Distributed parameter system

The basic concepts of the above lumped parameter and distributed parameter systems are shown in Fig. 1.7. 

Figure 1.7. Choosing balance regions for lumped and distributed parameter systems.

Optimization of a distributed parameter system can be posed in various ways. An example is a packed, tubular reactor with radial diffusion. Assume a single reversible reaction takes place. To set up the problem as a nonlinear programming problem, write the appropriate balances (constraints) including initial and boundary conditions using the following notation ... 

Flow patterns in a stirred tank (lumped parameter system) and a tubular reactor (distributed parameter system). 

Romagnoli, J. A., and Gani, R. (1983). Studies of distributed parameter systems Decoupling the state parameter estimation problem. Chem. Eng. Sci. 38, 1831-1843. 

Later development in singularity theory, especially the pioneering work of Golubitsky and Schaeffer [19], has provided a powerful tool for analyzing the bifurcation behavior of chemically reactive systems. These techniques have been used extensively, elegantly and successfully by Luss and his co-workers [6-11] to uncover a large number of possible types of bifurcation. They were also able to apply the technique successfully to complex reaction networks as well as to distributed parameter systems. 

V. V. Barelko and S.A. Zhukov, To the Stability of the CO Oxidation Processes on Platinum in the Lumped- and Distributed Parameter Systems, Preprint Otd. Inst. Khim. Fiz., Chernogolovka, 1979 (in Russian). ... 

Models are either dynamic or steady-state. Steady-state models are most often used to study continuous processes, particularly at the design stage. Dynamic models, which capture time-dependent behavior, are used for batch process design and for control system design. Another classification of models is in terms of lumped parameter or distributed parameter systems. A lumped parameter system... 

Computer Control of a Distributed Parameter System", presented at National AIChE Meeting, Houston, Texas,... 

For analysis of distributed-parameter systems, such as a tubular fixed bed reactor, numerical simulation of periodic operation at various values of control parameters is typically applied. Asymptotic models for quasisteady and relaxed steady states are valuable instruments for a substantial simplification of the original distributed-parameter system. A method allowing for... 

Except for the cases where the optimal temperature profile is of the bang-bang type, analytical solutions for axially varying optimal profiles are almost impossible. Denn et. al. (11) used a variational approach for a wide class of distributed parameter systems where the optimizing decisions may enter into the state equations or boundary conditions. When intermediate control is involved, one can only obtain numerical approximations to the optimal solution. 

This paper presents the application of a model based predictive control strategy for the primary stage of the freeze drying process, which has not been tackled until now. A model predictive control framework is provided to minimize the sublimation time. The problem is directly addressed for the non linear distributed parameters system that describes the dynamic of the process. The mathematical model takes in account the main phenomena, including the heat and mass transfer in both the dried and frozen layers, and the moving sublimation front. The obtained results show the efficiency of the control software developed (MPC CB) under Matlab. The MPC( CB based on a modified levenberg-marquardt algorithm allows to control a continuous process in the open or closed loop and to find the optimal constrained control. 

Keywords Freeze drying, moving boundary, non linear distributed parameter systems, model based predictive control, internal model control. 

Banks, H. T., and Kunisch, K., Estimation Techniques for Distributed Parameter Systems, Boston Birkhauser, 1989. 

Optimization requires that at/R have some reasonably high value so that the wall temperature has a significant influence on reactor performance. There is no requirement that 33 jR be large. Thus the method can be used for polymer systems that have thermal diffusivities typical of organic liquids but low molecular diffusivi-ties. The calculations needed to optimize distributed-parameter systems (i.e., sets of PDEs) are much longer than needed to optimize the lumped-parameter systems (i.e., sets of ODEs) studied in Chapter 6, but the numerical approach is the same and is still feasible using small computers. 

Example 9.1 used a distributed-parameter-system of simultaneous PDEs for the phthalic anhydride reaction in a packed bed. Axial dispersion is a lumped-parameter system of simultaneous DDEs that can also be used for a packed bed. Apply the axial dispersion model to the phthalic reaction using D as determined from Figure 9.7 and = 1.33 D. Compare your results to those obtained in Example 9.1. 

Every process is subject to the laws of thermodynamics and to the conservation laws for mass and momentum, and we can expect every dynamic simulation of an industrial process to need to invoke one or more of these laws. The interpretation of these laws as they apply to different types of processes leads to different forms for the describing equations. This chapter will begin by reviewing the thermodynamic relations needed for process simulation, and it will go on to derive the conservation equations necessary for modelling the major components found in industrial processes. Finally, the different equations arising from lumped-parameter and distributed-parameter systems containing fluids will be brought out. 

This is the equation of state that models the behavior of liquid s temperature (state variable) along the length of the exchanger. Since eq. (4.19) is a partial differential equation we say that the exchanger has been modeled as a distributed parameter system. Note that U is the overall heat transfer coefficient between steam and the liquid in the tube, and Ta is the temperature of saturated steam. 

Nonlinear, distributed parameter systems are described by partial differential equations of the following form ... 

Koda, M and Seinfeld, J. H. (1982) Sensitivity analysis of distributed parameter systems, IEEE Trans, on Automatic Control, AC-27, 951-955. 

By and large we can describe the results of the analysis of distributed parameter systems (i.e., flow reactors other than CSTRs) in terms of the gradients or profiles of concentration and temperature they generate. To a large extent, the analysis we shall pursue for the rest of this chapter is based on the one-dimensional axial dispersion model as used to describe both concentration and temperature fields within the nonideal reactor. The mass and energy conservation equations are coupled to each other through their mutual concern about the rate of reaction and, in fact, we can use this to simplify the mathematical formulation somewhat. Consider the adiabatic axial dispersion model in the steady state. 

In case of packed columns, a qualitatively different behavior can be found for finite and infinite intra-partide mass transfer resistance. For vanishing mass transfer resistance inside the catalyst a small number of solutions, typically three, can be observed. Note, that this is consistent with the TAME case discussed above. Instead, for finite transport inside the catalyst a very large number of solutions can be observed. An example is shown in Fig. 10.17, right. It was conjectured by Mohl et al. [74], that this behavior is caused by isothermal multiplidty of the single catalyst pellet and is therefore similar to the well-known fixed-bed reactor [38, 77]. However, further research is required to verify this hypothesis. Further, it was shown by Mohl et al. [74] that in both cases the number of solutions may crucially depend on the discretization of the underlying continuously distributed parameter system. A detailed discussion is given by Mohl et al. [74]. 

Adel Mhamdi and Wolfgang Marquardt, Jncremental Identification [Pg.325]

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