Dynamic model distillation column

Modelling a single tray is similar with a dynamic flash discussed before. The solution of the assembly of trays, increased with condenser, flash drum and reboiler, is a much more difficult problem, however. The equations presented below (for notations see Fig. 4.6) are known as MESH equations for modelling distillation columns at steady state enlarged with left hand terms for accumulation. 

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. 

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)... 

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

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 dynamic behavior of processes (pipe-vessel combinations, heat exchangers, transport pipelines, furnaces, boilers, pumps, compressors, turbines, and distillation columns) can be described using simplified models composed of process gains, dead times, and process dynamics. 

In what follows, we begin by introducing two examples of process systems with recycle and purge. First, we analyze the case of a reactor with gas effluent connected via a gas recycle stream to a condenser, and a purge stream used to remove the light impurity present in the feed. In the second case, the products of a liquid-phase reactor are separated by a distillation column. The bottoms of the column are recycled to the reactor, and the trace heavy impurity present in the feed stream is removed via a liquid purge stream. We show that, in both cases, the dynamics of the system is modeled by a system of stiff ODEs that can, potentially, exhibit a two-time-scale behavior. 

Once the large internal flow rates have been set via appropriate control laws, the index of the DAE system (7.21) is well defined, and a state-space realization (ODE representation) of the slow subsystem can be derived. This representation of the slow dynamics of the column can be used for the derivation of a model-based nonlinear controller to govern the input-output behavior of the column, namely to address the control of the product purity and of the overall material balance. To this end, the small distillate and bottoms flow rates as well as the setpoints of the level controllers are available as manipulated inputs. 

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. 

Lehtonen et al. (1998) considered polyesterification of maleic acid with propylene glycol in an experimental batch reactive distillation system. There were two side reactions in addition to the main esterification reaction. The equipment consists of a 4000 ml batch reactor with a one theoretical plate distillation column and a condenser. The reactions took place in the liquid phase of the reactor. By removing the water by distillation, the reaction equilibrium was shifted to the production of more esters. The reaction temperatures were 150-190° C and the catalyst concentrations were varied between 0.01 and 0.1 mol%. The kinetic and mass transfer parameters were estimated via the experiments. These were then used to develop a full-scale dynamic process model for the system. 

In Greaves et al. (2001) and Greaves (2003), instead of using a rigorous model (as in the methodology described above), an actual pilot plant batch distillation column is used. The differences in predictions between the actual plant and the simple model (Type III and also in Mujtaba, 1997) are defined as the dynamic process-model mismatches. The mismatches are modelled using neural network techniques as described in earlier sections and are incorporated in the simple model to develop the hybrid model that represents the predictions of the actual column. 

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. 

From a simulation viewpoint units SO, S6 and S7 may be considered blackboxes. On the contrary, SI to S5 are simulated by rigorous distillation columns, as sieve trays. In the steady state all the reactors can be described by a stoichiometric approach, but kinetic models are useful for Rl, R2 and R4 in dynamic simulation [7, 8]. As shown before, the reaction network should be formulated so as to use a minimum of representative chemical species, but respecting the atomic balance. This approach is necessary because yield reactors can misrepresent the process. 

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... 

Divided Wall Distillation Column Dynamic Modeling And Control... 

Keywords divided wall distillation column, dynamic modeling, optimal startup control... 

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... 

First of all, the plant flowsheet is split into units / group of units. The splitting it is done in units to which local control is applied the reactor (plus the heat exehangers around it), the distillation columns, mixing vessels, pumps. Since the mixers and the pumps are considered instantaneous (no dynamics) they are not interesting for the model reduction. Further, the units are individually reduced. 

For the distillation columns, linear model-order reduction will be used. The linear model is obtained in Aspen Dynamics. Some modifications to the previous study have been done to the linear models, in order to have the reboiler duty and the reflux ratio as input or output variables of the linear models. This is needed to have access to those variables in the reduced model, for the purpose of the dynamic optimization. A balanced realization of the linear models is performed in Matlab. The obtained balanced models are then redueed. The redueed models of the distillation columns are further implemented in gProms. When all the reduced models of the individual units are available, these models are further connected in order to obtain the full reduced model of the alkylation plant. The outeome of the model reduction procedure is presented in Table 1, together with some performances of the reduced model. 

The model of the distillation column used throughout the paper is developed by [10], composed by the mass, component mass and enthalpy balance equations used as basis to implement a SIMULINK model (figure 1) which describes the nonlinear column dynamics as a 2 inputs Q, Lvo/) and 2 output (Ad, Ab ). Implementations details for the overall column dynamics are given in [11]. 

Process dynamics is another important factor that must be considered. In a distillation column, for instance, the time elapsed between changing the reflux rate and observing a change in a product composition could be measured in hours. With this response time, and in the absence of dynamic prediction capability, the controller will start taking action hours after a disturbance occurs, and it would take even longer for the correction to take effect. Linear predictions are commonly used to forecast trends of process variables but many processes, particularly multistage separations, are often highly nonlinear. Substantial improvement can be achieved with a nonlinear model. 

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. 

All the equations above are state equations and describe the dynamic behavior of the distillation column. The state variables of the model are ... 

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