Conductivity Maxwell model

Returning to the Maxwell element, suppose we rapidly deform the system to some state of strain and secure it in such a way that it retains the initial deformation. Because the material possesses the capability to flow, some internal relaxation will occur such that less force will be required with the passage of time to sustain the deformation. Our goal with the Maxwell model is to calculate how the stress varies with time, or, expressing the stress relative to the constant strain, to describe the time-dependent modulus. Such an experiment can readily be performed on a polymer sample, the results yielding a time-dependent stress relaxation modulus. In principle, the experiment could be conducted in either a tensile or shear mode measuring E(t) or G(t), respectively. We shall discuss the Maxwell model in terms of shear. 

One of the first models proposed in order to calculate the generalized conductivity of a composite was the Maxwell model, namely a spherical insertion (component 1) in a continuous matrix (component 2). Thus the following equation was obtained [82] ... 

Many theoretical and empirical models have been proposed to predict the effective thermal conductivity of two phase mixtures. Comprehensive review articles have discussed the applicability of many of these models that appear to be more promising [34-36]. First, using potential theory. Maxwell [20] obtained a simple relationship for the conductivity of randomly distributed and non-interacting homogeneous spheres in a homogeneous medium. Maxwell model is good for low solid concentrations. Relative thermal conductivity enhancement (ratio of the effective thermal conductivity keffO nanofluid to base fluid kj) is. 

W. Yu and S.U.S. Choi, The role of interfacial layers in the enhanced thermal conductivity of nanofluids A renovated Maxwell model. Journal of Nanoparticle Research, 5, 167-171 (2003). 

Since then, a number of structural models have been proposed, some of which are given in Table 4.7. The perpendicular model assumes that heat conduction is perpendicular to alternate layers of the two phases, whereas the parallel model assumes that the two phases are parallel to heat conduction. In the mixed model, heat conduction is assumed to take place by a combination of parallel and perpendicular heat flow. In the random model, the two phases are assumed to be mixed randomly. The Maxwell model assumes that one phase is continuous, whereas the other phase is dispersed as uniform spheres. Several other models have been reviewed in Refs. [107,110,111], among others. 

The Maxwell model and Hamilton-Crosser model were established based on the assumptions that all the nanomaterials are completely dispersed and evenly distributed in the fluid phase. However anomalous increase of thermal conductivity with a small amount of CNT loading such as 0.001 vol.% addition (first data points for each line in Fig. 13) cannot be explained by the theoretical values. From our experimental analysis, thermal conductivities of CNT nanofluid showed a nonlinear increase with plasma-treated CNT... 

Figure 29 The conductivity dependence of the dielectric loss tangent and the dielectric constant of an ER fluid under the assumptions of e =2, s - 10, 0= 0.35, and/=1000 Hz predicted with Wagncr-Maxwell model. Reproduced with permission from T. llao, J. Colloid Interface Sci., 206(1998)240.

Maxwell model underestimates the thermal conductivity of PLA-eGR composites however Halpin-Tsai, Lewis and Nielsen and Hamilton models ensure a good fitting of experimental values. 

In order to incorporate the shape of the p>articles (e.g. cylinders) and the interaction between the particles, extensions of this Maxwell model were later developed by (Hamilton and Grosser, 1962) and (Hui et al., 1999). However, these classical models were found to be unable to accurately predict the anomalously high thermal conductivity of nanofluids (Murshed et al., 2008a). Thus, researchers have proposed several mechanisms to explain this phenomenon. For example, (Kebflnski et al., 2002) systematized the four different mechanisms for heat transfer to explain these enhancements, namely (i) Brownian motion of the nanoparticles (ii) liquid layering at the liquid/ particle interphase, (iii) the nature of the heat transport in the nanoparticles and (iv) the effect of nanoparticle clustering. From the analysis made in an exhaustive review paper on nanofluids (Murshed et al., 2008a) and other publications cited, therein, it is our belief that the effect of the particle surface chemistry and the structure of the interphase partide/fluid are the major mechanisms responsible for the unexpected enhancement in nanofluids. 

Figure 5.17 Heat conductivities of solids modeled by the series, the parallel, and the Maxwell formulation of spheres embedded in a matrix...

The energy-dependent speed of light is associated with the effect of the medium on the propagation of photon. The fluctuating refractive index of the medium induces this kind of the energy dependence. This kind of the medium has been considered, such as quantum gravity and the Maxwell vacuum with nonzero conductivity. So, it to make distinction between the contributions from standard model predictions as well as from other theories is needed. 

Figure 2.7 Thermal conductivity versus volume concentration of metallic particles of an epoxy resin. Solid lines represent predictions using Maxwell and Knappe models.

Criticism of the Stosszahlansatz and its corollaries arose as soon as it was recognized as paradoxical that the completely reversible gas model of the kinetic theory was apparently able to explain irreversible processes, i.e., phenomena whose development shows a definite direction in time. These nonstationary,51 irreversible processes were brought into the center of interest by the //-theorem of Boltzmann. In order to show that every non-Max-wellian distribution always approaches the Maxwell distribution in time, this theorem synthesizes all the special irreversible processes (like heat conduction and... 

Conducting particles held in a nonconducting medium form a system which has a frequency-dependent dielectric constant. The dielectric loss in such a system depends upon the build-up of charges at the interfaces, and has been modeled for a simple system by Wagner [8], As the concentration of the conducting phase is increased, a point is reached where individual conducting areas contribute and this has been developed by Maxwell and Wagner in a two-layer capacitor model. Some success is claimed for the relation... 

Two other approaches have been taken to modelling the conductivity of composites, effective medium theories (Landauer, 1978) and computer simulation. In the effective medium approach the properties of the composite are determined by a combination of the properties of the two components. Treating a composite containing spherical inclusions as a series combination of slabs of the component materials leads to the Maxwell-Wagner relations, see Section 3.6.1. Treating the composite as a mixture of spherical particles with a broad size distribution in order to minimise voids leads to the equation ... 

Another model of experimental interest concerns the case of a highly conductive shell around practically non-conductive material. It may be applied to macromolecules or colloidal particles in electrolyte solution which usually have counterion atmospheres so that the field may displace freely movable ionic charges on their surfaces. The resulting dielectric effect turns out to be equivalent to a simple Maxwell-Wagner dispersion of particles having an apparent bulk conductivity of... 

Metals form a class of solids with characteristic macroscopic properties. They are ductile, have a silver-white luster, and they conduct electricity and heat remarkably well. An early, but still relevant microscopic model aimed at explaining the electrical conductivity, heat conductivity, and optical properties was proposed by Drude [10]. His model incorporates two important successes of modem science the discovery of the electron in 1887 by J. J. Thomson, and the molecular kinetic gas theory put forward by Boltzmann and Maxwell in the second half of the 19th century. 

Time domain electromagnetic (EM) migration is based on downward extrapolation of the residual field in reverse time. In this section I will show that electromagnetic migration, as the solution of the boundary value problem for the adjoint Maxwell s equation, can be clearly associated with solution of the inverse problem in the time domain. In particular, I will demonstrate that the gradient of the residual field energy flow functional with respect to the perturbation of the model conductivity is equal to the vector cross-correlation function between the predicted field for the given... 

In most geophysical applications of electromagnetic methods, it is necessary to model geoelectrical structures of quite arbitrary shape and size, with anomalous conductivity varying in an arbitrary manner and not necessarily restricted to a local region. The most widely used approach to forward modeling of such problems is through the use of finite difference and finite element methods to find numerical solutions to Maxwell s equations written in differential form (Weaver, 1994 Zhdanov et ah, 1997). 

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