Dynamic Modeling of Distillation Columns

On the basis of mathematical models attempts are made to precalculate the dynamic behaviour of distilling columns. According to Kohler and Schober [264] column dynamics involves these problems ... 

The principle of the perfectly-mixed stirred tank has been discussed previously in Sec. 1.2.2, and this provides essential building block for modelling applications. In this section, the concept is applied to tank type reactor systems and stagewise mass transfer applications, such that the resulting model equations often appear in the form of linked sets of first-order difference differential equations. Solution by digital simulation works well for small problems, in which the number of equations are relatively small and where the problem is not compounded by stiffness or by the need for iterative procedures. For these reasons, the dynamic modelling of the continuous distillation columns in this section is intended only as a demonstration of method, rather than as a realistic attempt at solution. For the solution of complex distillation problems, the reader is referred to commercial dynamic simulation packages. 

Skogestad, S. Dynamic and Control of Distillation Columns—A Critical Survey. Modeling Identification Control 18 177-217 (1997). 

In a steady state continuous distillation with the assumption of a well mixed liquid and vapour on the plates, the holdup has no effect on the analysis (modelling of such columns does not usually include column holdup) since any quantity of liquid holdup in the system has no effect on the mass flows in the system (Rose, 1985). Batch distillation however is inherently an unsteady state process and the liquid holdup in the system become sinks (accumulators) of material which affect the rate of change of flows and hence the whole dynamic response of the system. 

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. 

Let us derive a dynamic model of the process with control structure CS2 included. A rigorous model of the reactor and the two distillation columns would be quite complex and of very high order. Because the dynamics of the liquid-phase reactor are much slower than the dynamics of the separation section in this process, we can develop a simple second-order model by assuming the separation section dynamics are instantaneous. Thus the separation section is always at steady state and is achieving its specified performance, i.e, product and recycle purities are at their setpoints. Given a flowrate F and the composition zA/zB of the reactor effluent stream, the flowrates of the light and heavy recycle streams D, and B-L can be calculated from the algebraic equations... 

Even at steady state, efficiencies vary from component to component and with position in a column. Thus, if the column is not at steady state, then efficiencies also must vary with time as a result of changes to flow rates and composition inside the column. Thus, equilibrium-stage models with efficiencies should not be used to model the dynamic behavior of distillation and absorption columns. Nonequilibrium models for studying column dynamics are described hy, e.g., Kooijman and Taylor [AlChE 41, 1852 (1995)], Baur et al. [Chem. 

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... 

The rigorous solution of batch distillation columns carries an extra dimension of complexity over continuous steady-state distillation because it is inherently a transient operation. The basic assumption of steady-state operation in the continuous column model obviously does not apply for batch distillation. The only possible steady-state operation in batch distillation is at total reflux, which is commonly used as the initial condition for the dynamic solution of the column. 

Distillation control schemes may be analyzed either on a steady-stale (sensitivity analysis) or on a dynamic basis. The latter requires a dynamic model that takes into account the dynamic response of the column and the control loope. An example of a dynamic model is described by McCune and Gailier,6 but il should be apparent from the material presented enrlier that the holdup characteristics of distillation columa devices can vary widely, and snch variation should be accommodated by the model. The development of the naw high-efficiency packings has caused a new look at the system dynamics when the liquid holdup in the column is quite low, and thus the existing models for trays may not be adjustable to application to packings. The use of a tray-type dynamic model is described in the article by Gailier and McCune 7 so work to date has been reported for packed column dynamic models. 

A dynamic model of a distillation column can be assembled from simpler units, as trays, heat exchangers (condenser, reboiler), reflux drum, valves and pumps (Fig. 4.5). Tray modelling has to answer two issues (1) accurate description of material and energy holdup, and (2) accurate pressure drop calculation. 

In order to discuss the mathematical modeling of multicomponent distillation, a dynamic model of a ternary equilibrium column of five stages including a partial condenser and a reboiler will be discussed. In particular, the distillation of benzene, toluene, and xylene will be considered. The system is shown in Figure 5.19. 

Two physical examples, a continuous blending system and a distillation column, have been used to introduce basic control concepts, notably, feedback and feedforward control. We also motivated the need for a systematic approach for the design of control systems for complex processes. Control system development consists of a number of separate activities that are shown in Fig. 1.10. In this book we advocate the design philosophy that for complex processes, a dynamic model of the process should be developed so that the control system can be properly designed. 

While we laud the virtue of dynamic modeling, we will not duphcate the introduction of basic conservation equations. It is important to recognize that all of the processes that we want to control, e.g. bioieactor, distillation column, flow rate in a pipe, a drag delivery system, etc., are what we have learned in other engineering classes. The so-called model equations are conservation equations in heat, mass, and momentum. We need force balance in mechanical devices, and in electrical engineering, we consider circuits analysis. The difference between what we now use in control and what we are more accustomed to is that control problems are transient in nature. Accordingly, we include the time derivative (also called accumulation) term in our balance (model) equations. 

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)... 

You may wonder why we would ever be satisfied with anything less than a very accurate integration. The ODEs that make up the mathematical models of most practical chemical engineering systems usually represent a mixture of fast dynamics and slow dynamics. For example, in a distillation column the liquid flow or hydraulic dynamic response occurs fairly rapidly, of the order of a few seconds per tray. The composition dynamics, the rate of change of hquid mole fractions on the trays, are usually much slower—minutes or even hours for columns with many trays. Systems with this mixture of fast and slow ODEs are called stiff systems. 

PID controller tunings for this model have been given by a number of researchers [9-13], Chen and Fruehauf [9] have given an industrial example of the level control in a distillation column where the open loop dynamics follows the IPTD model with parameters kp = 0.2 and d = 1A min. 

The modeling of RD processes is illustrated with the heterogenously catalyzed synthesis of methyl acetate and MTBE. The complex character of reactive distillation processes requires a detailed mathematical description of the interaction of mass transfer and chemical reaction and the dynamic column behavior. The most detailed model is based on a rigorous dynamic rate-based approach that takes into account diffusional interactions via the Maxwell-Stefan equations and overall reaction kinetics for the determination of the total conversion. All major influences of the column internals and the periphery can be considered by this approach. 

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