Mathematical Modeling steps

There are many methods that can be, and have been, used for optimization, classic and otherwise. These techniques are well documented in the literature of several fields. Deming and King [6] presented a general flowchart (Fig. 4) that can be used to describe general optimization techniques. The effect on a real system of changing some input (some factor or variable) is observed directly at the output (one measures some property), and that set of real data is used to develop mathematical models. The responses from the predictive models are then used for optimization. The first two methods discussed here, however, omit the mathematical-modeling step optimization is based on output from the real system. 

In this section, we will show the process of the construction of a mathematical model, step by step, in accordance with the procedure shown in Fig. 3.4. The case studied has already been introduced in Figs. 1.1 and 1.2 of Chapter 1. These figures are concerned with a device for filtration with membranes, where the gradient is given by the transmembrane pressure between the tangential flow of the suspension and the downstream flow. The interest here is to obtain data about the critical situations that impose stopping of the filtration. At the same time, it is important to, a priori, know the unit behaviour when some of the components of the unit, such as, for example, the type of pump or the membrane surface, are changed. 

The solution adopted by us is the use of computer simulations of mathematical models of the process and the mock-up situations. Eventually, simulation techniques will become so accurate, that the mock-up step can be discarded. For the time being it is reasonable to use such models to generate corrections for smaller differences between mock-up and process. 

The classical microscopic description of molecular processes leads to a mathematical model in terms of Hamiltonian differential equations. In principle, the discretization of such systems permits a simulation of the dynamics. However, as will be worked out below in Section 2, both forward and backward numerical analysis restrict such simulations to only short time spans and to comparatively small discretization steps. Fortunately, most questions of chemical relevance just require the computation of averages of physical observables, of stable conformations or of conformational changes. The computation of averages is usually performed on a statistical physics basis. In the subsequent Section 3 we advocate a new computational approach on the basis of the mathematical theory of dynamical systems we directly solve a... 

The model building step deals with the development of mathematical models to relate the optimized set of descriptors with the target property. Two statistical measures indicate the quality of a model, the regression coefficient, r, or its square, r, and the standard deviation, a (see Chapter 9). 

The production of acetic acid from butane is a complex process. Nonetheless, sufficient information on product sequences and rates has been obtained to permit development of a mathematical model of the system. The relationships of the intermediates throw significant light on LPO mechanisms in general (22). Surprisingly, ca 25% of the carbon in the consumed butane is converted to ethanol in the first reaction step. Most of the ethanol is consumed by subsequent reaction. 

The formulation step may result in algebraic equations, difference equations, differential equations, integr equations, or combinations of these. In any event these mathematical models usually arise from statements of physical laws such as the laws of mass and energy conservation in the form. 

No single method or algorithm of optimization exists that can be apphed efficiently to all problems. The method chosen for any particular case will depend primarily on (I) the character of the objective function, (2) the nature of the constraints, and (3) the number of independent and dependent variables. Table 8-6 summarizes the six general steps for the analysis and solution of optimization problems (Edgar and Himmelblau, Optimization of Chemical Processes, McGraw-HiU, New York, 1988). You do not have to follow the cited order exac tly, but vou should cover all of the steps eventually. Shortcuts in the procedure are allowable, and the easy steps can be performed first. Steps I, 2, and 3 deal with the mathematical definition of the problem ideutificatiou of variables and specification of the objective function and statement of the constraints. If the process to be optimized is very complex, it may be necessaiy to reformulate the problem so that it can be solved with reasonable effort. Later in this section, we discuss the development of mathematical models for the process and the objec tive function (the economic model). 

Cycles Design methods for cycles rely on mathematical modeling (or empiricism) and often extensive pilot plant experiments. Many cycles can be easily analyzed using the methods described above apphed to the collection of steps. In some cycles, however, especially those operated with short cycle times or in shallow beds, transitions may not be very fully developed, even at a periodic state, and the complexity may be compounded by multiple sorbates. 

On a prospective basis, an agency can project its source composition and location and their emissions into the future and by the use of mathematical models and statishcal techniques determine what control steps have to be taken now to establish future air quality levels. Since the future involves a mix of existing and new sources, decisions must be made about the control levels required for both categories and whether these levels should be the same or different. 

The first step is to define the objectives of the flow model, and to identify those flow aspects that are relevant for the performance of the reactor. Then, the engineer must identify and quantify the various times and space scales involved, as well as the geometry of the system. These actions allow the problem to be represented by a mathematical model. Creating this model accurately is the most crucial task in the flow modeling project. 

In this case, economic and technical considerations are incorporated with the results from the preceding steps to determine the final reactor system with respect to the size of the experimental reactor and its operating conditions. The data from the experimental reactor are used to make appropriate corrections for the mathematical model derived in the preceding steps. At this stage, it is essential to review the previous steps for revision of earlier results. 

The first step is to be certain of the basis of the published data and consider in what ways this will be affected by different conditions. Revised figures can then usually be determined. For extensive interpretation work, simple mathematical models of performance can be constructed [69]. 

As our first approach to the model, we considered the controlling step to be the mass transfer from gas to liquid, the mass transfer from liquid to catalyst, or the catalytic surface reaction step. The other steps were eliminated since convective transport with small catalyst particles and high local mixing should offer virtually no resistance to the overall reaction scheme. Mathematical models were constructed for each of these three steps. 

Gas-liquid-particle operations are of a comparatively complicated physical nature Three phases are present, the flow patterns are extremely complex, and the number of elementary process steps may be quite large. Exact mathematical models of the fluid flow and the mass and heat transport in these operations probably cannot be developed at the present time. Descriptions of these systems will be based upon simplified concepts. 

Previous reports on FMSZ catalysts have indicated that, in the absence of added H2, the isomerization activity exhibited a typical pattern when measured as a function of time on stream [8, 9], In all cases, the initial activity was very low, but as the reaction proceeded, the conversion slowly increased, reached a maximum, and then started to decrease. In a recent paper [7], we described the time evolution in terms of a simple mathematical model that includes induction and deactivation periods This model predicts the existence of two types of sites with different reactivity and stability. One type of site was responsible for most of the activity observed during the first few minutes on stream, but it rapidly deactivated. For the second type of site, both, the induction and deactivation processes, were significantly slower We proposed that the observed induction periods were due to the formation and accumulation of reaction intermediates that participate in the inter-molecular step described above. Here, we present new evidence to support this hypothesis for the particular case of Ni-promoted catalysts. 

In spite of all doubts, mathematical modelling in fine chemicals process development is strongly recommended. The following steps in mathematical modelling of chemical reactors can be distinguished ... 

There are two objectives of setting up a kinetic mathematical model for chiral products. The first is the elucidation of the reaction mechanism with identification of the rate-controlhng step. The second is to derive a mathematical expression for the selectivity in terms of the ratio of the major product to the minor product. Then, based upon this expression, the reaction conditions such as pressure or feed ratio are changed to increase the selectivity. However, when the enantiomeric purity is over 99%, the selectivity is extremely high hence, the reaction mechanism for the major manifold can be neglected to simplify the establishment of the kinetic model. 

Parameter estimation is one of the steps involved in the formulation and validation of a mathematical model that describes a process of interest. Parameter estimation refers to the process of obtaining values of the parameters from the matching of the model-based calculated values to the set of measurements (data). This is the classic parameter estimation or model fitting problem and it should be distinguished from the identification problem. The latter involves the development of a model from input/output data only. This case arises when there is no a priori information about the form of the model i.e. it is a black box. 

Parameter estimation and identification are an essential step in the development of mathematical models that describe the behavior of physical processes (Seinfeld and Lapidus, 1974 Aris, 1994). The reader is strongly advised to consult the above references for discussions on what is a model, types of models, model formulation and evaluation. The paper by Plackett that presents the history on the discovery of the least squares method is also recommended (Plackett, 1972). 

The third step is to find the values of the variables that give the optimum value of the objective function (maximum or minimum). The best techniques to be used for this step will depend on the complexity of the system and on the particular mathematical model used to represent the system. 

Alternative mechanisms have been recently proposed [78,79] based on a kinetic investigation of NO reduction by n-octane under isothermal (200°C) and steady-state conditions in the presence of H2. The authors built up a mathematical model based on supposed reaction pathways, which account for molecular adsorption of NO and CO and dissociative ones for H2 and 02. The elementary steps, which have been considered for modelling their results are reported in Table 10.3. Interesting kinetic information can be provided by the examination of this mechanism scheme in particular the fast bimolecular... 

Hougen-Watson Models for the Case of Equilibrium Adsorption. This section treats Hougen-Watson mathematical models for cases where the rate limiting step is the chemical reaction rate on the surface. In all cases it is assumed that equilibrium is established with respect to adsorption of all species. 

The main choice of these functions should be based on data from suitable laboratory experiments. This approach has been recently applied to the mathematical study of the allelopathic competition between Pseudokirchneriella subcapitata and Chlorella vulgaris in an open culture (a chemostat-like device), where the nutrient is a mixture of inorganic phosphates Pi. Here we will illustrate step by step the four mentioned features, observing that each equation of the mathematical model can be actually constructed as a balance equation. 

The next step in formulating the problem is to construct a mathematical model of the process by considering the fundamental chemical and physical phenomena and physical limitations that influence the process behavior. For the case of the blast furnace, typical features are... 

Many wastewater flows in industry can not be treated by standard aerobic or anaerobic treatment methods due to the presence of relatively low concentration of toxic pollutants. Ozone can be used as a pretreatment step for the selective oxidation of these toxic pollutants. Due to the high costs of ozone it is important to minimise the loss of ozone due to reaction of ozone with non-toxic easily biodegradable compounds, ozone decay and discharge of ozone with the effluent from the ozone reactor. By means of a mathematical model, set up for a plug flow reactor and a continuos flow stirred tank reactor, it is possible to calculate more quantitatively the efficiency of the ozone use, independent of reaction kinetics, mass transfer rates of ozone and reactor type. The model predicts that the oxidation process is most efficiently realised by application of a plug flow reactor instead of a continuous flow stirred tank reactor. 

---