Quasi-steady-state shape

U-shaped curve, we have mixtures that can be ignited for a sufficiently high spark energy. From Equation (4.25) and the dependence of the kinetics on both temperatures and reactant concentrations, it is possible to see why the experimental curve may have this shape. The lowest spark energy occurs near the stoichiometric mixture of XCUi =9.5%. In principle, it should be possible to use Equation (4.25) and data from Table 4.1 to compute these ignitability limits, but the complexities of temperature gradients and induced flows due to buoyancy tend to make such analysis only qualitative. From the theory described, it is possible to illustrate the process as a quasi-steady state (dT/dt = 0). From Equation (4.21) the energy release term represented as... 

Results from a quasi steady-state model (QSSM) valid for long crystals and a constant melt level (if some form of automatic replenishment of melt to the crucible exists) verified the correlation (equation 39) for the dependence of the radius on the growth rate (144) and predicted changes in the radius, the shape of the melt-crystal interface (which is a measure of radial temperature gradients in the crystal), and the axial temperature field with important control parameters like the heater temperature and the level of melt in the crucible. Processing strategies for holding the radius and solid-... 

The dynamic stability of the quasi steady-state process suggests that active control of the CZ system has to account only for random disturbances to the system about its set points and for the batchwise transient caused by the decreasing melt volume. Derby and Brown (150) implemented a simple proportional-integral (PI) controller that coupled the crystal radius to a set point temperature for the heater in an effort to control the dynamic CZ model with idealized radiation. Figure 20 shows the shapes of the crystal and melt predicted without control, with purely integral control, and with... 

Figure 24. Streamlines and isotherms for the growth of silicon in a prototype Czochralski system with self-consistent calculation of interface and crystal shapes by using the quasi steady-state thermal-capillary model and the condition that the crystal radius remains constant. Calculations are for decreasing melt volume. The Grashof number (scaled with the maximum temperature difference in the melt) varies between 1.0 X 107 and 2.0 X 107 with decreasing...

Equation (1) holds for a particle or a cell of any shape, even when the shape changes with the distance x, provided that it is reaching the surface mainly by a translational motion. Equation (1) also assumes a quasi-steady state of the motion of particles over the potential barrier. This approximation can be made because the region over which the potential acts is very thin and consequently the flux of particles through it can be considered practically constant with respect to the distance a at a given time t. 

Since the shape of one of the electrodes is varied during the machining, as is not known in advance, these problems belong to the class of problems with moving (free) boundaries, and their solution involves great difficulties [4-9], Therefore, approximate quasi-steady-state and local, one-dimensional methods, which enable one to reduce the ECM problems to those of known boundaries, are widely used [1-9]. 

In the quasi-steady-state approximation, which is also known as the step method [9], it is assumed that the rate of variation in the WP shape, that is, the anodic dissolution rate, is small compared with the rates of transfer processes in the gap therefore, for calculating the distribution of the current density, the WP surface can be considered as being immobile. This approximation can be used at not very high current densities. At very high current densities, ignoring the WP surface motion during anodic dissolution and the hydrodynamic flow induced by this motion causes a considerable error in the calculated distribution of current density [33]. 

In the general case, the TE and WP surfaces differ substantially. The determination of the ECM rate field requires consideration of the transfer processes in the IEG, which is a complex-shaped, three-dimensional channel. In this case, the quasi-steady-state approximation method is employed. 

In considering the evolution of the WP surface in the quasi-steady-state approximation, methods based on the explicit coordinate and parametrical description of the WP surface are most popular. These methods are described in detail in Refs 43, 56, 57. The explicit coordinate description is commonly used for sufficiently simple geometrical shapes of the WP surface,... 

Within the quasi-steady-state approximation, the most popular method is the embedding method involving the successive solving of a series of direct problems and corrections of the TE working surface until a desired shape of the WP is assured [45, 46, 50, 51]. Several methods of TE correction have been proposed they have a lot in common and can be presented in the following generalized form ... 

Figure 5-2 shows the effect of frequency on the concentration distribution. All three curves are for t= 100 d, which is the quasi steady-state case for a 500-cm long column. A low frequency of concentration change, co = 1/48 h l, is attenuated by dispersion, but propagates with only a small change in shape. By contrast, the highest frequency concentration loading with co = 1/12 h l approaches a quasi... 

To solve the inverse ECM problem in the local, one-dimensional approximation, the above-described method for the quasi-steady-state approximation can be used (see Sect. 12.5.2.3). However, especially for complex-shaped surfaces, it is advantageous to use a simpler and more efficient method. When correcting the TE surface, the error in the WP surface shape is conveniently estimated along the normal to the TE surface. This, on the one hand, does not lead to considerable error, because, in the quasi-equidistant ECM, the normals to the TE and WP surfaces differ little, and, on the other hand, enable one to significantly simplify the correction procedure. The TE correction is realized as follows ... 

It seems worthwhile to comment upon the quasi-steady state distribution function f(r, t) in this Ostwald ripening process. The mathematical verification of the fact that this distribution function/(r, t) can be written as a product/i(0 fii lr) (and f2(rlf) is a universal function even in cases where the initial distribution / (r, 0) is of Gaussian shape and of moderate width) is rather cumbersome and must be studied from the original work [33]. However, one may conceive the shape of the quasi-steady state distribution (which has a maximum between 0 r/F < 3/2 at r/F = 1.135 and is essentially zero at r/F > 3/2) by realizing that it is the interplay between the activity difference of the average activity of A in the solution matrix... 

Figure Bl.28.7. Schematic shape of steady-state voltaimnograms for reversible, quasi-reversible and irreversible electrode reactions.

Below we consider a quasi-one-dimensional model of flow and heat transfer in a heated capillary, with hydrodynamic, thermal and capillarity effects. We estimate the influence of heat transfer on steady-state laminar flow in a heated capillary, on the shape of the interface surface and the velocity and temperature distribution along the capillary axis. 

Fig. 4 Steady-state voltammograms of ET at the nanopipet-supported ITIES. (A) Voltam-mogram of reduction of 0.2 mM TCNQ by aqueous Fe(EDTA) at the 213-nm-radius silanized pipet (1) and a background curve obtained in the absence of TCNQ (2). (B) Background-subtracted voltammogram. (C) Experimental voltammogram (symbols) fitted to the theory for quasi-reversible ET at a disk-shaped interface. a = 164nm. v = 20 mV/s. For other parameters, see Cell 1.1. Reprinted with permission from ref. 15. Copyright 2006 American Chemical Society.

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