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Coupled Transport Processes
Published in John Newman, Vincent Battaglia, The Newman Lectures on Transport Phenomena, 2020
John Newman, Vincent Battaglia
The Onsager reciprocal relations are valuable because they are general and provide equalities among macroscopic quantities without any dependence on the molecular properties of the materials. For this reason, they belong in the study of transport phenomena. On the other hand, the Chapman-Enskog kinetic theory of gases had already led to thermal diffusion and the Dufour effect, with the same transport properties DiT appearing in both, but the theory was restricted to dilute mixtures of monatomic gases.
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Published in Splinter Robert, Illustrated Encyclopedia of Applied and Engineering Physics, 2017
[computational, mechanics, thermodynamics] Simple systems can be partitioned based on their respective quantum or energy states (Ei), which will follow the partition function: Zpart=∑ie−βTEi, where βT = 1/kbT is the “inverse temperature,” with kb = 1.3806488 × 10−23m2 kg/s2 K the Boltzmann coefficient, and T the temperature in kelvin. For the harmonic oscillator, this becomes, based on the energy E = [n + (1/2)] ħω, Zpart=e−(1/2)βTℏω/1−e−βTℏω, where ħ = h/2π, h = 6.62606957 × 10−34 m2 kg/s is Planck’s constant, ω = 2πν the angular velocity, ν the frequency, and n the quantum number. On a mathematical basis, specifically in probability theory, this also refers to a configuration integral, as a generalization, using the Hamiltonian (H) representing the potential function (which is measurable, is an observable): Zpart (βT) = ∫exp −{βTH(x1,x2,x3,…)}dx1,dx2,…. Specific applications are in information science, signal processing, and pertaining to dynamical systems. In statistical mechanics, the free entropies of a system can frequently be expressed as the logarithm of a partition function, specifically for the Onsager reciprocal relations. In thermodynamic systems, the Onsager reciprocal relations are used to express the equality of particular ratios between forces and flows that are not specifically in equilibrium; however, local equilibrium may exist under strict boundary conditions.
Wax deposition rate model for heat and mass coupling of piped waxy crude oil based on non-equilibrium thermodynamics
Published in Journal of Dispersion Science and Technology, 2018
Yang Liu, Chenlin Pan, Qinglin Cheng, Bing Wang, Xuxu Wang, Yifan Gan
According to Curie’s law, the “flow” and “force” that can be coupled could be sought out. Combining with the Onsager reciprocal relations, phenomenological equations of heat transfer and mass transfer in waxy crude oil are established:
Mass and thermal transport in liquid Cu-Ag alloys
Published in Philosophical Magazine, 2019
Ujjal Sarder, Tanvir Ahmed, William Yi Wang, Rafal Kozubski, Zi-Kui Liu, Irina V. Belova, Graeme E. Murch
In a binary system, the thermodynamic driving forces and depend on the gradients in temperature, pressure and concentration but the heat force depends only on the temperature gradient. In the laboratory reference frame (lrf) the driving forces can be defined as [10,11,31]:and similarly for . Here is the temperature, , (and ) are the partial enthalpies (per atom) of the atomic components and respectively and , (and ) are the gradients of chemical potentials. In the derivation of Eq. 2 we use the relation [31]:where are the partial entropies (per atom) of components and . For definiteness, the force is assumed to be the same in both the lrf and the centre of mass reference frame (cmrf). Let us denote the thermodynamic forces in cmrf as . They then can be expressed as:where is the gradient operator and are the partial enthalpies (per atom) obtained in the cmrf. When the thermodynamic forces are small enough, the atomic fluxes can be expressed as linear combinations of the forces [10,11,30–33]:where and are the atomic fluxes, and and are the Onsager phenomenological transport coefficients in the lrf and cmrf respectively. As the thermal force is the same in both reference frames, its conjugate flux should also be the same in both reference frames [30]:In the lrf the atomic fluxes are related simply as:Accordingly, using the Onsager reciprocal relations there are the following relations between the Onsager phenomenological coefficients [32,33]: