Time, digital simulations

The main process variables in differential contacting devices vary continuously with respect to distance. Dynamic simulations therefore involve variations with respect to both time and position. Thus two independent variables, time and position, are now involved. Although the basic principles remain the same, the mathematical formulation, for the dynamic system, now results in the form of partial differential equations. As most digital simulation languages permit the use of only one independent variable, the second independent variable, either time or distance is normally eliminated by the use of a finite-differencing procedure. In this chapter, the approach is based very largely on that of Franks (1967), and the distance coordinate is treated by finite differencing. 

The presence of a liquid layer on the surface of the filter cake will cause solute to diffuse from the top layer of cake into the liquid. Also if disturbed the layer of liquid will mix with the surface layer of filter cake. This effect can be incorporated into the digital simulation by assuming a given initial depth of liquid as an additional segment of the bed which mixes at time t=0 with the top cake segment. The initial concentrations in the liquid layer and top cake segment are then found by an initial mass balance. 

As many other industries, the fine chemical industry is characterized by strong pressures to decrease the time-to-market. New methods for the early screening of chemical reaction kinetics are needed (Heinzle and Hungerbiihler, 1997). Based on the data elaborated, the digital simulation of the chemical reactors is possible. The design of optimal feeding profiles to maximize predefined profit functions and the related assessment of critical reactor behavior is thus possible, as seen in the simulation examples RUN and SELCONT. 

Jfinr.- The easiest way to handle de time in a digital simulation is to set up an array for the variable to be delayed. At each point in time you use the variable at the bottom of the array as the delayed variable. Then each value is moved down one position in the array and the current undelayed value is stuffed into the top of the array. For fixed step sizes and fixed deodtimes, this is easy to program. For variable step sizes and variable deadtimes, the programming is more complex. 

When we run a digital simulation we must set up the Transient Analysis, as well as specify parameters for the digital simulation. We will first set up the Transient analysis. Select PSpice and then New Simulation Profile from the Capture menus, enter a name for the profile, and then click the Create button. By default the Time Domain (Transient) Analysis type is selected. Fill in the dialog box as shown to run the simulation for 20 ms ... 

A detailed treatment of the theoretical approach used in treating LSV and CV boundary value problems can be found in the monograph by MacDonald [23], More specific information on the numerical solution of integral equations common to electrochemical methods is available in the chapter by Nicholson [30]. The most commonly used method for the calculation of the theoretical electrochemical response, at the present time, is digital simulation which has been well reviewed by Feldberg [31, 32], Prater [33], Maloy [34], and Britz [35]. 

AAEP values calculated using eqn. (43) were observed to be within 0.13 mV of values obtained by digital simulation under conditions where the resolution of the simulation was 0.2 mV per time step for ip ranging from about 103 down to 0.75. In any event, eqn. (43) appears to describe AAEP to an accuracy well within experimental error. Some of the data are shown in Table 13. 

Spatially resolved absorbance spectroelectrochemistry has been used to observe the concentration profile of an absorbing species generated at the surface of a cylinder electrode with a radius of 6 /xm. The observed concentration profiles agreed very closely with those predicted by solution of Equation 12.7 by digital simulation [32]. As with the disk electrode, simple analytical expressions for the current-time and current-voltage relationships of cylinders and bands do not exist. 

To model this method using digital simulation techniques, one need only change the electrode boundary conditions after some predetermined number of time iterations (representing tf) have taken place. The electrode boundary conditions become... 

The real power of digital simulation techniques lies in their ability to predict current-potential-time relationships when the reactants or products of an electrode reaction participate in some intervening chemical reaction. These kinetic complications often result in a fairly difficult differential equation (when combined with the conditions for diffusion or convection encountered in electrochemical problems) that resists solution by ordinary means. Through simulation, however, the effect of any number of chemical steps may be predicted. In practice, it is best to limit these predictions to cases where the reactants and products participate in one or two rate-determining steps each independent step adds another dimensionless kinetics parameter that must be varied over the range of... 

Either method can be treated theoretically via digital simulation to obtain time dependencies of redox reaction rates, which are needed in some diagnostic and efficiency studies. 

In an efficient digital simulation, lumped loss factors of the form Gk (0)) are approximated by a rational frequency response Gk(c,mT). In general, the coefficients of the optimal rational loss filter are obtained by minimizing I Ylk (go) - Gk d r ) I with respect to the filter coefficients or the poles and zeros of the filter. To avoid introducing frequency-dependent delay, the loss filter should be a zero-phase, finite-impluse-response (FIR) filter [Rabiner and Gold, 1975], Restriction to zero phase requires the impulse response g.k(n) to be finite in length (i.e., an FIR filter) and it must be symmetric about time zero, i.e., ) k(-n) = gk(n). In most implementations, the zero-phase FIR filter can be converted into a causal, linear phase filter by reducing an adjacent delay line by half of the impulse-response duration. 

Kates, 1990] Kates, J. (1990). A time-domain digital simulation of hearing aid response. J. Rehab. Res. andDevel., 27 279-294. 

Fig. 18. (Left) Change in fluorescence spectra measured by streakscope at the different reaction times. The increasing count at 560 nm proved the formation of Zn(II)-ocqn complex at the interface of two-phase microsheath flow system. (Right) The result of the digital simulation of the Zn(II)-Hocqn complexation. The initial concentration of Hocqn was 1.1 x 10 4M.

Figure 14 shows the resulting grid of points. At each drawn point, there is a value of c. The digital simulation method now consists of developing rows of c values along x, (usually) one f-step at a time. Let us focus on the three filled-circle points Cj t, c.j and ei+1 at time tj. One of the various techniques to be described will compute from these three known points a new concentration value e = c (f = (J + 1 )St) (empty circle) at for the next time value fJ+1, by expressing (1.1) in discrete form ... 

In digital simulation, when discretising the diffusion equation, we have a first derivative with respect to time, and one or more second derivatives with respect to the space coordinates sometimes also spatial first derivatives. Efficient simuiation methods will always strive to maximise the orders. 

One of the main uses of digital simulation - for some workers, the only application - is for linear sweep (LSV) or cyclic voltammetry (CV). This is more demanding than simulation of step methods, for which the simulation usually spans one observation time unit, whereas in LSV or CV, the characteristic time r used to normalise time with is the time taken to sweep through one dimensionless potential unit (see Sect. 2.4.3) and typically, a sweep traverses around 24 of these units and a cyclic voltammogram twice that many. Thus, the explicit method is not very suitable, requiring rather many steps per unit, but will serve as a simple introduction. Also, the groundwork for the handling of boundary conditions for multispecies simulations is laid here. 

In the years since the 2nd Edition, much has happened in electrochemical digital simulation. Problems that ten years ago seemed insurmountable have been solved, such as the thin reaction layer formed by very fast homogeneous reactions, or sets of coupled reactions. Two-dimensional simulations are now commonplace, and with the help of unequal intervals, conformal maps and sparse matrix methods, these too can be solved within a reasonable time. Techniques have been developed that make simulation much more efficient, so that accurate results can be achieved in a short computing time. Stable higher-order methods have been adapted to the electrochemical context. 

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