Gian Paolo Beretta

Turbulent flame propagation and combustion in spark ignition engines

Abstract Pressure measurements synchronized with high-speed motion picture records of flame propagation have been made in a transparent piston engine. The data show that the initial expansion speed of the flame front is close to that of a laminar flame. As the flame expands, its speed rapidly accelerates to a quasi-steady value comparable with that of the turbulent velocity fluctuations in the unburned gas. During the quasi-steady propagation phase, a significant fraction of the gas behind the visible front is unburned. Final burnout of the charge may be approximated by an exponential decay in time. The data have been analyzed in a model independent way to obtain a set of empirical equations for calculating mass burning rates in spark ignition engines. The burning equations contain three parameters: the laminar burning speed sl, a characteristic speed uT, and a characteristic length lT. The laminar burning speed is known from laboratory measurements. Tentative correlations relating uT and lT to engine geometry and operating variables have been derived from the engine data.

Nonlinear model dynamics for closed-system, constrained, maximal-entropy-generation relaxation by energy redistribution

We discuss a nonlinear model for relaxation by energy redistribution within an isolated, closed system composed of noninteracting identical particles with energy levels with . The time-dependent occupation probabilities are assumed to obey the nonlinear rate equations where and are functionals of the s that maintain invariant the mean energy and the normalization condition . The entropy is a nondecreasing function of time until the initially nonzero occupation probabilities reach a Boltzmann-like canonical distribution over the occupied energy eigenstates. Initially zero occupation probabilities, instead, remain zero at all times. The solutions of the rate equations are unique and well defined for arbitrary initial conditions and for all times. The existence and uniqueness both forward and backward in time allows the reconstruction of the ancestral or primordial lowest entropy state. By casting the rate equations in terms not of the s but of their positive square roots , they unfold from the assumption that time evolution is at all times along the local direction of steepest entropy ascent or, equivalently, of maximal entropy generation. These rate equations have the same mathematical structure and basic features as the nonlinear dynamical equation proposed in a series of papers ending with G. P. Beretta, Found. Phys. 17, 365 (1987) and recently rediscovered by S. Gheorghiu-Svirschevski [Phys. Rev. A 63, 022105 (2001);63, 054102 (2001)]. Numerical results illustrate the features of the dynamics and the differences from the rate equations recently considered for the same problem by M. Lemanska and Z. Jaeger [Physica D 170, 72 (2002)]. We also interpret the functionals and as nonequilibrium generalizations of the thermodynamic-equilibrium Massieu characteristic function and inverse temperature, respectively.

Boiling regimes in a closed two-phase thermosyphon

Abstract Experimental results and an analytical model on the boiling mechanisms in a closed two-phase thermosyphon are presented in order to define the frontiers between the main boiling regimes and, in particular, the frontier between intermittent boiling and fully-developed boiling. The boiling regimes are classified on the basis of the frequency of bubble nucleation and the ratio of bubble diameter to device diameter. A criterion for the intermittent/developed-boiling frontier is based on the ratio of the bubble-nucleation waiting time and the bubble growth time. From this criterion a correlation between power throughput, working-fluid pressure and nucleation critical superheat are obtained. Experimental data on operating conditions, temperatures, and nucleation frequencies show the same functional dependence as the proposed correlation.

The Rate-Controlled Constrained-Equilibrium Approach to Far-From-Local-Equilibrium Thermodynamics

The Rate-Controlled Constrained-Equilibrium (RCCE) method for the description of the time-dependent behavior of dynamical systems in non-equilibrium states is a general, effective, physically based method for model order reduction that was originally developed in the framework of thermodynamics and chemical kinetics. A generalized mathematical formulation is presented here that allows including nonlinear constraints in non-local equilibrium systems characterized by the existence of a non-increasing Lyapunov functional under the system’s internal dynamics. The generalized formulation of RCCE enables to clarify the essentials of the method and the built-in general feature of thermodynamic consistency in the chemical kinetics context. In this paper, we work out the details of the method in a generalized mathematical-physics framework, but for definiteness we detail its well-known implementation in the traditional chemical kinetics framework. We detail proofs and spell out explicit functional dependences so as to bring out and clarify each underlying assumption of the method. In the standard context of chemical kinetics of ideal gas mixtures, we discuss the relations between the validity of the detailed balance condition off-equilibrium and the thermodynamic consistency of the method. We also discuss two examples of RCCE gas-phase combustion calculations to emphasize the constraint-dependent performance of the RCCE method.

Maximum entropy production rate in quantum thermodynamics

In the framework of the recent quest for well-behaved nonlinear extensions of the traditional Schrodinger-von Neumann unitary dynamics that could provide fundamental explanations of recent experimental evidence of loss of quantum coherence at the microscopic level, a recent paper [Gheorghiu-Svirschevski 2001 Phys. Rev. A 63 054102] reproposes the nonlinear equation of motion proposed by the present author [see Beretta G P 1987 Found. Phys. 17 365 and references therein] for quantum (thermo)dynamics of a single isolated indivisible constituent system, such as a single particle, qubit, qudit, spin or atomic system, or a Bose-Einstein or Fermi-Dirac field. As already proved, such nonlinear dynamics entails a fundamental unifying microscopic proof and extension of Onsagers reciprocity and Callens fluctuation-dissipation relations to all nonequilibrium states, close and far from thermodynamic equilibrium. In this paper we propose a brief but self-contained review of the main results already proved, including the explicit geometrical construction of the equation of motion from the steepest-entropy-ascent ansatz and its exact mathematical and conceptual equivalence with the maximal-entropy-generation variational-principle formulation presented in Gheorghiu-Svirschevski S 2001 Phys. Rev. A 63 022105. Moreover, we show how it can be extended to the case of a composite system to obtain the general form of the equation of motion, consistent with the demanding requirements of strong separability and of compatibility with general thermodynamics principles. The irreversible term in the equation of motion describes the spontaneous attraction of the state operator in the direction of steepest entropy ascent, thus implementing the maximum entropy production principle in quantum theory. The time rate at which the path of steepest entropy ascent is followed has so far been left unspecified. As a step towards the identification of such rate, here we propose a possible, well-behaved and intriguing, general closure of the dynamics, compatible with the nontrivial requirements of strong separability. Based on the time–energy Heisenberg uncertainty relation, we derive a lower bound to the internal-relaxation-time functionals that determine the rate of entropy generation. This bound entails an upper bound to the rate of entropy generation. By this extreme maximal-entropy-generation-rate ansatz, each indivisible subsystem follows the direction of steepest locally perceived entropy ascent at the highest rate compatible with the time– energy uncertainty principle.

Modeling Non-Equilibrium Dynamics of a Discrete Probability Distribution: General Rate Equation for Maximal Entropy Generation in a Maximum-Entropy Landscape with Time-Dependent Constraints

A rate equation for a discrete probability distribution is discussed as a route to describe smooth relaxation towards the maximum entropy distribution compatible at all times with one or more linear constraints. The resulting dynamics follows the path of steepest entropy ascent compatible with the constraints. The rate equation is consistent with the Onsager theorem of reciprocity and the fluctuation-dissipation theorem. The mathematical formalism was originally developed to obtain a quantum theoretical unification of mechanics and thermodinamics. It is presented here in a general, non-quantal formulation as a part of an effort to develop tools for the phenomenological treatment of non-equilibrium problems with applications in engineering, biology, sociology, and economics. The rate equation is also extended to include the case of assigned time-dependences of the constraints and the entropy, such as for modeling non-equilibrium energy and entropy exchanges.

Quantum thermodynamics of nonequilibrium. Onsager reciprocity and dispersion-dissipation relations

A generalized Onsager reciprocity theorem emerges as an exact consequence of the structure of the nonlinear equation of motion of quantum thermodynamics and is valid for all the dissipative nonequilibrium states, close and far from stable thermodynamic equilibrium, of an isolated system composed of a single constituent of matter with a finite-dimensional Hilbert space. In addition, a dispersion-dissipation theorem results in a precise relation between the generalized dissipative conductivity that describes the mutual interrelation between dissipative rates of a pair of observables and the codispersions of the same observables and the generators of the motion. These results are presented together with a review of quantum thermodynamic postulates and general results.

Energy and Entropy Balances in a Combustion Chamber: Analytical Solution

Abstract An analytical solution of the energy and entropy balance equations for a combustible gas mixture contained in an open combustion chamber, for example of an internal combustion engine, is presented. The solution is free of major assumptions and is in a form suitable for incorporating any detailed models for the effects of wall heat transfer, wall thermal boundary layer, non-uniform temperature distributions in the burnt mixture, crevice regions, mass exchange through the boundaries of the chosen control volume and other similar effects. Explicit expressions for the instantaneous mass of burnt mixture and for the entropy generated by irreversibility are presented as functions of pressure and volume history of the combustion chamber and properties of the gas mixtures.

Steepest entropy ascent in Quantum Thermodynamics

Quantum Thermodynamics [I] is a unified quantum theory that includes within a single uncontradictory nonstatistical structure the whole of Quantum Mechanics and Classical Equilibrium Thermodynamics, as well as a general description of nonequilibrium states, their entropy, and their irreversible motion towards stable equilibrium. Quantum Thermodynamics postulates that a system has access to a much broader set of states than contemplated in Quantum Mechanics. Specifically, for a system that is strictly uncorrelated from any other system, namely, a system for which Quantum Mechanics contemplates only states that are described by a state vector I~>, Quantum Thermodynamics postulates that in addition to the quantum mechanical states there exist many other states that cannot be described by a vector I~> but must be described by a self-adjoint, unit-trace, nonnegative-definite linear operator p that we call the state operator.

Steepest entropy ascent model for far-nonequilibrium thermodynamics: unified implementation of the maximum entropy production principle.

By suitable reformulations, we cast the mathematical frameworks of several well-known different approaches to the description of nonequilibrium dynamics into a unified formulation valid in all these contexts, which extends to such frameworks the concept of steepest entropy ascent (SEA) dynamics introduced by the present author in previous works on quantum thermodynamics. Actually, the present formulation constitutes a generalization also for the quantum thermodynamics framework. The analysis emphasizes that in the SEA modeling principle a key role is played by the geometrical metric with respect to which to measure the length of a trajectory in state space. In the near-thermodynamic-equilibrium limit, the metric tensor is directly related to the Onsagers generalized resistivity tensor. Therefore, through the identification of a suitable metric field which generalizes the Onsager generalized resistance to the arbitrarily far-nonequilibrium domain, most of the existing theories of nonequilibrium thermodynamics can be cast in such a way that the state exhibits the spontaneous tendency to evolve in state space along the path of SEA compatible with the conservation constraints and the boundary conditions. The resulting unified family of SEA dynamical models is intrinsically and strongly consistent with the second law of thermodynamics. The non-negativity of the entropy production is a general and readily proved feature of SEA dynamics. In several of the different approaches to nonequilibrium description we consider here, the SEA concept has not been investigated before. We believe it defines the precise meaning and the domain of general validity of the so-called maximum entropy production principle. Therefore, it is hoped that the present unifying approach may prove useful in providing a fresh basis for effective, thermodynamically consistent, numerical models and theoretical treatments of irreversible conservative relaxation towards equilibrium from far nonequilibrium states. The mathematical frameworks we consider are the following: (A) statistical or information-theoretic models of relaxation; (B) small-scale and rarefied gas dynamics (i.e., kinetic models for the Boltzmann equation); (C) rational extended thermodynamics, macroscopic nonequilibrium thermodynamics, and chemical kinetics; (D) mesoscopic nonequilibrium thermodynamics, continuum mechanics with fluctuations; and (E) quantum statistical mechanics, quantum thermodynamics, mesoscopic nonequilibrium quantum thermodynamics, and intrinsic quantum thermodynamics.