Mathematical modeling and analysis

J. E. Bailey, Mathematical modeling and analysis in biochemical engineering Past accom plishments and future opportunities. Biotech. Prog. 14, 8 20 (1998). 

Zeng AP (1996b), Mathematical modeling and analysis of monoclonal antibody production by hybridoma cells, Biotechnol. Bioeng. 50 238-247. 

Bailey, J.E. Mathematical modeling and analysis in chemical engineering past accomplishments and future opportunities, Biotechnol. Prog., 14, 8,1998. 

In the absence of complicating factors such as capillary condensation and competitive adsorption, the process of physical adsorption has no activation energy that is, it is diffusion-controlled and occurs essentially as rapidly as vapor molecules can arrive at the surface. The process will be reversible and equilibrium will be attained rapidly. Because the forces involved are the same as those involved in condensation, physical adsorption will generally be a multilayer process—that is, the amount of vapor that can be adsorbed onto a surface will not be limited simply by the available solid surface area, but molecules can stack up to a thickness of several molecules in a pseudoliquid assembly (Fig. 9.3). If the vapor pressure of the gas reaches saturation level, in fact, the condensation and adsorption processes overlap and become indistinguishable. The fact that physical adsorption can be a multilayer process is very important to the mathematical modeling and analysis of the process, as will be seen below. 

Progress in modelling and analysis of the crack problem in solids as well as contact problems for elastic and elastoplastic plates and shells gives rise to new attempts in using modern approaches to boundary value problems. The novel viewpoint of traditional treatment to many such problems, like the crack theory, enlarges the range of questions which can be clarified by mathematical tools. 

The work of Crank [38] provides a review of the mathematical analysis of well defined component transport in homogeneous systems. These mathematical models and measured concentration profile data may be used to estimate diffu-sivities in homogenized samples. The use of MRI measurements in this way will generate diffusivities applicable to models of large-scale transport processes and will thereby be of value in engineering analysis of these processes and equipment. 

As described above, a number of empirical and analytical correlations for droplet sizes have been established for normal liquids. These correlations are applicable mainly to atomizer designs, and operation conditions under which they were derived, and hold for fairly narrow variations of geometry and process parameters. In contrast, correlations for droplet sizes of liquid metals/alloys available in published literature 318]f323ff328]- 3311 [485]-[487] are relatively limited, and most of these correlations fail to provide quantitative information on mechanisms of droplet formation. Many of the empirical correlations for metal droplet sizes have been derived from off-line measurements of solidified particles (powders), mainly sieve analysis. In addition, the validity of the published correlations needs to be examined for a wide range of process conditions in different applications. Reviews of mathematical models and correlations for... 

For example. Figure 8 shows both RSD and RCB data for determination of chloride and lead in water. In Figure 8a, the least-squares curve of best fit closely fits the lead standard data, and the calibration process has little adverse effect on precision. RSD s and RGB s are almost equal. On the other hand, chloride standard data in Figure 8b does not closely fit the mathematical model, and the RSD data overstates the precision of the analysis by a factor of about two. 

Mathematical model and uncertainty propagation analysis of a hypothetical measurement method in the context of the three-step model. 

The extraction of more complex particle size distributions from PCS data (which is not part of the commonly performed particle size characterization of solid lipid nanoparticles) remains a challenging task, even though several corresponding mathematical models and software for commercial instruments are available. This type of analysis requires the user to have a high degree of experience and the data to have high statistical accuracy. In many cases, data obtained in routine measurements, as are often performed for particle size characterization, are not an adequate basis for a reliable particle size distribution analysis. 

There are two general types of aerosol source apportionment methods dispersion models and receptor models. Receptor models are divided into microscopic methods and chemical methods. Chemical mass balance, principal component factor analysis, target transformation factor analysis, etc. are all based on the same mathematical model and simply represent different approaches to solution of the fundamental receptor model equation. All require conservation of mass, as well as source composition information for qualitative analysis and a mass balance for a quantitative analysis. Each interpretive approach to the receptor model yields unique information useful in establishing the credibility of a study s final results. Source apportionment sutdies using the receptor model should include interpretation of the chemical data set by both multivariate methods. 

The term factor is a catch-all for the concept of an identifiable property of a system whose quantity value might have some effect on the response. Factor tends to be used synonymously with the terms variable and parameter, although each of these terms has a special meaning in some branches of science. In factor analysis, a multivariate method that decomposes a data matrix to identify independent variables that can reconstitute the observed data, the term latent variable or latent factor is used to identify factors of the model that are composites of input variables. A latent factor may not exist outside the mathematical model, and it might not therefore influence... 

Zaldivar, J.M., Hernandez, H. and Barcons, C. (1996) Development of a mathematical model and a simulator for the analysis and optimisation of batch reactors Experimental model characterisation using a reaction calorimeter. Thermochimica Acta, 289, 267-302. 

Once the best estimates of the adjustable parameters have been computed, an analysis of the results allows one to evaluate the quality of the correspondence between experimental data and mathematical model and to identify the best model among the available alternatives. This analysis consists of different steps, mainly based on... 

Sobol IM (1993) Sensitivity analysis for nonlinear mathematical models. Mathematical Modeling and Computational Experiment, 1(4) 407-414. 

Andres, T. H. and Hajas, W.C. (1993). Using iterated fractional factorial design to screen parameters in sensitivity analysis of a probabilistic risk assessment model. Proceedings of the Joint International Conference on Mathematical Models and Supercomputing in... 

Tel. 617-873-2669, e-mail prophet-info bbn.com Molecular building, molecular mechanics, simulations, and display. Statistical and mathematical modeling and display. Sequence analysis. Structural and sequence database retrieval. UNIX workstations, such as Sun, VAX (Ultrix), DECstations, and Macintosh Ilfx (A/UX). 

The statistical techniques which have been discussed to this point were primarily concerned with the testing of hypotheses. A more important and useful area of statistical analysis in engineering design is the development of mathematical models to represent physical situations. This type of analysis, called regression analysis, is concerned with the development of a specific mathematical relationship including the mathematical model and its statistical significance and reliability. It can be shown to be closely related to the Analysis of Variance model. 

In the first step, construction of the mathematical model, operations analysis quite naturally tends to concern itself with decisions not already highly rationalized by other professions. Thus the most successful applications of operations analysis have been in such unusual domains as military tactics (M3) and toll bridge management (El) where quantita-... 

---