Mathematical Treatment

The mathematical treatment was first developed by Lord Rayleigh in 1879, and a more exact one by Bohr has been reviewed by Sutherland [103], who gives the formula... 

The surface tension of a pure liquid should and does come out to be the same irrespective of the method used, although difficulties in the mathematical treatment of complex phenomena can lead to apparent discrepancies. In the case of solutions, however, dynamic methods, including detachment ones, often tend... 

As an example for the mathematical treatment, we take the bimolecular reaction... 

Attempts have been made to devise mathematical functions to represent the distributions that are found experimentally. The mathematical treatment is necessarily based on the assumption that the number of particles in the sample is large enough for statistical considerations to be applicable. With the SOO-member sample of the previous section one could not expect any more than approximate agreement between mathematical prediction and experiment. 

Earlier we noted that a response surface can be described mathematically by an equation relating the response to its factors. If a series of experiments is carried out in which we measure the response for several combinations of factor levels, then linear regression can be used to fit an equation describing the response surface to the data. The calculations for a linear regression when the system is first-order in one factor (a straight line) were described in Chapter 5. A complete mathematical treatment of linear regression for systems that are second-order or that contain more than one factor is beyond the scope of this text. Nevertheless, the computations for... 

The iatroduction of a plasticizer, which is a molecule of lower molecular weight than the resia, has the abiUty to impart a greater free volume per volume of material because there is an iucrease iu the proportion of end groups and the plasticizer has a glass-transition temperature, T, lower than that of the resia itself A detailed mathematical treatment (2) of this phenomenon can be carried out to explain the success of some plasticizers and the failure of others. Clearly, the use of a given plasticizer iu a certain appHcation is a compromise between the above ideas and physical properties such as volatiUty, compatibihty, high and low temperature performance, viscosity, etc. This choice is appHcation dependent, ie, there is no ideal plasticizer for every appHcation. 

The concept of functionaUty and its relationship to polymer formation was first advanced by Carothers (15). Flory (16) gready expanded the theoretical consideration and mathematical treatment of polycondensation systems. Thus if a dibasic acid and a diol react to form a polyester, assumiag there is no possibihty of other side reactions to compHcate the issue, only linear polymer molecules are formed. When the reactants are present ia stoichiometric amouats, the average degree of polymerization, follows the equatioa ... 

Liquid crystals stabilize in several ways. The lamellar stmcture leads to a strong reduction of the van der Waals forces during the coalescence step. The mathematical treatment of this problem is fairly complex (28). A diagram of the van der Waals potential (Fig. 15) illustrates the phenomenon (29). Without the Hquid crystalline phase, coalescence takes place over a thin Hquid film in a distance range, where the slope of the van der Waals potential is steep, ie, there is a large van der Waals force. With the Hquid crystal present, coalescence takes place over a thick film and the slope of the van der Waals potential is small. In addition, the Hquid crystal is highly viscous, and two droplets separated by a viscous film of Hquid crystal with only a small compressive force exhibit stabiHty against coalescence. Finally, the network of Hquid crystalline leaflets (30) hinders the free mobiHty of the emulsion droplets. 

With these kinetic data and a knowledge of the reactor configuration, the development of a computer simulation model of the esterification reaction is iavaluable for optimising esterification reaction operation (25—28). However, all esterification reactions do not necessarily permit straightforward mathematical treatment. In a study of the esterification of 2,3-butanediol and acetic acid usiag sulfuric acid catalyst, it was found that the reaction occurs through two pairs of consecutive reversible reactions of approximately equal speeds. These reactions do not conform to any simple first-, second-, or third-order equation, even ia the early stages (29). 

The mathematical treatment of engineering problems involves four basic steps ... 

It is emphasized that the delta L law does not apply when similar crystals are given preferential treatment based on size. It fails also when surface defects or dislocations significantly alter the growth rate of a crystal face. Nevertheless, it is a reasonably accurate generahzation for a surprising number of industrial cases. When it is, it is important because it simphfies the mathematical treatment in modeling real crystallizers and is useful in predicting crystal-size distribution in many types of industrial crystallization equipment. 

C, is the concentration of impurity or minor component in the solid phase, and Cf is the impurity concentration in the hquid phase. The distribution coefficient generally varies with composition. The value of k is greater than I when the solute raises the melting point and less than I when the melting point is depressed. In the regions near pure A or B the hquidus and solidus hues become linear i.e., the distribution coefficient becomes constant. This is the basis for the common assumption of constant k in many mathematical treatments of fractional solidification in which ultrapure materials are obtained. 

Not many operating data of large-scale hquid/hquid reactions are published. One study was made of the hydrolysis of fats with water at 230 to 260°C (446 to 500°F) and 41 to 48 atm (600 to 705 psi) in a continuous commercial spray tower. A small amount of water dissolved in the fat and reacted to form an acid and glycerine. Then most of the glycerine migrated to the water phase. Tlie tower was operated at about 18 percent of flooding, at which condition the HETS was found to be about 9 m (30 ft) compared with an expec ted 6 m (20 ft) for purely physical extrac tion (Jeffreys, Jenson, and Miles, Trans. In.st. Chem. Eng., 39, 389-396 [1961]). A similar mathematical treatment of a batch hydrolysis is made by Jenson and Jeffreys (In.st. Chem. Engrs. Symp. Ser, No. 23 [1967]). 

For the effective diffusivity in pores, De = (0/t)D, the void fraction 0 can be measured by a static method to be between 0.2 and 0.7 (Satterfield 1970). The tortuosity factor is more difficult to measure and its value is usually between 3 and 8. Although a preliminary estimate for pore diffusion limitations is always worthwhile, the final check must be made experimentally. Major results of the mathematical treatment involved in pore diffusion limitations with reaction is briefly reviewed next. 

Other cases, neglecting heat effects would cause serious errors. In such cases the mathematical treatment requires the simultaneous solution of the diffusion and heat conductivity equations for the catalyst pores. 

In this book, I have tried to assimilate the subject matter of various papers (and sometimes diverse views) into a comprehensive, unified treatment of gas turbines. Many illustrations, curves, and tables are employed to broaden the understanding of the descriptive text. Mathematical treatments are deliberately held to a minimum so that the reader can identify and resolve any problems before he is ready to execute a specific design. In addition, the references direct the reader to sources of information that will help him to investigate and solve his specific problems. It is hoped that this book will... 

There is no general explicit mathematical treatment of complicated rate equations. In Section 3.1 we describe kinetic schemes that lead to closed-form integrated rate equations of practical utility. Section 3.2 treats many further approaches, both experimental and mathematical, to these complicated systems. The chapter concludes with comments on the development of a kinetic scheme for a complex reaction. 

The next approximation involves expressing the jiiolecular orhiiah as linear combinations of a pre-defined set of one-electron functions kjiown as basis functions. These basis functions are usually centered on the atomic nuclei and so bear some resemblance to atomic orbitals. However, the actual mathematical treatment is more general than this, and any set of appropriately defined functions may be u.sed. 

It is usually necessary to graphically integrate the first terms of the above equations, although some problems do allow for mathematical treatment. 

The solution properties of polymers have been subjected to intensive study, in particular to highly complex mathematical treatment This section will, however, confine discussion to a qualitative and practical level . ... 

The explicit mathematical treatment for such stationary-state situations at certain ion-selective membranes was performed by Iljuschenko and Mirkin 106). As the publication is in Russian and in a not widely distributed journal, their work will be cited in the appendix. The authors obtain an equation (s. (34) on page 28) similar to the one developed by Eisenman et al. 6) for glass membranes using the three-segment potential approach. However, the mobilities used in the stationary-state treatment are those which describe the ion migration in an electric field through a diffusion layer at the phase boundary. A diffusion process through the entire membrane with constant ion mobilities does not have to be assumed. The non-Nernstian behavior of extremely thin layers (i.e., ISFET) can therefore also be described, as well as the role of an electron transfer at solid-state membranes. 

A mathematical treatment of the kinetic model shown in Scheme 2 gives a decay function as... 

The preceding oversimplified mathematical treatment really amounts to an evaluation of the absorption effect (6.1). The exponential term in Equation 6-4 is obviously a product of two exponential terms, each deriving from Beerks Law. One term governs the attenuation of the beam incident upon the volume element in question, and the other governs the attenuation of the characteristic line emerging frcJm this element. The films are so thin that the use of one value each for 6 and for 02 over the entire film thickness is justified. Finally, one must assume that the intensity measured by the detector remains proportional to the intensity of the source. An exact treatment of the problem would be so complicated that one is justified in seeing what can be done with the simple relationships obtained above. 

One aspect of the mathematical treatment of the quantum mechanical theory is of particular interest. The wavefunction of the perturbed molecule (i.e. the molecule after the radiation is switched on ) involves a summation over all the stationary states of the unperturbed molecule (i.e. the molecule before the radiation is switched on ). The expression for intensity of the line arising from the transition k —> n involves a product of transition moments, MkrMrn, where r is any one of the stationary states and is often referred to as the third common level in the scattering act. 

For further mathematical treatment of the topics of this chapter, the reader is referred to the author s text—N. Minorsky, Nonlinear OedUaHons, D. Van Nostrand Co., Inc., Princeton, N.J., 1962. 

R.C. Oliver et al, USDeptCom, Office Tech-Serv ..AD 265822,(1961) CA 60, 10466 (1969) Metal additives for solid proplnts formulas for calculating specific impulse and other proplnt performance parameters are given. A mathematical treatment of the free-energy minimization procedure for equilibrium compn calcns is provided. The treatment is extended to include ionized species and mixing of condensed phases. Sources and techniques for thermodynamic-property calcns are also discussed... 

Single-step nucleation, (ii) above, requires the unsatisfactory assumption that the generation of a single molecule (atom, ion-pair, etc.) of product constitutes the establishment of a nucleus. (It would seem to be more realistic to regard this as the outcome of several distinct chemical steps.) The mathematical treatment expressing the probability of the occurrence of this unimolecular process is... 

From this mathematical treatment we properly conclude that ki + k- is the natural rate constant. The data workup does not give the sum because of an artful algebraic manipulation. Instead, the sum k-i + k is simply the inverse of the intrinsic time constant for the single exponential that defines the approach to equilibrium. 

Many real reaction systems are not amenable to normal mathematical treatments that give algebraic expressions for concentration versus time, but by no means is the situation hopeless. Such systems need not be avoided. The numerical methods presented... 

A mechanical system, typified by a pendulum, can oscillate around a position of final equilibrium. Chemical systems cannot do so, because of the fundamental law of thermodynamics that at all times AG > 0 when the system is not at equilibrium. There is nonetheless the occasional chemical system in which intermediates oscillate in concentration during the course of the reaction. Products, too, are formed at oscillating rates. This striking phenomenon of oscillatory behavior can be shown to occur when there are dual sets of solutions to the steady-state equations. The full mathematical treatment of this phenomenon and of instability will not be given, but a simplified version will be presented. With two sets of steady-state concentrations for the intermediates, no sooner is one set established than the consequent other changes cause the system to pass quickly to the other set, and vice versa. In effect, this establishes a chemical feedback loop. 

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