Mathematical modeling example calculations

In each of the examples in this chapter, we followed three key steps. First we studied the process to gain a qualitative understanding. What trends are expected For example, how will the purity of the potable water depend on the amount of brine retained by the ice What behavior is expected at extremes, such as no water in the brine stream Second, we applied mathematical modeling to calculate a quantity or to yield operating equations. Third, we checked the result. Each step is important. 

They cannot be part of a mathematical model whose purpose would be to turn the classification into a continuous quantitative variable. In particular, the example of physical factors illustrates this. Whereas for the highest degree criteria are the same as those of the NFPA code, the simple fact of wanting to add in physical factors to these calculation models forced the originators of this technique to forget about the NFPA code. 

One must understand the physical mechanisms by which mass transfer takes place in catalyst pores to comprehend the development of mathematical models that can be used in engineering design calculations to estimate what fraction of the catalyst surface is effective in promoting reaction. There are several factors that complicate efforts to analyze mass transfer within such systems. They include the facts that (1) the pore geometry is extremely complex, and not subject to realistic modeling in terms of a small number of parameters, and that (2) different molecular phenomena are responsible for the mass transfer. Consequently, it is often useful to characterize the mass transfer process in terms of an effective diffusivity, i.e., a transport coefficient that pertains to a porous material in which the calculations are based on total area (void plus solid) normal to the direction of transport. For example, in a spherical catalyst pellet, the appropriate area to use in characterizing diffusion in the radial direction is 47ir2. 

Software sensors and related methods - This last group is considered because of the complexity of wastewater composition and of treatment process control. As all relevant parameters are not directly measurable, as will be seen hereafter, the use of more or less complex mathematical models for the calculation (estimation) of some of them is sometimes proposed. Software sensing is thus based on methods that allow calculation of the value of a parameter from the measurement of one or more other parameters, the measurement principle of which is completely different from an existing standard/reference method, or has no direct relation. Statistical correlative methods can also be considered in this group. Some examples will be presented in the following section. 

In addition to the temporal correlation coefficient, the spatial correlation coefficient was calculated approximately for fixed values of time. Except for one of the mathematical models, all techniques showed a better temporal correlation than spatial correlation. The two correlation coefficients are cross plotted in Figure 5-6. Nappo stressed that correlation coefficients express fidelity in predicting tends, rather than accuracy in absolute concentration predictions. Another measure is used for assessing accuracy in predicting concentrations the ratio of predicted to observed concentration. Nappo averaged this ratio over space and over time and extracted the standard deviation of the data sample for each. The standard deviation expresses consistency of accuracy for each model. For example, a model might have a predicted observed ratio near unity,... 

The previous sections have pointed out that mathematical models of the processes must be proved by experiments, or adapted to experimental results with the aid of pilot extractors. For economic reasons, pilot extractors are usually much smaller than large-scale industrial apparatus. Pulsed pilot columns, for example, have a diameter between 30 and 250 mm, whereas industrial-size columns are up to 2500 mm and more in size. Thus, the question arises of whether or not the calculations or pilot experiments may be used for the dimensions of large-scale apparatus. This is a general problem for engineers. 

Mathematical modelling of the dose-response relationship is an alternative approach to quantify the estimated response within the experimental range. This approach can be used to determine the BMD or benchmark concentration (BMC) for inhalation exposure, which can be used in place of the LOAEL or NOAEL (Crump, 1984). The BMD (used here for either BMD or BMC) is defined as the lower confidence limit on a dose that produces a particular level of response (e.g., 1%, 5%, 10%) and has several advantages over the LOAEL or NOAEL (Kimmel Gaylor, 1988 Kimmel, 1990 USEPA, 1995 IPCS, 1999). For example, (1) the BMD approach uses all of the data in fitting a model instead of only data indicating the LOAEL or NOAEL (2) by fitting all of the data, the BMD approach takes into account the slope of the dose-response curve (3) the BMD takes into account variability in the data and (4) the BMD is not limited to one experimental dose. Calculation and use of the BMD approach are described in a US EPA... 

Formulation of the mathematical model here adopts the usual assumptions of equimolar overflow, constant relative volatility, total condenser, and partial reboiler. Binary variables denote the existence of trays in the column, and their sum is the number of trays N. Continuous variables represent the liquid flow rates Li and compositions xj, vapor flow rates Vi and compositions yi, the reflux Ri and vapor boilup VBi, and the column diameter Di. The equations governing the model include material and component balances around each tray, thermodynamic relations between vapor and liquid phase compositions, and the column diameter calculation based on vapor flow rate. Additional logical constraints ensure that reflux and vapor boilup enter only on one tray and that the trays are arranged sequentially (so trays cannot be skipped). Also included are the product specifications. Under the assumptions made in this example, neither the temperature nor the pressure is an explicit variable, although they could easily be included if energy balances are required. A minimum and maximum number of trays can also be imposed on the problem. 

Throughout the data collection phase unexpected complications were encountered in the pilot application because calculations obtained from the ERP system were often not comparable across plants. Even if the same ERP system was used (which was not the case for all sites considered), different conventions were used for distributing costs across products. Additionally, recipes pulled from the ERP system often did not reflect the actual recipes employed in production. For example, significant differences in absolute raw material quantities required and relative relationship between the different raw materials were found. As similar problems are also reported in the literature discussing practical applications of mathematical modeling approaches (e.g., Kallrath 2000, p. 817 Lee and Billington 1995, p. 46), this appears to be the norm rather than a company-specific exception. 

The mathematical model of network formation in the pregel stage will focus on the prediction of the gel conversion and the evolution of number-and mass-average molar masses, Mn and Mw, respectively. For chainwise polymerizations, calculations will be restricted to the limit of a very low concentration of the polyfunctional monomer (A4 in the previous example). Thus, homogeneous systems will always be considered. 

Some mathematical models have been developed to predict the behavior of the pressure drop over the diesel particulate trap with time during the loading/regenera-tion cycles [62, 67-69], to calculate the effect of filter-medium properties on filter performance of fibrous filters [70] or to describe the flow and filtration process [71] and the regeneration process [72-75]. An illustrative example for the performance of such a pressure drop model is provided in Fig. 15.7. 

Numerous reports are available [19,229-248] on the development and analysis of the different procedures of estimating the reactivity ratio from the experimental data obtained over a wide range of conversions. These procedures employ different modifications of the integrated form of the copolymerization equation. For example, intersection [19,229,231,235], (KT) [236,240], (YBR) [235], and other [242] linear least-squares procedures have been developed for the treatment of initial polymer composition data. Naturally, the application of the non-linear procedures allows one to obtain more accurate estimates of the reactivity ratios. However, majority of the calculation procedures suffers from the fact that the measurement errors of the independent variable (the monomer feed composition) are not considered. This simplification can lead in certain cases to significant errors in the estimated kinetic parameters [239]. Special methods [238, 239, 241, 247] were developed to avoid these difficulties. One of them called error-in-variables method (EVM) [239, 241, 247] seems to be the best. EVM implies a statistical approach to the general problem of estimating parameters in mathematical models when the errors in all measured variables are taken into account. Though this method requires more information than do ordinary non-linear least-squares procedures, it provides more reliable estimates of rt and r2 as well as their confidence limits. 

Dente and Ranzi (in Albright et al., eds., Pyrolysis Theory and Industrial Practice, Academic Press, 1983, pp. 133-175) Mathematical modeling of hydrocarbon pyrolysis reactions Shah and Sharma (in Carberry and Varma, eds., Chemical Reaction and Reaction Engineering Handbook, Dekker, 1987, pp. 713-721) Hydroxylamine phosphate manufacture in a slurry reactor Some aspects of a kinetic model of methanol synthesis are described in the first example, which is followed by a second example that describes coping with the multiplicity of reactants and reactions of some petroleum conversion processes. Then two somewhat simplified industrial examples are worked out in detail mild thermal cracking and production of styrene. Even these calculations are impractical without a computer. The basic data and mathematics and some of the results are presented. 

Today contractors and licensors use sophisticated computerized mathematical models which take into account the many variables involved in the physical, chemical, geometrical and mechanical properties of the system. ICI, for example, was one of the first to develop a very versatile and effective model of the primary reformer. The program REFORM [361], [430], [439] can simulate all major types of reformers (see below) top-fired, side-fired, terraced-wall, concentric round configurations, the exchanger reformers (GHR, for example), and so on. The program is based on reaction kinetics, correlations with experimental heat transfer data, pressure drop functions, advanced furnace calculation methods, and a kinetic model of carbon formation [419],... 

Quantum mechanical calculations on small molecule association suggest that there are five major contributions to the energy of intermolecular interactions in the gas phase (3, 4). The sum of these is the dissociation energy of the intramolecular complex represented in Fig. 4.1. Table 4.1 contains some examples of magnitudes of the different energy components for different interactions. This section provides a qualitative introduction to these forces. Section gives and overview of mathematical models suitable for computer calculations. 

Mathematical models involve many different types of equations. These may be explicit, implicit, or differential equations. Dependent variables expressed as explicit equations are easily calculated. For example. 

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