Variable batch time

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. 

The yield that can be attained by a semibatch process is generally higher because the semibatch run starts from scratch, with maximum values of both variables Cg (o) = Cg and k] (o) = k . However, the yield from a continuous run in which t equals the batch time is governed by the product of Cg (t) and kj (t), so > and k (t) = k °. Because neither of these conditions is likely to be fulfilled completely, a continuous polymerization in a backmix reactor will probably always fail to attain the Y attainable by a semibatch reactor at the same t. However, several backmix reactors in series will approach the behavior of a plug flow continuous reactor, which is equivalent to a semibatch reactor. 

This messy result apparently requires knowledge of five parameters k, (A )o> Poo, and po- However, conversion to dimensionless variables usually reduces the number of parameters. In this case, set Y = Na/(Na)o (the fraction unreacted) and r = t/thatch (fractional batch time). Then algebra gives... 

The interval of the batch time is split in two equal intervals. Temperature and feed rate at the boundaries of sub-intervals are subjected to optimization together with the other variables. Temperature and feed rate between the boundaries of sub-intervals are assumed to be straight lines connecting the initial and final values. The optimum values of variables obtained in step two are taken as initial guesses for optimization. The new profiles consist of two ramps joining optimized points. 

At the start of optimization only the temperature profile, the batch time, and the feed time of G were optimized, while the other variables were kept constant. At the end all the variables specified above were relaxed and optimized. The optimization sequence is shown in Table 5.4-20. The changes in criterion J are shown in Fig. 5.4-38. 

Introduce a new variable re describing the net formation rate of B. Determine the optimum batch time and batch temperature. 

The reactor pressure is reduced to 0 psig to flash off any remaining water after a desired temperature is reached. Simultaneous ramp up of the heat source to a new setpoint is also carried out. The duration spent at this second setpoint is monitored using CUSUM plots to ensure the batch reaches a desired final reactor temperature within the prescribed batch time. The heat source subsequently is removed and the material is allowed to continue reacting until the final desired temperature is reached. The last stage involves the removal of the finished polymer as evidenced by the rise in the reactor pressure. Each reactor is equipped with sensors that measure the relevant temperature, pressure, and the heat source variable values. These sensors are interfaced to a distributed control system that monitors and controls the processing steps. 

Duration Constraints (Batch Time as a Function of Variable Batch Size)... 

Campaign processes have also variable production time. Different to continuous production, campaign production is related to multi-purpose assets, where different processes and products can run on the same production resource and change-over decisions between campaigns need to be taken. Finally, batch processes have a defined lot size, start and end time of production as well as throughput. 

Cost reductions usually arise out of improvements to the process control for both continuous and batch processes. Process analyzers enable chemical composition to be monitored essentially in real time. This in turn allows control of the process to be improved by shortening start-up and transition times (for continuous processes) or batch cycle times (for batch processes). This is accomplished by improving the ability to respond to process disturbances, by enabling process oscillations to be detected and corrected, and by reducing product variability. Real-time monitoring of chemical composition in a process allows a manufacturing plant to ... 

The classical batch reactor is a perfectly mixed vessel in which reactants are converted to products during the course of a batch cycle. All variables change dynamically with time. The reactants are charged into the vessel. Heat and/or catalyst is added to initiate reaction. Reactant concentrations decrease and product concentrations increase with time. Temperature or pressure is controlled according to some desired time trajectory. Batch time is also a design and operating variable, which has a strong impact on productivity. 

The optimum operation of this fed-batch reactor involves finding the trajectory of feed versus time that maximizes (or minimizes) some economic performance criterion. Of course, the batch time is also an operating optimization variable. 

Mathematical modeling of systems for which characteristic variables are time-dependent only and not space-dependent is done by ordinary differential equations (ODEs). The situation is found in a nearly well-mixed batch reactor. There one may find differences in temperature or concentrations from one site to another due to imperfect mixing. When space changes are not important to the model, the process variables can be approximated by means of lumped parameter models (LPMs). When the... 

The rate of cooling, or evaporation, or addition of diluent required to maintain specified conditions in a batch crystallizer often can be determined from a population-balance model. Moments of the population density function are used in the development of equations relating the control variable to time. As defined earlier, the moments are... 

The temperature-controlling features of this reaction scheme dominate selection and use of the reactor. However, the semibatch reactor does have some of the advantages of batch reactors temperature programming with time and variable reaction time control. 

The optimal control of a process can be defined as a control sequence in time, which when applied to the process over a specified control interval, will cause it to operate in some optimal manner. The criterion for optimality is defined in terms of an objective function and constraints and the process is characterised by a dynamic model. The optimality criterion in batch distillation may have a number of forms, maximising a profit function, maximising the amount of product, minimising the batch time, etc. subject to any constraints on the system. The most common constraints in batch distillation are on the amount and on the purity of the product at the end of the process or at some intermediate point in time. The most common control variable of the process is the reflux ratio for a conventional column and reboil ratio for an inverted column and both for an MVC column. 

Equality constraints h(D°, D°) = 0 may include, for example, a ratio between the amounts of two products, etc. Inequality constraints g(u, D°) < 0 for the overall operation include Equations 7.14-7.18 (the first two of which are easily eliminated when m and H are specified) and possibly bounds on total batch time for individual mixtures, energy utilisation, etc. Any variables of D° and D° which are fixed are simply dropped from the decision variable list. Here, Strategy II was adopted for the multiple duty specification, requiring B0 to be fixed a priori. Similar considerations hold for V, the vapour boilup rate. The batch time is inversely proportional to V for a specified amount of distillate. Also alternatively, for a given batch time, the amount of product is directly proportional to V. This can be further explained through Equations 7.24-7.26) ... 

The dynamic optimisation problem P2 now results in a single variable algebraic optimisation problem. The only variable to be optimised is the batch time t. The solution of the problem does no longer require full integration of the model equations. This method will solve the maximum profit problem very cheaply under frequently changing market prices of (CD/, CB0, C ) and will thus determine new optimum batch time for the plant. The optimal values of C, Dh r, QR, etc. can now be determined using the functions represented by Equations 9.2-9.5. 

Now, the terms D and Qr in Equation 9.1 (same as the functions gft) and g4(t) in Equation 9.6) can be simply replaced by the time-dependent polynomial equations presented in Figures 9.10 and 9.12. For a given set of cost parameters, batch time is the only remaining optimisation variable in Problem P2, which can be obtained with extremely little computational effort. 

Feed rate, Fp, kmol/hr = variable (maximised to minimise batch time)... 

Eqs. 1 to 3 relate the rate of production Rj of the balanced reaction component y to the molar amounts or their derivatives with respect to the time variable (reaction time or space time, see above). From the algebraic eq. 2 for the CSTR reactor the rate of production, Rj, may be calculated very simply by introducing the molar flow rates at the inlet and outlet of the reactor these quantities are easily derived from the known flow rate and the analytically determined composition of the reaction mixture. With a plug-flow or with a batch reactor we either have to limit the changes of conversion X or mole amount n7 to very low values so that the derivatives or dAy/d( //y,0) or dn7/d/ could be approximated by differences AXj/ (Q/Fj,0) or An7/A, (differential mode of operation), or to measure experimentally the dependence of Xj or nj on the space or reaction time in a broader region this dependence is then differentiated graphically or numerically. 

In Section 6.1 we saw. that the undesired product could be minimized by adjusting the reaction conditions (e.g., concentration) and by choosing the proper reactor. For series of consecutive reactions, the most important variable is time space-time for a flow reactor and real-timE for a batch reactor. To illustrate the importance of the time factor, we consider the sequence... 

The independent variable t is either the batch time or the plug flow residence time. 

For the batch tank reactor, with uniform concentrations and temperature, the independent variable is time, and the mathematical expression for the rate r is ... 

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