Convergence profile

As expected, the estimated values were found to be closer to the correct ones compared with the estimated values when the water-oil ratios are only matched. In the 2nd run, the horizontal permeabilities of layers 5 to 10 (6 zones) were estimated using the value of 200 md as initial guess. It was found necessary to use the pseudo-inverse option in this case to ensure convergence of the computations. The initial and converged profiles generated by the model are compared to the observed data in Figures 18.25a and 18.25b. 

All the methods described above can be used to try to improve the convergence profile for a free energy simulation. Statistical methods can be... 

One technique for measurement of extensional flow that has been used to study various doughs is that of Cogswell (1972, 1978) for entrance flows. The analysis is based on several assumptions (Padmanabhan and Bhattacharya, 1993) (1) The flow is isothermal and creeping (negligible inertial effects), (2) the fluid is incompressible and has a pressure-independent viscosity, (3) the shear viscosity follows the power law model, t]a = Ky" (4) there is no slip at the edge of the converging profile, and (5) that the entrance pressure drop (Ape) in converging flow from a circular barrel in to a circular die can be considered to be made up of that due to shear (Ape,s) and extensional flow (Ape,E) ... 

Different mesh sizes are used to perform mesh sensitivity analysis on the pressure profile. The number of elements in the x and y directions were kept constant (5x5) while the number of elements varies in the z direction such that 50, 100, 150 and 200 elements were used which results in a total number of 1250, 2500, 3750, and 5000 elements respectively. Fig 19 shows that as the number of element increases, the pressure profile converges to a unique value. The converged profile consists of a linearly increasing wave front, followed by a constant peak pressure plateau which is followed by the unloading. Further more, the mesh size affects wave propagation speed. Using coarse meshes result in underestimation of wave speed whereas when using fairly fine mesh, the wave speed reaches its theoretical value. 

A new one-dimensional mierowave imaging approaeh based on suecessive reeonstruetion of dielectrie interfaees is described. The reconstruction is obtained using the complex reflection coefficient data collected over some standard waveguide band. The problem is considered in terms of the optical path length to ensure better convergence of the iterative procedure. Then, the reverse coordinate transformation to the final profile is applied. The method is valid for highly contrasted discontinuous profiles and shows low sensitivity to the practical measurement error. Some numerical examples are presented. 

The described approach is suitable for the reconstruction of complicated dielectric profiles of high contrast and demonstrates good stability with respect to the noise in the input data. However, the convergence and the stability of the solution deteriorate if the low-frequency information is lacking. Thus, the method needs to be modified before using in praetiee with real microwave and millimeter wave sourees and antennas, whieh are usually essentially band-limited elements. 

By feeding the mixture through a converging nozzle, the velocity profile may be made nearly flat or uniform. A Bunsen flame in such a flow has a smaller range of stabiUty but the mechanism is essentially the same and the flame very closely approximates a cone. If the apex angle of the flame is , then S can be obtained from equation 21... 

A temperature profile plus a vapor-rate profile through the column must be assumed to start the procedure. These variables are referred to as tear variables and must be iterated on until convergence is achieved in which their values no longer change from iteration to iteration and all equations are satisfied to an acceptable degree of tolerance. Each iteration down and then up through the column is referred to as a column iteration. A set of assumed values of the tear variables consistent with the specifications, plus the component K values at the assumed temperatures, is as follows, using assumed end and middle temperatures and K values from Fig. 13-14. ... 

Convergence was achieved in 3 iterations. Converged values of temperatures, total flows, and component flow rates are tabulated in Table 13-14. Computed reboiler duty is 1,295,000 W (4,421,000 Btu/h). Computed temperature, total vapor flow, and component flow profiles, shown in Fig. 13-54, are not of the shapes that might be expected. Vapor and liquid flow rates for nC4 change dramatically from stage to stage. 

At this point in the inside-out method, the revised column profiles of temperature and phase compositions are used in the outer loop with the complex SRK thermodynamic models to compute updates of the approximate K and H constants. Then only one inner-loop iteration is required to obtain satisfactory convergence of the energy equations. The K and H constants are again updated in the outer loop. After one inner-loop iteration, the approximate K and H constants are found to be sufficiently close to the SRK values that overall convergence is achieved. Thus, a total of only 3 outer-loop iterations and 4 inner-loop iterations are required. 

Renormalized pair interactions (in mRy) for the (001) surface of fee Ag )r the converged inhomogeneous concentration profile. ... 

For the correlation bands obtained by a convolution of many shake-up lines, the size-consistency and size-intensivity requirements imply a convergence towards some asymptotic profile, when going to the polymer limit. This must ideally be achieved through a balance of the scaling properties of the individual shake-up lines, and the dispersion of the intensity of Ih lines over a rapidly increasing number of excited states with increasing system size. 

Perhaps the most unusual observation in this study, other than the unique pigment profile in S. angulatus, is the simple and identical flavonoid profile in the Kenyan, Madagascaran, and Canary Islands specimens. A close relationship between the two varieties from East Africa is not difficult to appreciate. The occurrence of this profile in specimens from the Canary Islands, however, points to a closer relationship than the distance between these areas might suggest. There is no way to know, at least from the data presented, whether this represents a case of convergence of flavonoid biosynthetic capacities involving unrelated species, whether it points to a relationship based... 

The method starts with an assumption of the column temperature and flow profiles. The stage equations are then solved to determine the stage component compositions and the results used to revise the temperature profiles for subsequent trial calculations. Efficient convergence procedures have been developed for the Thiele-Geddes method. The so-called theta method , described by Lyster et al. (1959) and Holland (1963), is recommended. The Thiele-Geddes method can be used for the solution of complex distillation problems,... 

If the equilibrium relationships and flow-rates are known (or assumed) the set of material balance equations for each component is linear in the component compositions. Amundson and Pontinen (1958) developed a method in which these equations are solved simultaneously and the results used to provide improved estimates of the temperature and flow profiles. The set of equations can be expressed in matrix form and solved using the standard inversion routines available in modem computer systems. Convergence can usually be achieved after a few iterations. 

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