Generalized Maxwell model

In order to design a zeoHte membrane-based process a good model description of the multicomponent mass transport properties is required. Moreover, this will reduce the amount of practical work required in the development of zeolite membranes and MRs. Concerning intracrystaUine mass transport, a decent continuum approach is available within a Maxwell-Stefan framework for mass transport [98-100]. The well-defined geometry of zeoHtes, however, gives rise to microscopic effects, like specific adsorption sites and nonisotropic diffusion, which become manifested at the macroscale. It remains challenging to incorporate these microscopic effects into a generalized model and to obtain an accurate multicomponent prediction of a real membrane. 

In order to obtain a general model of the creep and recovery functions we need to use a Kelvin model or a Kelvin kernel and retardation spectrum L. However, there are some additional subtleties that need to be accounted for. One of the features of a Maxwell model is that it possesses a high frequency limit to the shear modulus. This means there is an instantaneous response at all strains. The response of a simple Kelvin model is shown in Equation 4.80 ... 

Although the Maxwell-Wiechert model and the extended Burgers element exhibit the chief characteristics of the viscoelastic behaviour of polymers and lead to a spectrum of relaxation and retardation times, they are nevertheless of restricted value it is valid for very small deformations only. In a qualitative way the models are useful. The flow of a polymer is in general non-Newtonian and its elastic response non-Hookean. 

The reality, however, is not as simple as that. There are several possibilities to describe viscosity, 77, and first normal stress difference coefficient, P1. The first one originates from Lodge s rheological constitutive equation (Lodge 1964) for polymer melts and the second one from substitution of a sum of N Maxwell elements, the so-called Maxwell-Wiechert model (see Chap. 13), in this equation (see General references Te Nijenhuis, 2005). 

Fast and satisfactory mass transfer calculations are necessary since we may have to repeat such calculations many times for a rate-based distillation column model or two-phase flow with mass transfer between the phases in the design and simulation process. The generalized matrix method may be used for multicomponent mass transfer calculations. The generalized matrix method utilizes the Maxwell-Stefan model with the linearized film model for diffusion flux, assuming a constant diffusion coefficient matrix and total concentration in the diffusion region. In an isotropic medium, Fick s law may describe the multicomponent molecular mass transfer at a specified temperature and pressure, assuming independent diffusion of the species in a fluid mixture. Such independent diffusion, however, is only an approximation in the following cases (i) diffusion of a dilute component in a solvent, (ii) diffusion of various components with identical diffusion properties, and (iii) diffusion in a binary mixture. 

This section concerns the Cauchy problem or initial value problem, where initial data at time t = 0 are given. It was noticed by Rutkevitch [6,7], and systematized by Joseph et al. [8], Joseph and Saut [9], and Dupret and Marchal [10] that Maxwell type models can present Hadamard instabilities, that is, instabilities to short waves. (See [11] for a recent discussion of more general models.) Then, the Cauchy problem is not well-posed in any good class but analytic. Highly oscillatory initial data will grow exponentially in space at any prescribed time. An ill-posed problem leads to catastrophic instabilities in numerical simulations. For example, even if one initiates the solution in a stable region, one could get arbitrarily close to an unstable one. 

We now turn to local existence of solutions for Maxwell-type models. The situation is much trickier here since these models can display Hadamard instabilities (see Section 2.1), and no general results seem to be known so far. One has, in any case, to restrict initial data to Hadamard stable ones. A possible way to overcome the difficulty is to consider models satisfying an eUipticity condition, which will imply well-posedness. This approach was followed by Renardy [41], whose results are briefly described below. 

In [62] Renardy proves the linear stability of Couette flow of an upper-convected Maxwell fluid under the 2issumption of creeping flow. This extends a result of Gorodtsov and Leonov [63], who showed that the eigenvalues have negative real parts (I. e., condition (S3) holds). That result, however, does not allow any claim of stability for non-zero Reynolds number, however small. Also it uses in a crucial way the specific form of the upper-convected derivative in the upper-convected Maxwell model, aind does not generalize so far to other Maxwell-type models. 

This chapter is devoted to describe the impact of metallic nanosphere to the multi-photon excitation fluorescence of Tryptophan, and little further consideration to multi-photon absorption process will be given, as the reader can find several studies in [11-14]. In section II, the nonlinear light-matter interaction in composite materials is discussed through the mechanism of nonlinear susceptibilities. In section III, experimental results of fluorescence induced by multi-photon absorption in Tryptophan are reported and analyzed. Section IV described the main results of this chapter, which is the effect of metallic nanoparticles on the fluorescent emission of the Tryptophan excited by a multi-photon process. Influence of nanoparticle concentration on the Tryptophan-silver colloids is observed and discussed based coi a nonlinear generalization of the Maxwell Garnett model, introduced in section II. The main conclusion of the chapter is given in secticHi IV. 

Most of the general models for describing flux through porous media is given by the DGM. The term DGM, which was first described by Maxwell in 1860 [39], is relatively unfamiliar in the membrane separation field and has only recently been appearing in the membrane literature [33,39,41,60,61]. In this model, the porous medium is visualized as a collection of uniformly distributed dust particles, which are construed to be stationary. 

Multicomponent diffusion in pores is described by the dusty-gas model (DGM) [38,44,46 8]. This model combines molecular diffusion, Knudsen diffusion, viscous flux, and surface diffusion. The DGM is suitable for any model of porous structure. It was developed by Mason et al. [42] and is based on the Maxwell-Stefan approach for dilute gases, itself an approximation of Boltzmann s equation. The diffusion model obtained is called the generalized Maxwell-Stefan model (GMS). Thermal diffusion, pressmn diffusion, and forced diffusion are all easily included in the GMS model. This model is based on the principle that in order to cause relative motion between individual species in a mixture, a driving force has to be exerted on each of the individual species. The driving force exerted on any particular species i is balanced by the friction this species experiences with all other species present in the mixture. Each of these friction contributions is considered to be proportional to the corresponding differences in the diffusion velocities. 

Two other dependences were suggested by Kaczmarski et al. [123]. First, they used a model borrowed from gas-solid adsorption, the Maxwell-Stefan model, that is valid if the mobile phase is much less strongly adsorbed than the compoimd studied, which is generally true in RPLC. This model gives... 

Equation (21.10) is the general equation for the Maxwell mechanical model analogy for viscoelastic behavior. 

The generalized model consists of an arbitrary number of Maxwell elements in a parallel arrangement (Figure 14.11). 

As can be seen, the Maxwell-Weichert model possesses many relaxation times. For real materials we postulate the existence of a continuous spectrum of relaxation times (A,). A spectrum-skewed toward lower times would be characteristic of a viscoelastic fluid, whereas a spectrum skewed toward longer times would be characteristic of a viscoelastic solid. For a real system containing crosslinks the spectrum would be skewed heavily toward very long or infinite relaxation times. In generalizing, A may thus he allowed to range from zero to infinity. The concept that a continuous distribution of relaxation times should be required to represent the behavior of real systems would seem to follow naturally from the fact that real polymeric systems also exhibit distrihutions in conformational size, molecular weight, and distance between crosslinks. 

Maxwell-Stefan model. The Maxwell-Stefan model is generally agreed to be a better model than the Fickian model for nonideal binary and all ternary systems. However, it is not as widely understood by chemical engineers, data collected in terms of Fickian diffusivities need to be converted to Maxwell-Stefan values, and the model can be more difficult to use. Use this model, coupled with a mass-transfer model, when the Fickian model fails or requires an excessive amount of data. 

Fig. 19 Mechanical-viscoelastic model of Lin and Chen (1999) with two Maxwell models to describe SME in segmented PUs. (a) General model, (b) Change of the model in the shape-memory cycle, (c) Shape-memory behavior for two PU samples. Solid lines indicate the recoverable ration curves of the model. Taken from ref. [36], Copyright 1999. Reprinted with permission of John WUey Sons, Inc.

More complex arrangements of elements are often used, especially if multiple relaxations are involved or if accurate representations of engineering data are required. The Maxwell-Weichert model consists of a very large (or infinite) number of Maxwell elements in parallel (2). The generalized Voigt-Kelvin model places a number of Kelvin elements in series. In each of these models, a spring or a dashpot may be placed alone, indicating elastic or viscous contributions. 

Generally, the literature shows that, for magnetic composite systems, relative permittivity decreases as frequency increases, this behavior being attributed to the interfacial polarization predicted by Maxwell-Wagner model, derived for the permittivity of a homogeneous two-phase system. In addition, a decrease in electrical resistivity for these magnetic composites was recorded with an increase in frequency... 

Viscoelastic creep When a plastic material is subjected to a constant stress, it undergoes a time-dependent increase in strain. This behavior is called creep. It is a plastic for which at long times of applied stress, such as in creep, a steady flow is eventually achieved. Thus in a generalized Maxwell model, all the dashpot viscosities must have finite values and in generalized models must have zero stiffness. 

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