Rate of dissipation

The term j(t is the rate of dissipation of energy per unit volume by joule heating. This occurs within the working fluid, and so represents a departure... 

Liquids The rate of dissipation of charges in a liquid, assuming that its conductivity and dielectric permittivity are constant, can be expressed as ... 

The rate of charge generation must exceed the rate of dissipation, so charge can accumulate. 

The rate of dissipation of turbulent kinetic energy, s, is more difficult to measure. 

This response time should be compared to the turbulent eddy lifetime to estimate whether the drops will follow the turbulent flow. The timescale for the large turbulent eddies can be estimated from the turbulent kinetic energy k and the rate of dissipation e, Xc = 30-50 ms, for most chemical reactors. The Stokes number is an estimation of the effect of external flow on the particle movement, St = r /tc. If the Stokes number is above 1, the particles will have some random movement that increases the probability for coalescence. If St 1, the drops move with the turbulent eddies, and the rates of collisions and coalescence are very small. Coalescence will mainly be seen in shear layers at a high volume fraction of the dispersed phase. 

From a theoretical perspective, the object that is initially created in the excited state is a coherent superposition of all the wavefunctions encompassed by the broad frequency spread of the laser. Because the laser pulse is so short in comparison with the characteristic nuclear dynamical time scales of the motion, each excited wavefunction is prepared with a definite phase relation with respect to all the others in the superposition. It is this initial coherence and its rate of dissipation which determine all spectroscopic and collisional properties of the molecule as it evolves over a femtosecond time scale. For IBr, the nascent superposition state, or wavepacket, spreads and executes either periodic vibrational motion as it oscillates between the inner and outer turning points of the bound potential, or dissociates to form separated atoms, as indicated by the trajectories shown in Figure 1.3. 

The present analysis shows that when a thermodynamic gradient is first applied to a system, there is a transient regime in which dynamic order is induced and in which the dynamic order increases over time. The driving force for this is the dissipation of first entropy (i.e., reduction in the gradient), and what opposes it is the cost of the dynamic order. The second entropy provides a quantitative expression for these processes. In the nonlinear regime, the fluxes couple to the static structure, and structural order can be induced as well. The nature of this combined order is to dissipate first entropy, and in the transient regime the rate of dissipation increases with the evolution of the system over time. 

Air temperature and vapor density are two factors which influence the rate of dissipation of radiant energy received by plants from the sun, and they determine in a large measure the temperature of the plant part and consequently sunburn and sulfur burn (18), other factors, such as particle size of the sulfur, being constant. 

The k-s turbulence model was developed and described by Launder and Spalding (1972). The turbulent viscosity, pt, is defined in terms of the turbulent kinetic energy, k, and its rate of dissipation, z. 

Volatilization loss can be a significant dissipation pathway for organics applied to land. The rate of dissipation of organics is governed by the vapour pressure of the compound, and on soil and environmental conditions. Losses to the atmosphere may take place immediately if the organics are applied at the soil surface if the organics are incorporated with the surface soil layer or injected below surface, the rate of volatilization loss is significantly reduced and is dependent on the rate of transport to the soil surface. As an example, 90% of Heptachlor applied on the soil surface may be lost in 2-7 days, in comparison to a 7% loss in 167 days when incorporated to 7.5cm [14]. 

The second length scale characterizing turbulence is that over which molecular effects are significant it can be introduced in terms of a representative rate of dissipation of velocity fluctuations, essentially the rate of dissipation of the turbulent kinetic energy. This rate of dissipation, which is given by the symbol e0, is... 

We conclude that the growth of a new phase is controlled by the rate of dissipation at a moving kink. This dissipation is taking place at the microlevel and must be prescribed in order for the macro-description to be complete. The incompleteness of the continuum model manifests itself through the sensitivity of the solution to the singular (measure-valued) contributions describing fine structure of the subsonic jump discontinuities (kinks). 

Since the energy ofthe nucleus is identically zero, the integral impact of this localized contribution to the initial data can be measuredby the corresponding energy density which is finite. For our self-similar solution (2.5) one can equivalently calculate the rate of dissipation R (Dafermos, 1973)... 

The convective turbulence tend to dissipate large scale shears (wave length a > pressure scale height H ). The rate of dissipation... 

An intriguing aspect of these measurements is that the values of D determined from NMR and from sorption kinetics differ by several orders of magnitude. For example, for methane on (Ca,Na)-A the value of the diffusion coefficient determined by NMR is 2 x 10 5 cm2 sec-, and the value determined for sorption rates only 5 x 10"10 cm2 sec-1. The values from NMR are always larger and are similar to those measured in bulk liquids. The discrepancy, which is, of course, far greater than the uncertainty of either method, remained unexplained for several years, until careful studies (267,295,296) showed that the actual sorption rates are not determined by intracrystalline diffusion, but by diffusion outside the zeolite particles, by surface barriers, and/or by the rate of dissipation of the heat of sorption. NMR-derived results are therefore vindicated. Large diffusion coefficients (of the order of 10-6 cm2 sec-1) can be reliably measured by sorption kinetics... 

Quality factor (Q-factor) compares the frequency of oscillation to the rate of dissipation of energy of the oscillating system. Higher Q indicates less energy dissipation, relative to the oscillating frequency. 

If proper safeguards are to be maintained economically, it is essential to define the extent of the hazard and identify the problem areas. Research is needed to determine the sites and duration of exposure and to measure the amounts of residues and their rates of dissipation. Such measurements can be made with precision. The problem is to use knowledge gained in a particular situation to provide guidelines or models which can be applied more generally to field operations. Such extrapolations are controversial and they may also be dangerous if they are in error. The symposium includes descriptions of techniques for measurement of exposure, and some contributors indicate the controversial aspects of solutions that have been proposed. 

Here the first term to the right describes a rate of fluctuations for the given initial conditions, and the second term describes a rate of dissipation due to coupling back and forth between p- and s-regions. A similar equation can be written for the RDOp of the s-region, trp[P (t)], if needed. 

Hence, the collisional rate of dissipation and the collisional stress tensor are related to the coefficient of restitution by Eq. (5.284) and Eq. (5.277). 

Lawn, C.J. (1971). The determination of the rate of dissipation in turbulent pipe flow. J. Fluid Mech., 48, 477-505. 

Generally, one may establish that in some cases greatly enhanced concentration fluctuations occur under flow, in others, however, the size of concentration fluctuations is reduced and, obviously, flow promotes mutual miscibility of the polymers. Concentration fluctuations are accompanied by inhomogeneities of transport quantities as shear viscosity and diffusity. In a flow field the molecules are transferred into a non-equilibrium situation of extension. Two polymer molecules in a state of excess extension feel an additional repulsion due to the enhanced normal stress difference. Thus, the rate of dissipation by diffusion is low compared with the shear rate and the concentration fluctuations tend to grow. The opposite is true for a state of lower extension. In that case the dissipation of the concentration fluctuations is enhanced owing to an additional attraction between the chain molecules. 

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