Macroscopic kinetic energy

One of the primary goals of current research in the area of tribology is to understand how it is that the kinetic energy of a sliding object is converted into internal energy. These dissipation mechanisms detennine the rate of energy flow from macroscopic motion into the microscopic modes of the system. Numerous mechanisms can be... 

Equation 6-10 is the macroscopic energy balance equation, in which potential and kinetic energy terms are neglected. From tliermodynamics, the enthalpy per unit mass is expressed as... 

The application of dimensional analysis to Eq. (38) yields, under the assumption that heat and macroscopic kinetic energy are fundamentally different physical quantities (so that five units are required—heat, mass, length, time, and temperature), the expression... 

If we are going to relate the properties of our system to a physical situation, we need to be able to characterize the system s temperature, T. In a macroscopic collection of atoms that is in equilibrium at temperature T, the velocities of the atoms are distributed according to the Maxwell -Boltzmann distribution. One of the key properties of this distribution is that the average kinetic energy of each degree of freedom is... 

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... 

After manipulations systematically dropping higher order terms in x, the problem is reduced to one in classical calculus of variations. In taking the variations of, Q , certain dependencies exist. Thus Pax is proportional to the kinetic energy part of E. Our final end product will be explicit functional dependencies of Pap, Qa, on p,ua,E, whose approximations are the classical macroscopic relations and the Navier-Stokes equations. 

As two water molecules come together to form a hydrogen bond, attractive electric forces cause them to accelerate toward each other. This results in an increase in their kinetic energies (the energy of motion), which is perceived on the macroscopic scale as heat energy. 

The total eneigy V may be split into an internal energy, a potential energy, and a macroscopic kinetic energy. Eacli contribution taken separately does nol satisfy tin equation of the simple form of Equation 14) because of possible transformation or one form of energy to another... 

For the example of 198Hg, ( = 2.2 x 10-6 or vA = 670 m/s and corresponds to a kinetic energy of 0.92 eV. The magnitude of this difference is visible in Figure 9.9 because the separation between the two peaks is about twice the thermal width. Such a high velocity is difficult to attain with any macroscopic, that is, physical source. 

Turbulent velocity fluctuations ultimately dissipate their kinetic energy through viscous effects. Macroscopically, this energy dissipation requires pressure drop, or velocity decrease. The energy dissipation rate per unit mass is usually denoted e. For steady flow in a pipe, the average energy dissipation rate per unit mass is given by... 

All of the other phenomena associated with waves can also be observed in particles. For example, in 1927 Davisson and Germer accelerated a beam of electrons to a known kinetic energy and showed that these electrons could be diffracted off a nickel crystal, just as X-rays are diffracted (see Figure 3.8). Just as with photons, interference is not always seen if the wavelength spread or the slits are large, the fringes wash out. This also explains why interference is not seen with macroscopic objects, such as buckshot—the wavelength is far too small. 

The results of the last section showed that, for any macroscopic container at normal pressures, it is not reasonable to conclude that the molecules proceed from wall to wall without interruption. However, if the interaction potential energy between molecules at their mean separation is small compared to the kinetic energy, the speed distribution and the average concentration of gas molecules is about the same everywhere in the container. In this limit, the only real effect of collisions is the excluded volume occupied by the molecule, which effectively shrinks the size of the container. At 1 atm, only about 1/1000 of the space is occupied (remember the density ratio between gas and liquid), so each additional molecule sees only 99.9% of the container as free space. On the other hand, if the attractive part of the interaction potential cannot be totally neglected, the molecules which are very near the wall will be pulled slightly away from the wall by the other molecules. This tends to decrease the pressure. 

The internal energy of a substance does not include any energy that it may possess as a result of its macroscopic position or movement. Rather it refers to the energy of the molecules making up the substance, which are in ceaseless motion and possess kinetic energy of translation except for monatomic molecules, they also possess kinetic energy of rotation and of internal vibration. The addition of heat to a substance increases this molecular activity, and thus causes an increase in its internal energy. Work done on the substance can have the same effect, as was shown by Joule. 

Statistical methods represent a background for, e.g., the theory of the activated complex (239), the RRKM theory of unimolecular decay (240), the quasi-equilibrium theory of mass spectra (241), and the phase space theory of reaction kinetics (242). These theories yield results in terms of the total reaction cross-sections or detailed macroscopic rate constants. The RRKM and the phase space theory can be obtained as special cases of the single adiabatic channel model (SACM) developed by Quack and Troe (243). The SACM of unimolecular decay provides information on the distribution of the relative kinetic energy of the products released as well as on their angular distributions. 

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