Model equivalent box

Equity capital cost, 9 542 Equivalent box model (EBM) for polymer blends, 20 344 volume fraction calculation in, 20 345-346... 

Volume changes, by vitreous silica, 22 438 Volume flux, of droplets, 23 187 Volume fraction, in filtration, 11 328 Volume fraction calculation, in equivalent box model, 20 345—346 Volume mean diameter, 23 186 Volume of activation, 13 407-408... 

Fig. 19. Equivalent box model for a heterogeneous binary blend (schematically).

An agreement with experimental results was obtained by taking into account the increased effective fraction of the filler, Veff, due to the glassy interphase of the bound epoxide layer and assuming a co-continuous morphology of the epoxy-silica hybrid network. Mechanical properties in dependence on the phase continuity are treated by parallel and series models for bicontinuous morphology and discontinuous phases, respectively. The equivalent box model (EBM) developed by Takayanagi (13) (eqs 2-5) and Davies model (14) (eq. 6) were used to compare the experimental data with the theory (9). 

A model referred to as the equivalent box model (EBM) has shown promise in the ability to predict the modulus behavior over the entire composition. This model has similarities to earlier models by Takayangi et al. [6,7]. This model is a combination of the parallel and series models and has been developed by Kolarik [8-10]. This model will be described in detail because of its versatility for phase separated blends. This mechanical model is illustrated in Fig. 6.5. The modulus is calculated from... 

An analysis (equivalent box model) proposed by Kolarik, employing the universal constants predicted by the DeGennes percolation theory, can be used to predict the permeability of phase separated blends [162]. The universal constants can also be considered adjustable variables to fit the specific data. The EBM approach employs a series and parallel combination and is virtually identical to the EBM model described earlier in this chapter for modulus, except P is substituted for E. Thus ... 

The formalism employed for permeabihty relationships in the this section, such as the series model, parallel model. Maxwell s equation, and the equivalent box model (EBM) can be employed for thermal conductivity by replacing P with K. 

The prediction of conductivity, a (electrical, ionic or proton), based on pure component values for miscible and phase separated systems can employ the same models as used for modulus and permeability, where a (S/cm) can be substituted for modulus (E) or permeability (P) in the log additivity relationship for miscible systems and the parallel model, series model or equivalent box model for phase separated systems. While these expressions have not been generally employed for conductivity modeling, the principles on which they are based are analogous to modulus and permeability values. 

Many models have been presented to explain quantitatively the dependence of exciton energy on the cluster size [7, 11, 21-30]. This problem was first treated by Efros et al. [7], who considered a simple particle in a box model. This model assumes that the energy band is parabolic in shape, equivalent to the so-called effective mass approximation. The shift in absorption threshold, AE, is dependent upon the value of the cluster radius, R, Bohr radius of the electron, ae ( = h2v.jmce2), and Bohr radius of the hole, ah (= h2e/mhe2). When (1) R ah and R ae, and (2) ah R ac,... 

One way to present a gray box model of IPMC is to use an equivalent beam and an equivalent bimorph beam model and combine them with important physical properties of IPMCs Young s modulus and electromechanical coupling coefficient, determined from the rule of mixture bi-... 

Because of the complexity in modeling the real mold exterior surfaces, one may approximate the mold exterior surfaces as an equivalent box or sphere. 

A realistic model of a solution requires at least several hundred solvent molecules. To prevent the outer solvent molecules from boiling off into space, and minimizing surface effects, periodic boundary conditions are normally employed. The solvent molecules are placed in a suitable box, often (but not necessarily) having a cubic geometry (it has been shown that simulation results using any of the five types of space filling polyhedra are equivalent ). This box is then duplicated in all directions, i.e. the central box is suiTounded by 26 identical cubes, which again is surrounded by 98 boxes etc. If a... 

As a consequence, the presentation of the results will also differ from that in a MD or MC box, where a full set of molecules can be depicted (as snapshots). In an SCF model, all properties will be presented in, for example, (average) numbers of molecules per unit area of the membrane, or equivalent, i.e. the (average) densities of molecules as a function of the z-coordinate. The box thus consists, if one insists, only of one coordinate. For this reason, we can refer to this method as a one-gradient SCF theory or simply 1D-SCF theory. Extensions towards 2D-SCF are available, where lateral inhomogeneities in the bilayer can also be examined [80], There are even implementations of 3D SCF-like models, but here the interpretation is somewhat more delicate [78],... 

As pointed out by Peeters et al. (1996), based on their own experiments and on the reinterpretation of published field data, the adequate model to describe horizontal diffusion in lakes and oceans is the shear diffusion model by Carter and Okubo (1965). The model is described in Box 22.4. The most important consequence of this model is that the 4/3 law and the equivalent t3-power law for c2(t) expressed by Eq. 22-42 are replaced by an equation which corresponds to a continuous increase of the exponent m from 1 to 2 (Box 22.4, Eq.l) ... 

To shed more light on the structure of DNA, Rosalind Franklin and Maurice Wilkins used the powerful method of x-ray diffraction (see Box A-A) to analyze DNA fibers. They showed in the early 1950s that DNA produces a characteristic x-ray diffraction pattern (Fig. 8-14). From this pattern it was deduced that DNA molecules are helical with two periodicities along their long axis, a primary one of 3.4 A and a secondary one of 34 A The problem then was to formulate a three-dimensional model of the DNA molecule that could account not only for the x-ray diffraction data but also for the specific A = T and G = C base equivalences discovered by Chargaff and for the other chemical properties of DNA. 

Many methods for the fitting of data obtained from pulsed NMR have been described in the literature. The methods may, for example, be classified as either time domain (TD) or FD methods. Alternatively, they may be described as black-box methods or as interactive methods. An excellent review is given by de Beer and van Ormondt.20 There is now a consensus21 that FD and TD fitting methods are equivalent in terms of y2 parameter estimation if potential artifacts introduced by Fourier transformation are handled properly. TD and FD fitting can truly be equivalent (i.e. same 2 minima) if 2 is determined over the whole data range (a consequence of the power theorem on the Fourier transform) and if the model used to fit the experimental spectrum is correct. Very often, however, the model used is only an approximation. 

Since the unloaded QCM is an electromechanical transducer, it can be described by the Butterworth-Van Dyke (BVD) equivalent electrical circuit represented in Fig. 12.3 (box) which is formed by a series RLC circuit in parallel with a static capacitance C0. The electrical equivalence to the mechanical model (mass, elastic response and friction losses of the quartz crystal) are represented by the inductance L, the capacitance C and the resistance, R connected in series. The static capacitance in parallel with the series motional RLC arm represents the electrical capacitance of the parallel plate capacitor formed by both metal electrodes that sandwich the thin quartz crystal plus the stray capacitance due to the connectors. However, it is not related with the piezoelectric effect but it influences the QCM resonant frequency. 

Box and Meyer (1986) considered the identification of dispersion effects in unreplicated 2k p designs. The first step in their approach is to identify and estimate the active location effects. Let r,(i = 1,..., ) denote the residuals from the fitted location model. To examine whether factor j has a dispersion effect (or, equivalently, whether there is a dispersion effect associated with the jth main effect contrast), compute the sums of squared residuals at the two levels of this factor ... 

Box and Meyer also derived a useful result (which is applied in some of the subsequent methods in this chapter) that relates dispersion effects to location effects in regular 2k p designs. We present the result first for 2k designs and then explain how to extend it to fractional factorial designs. First, fit a fully saturated regression model, which includes all main effects and all possible interactions. Let /3, denote the estimated regression coefficient associated with contrast i in the saturated model. Based on the results, determine a location model for the data that is, decide which of the are needed to describe real location effects. We now compute the Box-Meyer statistic associated with contrast j from the coefficients 0, that are not in the location model. Let i o u denote the contrast obtained by elementwise multiplication of the columns of +1 s and—1 s for contrasts i and u. The n regression coefficients from the saturated model can be decomposed into n/2 pairs such that for each pair, the associated contrasts satisfy i o u = j that is, contrast i o u is identical to contrast j . Then Box and Meyer proved that equivalent expressions for the sums of squares SS(j+) and SS(j-) in their dispersion statistic are... 

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