Number Of Balance Equations Research Articles

Monatomic gas as a singular limit of polyatomic gas in molecular extended thermodynamics with many moments

The difference in the theoretical structure between monatomic and polyatomic gases in highly nonequilibrium states is discussed from the viewpoint of molecular extended thermodynamics (MET) of rarefied gases, which is free from the local equilibrium assumption. The MET theories of these two types of gases are based on the moment balance equations with different hierarchy structures due to whether the internal degrees of freedom of a molecule are incorporated in their distribution functions or not. In particular, the number of balance equations in the MET theory of polyatomic gases is greater than the number in the corresponding theory of monatomic gases. The closure procedure for the system of balance equations of polyatomic gases obtained in a recent paper (Arima et al., 2014) is adopted. We prove that the solutions for polyatomic gases converge, in the limit where the degrees of freedom of a molecule D tend to 3, to the ones for monatomic gases provided that we impose appropriate initial conditions compatible with monatomic gases. Thus a MET theory of rarefied monatomic gases can be identified as a singular limit of the corresponding MET theory of rarefied polyatomic gases. As illustrative examples, the asymptotic behaviors when D→3 in the dispersion relation of ultrasonic waves and in the shock wave structure are shown.

On Equilibrium Probabilities for the Delays in Deterministic Flow Lines With Random Arrivals

While flow line models have been studied for decades, excepting some cases with three servers or less, their equilibrium probabilities remain elusive. As such, approximations based on aggregation or decomposition methods are generally employed. In this paper, by focusing on flow lines with deterministic service durations and a renewal arrival process, we develop exact methods for steady-state analysis. Our starting point is the investigation of recursions for customer delay based on exact decomposition methods. We demonstrate that the delay a customer faces in each server possesses a Markovian property. For discrete-time flow lines, we obtain a multidimensional discrete-time time-homogeneous Markov chain for the delays; there are an infinite number of balance equations for the equilibrium probabilities. Exploiting a similarity between our system and the GI/D/1 queue allows us to reduce these to a finite number of balance equations that can be solved numerically. We also investigate the implications for continuous-time flow lines and consider an example inspired by production time windows in semiconductor manufacturing. To our knowledge, these are the first results that allow one to exactly obtain the equilibrium probabilities in flow lines consisting of more than three servers.

Control System of Substance and Energy Balances of Combined Heat-and-Power Plants Applying the Least Squares Adjustment Method

The set of substance and energy balance equations of water and steam collectors, together with the balance of boilers and turbines (including the regeneration system) is the base of the control system of the exploitation of a combined heat-and power plant. The initial values of this system are the results of measurements in the course of exploitation. Due to inevitable errors in the measurements the substance balances display discrepancies between the balance of steam and the balance of the feed water, and the calculated technical indices are uncertain. In order to increase the reliability of the results of the technical analysis of exploitation and to co-ordinate the balance equations we may use the least squares adjustment method. It may be applied on the condition that we have a sufficient surplus of measuring data. The number of balance equations must be higher than the number of unknown values. This makes it possible to apply the criterion of least squares warranting a maximum of the reliability function in an n dimensional. space of errors (n - number of measuring data). This method has been applied in the balance systems of the combined heat-and-power generating plant.

Conditions for applicability of a diffusion description for vibrational-translational relaxation of diatomic molecules

The description of relaxation by the methods of classical statistics and mechanics has a number of advantages, the chief one of which is obviously the opportunity of using in place of a large number of balance equations for individual levels a single transport equation for a distribution function over vibrational energy and, in a number of cases, the additional opportunity for more general and clearer consideration of the various factors characterizing intra- and intermolecular interactions. A version of the classical theory for vibrational relaxation during weak interaction of molecules with a gas is diffusion theory in which a Fokker-Planck diffusion equation serves as the transport equation. In particular, a detailed study of the influence of anharmonicity at various values of the adiabaticity parameter ξo on the kinetic characteristics of the process was carried out [2] within the limits of this theory on the basis of a solution of the diffusion equation from [1]; earlier [3], vibrational relaxation in a light inert-gas medium was considered in the diffusion approximation (for harmonic and anharmonic oscillators) which coresponds to relaxation for nonadiabatic interaction in a uniform temperature field (ξo → 0) as shown in [1]. The possibility of describing vibrational kinetics within the framework of diffusion theory involves two basic problems: 1) the possibility of approximating the transport equation for classical oscillators with a Fokker-Planck equation, and 2) the possibility of describing the relaxation of quantum oscillators by classical methods or the correspondence of classical and quantum theory of relaxation for weak interaction of molecules with a medium. The first of these problems is considered here.