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Academic literature on the topic 'Linear law of mixtures'
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Journal articles on the topic "Linear law of mixtures"
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Medellin, David, Vivek R. Ravi, and Carlos Torres-Verdín. "Nonlinear mixing law for magnetic resonance transverse-relaxation measurements of dispersed mixtures." GEOPHYSICS 84, no. 1 (2019): MR1—MR11. http://dx.doi.org/10.1190/geo2017-0342.1.
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Nuclear magnetic resonance (NMR) has been used to estimate the permeability of water-saturated rocks with uniform grain-surface composition. Geologic formations, however, usually consist of layers of rocks with different grain-surface compositions. In the absence of water diffusional coupling between adjacent layers, the NMR [Formula: see text] distribution measured by a borehole logging tool is a weighted average (i.e., it obeys a linear mixing law) of the [Formula: see text] distributions of each layer. However, when the layers are very thin (compared with the self-diffusion length of water) or the different grain types form a dispersed mixture (i.e., they are finely mixed), the resultant [Formula: see text] distribution does not obey a linear mixing law anymore. The question arises as to whether it is possible to differentiate between dispersed and layered formations using mixing laws. We have developed a nonlinear mixing law to calculate the NMR [Formula: see text] distribution of a water-saturated sample composed of a fine mixture of two different grain types from the [Formula: see text] distributions of the water-saturated samples for each grain type. Water-saturated mixtures of crushed Texas Cream limestone and white silica sand were prepared in different volume ratios; we used the new nonlinear mixing law to compute the [Formula: see text] distribution of the mixtures using the [Formula: see text] distributions of the pure water-saturated samples and their volume fractions. Conversely, we also applied the nonlinear mixing law to the mixture to calculate the volume fractions of pure components using the [Formula: see text] distributions of the pure components and the mixtures. Our results indicate that enough differences exist between the nonlinear and linear NMR mixing laws predictions that they can be used to establish whether in a given rock formation with two different grain types the pure components form layers or they are present in dispersed form.
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Mayerhöfer, Thomas G., and Jürgen Popp. "Beyond Beer's Law: Spectral Mixing Rules." Applied Spectroscopy 74, no. 10 (2020): 1287–94. http://dx.doi.org/10.1177/0003702820942273.
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Based on Beer's law, it is assumed that the absorbance of a mixture is that of the neat materials weighted by their relative amounts (linear mixing rule). In this contribution, we show that this is an assumption that holds only under various approximations for which no change of the chemical interactions is just one among several. To understand these approximations, which lead incrementally to different well known mixing rules, we finally derive the linear mixing rule from the Lorentz–Lorenz relation, with the first approximation that the local electric field is correctly described in this relation. Further levels of approximation are that the local field equals the applied field (Newton–Laplace mixing rule) and that the change of the index of refraction and, equivalently, absorption is weak (Gladstone–Dale/Arago–Biot mixing rule). Even then the linear mixing rule is only strictly valid if the indices of refraction in the transparency region at higher frequency than the absorption have the same value and the mixing is homogeneous relative to the resolving power of the light (“micro-homogeneous”). Under these preconditions, linear mixing of the individual absorbances is established. We illustrate the spectral differences between the different mixing rules, all of which are based on volume and not on mass fractions, with examples. For micro-heterogeneous samples, a different linear mixing rule governs the optical properties, which refers to the experimental quantities, reflectance, and transmittance. As a result, for such samples, mixtures of already comparably high content give only weak signals due to band flattening, which are hard to distinguish from baseline effects.
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DI MINO, Gaetano, Gordon AIREY, Mario DI PAOLA, Francesco Paolo PINNOLA, Giacomo D’ANGELO, and Davide LO PRESTI. "LINEAR AND NONLINEAR FRACTIONAL HEREDITARY CONSTITUTIVE LAWS OF ASPHALT MIXTURES." JOURNAL OF CIVIL ENGINEERING AND MANAGEMENT 22, no. 7 (2016): 882–89. http://dx.doi.org/10.3846/13923730.2014.914104.
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The aim of this paper is to propose a fractional viscoelastic and viscoplastic model of asphalt mixtures using experimental data of several tests such as creep and creep recovery performed at different temperatures and at different stress levels. From a best fitting procedure it is shown that both the creep one and recovery curve follow a power law model. It is shown that the suitable model for asphalt mixtures is a dashpot and a fractional element arranged in series. The proposed model is also available outside of the linear domain but in this case the parameters of the model depend on the stress level.
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Slater, Lee, Dimitrios Ntarlagiannis, and DeBonne Wishart. "On the relationship between induced polarization and surface area in metal-sand and clay-sand mixtures." GEOPHYSICS 71, no. 2 (2006): A1—A5. http://dx.doi.org/10.1190/1.2187707.
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Induced polarization (IP) measurements were conducted on saturated kaolinite-, iron-, and magnetite-sand mixtures as a function of varying percentage weight of a mineral constituent: 0%–100% for iron and magnetite and 0%–32% for kaolinite. We determined the specific surface area for each mineral using nitrogen gas adsorption, where the porosity of each mixture was calculated from weight loss after drying. We fit a Cole-Cole model (Cole and Cole, 1941) to the electrical data obtained for the magnetite and iron mixtures. In contrast, the kaolinite mixtures showed a power-law dependence of phase-on frequency. The global polarization magnitude we obtained from the Cole-Cole modeling of the iron and magnetite mixtures displays a single, near-linear dependence on the ratio of surface area to pore volume ([Formula: see text]) calculated for the mixtures. A similar relationship is found using a local measure of polarization (imaginary conductivity at 1 Hz) for the clay-sand mixtures. The [Formula: see text] appears to be a critical parameter for determining IP in both metallic- and clay-containing soils. This result is not easily reconciled with traditional models of induced polarization.
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Nourozieh, Hossein, Mohammad Kariznovi, and Jalal Abedi. "Measurement and Evaluation of Bitumen/Toluene-Mixture Properties at Temperatures Up to 190?°C and Pressures Up to 10 MPa." SPE Journal 21, no. 05 (2016): 1705–20. http://dx.doi.org/10.2118/180922-pa.
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Summary The viscosity of bitumen and heavy oil is extremely high at both reservoir and surface conditions, on the order of 1 million cp. Therefore, viscosity reduction is necessary for production from the reservoir, pipeline transportation, and oil processing. The aim of this study is to evaluate the effect of different parameters (temperature, pressure, and solvent-weight fraction) on the density and viscosity of bitumen-containing mixtures. Thus, the density and viscosity of mixtures are measured for a sample of Athabasca bitumen diluted with different fractions of toluene at pressures from 0.1 to 10 MPa and at temperatures from 22 to 190 °C. The mixture densities show a linear decrease with temperature, pressure, and solvent concentration. The viscosity of the mixtures indicates a curvilinear trend with respect to the solvent-weight fraction and temperature. The effect of pressure on the mixture viscosity is more pronounced at lower-solvent-weight fractions. The mixture-density data are evaluated with two different methods: no volume change upon mixing and excess volume. The excess-volume method predicts the mixture-density data with an overall average absolute relative deviation (AARD) of 0.34%. The viscosity data for mixtures are compared with different models: Arrhenius (1987), Cragoe (1933), Shu (1984), Lobe (1973), double-log (Yarranton et al. 2013), Lederer (1933), power-law (Kendall and Monroe 1917), and Bij (Yarranton et al. 2013). The Bij model (Yarranton et al. 2013) produces the most-reliable results for mixture viscosities, with 5.5% AARD.
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Carruthers, Chris, and Heshel Teitelbaum. "A simple reason for non-linear mixture rules in chemical kinetics. Part 1. Vibrational relaxation of diatomic molecules." Canadian Journal of Chemistry 63, no. 2 (1985): 381–93. http://dx.doi.org/10.1139/v85-064.
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The generalized rate law for the vibrational relaxation of diatomic molecules is extended to include inert collision partners. V–V energy transfer processes are accounted for explicitly as are thermal effects. The molecules are treated as Morse oscillators as far as energetics are concerned; however, the microscopic rate constants are Landau–Teller type. It is found that the phenomenon of non-linear mixture rules arises when experimental data are forced to fit a first-order rate law. The persistence of V–V processes at times well-advanced into the relaxation zone is responsible for deviations from linearity. The non-linearities are most pronounced at high temperatures, and can be avoided only by using extremely dilute mixtures. Several sources of ambiguity are pointed out. The type of excitation method influences the initial deviation from a Boltzmann distribution and plays a crucial role in determining the importance of V–V processes and hence the degree of non-linearity. Thus, when the initial distribution is Boltzmann as in shock waves, the mixture rule is found to be absolutely linear for the vibrational relaxation of diatomic molecules.Several examples, heretofore not recognized as such, are pointed out in the literature.
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Delmarre, David, Noboru Hioka, Ron Boch, Ethan Sternberg, and David Dolphin. "Aggregation studies of benzoporphyrin derivative." Canadian Journal of Chemistry 79, no. 5-6 (2001): 1068–74. http://dx.doi.org/10.1139/v01-043.
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The photophysical properties of a ring B benzoporphyrin derivative 1,3-diene dimethylester were investigated in solution (waterDMSO mixtures). At low proportions of water in DMSO, the diester obeys Beer's law at all pH values and ionic strengths investigated. For solvent systems containing more than 30% water, the extent of deviation from Beer's law as a function of porphyrin concentration is attributed to formation of porphyrin dimers. Dimerization equilibrium constants (Kd) have been determined in different percentage of solvent mixtures, using absorption and fluorescence spectroscopies, and were found to exhibit a linear correlation between log (Kd) and the inverse of the dielectric constant (log (Kd) = 24.6 1060.4 × 1/εmix). This equation permits the calculation of Kd in mixtures of waterDMSO.Key words: porphyrins, dimerization, aggregation, fluorescence, benzoporphyrin derivative, photodynamic therapy, photosensitizers.
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González, Juan Antonio, Ismael Mozo, Isaías García de la Fuente, and José Carlos Cobos. "Thermodynamics of binary mixtures with strongly negative deviations from Raoult's Law. X. linear alkanoate + CHCl3or + 1,1,2,2-tetrachloroethane." Physics and Chemistry of Liquids 43, no. 4 (2005): 317–32. http://dx.doi.org/10.1080/00319100500062496.
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Eringen, A. C. "A mixture theory for geophysical fluids." Nonlinear Processes in Geophysics 11, no. 1 (2004): 75–82. http://dx.doi.org/10.5194/npg-11-75-2004.
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Abstract. A continuum theory is developed for a geophysical fluid consisting of two species. Balance laws are given for the individual components of the mixture, modeled as micropolar viscous fluids. The continua allow independent rotational degrees of freedom, so that the fluids can exhibit couple stresses and a non-symmetric stress tensor. The second law of thermodynamics is used to develop constitutive equations. Linear constitutive equations are constituted for a heat conducting mixture, each species possessing separate viscosities. Field equations are obtained and boundary and initial conditions are stated. This theory is relevant to an atmospheric mixture consisting of any two species from rain, snow and/or sand. Also, this is a continuum theory for oceanic mixtures, such as water and silt, or water and oil spills, etc.
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Sardeshmukh, Prashant D., and Philip Sura. "Reconciling Non-Gaussian Climate Statistics with Linear Dynamics." Journal of Climate 22, no. 5 (2009): 1193–207. http://dx.doi.org/10.1175/2008jcli2358.1.
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Abstract Linear stochastically forced models have been found to be competitive with comprehensive nonlinear weather and climate models at representing many features of the observed covariance statistics and at predictions beyond a week. Their success seems at odds with the fact that the observed statistics can be significantly non-Gaussian, which is often attributed to nonlinear dynamics. The stochastic noise in the linear models can be a mixture of state-independent (“additive”) and linearly state-dependent (“multiplicative”) Gaussian white noises. It is shown here that such mixtures can produce not only symmetric but also skewed non-Gaussian probability distributions if the additive and multiplicative noises are correlated. Such correlations are readily anticipated from first principles. A generic stochastically generated skewed (SGS) distribution can be analytically derived from the Fokker–Planck equation for a single-component system. In addition to skew, all such SGS distributions have power-law tails, as well as a striking property that the (excess) kurtosis K is always greater than 1.5 times the square of the skew S. Remarkably, this K–S inequality is found to be satisfied by circulation variables even in the observed multicomponent climate system. A principle of “diagonal dominance” in the multicomponent moment equations is introduced to understand this behavior. To clarify the nature of the stochastic noises (turbulent adiabatic versus diabatic fluctuations) responsible for the observed non-Gaussian statistics, a long 1200-winter simulation of the northern winter climate is generated using a dry adiabatic atmospheric general circulation model forced only with the observed long-term winter-mean diabatic forcing as a constant forcing. Despite the complete neglect of diabatic variations, the model reproduces the observed K–S relationships and also the spatial patterns of the skew and kurtosis of the daily tropospheric circulation anomalies. This suggests that the stochastic generators of these higher moments are mostly associated with local adiabatic turbulent fluxes. The model also simulates fifth moments that are approximately 10 times the skew, and probability densities with power-law tails, as predicted by the linear theory.