Nielsen equation

Figure 6-24 Estimated Reinforcement (G /Gq) by Spherical Particles in a Matrix using the Kemer-Nielsen Equation G is the Modulus of the Composite, Gq of the Amylose Matrix, and G of the Starch Granule.

Lewis-Nielsen equation An equation derived from the theory of mixtures that provides an estimate of the modulus of a short-fiber, thermoplastic composite. It is given below. 

The Kerner-Nielsen equation (119) is a useful alternative to Eqs. (11.4) and (11.5) in the case when the elastic modulus of an interphase is substantially greater than that of the matrix. Most often, interphase elastic modulus is assumed to be equal to that of the filler resulting in a slightly overestimated elastic modulus of the compound. The effect of an interphase is then introduced in the Kerner-Nielsen equation substituting an effective filler volume fraction, Veff = (Vf + v,), for the filler volume fraction, Vt, (71) ... 

In boundary regions, ie, < oicr or V2 < i 2cr, the blend modulus can be calculated from the Kerner - Nielsen equation (216,217,219), which for dispersed component 2 with E2 > E reads... 

The second explanation for the deviation involves the randomization of the arrangement of the clay particles in the polymer nanocomposite, a disorder factor. This disorder factor attempts to account for misalignment of the clay nanoparticles that lowers the cross-section that a diffusing molecule encounters traversing the composite film. This approach has been applied to the Nielsen equation by Bharadwaj [5]. 

The general effect of the orientation factor is to lower the effective aspect ratio. When 5=1, the equation reduces to the Nielsen equation. Fredrickson and Bicerano [15] applied the same logic with nanoparticles that were modeled as disks. An excellent review of these models, including the Cussler-Aris model [16], is provided by Manias et al. [17]. 

The Nielsen equation above with the orientation factor correction is preferred for polymer-clay nanocomposites. The assumption by the... 

Figure 24 shows a plot of relative storage modulus versus volume fraction of iron powder in the feedstock. The relative storage modulus obtained at various dynamic stress frequency are shown. The data obtained at various frequency were very close and seems to be independent of the frequency. Both the Nielsen equation and modified Kemer equation were used in an attempt to fit the experimental data. The modified Kemer equation fits the experimental data at low powder loading very well. However, it failed to predict the data at powder volmne fraction of 0.58. In contrast, the Nielsen equation fits the data for the entire range of volume fraction. 

Figure 24. Plot of Nielsen equation and modified Kemer equation with experimental relative storage modulus data obtained at various dynamic stress frequency.

Some researchers (e.g., Abramovich,Baturin,Rajaratnam,- and Nielsen and Moller ) consider x to be the distance from a point located at some distance Xq upstream from the diffuser face. Equations for the jet boundaries and velocity profile used in the centerline velocity derivation assume that the jet is supplied from the point source. Addition of the distance Xq to the distance from the outlet corrects for the influence of the outlet size on the jet geometry. For practical reasons some researchers neglect Xq. 

Nielsen suggested a more exact equation relating viscosity and elasticity modulus [27] ... 

This equation is accurate at low damping (A < 1), but the error becomes large at high damping. More exact equations have been discussed by Struik (II) and Nielsen (4). The standard ASTM test is D2236-69. 

Nielsen (88) averaged the data in the literature and arrived at the ap proximate empirical equation... 

DiMarzjo (93), Nielsen (88), DiBenedetto (94), and others (89) have derived theoretical equations relating the shift in Tg en used by cross-linking ... 

On the surface of the Ni catalyst, carbon is normally produced in a whisker (or filamentous) form. According to Rostrup-Nielsen, carbon formation is avoided when the concentration of carbon dissolved in Ni crystal is smaller than that at the equilibrium. The steady-state activity is proportional to [C ], which can be expressed by the following equation ... 

Laboratory and mixed field/laboratory studies have confirmed that half-order kinetics for DO surface removal rates may be a reasonable approximation for sewer biofilm (Raunkjaeretal., 1997 Bjerre etal., 1998b). These results also showed the influence of readily biodegradable substrate. Furthermore, temperature dependency limited by diffusion is included (Nielsen et al., 1998). The following equation for the aerobic growth rate was therefore used ... 

Recently, the effect of dual continuity on a general property, P, of the system was demonstrated by Lipatov et al. [30,31]. They used an equation proposed by Nielsen [32]. 

Nielsen treats the solute concentration c(p,t) in the space surrounding a particle as a function of distance (radius p from the center) and time t, and the diffusion process obeys the following equations for dc/dt and (defined as the amount of solute diffusing in the direction of the solute concentration gradient per unit area of a cross section lying perpendicular to the gradient) ... 

Nielsen (2000) has recently attempted to preserve the equation REM = dreaming by suggesting that NREM dreams may be associated with dissociated (or covert ) REM events. However this suggestion is entirely speculative (Solms, 2000a). 

The reaction between aldehydes and ammonia leads to compounds usually called aldehyde ammonias (equation 80). The first reported compound ( acetaldehyde ammonia ) was synthesized by Liebig (1835LA(14)133). Nielsen et al. (73JOC3288) have shown that most of these compounds are hexahydro-l,3,5-triazines, which are usually isolated as the trihydrates (Table 17). When aldehydes containing a branched alkyl substituent are used, the free hexahydro-l,3,5-triazines may be formed (Table 17). The kinetics of the reaction between acetaldehyde and ammonia have been studied by H NMR spectroscopy (73JOC2931), and shown to be rather complex. It is beyond the scope of this review to discuss the reaction fully. 

According to Nielsen s method, Young s modulus E can also be calculated using the well-known equation... 

As a first approximation, the temperature depression for hydrate inhibition might be considered to be similar to the depression of the freezing point of ice by an equivalent mass fraction of the inhibitor. However, Nielsen and Bucklin (1983) derived an equation indicating that the hydrate depression temperature will always be less than the ice depression temperature by a factor equal to [(heat of fusion of ice)/(heat of hydrate dissociation)], which has a numerical value between 0.6 and 0.7 as a function of the hydrate structure. This is illustrated in Figure 4.2d, by the fact that at constant pressure, the ice depression temperature (i.e., distance between... 

Pieroen (1955) and Nielsen and Bucklin (1983) presented derivations to show the theoretical validity of the Hammerschmidt equation. The latter work suggested that the equation applies only to typical natural gases, and to methanol concentrations less than 0.20 mole fraction (typically for system operation at temperatures above 250 K). It may easily be shown (Yamanlar et al., 1991) that the Hammerschmidt equation should not apply to high concentrations of an inhibitor that might vaporize. Nixdorf and Oellrich (1996) have shown that the Hammerschmidt equation under-predicts natural gas systems inhibited with TEG. 

Nielsen and Bucklin (1983) presented an improved version of the Hammerschmidt equation which is accurate over a wider range, that is, to concentrations as large as 0.8 mole fraction. They suggested that Equation 4.8 may be effectively used to design methanol injection systems operating as low as 165 K... 

Nielsen, J. E. and Vriend, G. (2001) Optimizing the hydrogen-bond network in Poisson-Boltzmann equation-based pKa calculations. Proteins 43,403-412. 

Many of the equations employed in unstructured and non-segregated models derive from those of enzymatic kinetics (Sinclair and Kristiansen, 1987 Nielsen and Nikolajsen, 1988). Cells are considered as chemical reactors that support thousands of complex reactions catalyzed by enzymes that allow the conversion of substrates into secreted products. The equation formulated by Michaelis and Menten represents the enzymatic conversion rate of a unique substrate into one product (Equation 14). 

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