Tokamak plasmas are inherently comprised of multiple ion species. This is due to wall-bred impurities and, in future reactors, will result from fusion-born alpha particles. Relatively small densities nI, of highly charged non-hydrogenic impurities can strongly influence plasma transport properties whenever . The determination of the complete neoclassical Onsager matrix for a toroidally confined multispecies plasma, which provides the linear relation between the surface averaged radial fluxes and the thermodynamic forces (i.e. gradients of density and temperature, and the parallel electric field), is reviewed. A closed set of one-dimensional moment equations is presented for the time evolution of thermodynamic and magnetic field quantities which results from collisional transport of the plasma and two-dimensional motion of the magnetic flux surface geometry. The effects of neutral-beam injection on the equilibrium and transport properties of a toroidal plasma are consistently included.

https://iopscience.iop.org/article/10.1088/0029-5515/21/9/003/pdf

Neoclassical transport of impurities in tokamak plasmas

Advanced models of fuel droplet heating and evaporation

General Theory: The Entropy Production for a Homogeneous Phase The Excess Entropy Production for the Surface Flux Equations and Onsager Relations Transport of Heat and Mass Transport of Mass and Charge Applications: Evaporation and Condensation A Non-Isothermal Concentration Cell Adiabatic Electrode Reactions The Formation Cell Modeling the Polymer Electrolyte Fuel Cell The Impedance of an Electrode Surface The Non-Equilibrium Two-Phase van der Waals Model and other chapters.

Non-Equilibrium Thermodynamics of Heterogeneous Systems

Although the Hertz-Knudsen (HK) relation is often used to correlate evaporation data, the relation contains two empirical parameters (the evaporation and condensation coefficients) that have inexplicably been found to span 3 orders of magnitude. Explicit expressions for these coefficients have yet to be determined. This review will examine sources of error in the HK relation that have led to the coefficients’ scatter. Through an examination of theoretical, experimental, and molecular dynamics simulation studies of evaporation, this review will show that the HK relation is incomplete, since it is missing an important physical concept: the coupling between the vapor and liquid phases during evaporation. The review also examines a modified HK relation, obtained from the quantum-mechanically based statistical rate theory (SRT) expression for the evaporation flux and applying a limit to it in which the thermal energy is dominant. Explicit expressions for the evaporation and condensation coefficients are define...

Expressions for the Evaporation and Condensation Coefficients in the Hertz-Knudsen Relation.

Evaporation is ubiquitous in nature. This process influences the climate, the formation of clouds, transpiration in plants, the survival of arctic organisms, the efficiency of car engines, the structure of dried materials and many other phenomena. Recent experiments discovered two novel mechanisms accompanying evaporation: temperature discontinuity at the liquid-vapour interface during evaporation and equilibration of pressures in the whole system during evaporation. None of these effects has been predicted previously by existing theories despite the fact that after 130 years of investigation the theory of evaporation was believed to be mature. These two effects call for reanalysis of existing experimental data and such is the goal of this review. In this article we analyse the experimental and the computational simulation data on the droplet evaporation of several different systems: water into its own vapour, water into the air, diethylene glycol into nitrogen and argon into its own vapour. We show that the temperature discontinuity at the liquid-vapour interface discovered by Fang and Ward (1999 Phys. Rev. E 59 417-28) is a rule rather than an exception. We show in computer simulations for a single-component system (argon) that this discontinuity is due to the constraint of momentum/pressure equilibrium during evaporation. For high vapour pressure the temperature is continuous across the liquid-vapour interface, while for small vapour pressures the temperature is discontinuous. The temperature jump at the interface is inversely proportional to the vapour density close to the interface. We have also found that all analysed data are described by the following equation: da/dt = P(1)/(a + P(2)), where a is the radius of the evaporating droplet, t is time and P(1) and P(2) are two parameters. P(1) = -λΔT/(q(eff)ρ(L)), where λ is the thermal conductivity coefficient in the vapour at the interface, ΔT is the temperature difference between the liquid droplet and the vapour far from the interface, q(eff) is the enthalpy of evaporation per unit mass and ρ(L) is the liquid density. The P(2) parameter is the kinetic correction proportional to the evaporation coefficient. P(2) = 0 only in the absence of temperature discontinuity at the interface. We discuss various models and problems in the determination of the evaporation coefficient and discuss evaporation scenarios in the case of single- and multi-component systems.

https://iopscience.iop.org/article/10.1088/0034-4885/76/3/034601/pdf

Evaporation of freely suspended single droplets: experimental, theoretical and computational simulations

The motion of a vapor gas in contact with interphase surfaces is studied on the basis of kinetic theory of gases. The mass and energy fluxes are determined from the condition of the vapor far away from the interfaces. The result for a vapor gas between two interfaces shows that the temperature gradient in the vapor is opposite in sign to the externally maintained temperature difference.

Application of Kinetic Theory to the Problem of Evaporation and Condensation

The problem of evaporation from an interphase and the problem of heat conduction in a gas are studied on the basis of the linearized single‐relaxation model equation. Both problems lead to the same set of integral equations. Exact values are obtained for the microscopic temperature and density jumps. It is found that the macroscopic temperature and density jumps of the evaporation problem can be expressed in terms of the macroscopic jumps of the heat conduction problem and vice versa. A proof for the existence and uniqueness of the solutions is also given.

Temperature and Density Jumps in the Kinetic Theory of Gases and Vapors

Time‐dependent classical diffusion in axisymmetric toridal plasmas with finite aspect ratio and finite beta is examined. The velocity component normal to the flux surfaces can be solved from a partial differentio‐integral equation in terms of instantaneous equilibrium quantities. A method for determining the other two velocity components is given which involves the use of convective and viscous terms in the momentum equation. Equations describing the evolution of magnetic field, plasma pressure, and density are presented.

Classical diffusion in toroidal plasmas

The spectrum of stable resistive magnetohydrodynamic modes is shown to be determined by the geometry of anti‐Stokes lines. Behavior of the eigenfunctions is also examined.

Analytic theory of stable resistive magnetohydrodynamic modes

It can be shown that for linearly unstable magnetohydrodynamic modes near the threshold of linear instability, the mode amplitude A (t) evolves according to ∂2A/Λt2 =λ2A+αA3 which can lead to nonlinear stabilization, explosive instability, or eventual decay, depending on the sign of α This is demonstrated explicitly for the m=1 kink modes in a sharp boundary plasma pinch

Young‐ping Pao

Papers

Application of Kinetic Theory to the Problem of Evaporation and Condensation

Temperature and Density Jumps in the Kinetic Theory of Gases and Vapors

Classical diffusion in toroidal plasmas

Analytic theory of stable resistive magnetohydrodynamic modes

Nonlinear behavior of linearly unstable magnetohydrodynamic modes