Batch time minimum

The parameter for variable batch time is defined by constraint (2.9). This gives the amount of time required to process a unit amount of a batch corresponding to a particular effective state in a corresponding unit operation. Constraint (2.10) denotes the minimum processing time for the effective state in the corresponding unit operation. This is, in essence, the minimum residence time of a batch within a unit operation. In constraints (2.10) and (2.11), v (j n is the percentage variation in processing time based on operational experience. 

Typically, orders are simpler to explode than anonymous demands, because for an anonymous demand it might be required to define a distribution of the forecast quantities over the forecast interval only one quant, quants with equal quantities, due dates at the beginning or end or equally distributed. At this time it can be defined for each product if lot sizes, batch sizes, minimum quantities, maximum quantities of quants should be considered. The quantities are broken into predefined equal parts and then assembled until they meet the mentioned constraints. 

In general, an objective function in the optimization problem can be chosen, depending on the nature of the problem. Here, two practical optimization problems related to batch operation maximization of product concentration in a fixed batch time and minimization of batch operation time given amount of desired product, are considered to determine an optimal reactor temperature profile. The first problem formulation is applied to a situation where we need to increase the amount of desired product while batch operation time is fixed. This is due to the limitation of complete production line in a sequential processing. However, in some circumstances, we need to reduce the duration of batch run to allow the operation of more runs per day. This requirement leads to the minimum time optimization problem. These problems can be described in details as follows. 

The purpose of this optimization problem is to determine the optimal temperature profiles to achieve the desired final product concentration in minimum batch time, thus the performance index is the final time whereas the desired production concentration is defined as a terminal constraints. The formulation of the minimum batch time problem can be shown as... 

Several simulations have been carried out under process parameter uncertainties e.g. in pre-exponential rate constant (ko) and activation energy (Ea). In all case studies we considered 10 time intervals when reactor temperature and switching time are optimized while minimizing the final batch operation time. Results, reported in the value of minimum batch time to obtain the desired product C and the amount of the desired product C at the end of batch operation, from on-line dynamic optimization strategy are also compared with those from the off-line strategy. 

The results for the remaining case studies are summarized in Table 6. An important aspect obtained from these results is that in all cases, the minimum batch time to obtain the desired product concentration in the on-line set point... 

Greaves et al, (2001) developed an experiment-based algorithm for finding the minimum batch time for the column and mixture described in section 3.3.4. 

The optimum reflux ratio and the minimum batch time for separation task 1 are 3 and 80.62 min (Table 3.1). The separation task 2 could be achieved using 3 different reflux ratio (Table 3.2) but however, / exp = 2 gives the true minimum batch time which is about 40% lower than the batch time that would be required to achieve the same separation with Rexp = 4. 

Figure 3.17. Experiment Based Algorithm for Calculating Minimum Batch Time. et is a small positive number.

The effects of column holdup can be easily correlated in terms of q and of the minimum batch time required to achieve a given separation task. 

Ti = minimum batch time for the separation using lowest plate holdup (2%) T2 = minimum batch time for the separation using optimal plate holdup Ts =% time saved = (7) - T2) x 100/7)... 

Figure 3.18a. Minimum Batch Time vs Column Holdup at different q...

A series of minimum time problems (Chapter 5) were solved at different values of q with increasing holdup for each case. Figures 3.18a and 3.18b show the minimum time solution vs. percent total holdup in the column for different mixtures at different q and Figures 3.19a and 3.19b show the corresponding optimum reflux ratio (required to get the separation in minimum time) vs. percent total holdup of the column. The results are summarized in Table 3.3 which shows, for each given separation, the optimum value of holdup to achieve the best performance out of the given column. The corresponding best minimum batch time and the optimum reflux ratio to achieve that are also presented in the table for each case. 

The last column of Table 3.3 shows the percent reduction in batch time achieved using the optimum plate holdup compared to the batch time with the lowest plate holdup (2%). It clearly shows that the column performance in terms of minimum batch time is improved significantly with increasing plate holdup for easy separation (q < 0.60). For one case 23% batch time saving is observed (case 10). 

However, for difficult separation (q > 0.60) this is reversed. Figures 3.18a and 3.18b clearly show that for q > 0.60 the column performance, in terms of minimum batch time, is improved significantly with decreasing plate holdup and suggests that for difficult separations the column holdup should be kept as minimum as possible. This is also clear from the results presented in Table 3.3 which show that for difficult separations optimum column holdup is very close to the minimum (Mujtaba and Macchietto, 1998 used 2% as the minimum column holdup). The minimum batch times for both cases (using minimum and optimum holdup) are almost alike and no time saving could be realized when compared to one another (last column of Table 3.3). The results discussed so far clearly show that holdup may have a dramatic effect on the operation. 

Figure 3.24 shows the effect of the condenser holdup on the performance of the column. Since, for a total condenser holdup only plays as an accumulator of material but not as a separation stage, larger condenser holdup means longer batch time is always required to achieve a given separation. This is quite clear from Figure 3.24. In practice there must always be a certain amount of condenser holdup to ensure a neat reflux operation, however this should be kept to a minimum. Luyben (1971) also arrived at similar conclusions.

The minimum time problem is also known as the time optimal control problem. Coward (1967), Hansen and Jorgensen (1986), Robinson (1970), Mayur et al. (1970), Mayur and Jackson (1971), Mujtaba (1989) and Mujtaba and Macchietto (1992, 1993, 1996, 1998) all minimised the batch time to yield a given amount and composition of distillate using conventional batch distillation columns. The time optimal operation is often desirable when the amount of product and its purity are specified a priori and a reduction in batch time can produce either savings in the operating costs of the column itself or permit improved scheduling of other batch operations elsewhere in a process. Mathematically the problem can be written as ... 

Application to Batch Distillation Minimum Time Problem... 

Reduction in hatch time For a given fresh feed and a given separation, the column performance is measured in terms of minimum batch time required to achieve a desired separation (specified top product purity (x D]) and bottom product purity (x B2) for binary mixture). Then an optimal amount and composition of recycle, subject to physical bounds (maximum reboiler capacity, maximum allowable purity of the off-cut) are obtained in an overall minimum time to produce the same separation (identical top and bottom products as in the... 

Mayur et al. (1970) formulated a two level dynamic optimisation problem to obtain optimal amount and composition of the off-cut recycle for the quasi-steady state operation which would minimise the overall distillation time for the whole cycle. For a particular choice of the amount of off-cut and its composition (Rl, xRI) (Figure 8.1) they obtained a solution for the two distillation tasks which minimises the distillation time of the individual tasks by selecting an optimal reflux policy. The optimum reflux ratio policy is described by a function rft) during Task 1 when a mixed charge (BC, xBC) is separated into a distillate (Dl, x DI) and a residue (Bl, xBi), followed by a function r2(t) during Task 2, when the residue is separated into an off-cut (Rl, xR2) and a bottom product (B2, x B2)- Both r2(t)and r2(t) are chosen to minimise the time for the respective task. However, these conditions are not sufficient to completely define the operation, because Rl and xRI can take many feasible values. Therefore the authors used a sequential simplex method to obtain the optimal values of Rl and xR which minimise the overall distillation time. The authors showed for one example that the inclusion of a recycled off-cut reduced the batch time by 5% compared to the minimum time for a distillation without recycled off-cut. 

The minimum batch times for the individual cuts and for the whole multiperiod operation are presented in Table 8.8 together with the optimal amount of recycle and its composition for each cut. The percentage time savings using recycle policies are also shown for the individual cuts and also for the whole operation. Figure 8.18 shows the accumulated distillate and composition profile with and without recycle case for the operation. These also show the optimal reflux ratio profiles. Please see Mujtaba (1989) for the solution statistics for this example problem. 

The final state of Task 2 becomes the initial state of Task 3. Therefore, Hi of Task 2 becomes the initial feed amount B2 for Task 2 (Figure 10.6). In Task 3 the increase in B0 increases the amount of distillate. This is due to fixed distillate and bottom compositions. However, since the initial composition of the mixture at the beginning of Task 3 was not identical, the minimum batch time for Task 3 does not necessarily increase with the increase in B0 and Dj. 

Mujtaba (1997) used the minimum time to evaluate the performance of continuous column operation under multiple separation duties. However, time does not explicitly appear in continuous column model equations but the feed rate is a measure of the batch time (t = BJF). Note, maximisation of the feed rate will therefore ensure minimisation of the batch time. 

Here the feed rate is maximised while the reflux ratio is optimised. The bottom product composition imposes an additional constraint to the problem. The results are summarised in Table 11.8 which gives the maximum feed rate, minimum batch time, optimum reflux ratio, and total number of batches for each mixture and total yearly profit. 

The performance of continuous columns undergoing multiple separation duties is evaluated in terms of minimum batch time and yearly profit. By choosing an... 

Greaves et al. (2001) and Greaves (2003) presented two simple algorithms which is computationally less expensive to obtain the minimum batch time for a given separation task. These algorithms are the results of the application of some of the unique properties of batch distillation process to the general optimisation framework discussed earlier. The algorithm-1 is based on experiment and is presented in section 3.4 of Chapter 3. 

Another consequence of Eq. (4.91) is that if we arrange the n subsystems in time instead of in space, then the collection of subsystems constitutes the reaction path of a batch reactor where Vk is the volume of subsystem k. For a specified conversion and time, we should minimize the sum of Jk(AGk/T)Vk. This minimization leads to results similar to Eq. (4.91), and supports the principle of equipartition of forces. Hence, for a given total conversion and reaction time, minimum entropy production results when the driving force A GIT is equal in all n time intervals. Similarly, the conversion is maximum for a given entropy production and reaction time when the driving forces are uniform. 

To run a batch reactor at its optimum (minimum) batch time, the heat generation rate at any point during polymerization should equal the heat removal rate, with some allowance for a safety factor. This would translate to a monomer conversion profile varying almost linearly with time. 

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