study the power output of the system and find increases (∼7% in power) in these quantities near the autonomous frequency. The experimental results are compared to the predictions of a numerical model specific to our system and to the analytic solution of a linear set of differential equations with a stable focus. We find good agreement with both, but there is an aspect of the experimental results which requires additional hypotheses.

Notes: We prove that Liapunov functions for a single reactive intermediate evolving toward a nonequilibrium steady state can be obtained from a global potential φ. We consider reactions occurring in a chamber containing a reactant, the intermediate, and a product. Reservoirs connected to the chamber serve to hold the reactant and product concentrations constant, in nonequilibrium proportions. The Liapunov property of φ is significant because of the role it plays in the thermodynamic and stochastic analysis of nonequilibrium systems: φ is defined in terms of the reactive flux to produce the intermediate and the flux to remove the intermediate. The derivative of φ with respect to the concentration of the intermediate yields an effective chemical driving force that is specific to the intermediate, and its time derivative yields a species-specific component of the dissipation that is minimized at steady states. These results hold both near to equilibrium and far from equilibrium for systems with one intermediate, independent of the number of steady states. Local Liapunov functions are also provided by the "excess dissipation,'' the second variation in the entropy or in the Helmholtz free energy for the reaction chamber, and quadratic functions introduced in Keizer's fluctuation–dissipation theory. Linearization of the force and flux expansions for nonequilibrium systems yields an idealized model in which the dissipation decreases monotonically in time and thus provides a Liapunov function for evolution to steady states. This result does not hold for a chemical system approaching a steady state with an arbitrarily small, but macroscopic displacement from equilibrium, even though the series expansions of the reactive fluxes and conjugate thermodynamic forces are closely approximated by truncation at the linear terms. There are always small regions in the immediate vicinity of nonequilibrium steady states where the dissipation increases in time while the system relaxes.