Process Mathematical Models

Development of Process (Mathematical) Models Constraints in optimization problems arise from physical bounds on the variables, empirical relations, physical laws, and so on. The mathematical relations describing the process also comprise constraints. Two general categories of models exist  

Those based on strictly empirical descriptions Mathematical models based on physical and cnemical laws (e.g., mass and energy balances, thermodynamics, chemical reaction kinetics) are frequently employed in optimization applications. These models are conceptually attractive because a general model for any system size can be developed before the system is constructed. On the other hand, an empirical model can be devised that simply correlates input/output data without any physiochemical analysis of the process. For these models, optimization is often used to fit a model to process data, using a procedure called parameter estimation. The well-known least squares curve-fitting procedure is based on optimization theory, assuming that the model parameters are contained linearly in the model. One example is the yield matrix, where the percentage yield of each product in a unit operation is estimated for each feed component 

---

Various methods 1. Deterministic process mathematical model especially given by differential equations 2. With or without inequality type constraints... 

The combined methods Various variants 1. Process mathematical models with distributed inputs 2. Capacity to be associated with a Kalman filter 3. Without inequality type constraints The maximum likelihood method... 

Before the era of modern computers, the EVOP process investigation was used successfully to improve the efficiency of many chemical engineering processes. Now its use is receding due to the competition from process mathematical modelling and simulation. However, biochemical and life processes are two large domains where the use of the EVOP investigation can still bring spectacular results. 

The mathematical model gives the input and output vectors for the ANN, which, in normal cases, are represented by the measured data. When the learning process has been completed, the process mathematical model (PMM) and the optimizing algorithm (OA) are decoupled and the ANN is ready to produce the simulation results for the process. This procedure is also used to produce the ANN simulators needed for the control of the processes or their usual automatic operation. 

Such a relationship in turn is not possible if we do not have a mathematical representation of the process (mathematical model). Once the value of the control objective can be estimated from a relationship such as the above, it can be compared to the desired value (set point) and the controller can be activated for appropriate action as in feedback control. 

Differential equations. 2. Chemical processes—Mathematical models. 3. Chemical engineering—Mathematics. I. Duong, D. Do. 

There are a number of modeling approaches that can be used with process control systems. Whereas mathematical models based on the chemistry and physics of the system represent one alternative, the typical process control model utilizes an empirical input/output relationship, the so-called black-box model. These models are found by experimental tests of the process. Mathematical models of the control system may include not only the process but also the controller, the final control element, and other electronic components such as measurement devices and transducers. Once these component models have been determined, one can proceed to analyze the overall system dynamics, the effect of different controllers in the operating process configuration, and the stability of the system, as well as obtain other usefid information. 

In our case, non-stationary processes are analyzed, i.e. the values of factors depend on time t. Classical apphcation of Bayesian method is not correct in this case, because observations obtained in different time moments represent the other state of the indicators. The modified application of Bayesian method for the calculation estimates of parameters of non-stationary process mathematical models is presented in research papers (Augutis et al, 2012 Zutautaite-Seputiene et al, 2010). 

The determination of the process mathematical model is often the most difficult and time-consuming step in control system analysis. This is a result of the dynamic nature of the process in other words, how the system reacts during upsets or disturbances. The problem is further complicated by process nonlinearities and time-varying parameters. To illustrate the modelling procedure we will look at developing a model for a shell and tube heat exchanger with temperature control [7], shown in Figure 3.29. 

Chemical processes—Mathematical models. 2. Chemical processes—Computer simulation. 3. Microsoft Visual BASIC. I. Title. 

---