Building Simple Mathematical Models

In this section, we will attempt to develop mathematical models for different situations that you will face as a process or bioprocess engineer. Although these examples are limited and tailored to your mathematical background, it will be interesting and rewarding for you to discover that you can be involved with fascinating and ever-challenging examples. 

Adapting a reactor batch-operation stage to a continuous processing line (can also be applied to, for example, bioreactors, autoclaves, and dryers). 

A Gantt chart showing the temporal programming schedule of a battery reactor system (Fig. 9.3) can be used as a first step in determining the number of reactors as a function of its effective volume (Fr) and volumetric flow rate (Fp) from the processing line. The continuous operation of the reactor 

According to (9.5), the minimum number of reactors for a continuous processing line operation is three (R 3). In addition, if we take into accotmt that the effective volume of a reactor is the loading time (tx) multiplied by the volumetric flow rate of the processing line (Fp), then 

This procedure, developed for chemical reactors, can be easily adapted and extended to other processes, like winemaking, canned food production, and batch dehydration. 

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The theory behind molecular vibrations is a science of its own, involving highly complex mathematical models and abstract theories and literally fills books. In practice, almost none of that is needed for building or using vibration spectroscopic sensors. The simple, classical mechanical analogue of mass points connected by springs is more than adequate. 

Calibration is the process of measuring the instrument response (y) of an analytical method to known concentrations of analytes (x) using model building and validation procedures. These measurements, along with the predetermined analyte levels, encompass a calibration set. This set is then used to develop a mathematical model that relates the amount of sample to the measurements by the instrument. In some cases, the construction of the model is simple due to relationships such as Beer s Law in the application of ultraviolet spectroscopy. 

The term microkinetics is understood to mean the kinetics of a reaction that are not masked by transport phenomena and to refer to a series of reaction steps. For the investigation of intermediary metabolism, idealized conditions are chosen that often do not correspond to the real conditions of engineering processes. This fact makes it difficult to transfer microkinetic data to technical processes. For the purposes of technologically oriented research and the development of a process to technical ripeness, it is often sufficient to know quantitatively how a process runs without necessarily knowing why. (Macrokinetics, however, must be avoided, as they are scale dependent). Mathematical formulations are needed that reproduce the kinetics adequately for the purpose but are as simple and have as few parameters as possible. Today, even when electronic computers greatly reduce the labor of computation, the criterion of simplicity remains important due to the problem of experimental verification. The iterative nature of the process of building an adequate model is an important point that will be considered in greater detail in Sect. 2.4. 

Importance of Building Mathematical Models and Constructing Simple Models... 

Figure 9 demonstrates how a data matrix X of a given class of objects is mathematically modeled by a set of principal components. The optimum number of components can be determined by classifying a set of objects which have not been used for model building or by cross-validation. The sum of squared residuals usually has a minimum for a medium number of model components. If the number of components is too small then the model is too simple and the prediction therefore is bad if the number is too high then the model fits... 

Reducing the dimensionality of the descriptor space not only facilitates model building with molecular descriptors but also makes data visualization and identification of key variables in various models possible. Notice that while a low dimension mathematically simplifies a problem such as model development or data visualization, it is usually more difficult to correlate trends directly with physical descriptors, and hence the data become less interpretable, after the dimension transformation. Trends directly linked with physical descriptors provide simple guidance for molecular modifications during potency/property optimizations. 

There are many chemometric methods to build initial estimates some are particularly suitable when the data consists of the evolutionary profiles of a process, such as evolving factor analysis (see Figure 11.4b in Section 11.3) [27, 28, 51], whereas some others mathematically select the purest rows or the purest columns of the data matrix as initial profiles. Of the latter approach, key-set factor analysis (KSFA) [52] works in the FA abstract domain, and other procedures, such as the simple-to-use interactive self-modeling analysis (SIMPLISMA) [53] and the orthogonal projection approach (OPA) [54], work with the real variables in the data set to select rows of purest variables or columns of purest spectra, that are most dissimilar to each other. In these latter two methods, the profiles are selected sequentially so that any new profile included in the estimate is the most uncorrelated to all of the previously selected ones. 

Schilling No, I just use my laptop—but I m working on relatively simple problems now using basic linear algebra. In the future, as we try to build models of more complex organisms—such as humans—and apply more intensive mathematical... 

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