Quadratic profile

With this result, the reactor is treated as if composed of two sections. In addition, from the form of the profile in Murase et al, U t) for the latter section can be approximated by a quadratic profile ... 

However, a quadratic representation of the radial concentration profile may not be adequate since application of zero flux conditions at the inner thermal well and outer cooling wall with a quadratic profile reduces to an assumption of uniform radial concentrations. Although additional radial... 

FIGURE 6.4 Quadratic profiles used in QUICK scheme. 

Figure 1.22 Sketch of the top view of the suspended channel with the exception of the front end of the flow near the interface where the flow is not estabhshed, the velocity profile is close to the Poiseuille quadratic profile.

Ren/ in collaboration with Nobbs, developed a theoretieal model of the package dyeing process, which can be used to quantify the diffeienee between the amount of dye inside and outside of the paekage over the whole of a dyeing cycle, which leads to the proposal of a new type of optimum exhaustion profile, the quadratic profile. 

Experimental work on dyebath exhaustion control was also carried out using a pilot-scale radial flow package dyeing machine, and the results supported the flndings of the theoretical model. In particular, it was found that a quadratic profile was preferable to an exponential profile, which in turn was preferable to a linear profile. All controlled exhaustion dyeings gave better levelness than a standard (constant temperature ramp) dyeing method. This work was continued by Illett, who improved the mathematical solution of the Nobbs-Ren model and applied it to axial flow machines. 

Modelling, Simulation and Control of the Dyeing Process Quadratic profile ... 

Turbulent or unbalanced media flow (i.e., aerodynamic or hydraulic instability) does not have the same quadratic impacts on the vibration profile as that of load change, but it increases the overall vibration energy. This generates a unique profile that can be used to quantify the level of instability present in the machine. The profile generated by unbalanced flow is visible at the vane or blade-pass frequency of the rotating element. In addition, the profile shows a marked increase in the random noise generated by the flow of gas or liquid through the machine. 

Fig. 12. Interface width a as a function of annealing time x during initial stages of interdiffusion of PS(D)/PS(H) [95]. Data points are obtained by a fit with error function profiles of neutron reflectivity curves as shown in Fig. 11. Different symbols correspond to different samples. The interface width a0 prior to annealing is also indicated (T) and is subtracted quadratically from the data (a = [

The heptane water and toluene water interfaces were simulated by the use of the DREIDING force field on the software of Cerius2 Dynamics and Minimizer modules (MSI, San Diego) [6]. The two-phase systems were constructed from 62 heptane molecules and 500 water molecules or 100 toluene molecules and 500 water molecules in a quadratic prism cell. Each bulk phase was optimized for 500 ps at 300 K under NET ensemble in advance. The periodic boundary conditions were applied along all three directions. The calculations of the two-phase system were run under NVT ensemble. The dimensions of the cells in the final calculations were 23.5 A x 22.6 Ax 52.4 A for the heptane-water system and 24.5 A x 24.3 A x 55.2 A for the toluene-water system. The timestep was 1 fs in all cases and the simulation almost reached equilibrium after 50 ps. The density vs. distance profile showed a clear interface with a thickness of ca. 10 A in both systems. The result in the heptane-water system is shown in Fig. 3. Interfacial adsorption of an extractant can be simulated by a similar procedure after the introduction of the extractant molecule at the position from where the dynamics will be started. 

The Grashof number given by Eq. (40) appears to have a weaker theoretical basis than that given by Eq. (37), since it is based on an analysis that approximates the profile of the vertical velocity component in free convection, for example, by a quadratic function of the distance to the electrode. The choice of an appropriate Grashof number, as well as the experimental conditions in the work of de Leeuw den Bouter et al. (DIO) and Marchiano and Arvia (M3), has been reviewed critically by Wragg and Nasiruddin (W10). They measured mass transfer by combined thermal and diffusional, turbulent, free convection at a horizontal plate [see Eq. (31) in Table VII], and correlated their results satisfactorily with the Grashof number of Eq. (37). 

Yethiraj and Hall [94] studied the density profiles, surface forces, and partition coefficient of freely jointed tangent hard-sphere chains between hard walls. The theory was able to capture the depletion of chain sites at the surface at low densities and the enhancement of chain sites at the surface at high densities. This theory is in qualitative agreement with simulations for the density profiles and partitioning of 4 and 20 bead chains, although several quantitative deficiencies are present. At low densities the theory overestimates the value of the density profile near the surface. Furthermore, it predicts a quadratic variation of density with distance near the surface, whereas in reality the density profile should be linear in distance, for long chains. At high densities the theory underestimates the value of the density near the surface. The theory is quite accurate, however, for the partition coefficient for hard chains in slit-like pores. 

From point (1), the velocity profile is parabolic that is, the linear (axial) velocity u depends quadratically on radial position r, as described by fluid mechanics (see, e.g., Kay and Nedderman, 1974, pp. 69-71) ... 

The quadratic rate equation [Eq. (1)] of the continuum theory arises because it implicitly assumed the parabolic dependence of the free energy profile on the solvent coordinate q. One of the consequences of this quadratic equation is the generation of a maximum in the dependence of the rate of reaction on the free energy of reaction and also in current density-overpotential dependence. 

The second-order rate constants for thiocyanate anation vs pH are shown in Fig. 1.13. The full line represents (1.216) with the values shown in scheme (1.217). This profile had been earlier recognized in the ring closure of the three analogous pH-related forms of Co(III)-edta to give Co(edta) in which the edta is completely coordinated.In the Co(lll) case the reactivities of the three forms are much closer. A plot of A [H+] -1-[H+] is a quadratic curve from which / ah2> ah be obtained. 

Fig. 28 a Single-photon fluorescence readout of data recorded by single-photon writing scale bar 100 ixm) b intensity profile (the direction is shown hy the arrows) of (a) c two-photon fluorescence readout of data recorded by single-photon writing (scale bar 100 xm) d intensity profile (the direction is shown by the arrows) of (c) e quadratic dependence of up-converted fluorescence of fluorene 17 on the input intensity. The smallest readout pattern achieved in this system was 3.5 xm... 

Prior to deconvolution, the background was subtracted and the data were smoothed with a 15-point quadratic least-squares polynomial followed by a 19-point quartic least-squares polynomial. The data were then scaled from 0 to 1. The S3 profile was deconvolved using a weight constraint of the form... 

The first step in the solution procedure is discretization in the radial dimension, which involves writing the three-dimensional differential equations as an enlarged set of two-dimensional equations at the radial collocation points with the assumed profile identically satisfying the radial boundary conditions. An examination of experimental measurements (Valstar et al., 1975) and typical radial profiles in packed beds (Finlayson, 1971) indicates that radial temperature profiles can be represented adequately by a quadratic function of radial position. The quadratic representation is preferable to one of higher order since only one interior collocation point is then necessary,6 thus not increasing the dimensionality of the system. The assumed radial temperature profile for either the gas or solid is of the form... 

For radial concentration profiles, a quadratic representation may not be adequate since application of the zero flux boundary conditions at r, = cp0 and r, = 1.0 leads to d2 = d3 = 0. Thus a quadratic representation for the concentration profiles reduces to the assumption of uniform radial concentrations, which for a highly exothermic system may be significantly inaccurate. 

The quadratic dependence of the Gibbs energy of activation on the driving force implies that the transfer coefficient a is no longer a constant. Instead, it depends linearly on the overpotential. Figure 6.7 illustrates the free energy profile curves along the reaction coordinates for the reaction... 

In the framework of the impact approximation of pressure broadening, the shape of an ordinary, allowed line is a Lorentzian. At low gas densities the profile would be sharp. With increasing pressure, the peak decreases linearly with density and the Lorentzian broadens in such a way that the area under the curve remains constant. This is more or less what we see in Fig. 3.36 at low enough density. Above a certain density, the l i(0) line shows an anomalous dispersion shape and finally turns upside down. The asymmetry of the profile increases with increasing density [258, 264, 345]. Besides the Ri(j) lines, we see of course also a purely collision-induced background, which arises from the other induced dipole components which do not interfere with the allowed lines its intensity varies as density squared in the low-density limit. In the Qi(j) lines, the intercollisional dip of absorption is clearly seen at low densities, it may be thought to arise from three-body collisional processes. The spectral moments and the integrated absorption coefficient thus show terms of a linear, quadratic and cubic density dependence,... 

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