Distillation optimizing control

The indices of performance used in batch distillation optimal control problems... 

The following example of batch distillation optimal control (optimal reflux policy problem illustrates this. 

Spreadsheet Applications. The types of appHcations handled with spreadsheets are a microcosm of the types of problems and situations handled with fuU-blown appHcation programs that are mn on microcomputers, minis, and mainframes and include engineering computations, process simulation, equipment design and rating, process optimization, reactor kinetics—design, cost estimation, feedback control, data analysis, and unsteady-state simulation (eg, batch distillation optimization). 

Other synonyms for steady state are time-invariant, static, or stationary. These terms refer to a process in which the values of the dependent variables remain constant with respect to time. Unsteady state processes are also called nonsteady state, transient, or dynamic and represent the situation when the process-dependent variables change with time. A typical example of an unsteady state process is the operation of a batch distillation column, which would exhibit a time-varying product composition. A transient model reduces to a steady state model when d/dt = 0. Most optimization problems treated in this book are based on steady state models. Optimization problems involving dynamic models usually pertain to optimal control or real-time optimization problems (see Chapter 16)... 

The embedded model approach represented by problem (17) has been very successful in solving large process problems. Sargent and Sullivan (1979) optimized feed changeover policies for a sequence of distillation columns that included seven control profiles and 50 differential equations. More recently, Mujtaba and Macchietto (1988) used the SPEEDUP implementation of this method for optimal control of plate-to-plate batch distillation columns. 

While the reduced SQP algorithm is often suitable for parameter optimization problems, it can become inefficient for optimal control problems with many degrees of freedom (the control variables). Logsdon et al. (1990) noted this property in determining optimal reflux policies for batch distillation columns. Here, the reduced SQP method was quite successful in dealing with DAOP problems with state and control profile constraints. However, the degrees of freedom (for control variables) increase linearly with the number of elements. Consequently, if many elements are required, the effectiveness of the reduced SQP algorithm is reduced. This is due to three effects ... 

Mujtaba, I. M., and Macchietto, S., Optimal control of batch distillation, presented at 12th IMACS World Congress, Paris, July (1988). 

Dynamic Optimisation (Optimal Control) of Batch Distillation... 

Batch distillation is inherently a dynamic process and thus results to optimal control or dynamic optimisation problems (unless batch distillation task is carried out in a continuous distillation column). 

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. 

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 ... 

The application of Equations P.1-P.7 to batch distillation enables the solution of time optimal control problems. 

The one level optimal control formulation proposed by Mujtaba (1989) is found to be much faster than the classical two-level formulation to obtain optimal recycle policies in binary batch distillation. In addition, the one level formulation is also much more robust. The reason for the robustness is that for every function evaluation of the outer loop problem, the two-level method requires to reinitialise the reflux ratio profile for each new value of (Rl, xRI). This was done automatically in Mujtaba (1989) using the reflux ratio profile calculated at the previous function evaluation in the outer loop so that the inner loop problems (specially problem P2) could be solved in a small number of iterations. However, experience has shown that even after this re-initialisation of the reflux profile sometimes no solutions (even sub-optimal) were obtained. This is due to failure to converge within a maximum limit of function evaluations for the inner loop problems. On the other hand the one level formulation does not require such re-initialisation. The reflux profile was set only at the beginning and a solution was always found within the prescribed number of function evaluations. 

Mujtaba and Hussain (1998) implemented the general optimisation framework based on the hybrid scheme for a binary batch distillation process. It was shown that the optimal control policy using a detailed process model was very close to that obtained using the hybrid model. 

The application of the design methods developed in the previous sections has been demonstrated on a CSTR, a distillation column, fluid catalytic cracking units and a gasoline polymerization plant (20). Here, we will discuss optimizing control of the fluid catalytic cracker. 

The digital simulation of an extractive distillation column was performed in order to understand the dynamic behaviour of the system. Based on this results a considerably simplified dynamic model of sufficient accuracy could be developed. This model was employed in the design of a state observer and of an optimal control. After implementation in the large scale plant this new control system has proved to be highly efficient and reliable. 

I. J. Halvorsen, S. Skogestad, 2004, Shortcut Analisys of Optimal Operation of Petliuk Distillation, Ind. Eng. Chem. Res., vol. 43, p.3994 R. Luus, 1989, Optimal Control by Dynamie Programming Using Accessible Grid Points and Region Reduetion, Hung. J. Ind. Chem., vol. 17, p.523... 

I. Development of the Mathematical Model and Algorithm, Rev. Chim., vol. 37, p. 697 A. Woinaroschy, 2007, Time-Optimal Control of Distillation Columns by Iterative Dynamic Programming, Chem. Eng. Trans., vol. 11, p. 253... 

Figure 1.3 shows a schematic of the batch distillation process for the separation of a volatile compound from a binary liquid mixture. It is heated in the bottom still to generate vapors, which condense at the top to yield distillate having a higher concentration of a the volatile compound. A part of the distillate is withdrawn as product while the rest is recycled to the still. An optimal control problem is to maximize the production of distillate of a desired purity over a fixed time duration by controlling the distillate production rate with time (Converse and Gross, 1963). 

However, the steam flow set point signal coming from the TC is also sent as the process variable signal to a valve-position controller (VPC) whose set point signal is the minimum steam flow rate. Valve-position control is a type of optimizing control suggested over three decades ago by Shinskey who used it to achieve floating pressure control in distillation columns. Many other practical applications have been described. ... 

The synthesis of fine chemicals or pharmaceuticals, widely carried out in batch processes, implies many successive reaction and separation steps. Thus, synthesis optimisation is often restricted to the determination of the optimal operating conditions of each step separately. This approach is based on the use of reliable optimisation tools and has involved the development of various optimal control studies in reaction and distillation (Toulouse, 1999 Furlonge, 2000). 

Optimal control of a batch distillation column consists in the determination of the suitable reflux policy with respect to a particular objective function (e.g. profit) and set of constraints. In the purpose of the present work, the optimisation problem is defined with an operating time objective function and purity constraints set on the recovery ratio (90%) and on the propylene glycol final purity (80% molar). Different basis fimctions have been adopted for the control vector parameterisation of the problem piecewise constant and linear, hyperbolic tangent function. Optimal reflux profiles are determined with the final conditions of the previous optimal reactions as initial conditions. The optimal profiles of the resultant distillations are presented on figure 2. 

Here, we follow a later, simpler formulation that illustrates the power of optimal control for finite-time thermodynamic processes [11]. We take as the control variable the set of temperatures at a given number of equally spaced heat-exchange points along the length of the distillation column. The (assumed) binary mixtme comes in as a feed at rate F and is separated into the less volatile bottom at rate B and the distillate, at rate D, that collects at the top of the colmrm. Let x be the mole fraction of the more volatile component in the liquid and y, the corresponding mole fi action in the vapom, and their subscripts, the indications of the respective points of reference. Thus the total flow rates, for steady flow, must satisfy F = D + B, and xpF = x D + xbB. We index the trays from 0 at the top to N at the bottom. Mass balance requires that the rate V +i of vapour coming up from tray n + 1, less the rate of liquid dropping from tray n, L , must equal D for trays above the feed point at which F enters, and must equal —B below the feed point. Likewise the mole fractions must satisfy the condition that Vn+iVn+i —XnLn = xpD above the feed and —xpB below the feed. The heat required at each nth tray is... 

Pollard GP, Sargent RWH. Off line computation of optimal controls for a plate distillation column. Automatica 1970 6 59-76. 

Most sampled-data control systems employ discrete versions of PI and PID control algorithms although computers are certainly not limited to only these types. Special-purpose algorithms can be construaed in the software to deal with the multivariable, nonlinear nature of a distillation column. Adaptive control, for example, updates the parameters in control algorithms and sampling rates to compensate for nonlinearities in the process. Optimal control is a... 

Batch Distillation Chapter 4 is devoted to batch distillation. This is one of the most important and one of the most studied unit operations in batch industries. Separation is based on vapor-liquid equilibria. There are a number of configurations possible in conventional batch colmnn, namely, the constant reflux mode, the variable reflux mode, and the optimal reflux mode. There are a number of new configurations that have emerged in the literature for batch distillation. This chapter describes aU these operating modes and configurations. Various levels of models are available for different analysis. Different numerical integration techniques are needed to solve equations of these different models. Optimization and optimal control are well studied for this unit operation. 

Batch Crystallization Chapter 10 is devoted to batch crystallization where a phase diagram is used to find the supersaturation at which point material crystallizes. This is again one of the most studied batch operations. Similar to batch distillation, various modeling techniques are used to describe the operation of batch crystallizer, and optimization and optimal control problems are well studied. 

As stated earlier, the two basic modes of batch distillation are (l)constant reflux, and (2)variable reflux, resulting in variable distillation composition and constant distillate composition of the key component, respectively. The third operating mode, optimal reflux or optimal control is the trade-off between the two operating modes. 

The rigorous model of batch distillation operation involves a solution of several stiff differential equations and the semirigorous model involves a set of highly nonlinear equations. The computational intensity and memory requirement of the problem increase with an increase in the number of plates and components. The computational complexity associated with these models does not allow us to derive global properties such as feasible regions of operation, which are critical for optimization, optimal control, and synthesis problems. Even if such information is available, the computational costs of optimization, optimal control, or synthesis using these models are prohibitive. One way to deal with these problems associated with these models is to develop simphfied models such as the shortcut model. 

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