Theoretical models development mathematical model

Theoretical models and mathematical derivations are developed in enough detail to be comprehensible to the student reader. Only rarely do 1 pull results out of a hat, and 1 scrupulously avoid saying it is obvious that... ... 

A number of theories of the contribution of interdomain polymeric material to the stress-strain, modulus, and swelling behavior of block copolymers and semicrystalline polymers are examined. The conceptual foundation and the mathematical details of each theory are summarized. A critique is then made of each theory in terms of the validity of the theoretical model, the mathematical development of the theory, and the ability of the theory to explain experimental findings. 

Fig. 6. (a) Interstitial pressure gradients in the mammary adenocarcinoma R3230AC as a function of radial position. The circles ( ) represent data points (Boucher et al., 1990), and the solid line represents the theoretical profile based on our previously developed mathematical model (Jain and Baxter, 1988 Baxter and Jain, 1989). Note that the pressure is nearly uniform in most of the tumor, but drops precipitously to normal tissue values in the periphery. Elevated pressure in the central region retards the extravasation of fluid and macromolecules. In addition, the pressure drop from the center to the periphery leads to an experimentally verifiable, radially outward fluid flow. (Reproduced from Boucher et al., 1990, with permission.) (b) Microvascular pressure (MVP) in the peripheral vessels of the mammary adenocarcinoma R3230AC is comparable to the central interstitial fluid pressure (IFP) (adapted from Boucher and Jain, 1992). These results suggest that osmotic pressure difference across vessel walls is small in this tumor. 

Another approach to optimizing a method is to develop a mathematical model of the response surface. Such models can be theoretical, in that they are derived from a known chemical and... 

Theoretically based correlations (or semitheoretical extensions of them), rooted in thermodynamics or other fundamentals are ordinarily preferred. However, rigorous theoretical understanding of real systems is far from complete, and purely empirical correlations typically have strict limits on apphcabihty. Many correlations result from curve-fitting the desired parameter to an appropriate independent variable. Some fitting exercises are rooted in theory, eg, Antoine s equation for vapor pressure others can be described as being semitheoretical. These distinctions usually do not refer to adherence to the observations of natural systems, but rather to the agreement in form to mathematical models of idealized systems. The advent of readily available computers has revolutionized the development and use of correlation techniques (see Chemometrics Computer technology Dimensional analysis). 

Theoretical representation of the behaviour of a hydrocyclone requires adequate analysis of three distinct physical phenomenon taking place in these devices, viz. the understanding of fluid flow, its interactions with the dispersed solid phase and the quantification of shear induced attrition of crystals. Simplified analytical solutions to conservation of mass and momentum equations derived from the Navier-Stokes equation can be used to quantify fluid flow in the hydrocyclone. For dilute slurries, once bulk flow has been quantified in terms of spatial components of velocity, crystal motion can then be traced by balancing forces on the crystals themselves to map out their trajectories. The trajectories for different sizes can then be used to develop a separation efficiency curve, which quantifies performance of the vessel (Bloor and Ingham, 1987). In principle, population balances can be included for crystal attrition in the above description for developing a thorough mathematical model. 

Mao and White developed a mathematical model for discharge of an Li / TiS2 cell [39]. Their model predicts that increasing the thickness of the separator from 25 to 100 pm decreases discharge capacity from 95 percent to about 90 percent further increasing separator thickness to 200 pm reduced discharge capacity to 75 percent. These theoretical results indicate that conventional separators (25-37 pm thick) do not significantly limit mass transfer of lithium. 

To illustrate the actual importance of dynamic properties for the functioning of metabolic networks, we briefly describe and summarize a recent computational study on a model of human erythrocytes [296]. Erythrocytes play a fundamental role in the oxygen supply of cells and have been subject to extensive experimental and theoretical research for decades. In particular, a variety of explicit mathematical models have been developed since the late 1970s [108, 111, 114, 123, 338 341], allowing us to test the reliability of the results in a straightforward way. 

Throughout the chapter, the importance of network formation theories in understanding and predicting structural development is stressed. Therefore, a short expose on network formation theories is given in this chapter. Although the use of theoretical modeling of network build-up and comparison with experiments play a central role in this chapter, most mathematical relations and their derivation are avoided and only basic postulates of the theories are stated. The reader can always find references to literature sources where such mathematical relations are derived. 

The development of an adequate mathematical model representing a physical or chemical system is the object of a considerable effort in research and development activities. A technique has been formalized by Box and Hunter (B14) whereby the functional form of reaction-rate models may be exploited to lead the experimenter to an adequate representation of a given set of kinetic data. The procedure utilizes an analysis of the residuals of a diagnostic parameter to lead to an adequate model with a minimum number of parameters. The procedure is used in the building of a model representing the data rather than the postulation of a large number of possible models and the subsequent selection of one of these, as has been considered earlier. That is, the residual analysis of intrinsic parameters, such as Cx and C2, will not only indicate the inadequacy of a proposed model (if it exists) but also will indicate how the model might be modified to yield a more satisfactory theoretical model. 

The flow patterns in the hydrocyclone are complex, and much development work has been necessary to determine the most effective geometry, as theoretical considerations alone will not allow the accurate prediction of the size cut which will be obtained. A mathematical model has been proposed by Rhodes et alP6), and predictions of streamlines from their work are shown in Figure 1.38. Salcudean and Gartshore137 have also carried out numerical simulations of the three-dimensional flow in a hydrocyclone and have used the results to predict cut sizes. Good agreement has been obtained with experimental measurements. 

The dynamic relationships discussed thus far in this book were determined from mathematical models of the process. Mathematical equations, based on fundamental physical and chemical laws, were developed to describe the time-dependent behavior of the system. We assumed that the values of all parameters, such as holdups, reaction rates, heat transfer coeflicients, etc., were known. Thus the dynamic behavior was predicted on essentially a theoretical basis. 

A third and very important use of dynamic experiments is to confirm the predictions of a theoretical mathematical model. As we indicated in Part I, the verification of the model is a very desirable step in its development and application. 

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