Fig. 3.6. A schematic representation of the 3t model describing the motion of the C—H internuclear vector. |

This equation corresponds to Equation (3.35) for the 3t model as well as Equation (3.36) for p = 3 for the first-order model-free treatment. In contrast, Clore et al. [18, 19] have empirically derived the total correlation function similar to Equation (3.40) but Equation (3.40) is expressed more explicitly in terms of the respective order parameters. [Pg.65]

Fig. 3.13 Step function model for the gas-phase reaction that starts at x = X and ends 3t X = |

M. R. McHenry and B. Laub, Ablative Radome Materials Thermal-Ablation and Erosion Modelling( 3t]i Intersociety Conf. on Environmental Systems, [Pg.7]

Dejean et al. [7], proposed another conformation jump model (referred to as the DLM model), in which a librational motion of the jump axis was introduced as the third motion. This model will correspond to the 3t model described below but the derivation of G (t) was empirically made in this case. [Pg.59]

The percolation probability (q) for the lattice models is defined as the probability that a given site (or bond) belongs to an infinite open cluster (47). It is fundamental to percolation theory that there exists a critical value qc of q such that 9(q) = 0 3t q < qc, and (q) > 0 if > qc. The value qc is called the critical probability or the percolation threshold. Mathematical methods of calculating this threshold are so far restricted to two dimensions, consistent with the experience in the field of phase transitions that three-dimensional problems in general cannot be solved exactly (12,13). Almost all quantitative information available on the percolation properties of specific lattices has come from Monte Carlo calculations on finite specimens (8,11,12). In particular. Table I summarizes exactly and approximately known percolation thresholds for the most important two- and three-dimensional lattices. For the bond problem, the data presented in Table I support the following well-known empirical invariant (8) [Pg.11]

A first principle mathematical model of the extruder barrel and temperature control system was developed using time dependent partial differential equations in cylindrical coordinates in two spatial dimensions (r and z). There was no angular dependence in the temperature function (3T/30=O). The equation for this model is (from standard texts, i.e. 1-2) [Pg.493]

The ratio A Cp D)/A Cp(N D) should be a measure of the relative burial of hydrophobic surface areas in the transition and folded states, if the hydrophobic model is correct. The ratio is 0.51 for CI2, compared with a value of /3T of 0.6,27 which is a measure of the overall change in surface area (equation 18.9). [Pg.612]

Transient Heat Conduction. Our next simulation might be used to model the transient temperature history in a slab of material placed suddenly in a heated press, as is frequently done in lamination processing. This is a classical problem with a well known closed solution it is governed by the much-studied differential equation (3T/3x) - q(3 T/3x ), where here a - (k/pc) is the thermal diffuslvity. This analysis is also identical to transient species diffusion or flow near a suddenly accelerated flat plate, if q is suitably interpreted (6). [Pg.274]

Let us now consider how the limitations imposed by the size of these clusters may influence the results for the chemical reaction step. In spite of the fact that only one type of acid site can be modeled by the 3T and 5T clusters, it must be remembered that the difference in acidic strength of the different sites is much smaller than the activation energy of most reactions. Thus, in the absence of other effects, the fact that the clusters used cannot distinguish different [Pg.57]

With this change to the Jacobian generator and a user supplied routine to evaluate the extents of reaction and at the user s option the partial derivatives 3r/3x, 3r/3l, and 3R/3T along with the ECES generated subroutines, this form of process can be successfully modeled. This type of model has been successfully used to simulate proprietary processes developed by our clients. [Pg.317]

For an experiment concerning the investigation of the dependence on the concentration of a catalyst Y, [Y], the molar concentration of the catalyst could be varied in a linear way, that is, M [Y] = (3t, where p is the concentration gradient. If the dependence function is known, for example, D kobs = /cy[Y], the mathematical model will assume the forms [Pg.707]

Figures 3.7 and 3.8 shows the frequency dependencies of Ti and NOE measured for the CH2 (rrr) carbon of poly(methyl methacrylate) (PMMA) in a deuterated chloroform solution at 55°C [10]. Different curves indicate the simulated results obtained by using the box-type distribution, log- distribution, 2r and 3t models described above. As is clearly seen in Fig. |

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