Transient calculation

Calculated transient voltage (c) Bank-A voltages, and (d) Bank-B voltages. 

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Figure 9. Arc length as a function of time for three time transient calculations, (a) Evolution of interface from an unstable planar interface to a shape in the (lA -family. (b) Evolution of perturbation to a shape in the (up-family for P < Continued on next page.

Fig, 7. Laser-induced heating model. The solid line represents the temperature transient calculated from Eq. (3) for a S ns FWHM laser pulse (dott trace). The instantaneous desorption rate calculated from Eq. (4) is represented by the... 

Table 8.7 shows some data characteristics of the experimental system and parameters used for the transient calculations. The system was operated as a conventional constant speed machine, with extensive piping between compressor and turbine. 

Figure 8.27 illustrates the theoretical electron density profiles and photocurrent transients calculated by Solbrand et al. The transients exhibit a maximum at a time fpeak = d2/6D. The inset in Fig. 8.26 shows that a plot of fpeak VS. d2 is linear as predicted (the authors use W rather than d to denote the film thickness), and the slope of the plot gives a value of 1.5 x 10-scm-2s-1 for the electron diffusion coefficient. 

Fig. 8.27. Illustration of electron density profiles at different times /t c /2 c /3 following pulsed excitation from the electrolyte side. W is the film thickness. The lower part of the figure.shows transients calculated for different film thicknesses. Taken from Ref. [78].

An IBM 370/158 computer was used for all of the calculations. As would be expected, the calculation speed was different for each of the three dynamic models. The distance method of lines model ran at a speed of 0.23 times real time (4.3 times slower than real time) which was very slow. For short time transient calculations, the speed could be increased to real time speed, but long term numerical stability required the slower speed. The time method of lines model ran 3.92 times faster than real time. However, the calculated transient responses were incorrect and the model could not be used for that purpose. For a time slice of 1.5 minutes, the method of characteristics model ran 1.56 times faster than real time. If the time slice was increased to 5.0 minutes (fewer nodes), the speed increased to 4.75 times real time but the gas stream accuracy was reduced. Therefore, the 1.5 minute time slice was used for the calculations shown here. 

In addition to the Navier-Stokes equations, the convective diffusion or mass balance equations need to be considered. Filtration is included in the simulation by preventing convection or diffusion of the retained species. The porosity of the membrane is assumed to decrease exponentially with time as a result of fouling. Wai and Fumeaux [1990] modeled the filtration of a 0.2 pm membrane with a central transverse filtrate outlet across the membrane support. They performed transient calculations to predict the flux reduction as a function of time due to fouling. Different membrane or membrane reactor designs can be evaluated by CFD with an ever decreasing amount of computational time. 

Figure 5.21 Anodic current transients calculated (a) At the potential step across the disorder-to-order and the order-to-disorder transition points, with and without considering constraint with/= 0.2. (Reproduced with...

Similarly, the application of isochronal analysis to transient data (20) assumes that the time dependence of the transient response is independent of the parameter being varied. For example, plotting current at a particular time against temperature yields useful data only if the transient proceeds at a rate independent of temperature. However, the example of Figure 1 and transients calculated from the model have features which would appear as peaks in an isochronal plot, yet are intrinsic to the conduction mechanism. 

The pressure transient calculated by VHIM is shown in Figure 10.8, which matches closely that plotted in Figure 10.4 for SVHIM. In particular, it will be seen that the pipe outlet pressure, p3, has reduced to atmospheric, p4, by 20 seconds, in agreement with SVHIM, indicating that the outlet flow is subsonic at this time. 

The resulting flow transient calculated by ASVAM is shown in Figure lO.lOcompared with that of VHIM. The flows are almost identical over the subsonic region up to time = 16 seconds, and come back together again aher 18 seconds. It is noticeable that the discontinuity between subsonic valve flow and sonic valve flow that characterizes VHIM disappears under ASVAM. This is because of the transition in ASVAM takes a very simple form, namely the maximum selection of equation (10.62). 

Figure 10.11 compares the flow transient calculated by ASVAM with the standard transient calculated by SVHIM. ASVAM, like VHIM, relies implicitly on the C value to characterize valve flow in the subsonic region via the calculation of AT, and then Kp. As a result, ASVAM underestimates the flow by about 3% at the beginning of the transient in the same way as VHIM. But ASVAM produces essentially the same value as SVHIM for flow by time = 15 seconds. In fact, ASVAM predicts sonic flow in the valve at time = 16 seconds, a second in advance of SVHIM, but the difference in flow is very small. 

Within the scope of this work, the initial spray breakup process, providing information about the dense spray core, will be investigated. The formation of fuel drops will be simulated based on first-principles and will offer detailed insight into primary atomization. The three-dimensional, transient calculation will track the interface evolution through droplet formation and breakup. Because the results will be based on conservation laws, they will be extremely general. This will lead to better models that can be used with confidence in the engine design process. 

Fig. 39 illustrates the theoretical electron density profiles and photocurrent transients calculated from Eq. 85 and 87. If the RC time constant is small, differentiation of Eq. 87 with respect to t shows that the photocurrent transients exhibit a maximum at a time tpeak = d 6D. In the limit of small perturbations, this time is expected to be inversely related to the characteristic frequency in the IMPS response. 

Fig. 4Q. Theoretical dimensionless photocurrent transients calculated from Eq. 88 for two different values of a d, where a is the absorption coefficient and d is the film thickness. The dimensionless time t is defined by t = Dt/d (taken from [80]).

The phenomena occurring on the surface and in the liquid active phase of the vanadium catalyst have undoubtedly a major effect on the transients calculated by the dynamic models of catalytic reactors. The physicochemistry of these phenomena has not, so far, been satisfactorily elucidated. The introduction of an effective specific heat and effective molar capacity of the bed as parameters to be identified by comparing the actual transients with those predicted by the model is only a temporary measure. It cannot be expected that the phenomena occurring on the catalyst surface can be thoroughly explained without expensive and tedious studies. The results of the present investigations can be... 

Figure 21. (a) Logarithmic cathodic current transients, (b) Logarithmic anodic current transients of a Lii.sNi02 electrode, determined theoretically by means of numerical analysis based upon the cell-impedance controlled lithium intercalation, where the solid line and the dotted line indicate the current transients calculated theoretically under the two conditions that Rcdi is constant and that Rcdi is varied with E, respectively. 

Surface area of third group of concrete slabs Hot metal surface area Cold metal surface area Current time Containment atmosphere temperature before accident Temperature of the external air Time after rupture at which transient calculation is terminated Initial temperature of the containment mixture after efflux Hot metals initial temperature Internal free volume of the containment... 

In this study, the time step used in transient calculation is 0.001 s for both ANSYS POLYFLOW model and hybrid model (FLUENT/POLYFLOW). Numerical computations were run with 8 cores on a workstation with Intel Xeon CPUs ES-2640 2.5GHz. 

Fluorescence transients calculated with the values of k3 = 20 10 and k4 = 5-10 (equivalent to I = 200 jxs) are similar to experimentally recorded curves in sunflower leaves (Fig.2). 

To finish, it is worth mentioning that temperature transients calculated with different degrees of approximation were compared in reference showing that at i > 0.05 ps the results agree well. 

Fig. 9.20 - (a) Current transients calculated for the growth of hemispherical centres formed by instantaneous nucleation (b) Current transients calculated for the growth of hemispherical centres formed by progressive nucleation. Reproduced with permission from E. Bosco S. K. Rangarajan,Electroanal. 134,(1982), 213. 

Fig. 9. 22 - (a) Current transients calculated from Equation (9.55) for different overpotentials. The transients show the typical behaviour expected for the growth of a single centre under diffusion control, (b) ExperimenUl transient for the growth of a single mercury centre on a carbon fibre microelectrode from Hg2(N03)2 solution (taken from B. Scharifker, PhD thesis, University of Southampton, 1979). 

Accumulators provide a very high flow for a limited time, which depends on the initiating fault (Section 15. 6 of Reference 6.1 For example, in the event of a large-break LOCA transient, calculations demonstrate that accumulators provide sufficient makeup water. These show that the lower plenum fills to the point where water begins to re-flood the core from below after approximately 54 seconds (Section 15.6.5.4A.6 of Reference 6.1). 

D Eulerian-Eulerian simulation of fluid catalytic cracking of gas oil. Snapshot of fluctuations around the statistically stationary species field. Mass fraction in the gas phase (a) gasoil fraction of the feed, (b) gasoline fraction in the product, (c) Transient calculation of the meso-scale fluctuations around a statistically stationary state solids volume fraction and axial velocity at 0.1 m from the wall and 0.25 m from the bottom. 

Transient calculation of the meso-scale fluctuations around a statistically stationary state, (a) Center-line liquid velocity (b) concentration of A in the gas phase (c) concentration of A in the liquid phase. Values at 4.5 m height at the center of the column. 50% inerts case. From van Baten and Krishna [2004]. 

Examples of transient simulations for different values of L, starting with a square shaped electrode (L = 1) are shown in [277] and [276]. The percentage deviation between transients calculated for two-dimensional diffusion from the analytical equation of Aoki et al. [12, 288] and three-dimensional simulation with decreasing value of L has been presented [277]. An Aoki type equation where the coefficients were evaluated by fitting to three-dimensional numerical simulations to describe transients at square and rectangular electrodes has been published [276]. 

The simulation should be carried out in a step-by-step procedure developing from a lower to a higher level of complexity. For example, radiation was taken into account in the late stages of the calculation starting from PI to discrete ordinates model. Initially, the numerical calculations are made on a stationary time scale. If the convergence of the overall case shows a periodical behavior, the stationary solution can be used to initialize the transient calculation as in the case of INCI simulation. 

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