Kinetic weight function

Variously defined functions have been proposed and used for W, in Eq. (4). These include (6) the static weight function - the solution of the equation adjoint to Eq. (3) the kinetic weight function - - the solution of the equation adjoint to the w-mode Boltzmann equation [Eq. (17)] and the dynamic weight function, which equals unity If the weighting function is selected as the importance function, i.e., the solution of the adjoint equation... 

Chemostat kinetics Weighted divergence as a function of square wave frequency for eft) at the grid point (1, 0 20). 

The authors applied this model to the situation of dissolving and deposited interfaces, involving chemically interacting species, and included rate kinetics to model mass transfer as a result of chemical reactions [60]. The use of a stochastic weighting function, based on solutions of differential equations for particle motion, may be a useful method to model stochastic processes at solid-liquid interfaces, especially where chemical interactions between the surface and the liquid are involved. 

The data in Table II pertaining to pyrolysis conditions shows that all four feedstocks were pyrolyzed under substantially similar conditions, namely steam-to-hydrocarbon weight ratios of 0.9 0.1, residence times of 0.3 sec, reactor exit pressures of 2.0 bar absolute, and reactor exit temperatures of 835°C. Care also was taken to maintain identical axial temperature profiles in the reactor for each of these runs. No unambiguous measure of substrate conversion during pyrolysis is possible for distillate feedstocks of the type used in the present experiments in terms of the empirical kinetic severity function of Zdonik et al. (5), all of the present experiments were conducted at a severity of about 2. 

We now move to the general case of a continuous description the pragmatical usefulness of such a type of description in real-life kinetic problems has been discussed by Krambeck (1991a,b). Let c(x,0) be the initial distribution of reactant concentrations in the batch reactor, and let c(x,t) be the concentration distribution at any subsequent (dimensional) time t. We assume that both the label x and the concentration c have already been normalized so that = = 1. Furthermore, we assume that a (dimensional) frequency factor k(x) can be identified, and that x has been normalized so that k(x) = k x, where k is the average value of k(x) at r = 0. One then normalizes the time scale as well by defining the dimensionless time t as k t. The overall concentration C(t) is defined with a weighting function that is identical to unity, C(t) = , 0(0) = 1. 

Tahara Y, Shima K. Kinetics of HbAlc, glycated albumin, and fructosamine and analysis of their weight functions against preceding plasma glucose level. Diabetes Care 1995 18 440-7. 

If one can model the noninteracting one-matrix yj(r,r ) as a functional of the electron density, then, using Equation 1.47 or Equation 1.48, one can compute the kinetic energy. This is the most straightforward approach to deriving kinetic energy functionals, and the Thomas-Eermi functional and the Weizsacker functional can both be derived in this way. (Indeed, all of the most popular functionals can be derived in several different ways.) The one-matrix can also be modeled based on weighted density approximation (WDA), which we will discuss subsequently. 

The symmetrized form seems to be the best for recovering shell structure in the electron density. " Taking the Fermi momentum to be constant has computational advantages because the kinetic energy functional can be evaluated very quickly as a double convolution of densities with the weighting function. In this case, the form of the functional also simplifies, becoming... 

We note that Hoover defines y kT as the instantaneous kinetic energy per particle, KEqj /N, and for a "constant temperature simulation," Hoover constrains the kinetic energy to be strictly constant. This produces an ensemble whose weight function contains 6(KEqj - constant), and hence it is different from the MD ensemble. Several articles on nonequilibrium fluids appear in the January, 1984, issue of Physics Today, and of particular interest for nonequilibrium molecular dynamics are the articles by Evans, Hanley, and Hess, by Hoover, and by Alder and Alley. ... 

In reaction kinetics simulations, we might wish that the relative accuracy of the fitted concentrations should be equal for both low and high concentrations (e.g. radicals and products). For this reason, a weighting function w,= l/F Cx ) is... 

Miscible blends of high molecular weight polymers often exhibit LOST behavior (3) blends that are miscible only because of relatively low molecular weights may show UCST behavior (11). The cloud-point temperatures associated with Hquid—Hquid phase separation can often be adequately determined by simple visual observations (39) nevertheless, instmmented light transmission or scattering measurements frequendy are used (49). The cloud point observed maybe a sensitive function of the rate of temperature change used, owing to the kinetics of the phase-separation process (39). 

In actual experiments we do not usually observe directly the desorbed amount, but rather the derived read-out quantities, as is the time dependence of the pressure in most cases. In a closed system, this pressure is obviously a monotonously increasing function of time. In a flow or pumped system, the pressure-time dependence can exert a maximum, which is a function of the maximum desorption rate, but need not necessarily occur at the same time due to the effect of the pumping speed S. If there are particles on the surface which require different activation energies Ed for their desorption, several maxima (peaks) appear on the time curve of the recorded quantity reflecting the desorption process (total or partial pressure, weight loss). Thereby, the so-called desorption spectrum arises. It is naturally advantageous to evaluate the required kinetic parameters of the desorption processes from the primarily registered read-out curves, particularly from their maxima which are the best defined points. 

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