Transport of Kinetic Energy

The average of the second term on the RHS, (m -C)M, vanishes, whereas the average of the last term on the RHS, denotes the kinetic energy of the macroscopic 

The translational kinetic gas temperature, T, is defined in terms of e by the relation  

It is noted that for point particles equipartition of energy gives the relation jm(C )M = kT. k is the universal Boltzmann constant, k = 1.380 x 10 (J/K). 

The equipartition principle was initially proposed by Maxwell [96] in 1867 who stated that the energy of a gas is equally divided between linear and rotational energy. The original theorem was later generalized by Boltzmann [10] in 1872 by showing that the internal energy is actually equally divided among all the independent components of motion in the system. 

6 The Equation of Change in Terms of Mean Molecular Properties 

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The present investigation applies deterministic methods of continuous mechanics of multiphase flows to determine the mean values of parameters of the gaseous phase. It also applies stochastic methods to describe the evolution of polydispersed particles and fluctuations of parameters [4]. Thus the influence of chaotic pulsations on the rate of energy release and mean values of flow parameters can be estimated. The transport of kinetic energy of turbulent pulsations obeys the deterministic laws. 

Equations of Change for Multi-Component Mixtures 47 Transport of kinetic energy... 

The species s heat flux vector associated with the transport of kinetic energy is obtained if we let V s = rnsC, hence the mixture heat flux vector is determined by the sum over all the components ... 

Therefore, a kinetic energy correction factor, a, can be defined as the ratio of the true rate of kinetic energy transport relative to that which would occur if the fluid velocity is everywhere equal to the average (plug flow) velocity, e.g.,... 

Reynolds (Rl, R2) was one of the earlier investigators to appreciate the random nature of turbulence. The dimensionless parameter bearing his name is widely used as a measure of the physical characteristics of steady, uniform flow. Such a measure is essentially macroscopic and does not describe the local or transient behavior at a point in the stream. In recent years much effort has been devoted to understanding the basic mechanism of momentum transport by turbulence. The early work of Prandtl (P6), Taylor (Tl), Karmdn (Kl), and Howarth (K4) laid a basis for the statistical theory of turbulence which is apparently in reasonable agreement with experiment. More recently Onsager (03), Corrsin (C6), and Kolmogoroff (K10) extended the statistical theory of turbulence to describe the available experimental data in terms of kinetic-energy... 

An electrostatic mirror can be produced by an electrode at a potential energy that is greater than the kinetic energy divided by the charge of the particle. The bending and focusing power of electrostatic systems are limited by the maximum electric fields that can be applied across the electrodes. Extensive electrostatic systems have been constructed for the transport of low-energy beams, KE < 50 keV, for example, beams extracted from ion sources are usually transported with electrostatic elements. 

The x component of kinetic energy transported by these molecules is... 

To evaluate the heating, a relativistic 1-D Fokker-Planck code was used. The configuration space is 1-D but the momentum space is 2-D, with axial symmetry. This code is coupled to a radiation-hydrodynamic simulation in order to include energy dissipation via ionization processes, hydrodynamic flow, the equation-of-state (EOS), and radiation transport. The loss of kinetic energy from hot electrons is treated through Coulomb and electromagnetic fields. 

The transport of thermal energy can be broken down into one or more of three mechanisms conduction--heat transfer via atomic vibrations in solids or kinetic interaction amongst atoms in gases1 convection - - heat rapidly removed from a surface by a mobile fluid or gas and radiation—heat transferred through a vacuum by electromagnetic waves. The discussion will begin with brief explanations of each. These concepts are important background in the optical measurement of temperature (optical pyrometry) and in experimental measurement of the thermally conductive behavior of materials. 

Kinetic energy, rate of kinetic energy transport by a flowing stream. 

As mentioned in the introductory remarks, ultrasound waves transport both kinetic energy (particles of the medium oscillate and displace from their equilibrium position in the direction of propagation) and potential energy (fluid compression), as fluids can support negative pressure for short times. When a sufficiently large... 

Viscoelasticity. When in addition, the flnid or the material stores some capacitive energy, an elastic property comes in snpplement to the dynamic and kinematic viscosities for featuring the system. The storage and the transport of capacitive energy is then supported by a combined property of viscoelasticity, analogous to a kinetic constant in physical chemistry (see case study All Reactive Chemical Species in Chapter 4 and case stndy 116 Viscoelastic Relaxation in Chapter 11). 

Our treatment is limited to a very simple model, a dilute gas of hard spheres. Nonetheless the results are only approximate. The expressions for Y and Z), (2.35) and (2.41), differ from the results of an exact theory of hard-sphere transport by a numerical factor. The expression for (2.38), is in error in an additional way. The derivation does not account for possible differences in the rate of transport of translational energy (center-of-mass kinetic energy) and of rotational and vibrational energy (internal energy). Exact theory does the results, assuming internal energy is also equilibrated with each collision, are summarized in Table 2.1. 

Comparison of and indicates that the transport of kinetic and internal energy involves different averaging processes. Thus x is not simply proportional to the total molecular heat capacity, 6 + 1.5/ . 

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