One-dimensional Planar Model Research Articles

Application of particle trajectory model in 1D planar ejection

A simple one-dimensional planar model for ejection was set up based on experiments. And numerical simulation was performed on this model with particle trajectory model method. An Eulerian finite volume method was conducted to resolve gas field. And Lagrangian method was imposed to track each particle. The interaction between gas and particles was responded as source terms in governing equations which were induced by forces. The effects of total spraying mass, particle size and other factors on the mixture of particles and gas were investigated. The spatial distributions of particle mass and velocity at different time were presented. The result shows that the numerical results are qualitatively consistent to those of experiments.

A radiation transport coupled particle-in-cell simulation. II. Simulation results in a one-dimensional planar model

The radiation-transport coupled particle-in-cell model is applied to one-dimensional planar model to investigate the radiation trapping phenomena in relatively high- and low-pressure Ar glow discharges. The radiation intensity spectra and the total radiation flux are calculated from the radiative excited-state profile. The simulation results for Doppler and resonance collision broadenings are compared with the eigenmode description of the Holstein equation, with good agreement. Observed are the effects of radiation trapping of photons, diffusion, collisional quenching, and step ionization from excited states on the discharge properties.

A radiation transport coupled particle-in-cell simulation. I. Description of the model

A one-dimensional radiation transport model is coupled with a particle-in-cell simulation in order to incorporate the resonance trapping effect of photons and the kinetic effect of plasma in fluorescent-lamp-like discharges. Electrons and ions are treated with the conventional particle-in-cell method, and neutral species such as ground, radiative, and metastable state atoms are treated with a fluid model combined with the Holstein–Biberman equation. Also included are the atomic collisions among neutral species and the effect of nonuniform ground state density on photon transport. The general scheme of the model and the numerical methods for Doppler and pressure line broadenings are described in a one-dimensional planar model for the nonuniform ground-state density.

Stability of commensurate phases of a planar model with competing interactions

We study a one-dimensional planar model with competing interactions at zero temperature in the limit of low and high external magnetic fields. The stability of commensurate phases with wave number ${q}_{0}=2\ensuremath{\pi}(P/Q)$, where $P/Q$ is an irreducible fraction, is discussed in the continuum approximation. We show that for small fields $H\ensuremath{\rightarrow}0+$, the helical phase with period $Q>4$ has a width proportional to ${H}^{Q/2}$. Similarly, for high fields $H\ensuremath{\rightarrow}{H}_{c}\ensuremath{-}$, where ${H}_{c}$ is the critical field for the parafan transition, the fan phase with period $Q>~4$ has a width proportional to $|H\ensuremath{-}{H}_{c}{|}^{(Q\ensuremath{-}2)/4}$ if $Q$ is an even number and $|H\ensuremath{-}{H}_{c}{|}^{(Q\ensuremath{-}1)/2}$ if $Q$ is an odd number.

A spherical model of the clark electrode in a flowing liquid medium

A spherical model has been developed for a Clark electrode immersed in a flowing liquid medium. This model relates the sensitivity of the electrode to cathode radius, to electrolyte layer permeability and thickness, to membrane permeability and thickness and to the concentration boundary-layer thickness established by the flowing medium. The model was tested with sensitivity values obtained from a 1.3 cm diameter Teflon-covered electrode assembly which was placed in a 2.5 cm i.d. flow loop such that the centerline streamline stagnated at the cathode. The data set spanned a range of cathode radii from 65 to 677 μm, membrane thicknesses of 6 μm and 76 μm and a range of mean axial tube velocities ( V) from 0.5 to 62 cm s −1 corresponding to tube Reynolds numbers of 125–16,000. The boundary-layer thickness was assumed to be of the form (γ V − n ) and the constants γ and n were determined by a non-linear least-squares regression of the sensitivity data to the spherical model. For the purposes of this regression, the sensitivity values were grouped into a laminar and a turbulent data set and (γ, n) values were determined for each set separately. The parameters P e/ P m (electrolyte to membrane permeability ratio) and ω e (electrolyte layer thickness) were treated as constants, with their values set at the regressed values previously obtained from fitting the gas-phase sensitivity values. Here, γ and n were both treated as free parameters. The spherical model fits the experimental current values in the laminar and turbulent regions with average relative errors of 8.74% and 13.1% respectively. Residuals were randomly distributed over the range of cathode radii and the two membrane thicknesses used. The final regressed values of γ and n were 0.0307 and 0.518 respectively for the laminar region, and 0.0317 and 0.451 respectively for the turbulent region. The n value for the laminar region is in reasonable agreement with the theoretically predicted value of 0.5. For comparison, a one-dimensional planar model was used to correlate the laminar and the turbulent data sets. Here again the parameters P e/ P m and ω e were assigned those values previously obtained from fitting gas-phase sensitivity values by the planar model. The relative average errors, 34.7% and 37.6% for the laminar and turbulent regions respectively, were excessively large and the final regressed value of n, 0.387 for the laminar data set, was unrealistic with respect to the theoretical prediction.

A spherical model of the clark electrode

A spherical model that relates the sensitivity ( S) of a Clark electrode to cathode radius ( a), to electrolyte layer permeability ( P c) and thickness ( ω c), and to membrane permeability ( P m) and thickness ( ω m) has been developed. This model was used to correlate two different data sets, one obtained from polypropylene-covered ( P m=1.7×10 −11cm 3 O 2/cm.s.Torr) and the other obtained from Teflon-covered ( P m=6.0×10 −11 cm 3 O 2/cm.s.Torr) Clark electrodes. The two data sets spanned a range of cathode radii from 10 to 677.5 μm and a range of membrane thickness from 6.35 to 76.2 μm. The spherical model fits the experimental sensitivity values with average relative errors of 13.3 and 23.9% for the polypropylene and Teflon data sets respectively; individual errors were randomly distributed over the range of membrane thicknesses, membrane permeabilities and cathode radii. The final regressed values of the free parameters ( P c/ P m and ω c) were found to be 69.22 and 27.7 μm, respectively, for the polypropylene data, and 6.13 and 154.6 μm, respectively, for the Teflon data. Although the P c/ P m values are consistent with reported values in the literature, ω c values are larger than expected. For purposes of comparison, a one-dimensional planar model and a pseudo-two-dimensional model were also used to correlate the data sets. The planar model was incapable of fitting sensitivity values corresponding to small cathode radii, and the pseudo-two-dimensional model was incapable of fitting sensitivity values for the Clark electrodes covered with the more permeable (Teflon) membrane. Final regressed values of the two free parameters ( P c/ P m and ω c) were generally unrealistic for these two models.

A test of the collective field method for the N → ∞ limit

The collective field method applied to the one-dimensional planar model is shown to give exactly the entire epectrum of SU( N) symmetric states in the large- N limit.

Thermodynamic and static properties of the one-dimensional planar model with symmetry-breaking fields

Thermodynamic and static properties of the one-dimensional planar model with symmetry-breaking fields