Energy density, equations

Based on the energy density equation (Section 4), it is theoretically possible to increase the energy density of BOPP capacitors by increasing the PP dielectric constant (e) and/or the applied electric field ( ) controlled by breakdown strength. However, recent experimental results, studying several high dielectric... 

We may also define the helical wave vector k-lTzIp, which then appears in the twist term of the free energy density equation for an N phase, i.e.,... 

The value of at zero temperature can be estimated from the electron density ( equation Al.3.26). Typical values of the Femii energy range from about 1.6 eV for Cs to 14.1 eV for Be. In temis of temperature (Jp = p//r), the range is approxunately 2000-16,000 K. As a consequence, the Femii energy is a very weak ftuiction of temperature under ambient conditions. The electronic contribution to the heat capacity, C, can be detemiined from... 

C and I account for gradients of the smectic order parameter the fifth tenn also allows for director fluctuations, n. The tenn is the elastic free-energy density of the nematic phase, given by equation (02.2.9). In the smectic... 

A phase itself, the amplitude of the density modulation is constant and twist and splay distortions are forbidden, thus the expression for the free energy density simplifies to equation (C2.2.10). 

When the iateraction energy density is positive, equation 5 defines a critical temperature of the UCST type (Fig. la) that is a function of component molecular weights. The LCST-type phase diagram, quite common for polymer blends, is not predicted by this simple theory unless B is... 

The permachor method is an empirical method for predicting the permeabiUties of oxygen, nitrogen, and carbon dioxide in polymers (29). In this method a numerical value is assigned to each constituent part of the polymer. An average number is derived for the polymer, and a simple equation converts the value into a permeabiUty. This method has been shown to be related to the cohesive energy density and the free volume of the polymer (2). The model has been modified to liquid permeation with some success. 

In these equations, is the molar volume and 5 is the solubiUty parameter of the pure Hquid k. The solubiUty parameter is the square root of an energy density, defined as... 

The energy densities of laser beams which are conventionally used in the production of thin films is about 10 — 10 Jcm s and a typical subsU ate in the semiconductor industry is a material having a low drermal conductivity, and drerefore dre radiation which is absorbed by dre substrate is retained near to dre surface. Table 2.8 shows dre relevant physical properties of some typical substrate materials, which can be used in dre solution of Fourier s equation given above as a first approximation to dre real situation. 

The mixture cohesive energy density, coh-m> was not to be obtained from some mixture equation of state but rather from the pure-component cohesive energy densities via appropriate mixing rules. Scatchard and Hildebrand chose a quadratic expression in volume fractions (rather than the usual mole fractions) for coh-m arid used the traditional geometric mean mixing rule for the cross constant ... 

The stiffness matrix, Cy, has 36 constants in Equation (2.1). However, less than 36 of the constants can be shown to actually be independent for elastic materials when important characteristics of the strain energy are considered. Elastic materials for which an elastic potential or strain energy density function exists have incremental work per unit volume of... 

The difference between the two ordinate origins represents the log of the coefficient in the equation between abrasion resistance, defined by the energy required to remove unit volume of mbber (J/mm ) and the energy density at break in the same units. 

The Navier-Stokes equation and the enthalpy equation are coupled in a complex way even in the case of incompressible fluids, since in general the viscosity is a function of temperature. There are, however, many situations in which such interdependencies can be neglected. As an example, the temperature variation in a microfluidic system might be so small that the viscosity can be assumed to be constant. In such cases the velocity field can be determined independently from the temperature field. When inserting the computed velocity field into Eq. (77) and expressing the energy density e by the temperature T, a linear equahon in T is... 

The electron energy density We = tiff (the product of the electron density and average electron energy e) is calculated self-consistently from the second moment of the Boltzmann equation ... 

The fluid model is a description of the RF discharge in terms of averaged quantities [268, 269]. Balance equations for particle, momentum, and/or energy density are solved consistently with the Poisson equation for the electric field. Fluxes described by drift and diffusion terms may replace the momentum balance. In most cases, for the electrons both the particle density and the energy are incorporated, whereas for the ions only the densities are calculated. If the balance equation for the averaged electron energy is incorporated, the electron transport coefficients and the ionization, attachment, and excitation rates can be handled as functions of the electron temperature instead of the local electric field. 

In addition, Turner and Trimble defined a slip equation of state combination as the specification of mass flux, momentum flux, energy density, and energy flux as single-valued functions of the geometric parameters (area, equivalent diameter, roughness, etc.) at any z location, and of mass flux, pressure, and enthalpy,... 

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