One Time Interval

A particular problem with GRAFA and RBL is the reproducibility of the retention data. The retention time axes should be perfectly synchronized. Small shifts of one time interval (thus the ith spectrum in X, corresponds with the i+lth spectrum in X ) already introduce major errors (> 5%) when the chromatographic resolution is less than 0.6. The results of an extensive study on the influence of these factors on the accuracy of the results obtained by GRAFA and RBL have been reported in Ref. [37]. Although some practical applications have been reported [38,39], the lack of robustness of RBL and GRAFA due to artifacts mentioned above has limited their widespread application in chromatography. 

If the rate of feed addition, rate of product takeoff, temperature and pressure are known in each time interval, a simulation of the reactor can be carried out in that time interval. The problem is that the conditions will change from one time interval to subsequent time intervals. The profile of the dynamic variables (feed addition, product takeoff, temperature and pressure) need to be known through time. In the approach described in Chapter 3 for profile optimization5, a shape can be imposed for a given variable through time and the dynamic variables optimized in conjunction with the temporal superstructure. One profile for each dynamic variable is assigned to the... 

Simulation results with different time interval (P) are reported in Table 4. Optimal control policy in reactor temperature for each case is shown in Fig. 4. As shown in Table 4, when one time interval (P = 1) is used, the amount of product C obtained at the final time (tf = 200 min) is 7.0171 kmol and the optimal temperature (isothermal operation) setpoint is 88.01 °C whereas usingP = 20, the amount of product C achieved is 7.0379 kmol. It was found from... 

In the following, the use of periodical potentials will be described the periodicity will be denoted T[28]. Switching between two ore more flow patterns is performed inducing chaotic advection. One flow field is maintained in one time interval and another flow field in a second interval. This is repeated with the period T. The switching of the flow fields is accomplished by controlling the distribution of the C, potential created by the electrodes. By flow field alternation, particles virtually expose a zig-zag path, thereby distributing material all over the channel s cross-section. Such transport is similar to efficient stirring. 

The reflux ratio is discretised into two time intervals for task 1 and one time interval for task 2. Thus a total of 3 reflux ratio levels and 3 switching times are optimised for the whole multiperiod operation. Three cases are considered, corresponding to different values of the main-cut 1 product. For each case the... 

Figure 10.4. Typical Reboiler Holdup Profile with One Time Interval. [Mujtaba, 1999]b...

For cases 2-4, the optimal values of R in both intervals were less than Rmax meaning the column was never flooded during the operation. This is also evident from the reboiler holdup values at the end of each interval. Also for cases 2-4 the constraint given by Equation 10.5 was not active. Mujtaba (1999) noted that cases 2-4 were re-run with only one time interval for the distillate Task. The constraint given by Equation 10.5 was imposed on the optimisation problem. For cases 2-3 the maximum productivity obtained was about 0.55 and for case 4 no solution was obtained. This shows the importance of having time-sequenced operation. 

For Tasks 1 and 3 two time intervals are used. For Task 2 only one time interval is used. Within each interval the reflux ratio, the solvent feed rate and the length of the interval are optimised. Vapour liquid equilibrium is calculated using UNIQUAC model. A number of cases were studied for different amount of initial feed charge (Bo) to the reboiler. For each case, the optimal reflux and solvent feed profiles for each Task, percentage of Acetone and solvent recovered and the overall profit of the operation are shown in Table 10.10. 

Numeric dispersion can be eliminated largely by a high-resolution discretisation. The Grid-Peclet number helps for the definition of the cell size. Pinder and Gray (1977) recommend the Pe to be < 2. The high resolution discretisation, however, leads to extremely long computing times. Additionally the stability of the numeric finite-differences method is influenced by the discretisation of time. The Courant number (Eq. 104) is a criterion, so that the transport of a particle is calculated within at least one time interval per cell. 

Partial-Feed processes switch the feed stream on and off within one time interval while the other inlet and outlet lines are active all the time. ISMB (Improved SMB) processes partition the switching interval in a different way. In the first part of the period (Fig. 5.24a), all external lines (desorbent and feed inlets as well as extract and raffinate outlets) operate. However, in contrast to a classical SMB unit, the outlet of section IV is not recycled during this part of the switching interval and, consequently, the flow rate in section IV is zero. The first. .injection period" is followed by a recirculation period in the second part of the switching interval (Fig. 5.24b). During this time all external ports are closed and recirculation is performed with a constant... 

DeHnition. The one-step transition probability function pjk for a Markov chain is a function that gives the probability of going from state j to state k in one step (one time interval) for each j and k. It will be denoted by ... 

Blocking, dividing the taking of data into distinct time intervals and allowing for changes from one time interval to another in the fitting ... 

For time, typical units are seconds, minutes, hours, and so on. If, for example, the physical quantity was distance then typical units could be metres and the rate of change would correspond to speed measured in, say, metres per second (m s ). Since the physical quantity changes linearly with time this means that the change in any one time interval is exactly the same as that in any other equal interval. In other words a plot of physical quantity versus time will be a straight line and there is a uniform, or constant, rate of change. 

Suppose we consider the proposition The nucleus will undergo fission in the time interval t to t + At. This is conditional, of course, since it presupposes that the nucleus has not undergone fission in earlier time intervals. As a model for calculating the probability of the proposition, adopt the following. The conditional probability will certainly depend on the duration of the time interval. At it should be zero if At is zero. The probability should be continuous in At, so that the probability does not change by a finite amount in an infinitesimal interval of time. Moreover, the probability should be independent of absolute time t, since we have no reason to expect fission in one time interval to be more likely than fission in any other time interval. Hence, the required probability will be of the form /(At), and Taylor series expansion about At = 0 subject to hypotheses stated gives... 

Even if the experimental design and execution are perfect so that the determinant error is eliminated in experiments involving radioactivity there is always a random error due to the statistical nature of radioactive decay. Each radioactive atom has a certain probability of decay within any one time interval. Consequently, since this probability allows unlikely processes to occur occasionally and likely processes not to occur in any particular time interval, the number of decays may be more or less than the number in another similar... 

Let us suppose that we want to separate a mixture of a blue dye and a red dye. The blue dye has a lower retention factor than the red dye. llius the blue dye will move into the same direction as the mobile phase, and the red dye will move in the direction of the bed. For quantitative purposes, we assume that the blue dye migrates column lengths during one time interval, while the red dye moves column length. 

In another study (53). sodium fluorescein was applied in liposomes prepared from soya lecithin as formulation, as pellet, as supemaiani. and in a buffer solution. When applied on the skin for 30 min the skin was stripped and in each strip the amount of fluorescein was measured. Although the authors claimed a weak enhanced penetration of the label, the differences observed were very small. Funhemiore, since only one time interval was chosen and not 100% of the label was recovered, some doubts arise as to whether the conclusion of the authors that vesicles enhance the penetration of the fluorescent label in the skin is justified. 

Of the 230 or so genera of short-looped, terebratuloid brachiopods of Jurassic to Recent age which have been described, about 130 occur in the Jurassic 70 in the Cretaceous 20 in the Cenozoic and 20 in modern seas (some genera span more than one time interval). Many of these short-looped terebratuloid genera are of medium size, smooth, oval in outline, biconvex, and with a moderately large foramen. The majority are distinguished on the basis of small differences in loops and cardinalia (Cooper, 1983). 

What happens in one time interval is independent of what happens in another non-overlapping time interval, independent increments)... 

Values presented as mean of all samples of one time interval standard deviation. 

For b greater than 2 the correlations are persistent, because upwards and downwards trends tend to maintain themselves. A large excursion in one time interval is likely to be followed by another large excursion in the next time interval of the same length. The time series seem to have a long-term memory. With b less than 2 the correlations are antipersistent in the sense that an upswing now is likely... 

The experimental desideratum is that the probe laser follows the pump laser within one time interval between two successive collisions. Either laser pulse needs therefore to be individually shorter than that time interval. The frequency of collisions of course depends on the pressure in the bulk sample. The lower the pressure the less frequent are the collisions and the less demanding are the conditions on the duration of the laser pulses. On the other hand, the lower the pressme the fewer the molecules in the sample and the lower is their measured response to the probe laser. Chapter 2 will demonstrate that laser pulses below about 50 ns are sufficiently short. 

Listing 13.7. Code segment for implementing a 2D PDE with time variable through one time interval using the FE approach. (Some eomment lines are missing from listing). 

---