Mathematical dynamic model development applications

A third and very important use of dynamic experiments is to confirm the predictions of a theoretical mathematical model. As we indicated in Part I, the verification of the model is a very desirable step in its development and application. 

The formulation described above provides a useful framework for treating feedback control of combustion instability. However, direct application of the model to practical problems must be exercised with caution due to uncertainties associated with system parameters such as and Eni in Eq. (22.12), and time delays and spatial distribution parameters bk in Eq. (22.13). The intrinsic complexities in combustor flows prohibit precise estimates of those parameters without considerable errors, except for some simple well-defined configurations. Furthermore, the model may not accommodate all the essential processes involved because of the physical assumptions and mathematical approximations employed. These model and parameter uncertainties must be carefully treated in the development of a robust controller. To this end, the system dynamics equations, Eqs. (22.12)-(22.14), are extended to include uncertainties, and can be represented with the following state-space model ... 

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. 

For a process already in operation, there is an alternative approach based on experimental dynamic data obtained from plant tests. The experimental approach is sometimes used when the process is thought to be too complex to model from first principles. More often, however, we use it to find the values of some model parameters that are unknown. Although many of the parameters can be calculated from steady-state plant data, some must be found from dynamic tests (e.g., holdups in nonreactive systems). Additionally, we employ dynamic plant experiments to confirm the predictions of a theoretical mathematical model. Verification is a critical step in a model s development and application. 

The theoretical models proposed in Chapters 2-4 for the description of equilibrium and dynamics of individual and mixed solutions are by part rather complicated. The application of these models to experimental data, with the final aim to reveal the molecular mechanism of the adsorption process, to determine the adsorption characteristics of the individual surfactant or non-additive contributions in case of mixtures, requires the development of a problem-oriented software. In Chapter 7 four programs are presented, which deal with the equilibrium adsorption from individual solutions, mixtures of non-ionic surfactants, mixtures of ionic surfactants and adsorption kinetics. Here the mathematics used in solving the problems is presented for particular models, along with the principles of the optimisation of model parameters, and input/output data conventions. For each program, examples are given based on experimental data for systems considered in the previous chapters. This Chapter ean be regarded as an introduction into the problem software which is supplied with the book an a CD. 

Feedforward Control If the process exhibits slow dynamic response and disturbances are frequent, then the application of feedforward control may be advantageous. Feedforward (FF) control differs from feedback (FB) control in that the primary disturbance or load (L) is measured via a sensor and the manipulated variable (m) is adjusted so that deviations in the controlled variable from the set point are minimized or eliminated (see Fig. 8-29). By taking control action based on measured disturbances rather than controlled variable error, the controller can reject disturbances before they affect the controlled variable c. In order to determine the appropriate settings for the manipulated variable, one must develop mathematical models that relate ... 

Mathematical models are useful to better understand the processes that occur under electric field and predict remedial performance in field application. Compared with laboratory studies, only few studies have been reported on the mathematical modeling of electrochemical processes and remediation. Generally, electrochemical remediation models should incorporate the contaminant transport, transfer, and transformation processes and dynamic changes in electrical conductivity, pH, and geochemical reactions. Recognizing this as a complex task, researchers have developed some simple models based on a set of simplified assumptions (Chapters 25 and 26). 

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