Theoretical Treatment of the Process

The assumptions are the same as those that have been made for the MDR in scanning mode, with the exception of the temperature which varies with the square root of time. 

Integrating between and T, leads the function Z as a function of temperature T. 

By plotting Ln versus Ln(l - 7) in Equation 3.26, the order n of the overall cure reaction can be obtained, being equal to the slope. 

In the same way, by plotting Ln (l-F) versus the reciprocal temperature in Equation 3.26, the slope gives the value of the energy of activation 

This important problem has been already studied a few years earlier [7], by considering the following operational conditions  

The first condition at the polymer-mould interface is expressed by  

The heat transfer by conduction in each medium (mould, PET) is expressed by  

The condition on the external surface of the mould when the polymer is in it, as well as on the external and the internal surfaces of the bottle when it has been removed from the mould is written by using Equation (3.5), where the temperature (Tju y jj g) is kept constant far away from the mould and from the bottle. 

The condition on the internal surface of the bottle, leads to the following equation expressing the fact that the volume of air (V) enclosed in the bottle is heated - the variation of the temperature of the air located inside the bottle with time is obtained from this relationship  

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Raffi et a/.102 132 have extended a theoretical treatment of the processes taking place on irradiation of starch. This treatment provides essential predictive power and permits manipulation of certain parameters of the radiation, depending on the properties of source and the properties desired for the products. This treatment was later proved experimentally.96 The fundamental relation between the dose of radiation and the reciprocal of degree of depolymerization (DP) is linear. Likewise, the reciprocal of intrinsic viscosity is linear in the radiation dose (that is, also in DP)... 

The theoretical treatment of the process of mass transfer controlled by diffusion-convection with liquid food or even by diffusion-diffnsion when the food is solid is described precisely. Emphasis is placed on the fact that an infinite rate of convection cannot exist at the package-liquid interface in our finite world, and conseqnently we have to deal with more complex equations. The question of the volume of the package as a fraction of the volume of the liquid is also considered. In fact, the problems of diffnsion are not simple to resolve, either by considering the experiments or by making calcnlations. 

In addition to the papers referred to above several others have recently dealt with the theoretical treatment of the processes during the migration of the zones in an adsorption column in order to estimate the possibilities of the separation of different substances. The concentration distribution for different quantities of the eluent when two components migrate through an adsorption column can be calculated if the adsorption isotherms of both components are known as well as the mutual influence... 

Theoretical treatment of the reaction as a conrotatory process proceeding through the very unstable Z,Z, -isomer of benzene satisfactorily accounts for the observed activation barrier. ... 

The complex distribution system that results from the frontal analysis of a multicomponent solvent mixture on a thin layer plate makes the theoretical treatment of the TLC process exceedingly difficult. Although specific expressions for the important parameters can be obtained for a simple, particular, application, a general set of expressions that can help with all types of TLC analyses has not yet been developed. One advantage of the frontal analysis of the solvent, however, is to produce a concentration effect that improves the overall sensitivity of the technique. 

There is no generally acceptable comprehensive theory of melting. A feature of the fusion process, which is usually regarded as important in theoretical treatments of the subject, is the inability of a solid to superheat, and only a very small number of exceptions to this generalization are known [2], This almost universal onset of liquefaction immediately upon reaching the melting point is in sharp contrast with the reverse process since supercooling of the vast majority of liquids can be demonstrated under appropriate conditions. 

The concentrations of reactants are of little significance in the theoretical treatment of the kinetics of solid phase reactions, since this parameter does not usually vary in a manner which is readily related to changes in the quantity of undecomposed reactant remaining. The inhomogeneity inherent in solid state rate processes makes it necessary to consider always both numbers and local spatial distributions of the participants in a chemical change, rather than the total numbers present in the volume of reactant studied. This is in sharp contrast with methods used to analyse rate data for homogeneous reactions in the liquid or gas phases. 

Finally, it must be recalled that the transport properties of any material are strongly dependent on the molecular or ionic interactions, and that the dynamics of each entity are narrowly correlated with the neighboring particles. This is the main reason why the theoretical treatment of these processes often shows similarities with models used for thermodynamic properties. The most classical example is the treatment of dilute electrolyte solutions by the Debye-Hiickel equation for thermodynamics and by the Debye-Onsager equation for conductivity. 

A complete theoretical treatment of the + He collision should therefore take into account, first the single-electron capture process from the ground state entry channel... 

The value of EM for a cooperative self-assembled structure provides a measure of the monomer concentration at which trivial polymeric structures start to compete, and therefore EM represents the upper limit of the concentration range within which the cooperative structure is stable (Scheme 2). The lower limit of this range is called the critical self-assembly concentration (csac) and is determined by the stoichiometry of the assembly and the strength of the non-covalent binding interactions weaker interactions and larger numbers of components raise the csac and narrow the stability window of the assembly (8). Theoretical treatments of the thermodynamics of the self-assembly process have been reported by Hunter (8), Sanders (9), and Mandolini (10). The value of EM is lowered by enthalpic contributions associated with... 

Many individuals have developed more elegant theoretical treatments of adsorption processes since Brunauer, Emmett, and Teller published their classic paper. Nonetheless, the BET and Langmuir isotherms are the most significant ones for chemical engineering applications. 

It has also to be remembered that the band model is a theory of the bulk properties of the metal (magnetism, electrical conductivity, specific heat, etc.), whereas chemisorption and catalysis depend upon the formation of bonds between surface metal atoms and the adsorbed species. Hence, modern theories of chemisorption have tended to concentrate on the formation of bonds with localized orbitals on surface metal atoms. Recently, the directional properties of the orbitals emerging at the surface, as discussed by Dowden (102) and Bond (103) on the basis of the Good-enough model, have been used to interpret the chemisorption behavior of different crystal faces (104, 105). A more elaborate theoretical treatment of the chemisorption process by Grimley (106) envisages the formation of a surface compound with localized metal orbitals, and in this case a weak interaction is allowed with the electrons in the metal. 

In temperature programmed reduction one follows the degree of reduction of the catalyst as a function of time, while the temperature increases at a linear rate. We follow the theoretical treatment of the TPR process as given by Hurst et al. [1] and by Wimmers et al. [8]. Formally, one can write the rate expression for the reduction reaction in (2-1), under conditions where the reverse reaction from metal to oxide can be ignored, as... 

Atomization, or generally speaking droplet generation, is an extremely complex process that cannot yet be precisely predicted theoretically. The lack of general theoretical treatment of droplet processes has led to the development of numerous empirical correlations for droplet properties as a function of process parameters and material properties. In this chapter, empirical and analytical correlations for the prediction of droplet properties, such as droplet size distribution and droplet deformation characteristics will be summarized from experimental observations and theoretical analyses in available literature. 

In addition to the Hopf cydization of 176, there is a second pericydic reaction leading to 162, that is, the dehydro Diels-Alder reaction of butenyne with acetylene (Scheme 6.47). The theoretical treatment of this process by Johnson et al. [59] predicted a free reaction enthalpy and a free activation enthalpy, both at 25 °C, of -13.4and 42.0kcalmol-1, respectively. Ananikov [116] arrived at a similar result for the intramolecular case of non-l-en-3,8-diyne (202) and calculated the same quantities to be -15.3 and 30.9 kcal mol-1 for the formation of the isoindane 203. As already discussed regarding Scheme 6.40, the conversion of 162 into benzene and likewise that of 203 into indane have to be considered as a sequence of two [1,2]-H shifts 116, 117], whose highest transition state has a significantly lower energy than that for the formation of 162 and 203 by the dehydro Diels-Alder reaction. 

Here we will consider photochemical processes initiated by multiphoton absorption. A theoretical treatment of multiphoton processes was developed as early as 1931 by Goeppert-Mayer i 9) but experimental verification was not possible until lasers were available to provide light of the necessary intensity. 

In some theoretical treatments of the growth-decay process of clusters, growth is considered to proceed by the gain or loss of single adions, Aj. Thus, allowed reactions for the model system are... 

The build-up of the laser pulse from the statistical fluctuation is beautifully demonstrated in 31), where a short period of statistical fluctuations was registered with a streak camera while they were traveling back and forth in the resonator. A detailed theoretical treatment of this process is given in 32> for the case of solid-state lasers and in 33> for the case of dye lasers in which the saturation of the active medium plays an important role. The halfwidth of the pulse which is finally reached in this mode-locking process is theoretically determined by the inverse spectral bandwidth of the active medium provided that the dye relaxation time rp is sufficiently short. 

The problem of turbulent flow in thin films has received comparatively little attention. Because of the great complexity of the flow processes involved, there are no theoretical treatments of the problem of wavy turbulent flow, and the usual procedure is to neglect the surface waves and obtain solutions for the case of smooth turbulent flow. 

Shinskey has discussed in detail both interaction and decoupling in general(33) and in respect to the distillation process<34). ROSENBROCK 35 has presented a more theoretical treatment of the problem. 

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