Multiple-variable process model

Process models are unfortunately often oversold and improperly used. Simulations, by definition, are not the actual process. To model the process, assumptions must be made about the process that may later prove to be incorrect. Further, there may be variables in the material or processing equipment that are not included in the model. This is especially true of complex processes. It is important not to confuse virtual reality with reality. The claim is often made that the model can optimize a cure cycle. The complex sets of differential equations in these models cannot be inverted to optimize the multiple properties they predict. It is the intelligent use of models by an experimenter or an optimizing routine that finds a best case among the ones tried. As a consequence, the literature is full of references to the development of process models, but examples of their industrial use in complex batch processes are not common. 

It is a well-understood phenomenon that a prediction model built from a highdimensional dataset consisting of thousands of available predictor variables and a relatively small sample size can be quite unstable (Miller, 2002). Models that are developed using an intense selection process are highly prone to change with a new training sample. Eurthermore, there is a multiplicity of good models when n< m, as observed by examination of model fit, where n is the sample size and m is the number of predictor variables. 

All climate models are mathematical models derived from differential equations defining the principles of physics, chemistry, and fluid dynamics driving the observable processes of the Earth s climate. Climate models can be highly complex three-dimensional computer simulations using multiple variables and tens of thousands of differential equations requiring trillions of computations or fairly simple two-dimensional projections with a single equation defining a sole observable process. In all climate models, each additional physical process incorporated into the model increases the level of complexity and escalates the mathematical parameters needed to define the process s potential effects. 

The purpose of the multiple-variable model is to provide an apprentice technician with a comprehensive view of the large scope of operations he or she will be exposed to in the chemical processing industry. When all of the equipment pieces are combined into a full-scale plant, it is easier to see how each system operates and the potential problems that troubleshooters will encounter. Nine of the troubleshooting models have been combined to make the multivariable model shown in Figure 17-15. 

Model-predictive control (MPC) uses process models derived from past process behavior to predict future process behaviour. The predictions are used to control process imits dynamically at optimum steady-state targets. MPC applications may also include the use of predicted product properties (inferential analyzers) and certain process calculations. Model-predictive controllers almost always include multiple independent variables. 

Temperature and composition are both properties of a flowing stream. Heat and material balances involve multiplication of these variables by flow, producing a characteristic nonlinear process model. Feedforward systems for control of these variables are similarly characterized by multiplication and division. The general form of process model for the applications is... 

In this chapter we consider model predictive control (MPC), an important advanced control technique for difficult multivariable control problems. The basic MPC concept can be summarized as follows. Suppose that we wish to control a multiple-input, multiple-output process while satisfying inequality constraints on the input and output variables. If a reasonably accurate dynamic model of the process is available, model and current measurements can be used to predict future values of the outputs. Then the appropriate changes in the input variables can be calculated based on both predictions and measurements. In essence, the changes in the individual input variables are coordinated after considering the input-output relationships represented by the process model. In MPC applications, the output variables are also referred to as controlled variables or CVs, while the input variables are also called manipulated variables or MVs. Measured disturbance variables are... 

In PCR, a principal component analysis (PCA) is first made of the X matrix (properly transformed and scaled), giving as the result the score matrix T and the loading matrix P. Then in a second step a few of the first score vectors tg are used as predictor variables in a multiple linear regression with Y as the response matrix. In the case that the few first components of PCA indeed contain most of the information of X related to Y, PCR indeed works as well as PLS. This is often the case in spectroscopic data, and here PCR is an often used alternative. In more complicated applications, however, such as QSAR and process modeling, the first few principal components of X rarely contain a sufficient part of the relevant information, and PLS works much better than PCR. ... 

A sequential approach is often followed for geostatistical modeling. The overall geometry and major layering or zones are defined first, perhaps deterministically. The rock types are modeled within each major layer or zone. Continnons variables are modeled within homogeneous rock types. Repeating the entire process creates multiple equally probable realizations. 

A wide variety of complex process cycles have been developed. Systems with many beds incorporating multiple sorbents, possibly in layered beds, are in use. Mathematical models constructed to analyze such cycles can be complex. With a large number of variables and nonlinear equilibria involved, it is usually not beneficial to make all... 

This review is structured as follows. In the next section we present the theory for adsorbates that remain in quasi-equilibrium throughout the desorption process, in which case a few macroscopic variables, namely the partial coverages 0, and their rate equations are needed. We introduce the lattice gas model and discuss results ranging from non-interacting adsorbates to systems with multiple interactions, treated essentially exactly with the transfer matrix method, in Sec. II. Examples of the accuracy possible in the modehng of experimental data using this theory, from our own work, are presented for such diverse systems as multilayers of alkali metals on metals, competitive desorption of tellurium from tungsten, and dissociative... 

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