Application of approximations for joint cumulative k-distributions for mixtures to FSK radiation heat transfer in multi-component high temperature non-LTE plasmas

Approximations for joint cumulative k-distribution for mixtures are efficient for full spectrum k-distribution (FSK) computations. These approximations provide reduction of the database that is necessary to perform FSK computation when compared to the direct approach, which uses cumulative k-distributions computed from the spectrum of the mixture, and also less computational expensive when compared to techniques in which RTE's are required to be solved for each component of the mixture. The aim of the present paper is to extend the approximations for joint cumulative k-distributions for non-LTE media. For doing that, a FSK to non-LTE media formulation well-suited to be applied along with approximations for joint cumulative k-distributions is presented. The application of the proposed methodology is demonstrated by solving the radiation heat transfer in non-LTE high temperature plasmas composed of N, O, N₂, NO, N₂⁺ and mixtures of these species. The two more efficient approximations, that is, the superposition and multiplication are employed and analyzed.     

The spectral-line moment-based (SLMB) modeling of the wide band and global blackbody-weighted transmission function and cumulative distribution function of the absorption coefficient in uniform gaseous media

The spectral-line moment-based (SLMB) modeling is proposed for the calculation of radiative properties of gases on any spectral width. The associated mathematical formulation is obtained by applying several concepts of the k-distribution methods such as the reordering of the wavenumber scale by monotonic variations of the absorption coefficient, together with the application of the k-moment method's principles. This approach gives both a general formula for the BTF and a simple and readily applicable approximation for the blackbody-weighted cumulated k-distribution function of the absorption coefficient. The model is applied for the computation of wide band BTFs and cumulative k-distributions for uniform columns of CO₂ and H₂O in the temperature range (300-2400 K) at atmospheric pressure. Model parameters are deduced from line-by-line (LBL) spectra calculated using the HITEMP database. Comparisons with LBL reference data as well as with contemporary modeling approaches (SLW, FSK, SNB) are performed and discussed.     

A full spectrum k-distribution based non-gray radiative property model for unburnt char

By using the concept of weighted sum of four gray particles and spectrum k-distribution (WSGP-SK), a non-gray radiative property model for unburnt char particles is developed. Based on the carbon burnout kinetic model for structure during oxidation, and the linear mixed approximation theory for complex index of refraction, spectral radiative properties of unburnt char particles are first calculated as function of the burnout ratio by Mie theory. Referring to the full spectrum k-distribution model, k-distribution is applied to reorder absorption and scattering efficiencies of particles. Then, weighting factors and efficiency factors of the non-gray radiative property model are directly obtained from Gaussian integral points of k-distribution. The model is validated against the benchmark solutions of line-by-line (LBL) model. Maximum relative errors of this model are 3% and 15% for radiative heat fluxes and source terms in non-isothermal inhomogeneous particulate media, respectively. The assumption of linearly varying radiative properties with burnout ratio (Lockwood et al. 1986) will result in a predicted deviation of 53% for radiative source terms. Results also show that this non-gray model is remarkably better than the Planck mean method. Moreover, a satisfactory comparison with LBL solutions is achieved in the gas and particle mixture by combining the non-gray WSGG-SK model (Guo et al. 2015). As a radiation sub-model, this non-gray radiative property model can significantly improve prediction accuracy of radiative heat transfer in oxy-fuel combustion.     

Extension of the weak-line approximation and application to correlated-k methods

Global climate models require accurate and rapid computation of the radiative transfer through the atmosphere. Correlated-k methods are often used. One of the approximations used in correlated-k models is the weak-line approximation. We introduce an approximation T_γ which reduces to the weak-line limit when optical depths are small, and captures the deviation from the weak-line limit as the extinction deviates from the weak-line limit. This approximation is constructed to match the first two moments of the gamma distribution to the k-distribution of the transmission. We compare the errors of the weak-line approximation with T_γ in the context of a water vapor spectrum. The extension T_γ is more accurate and converges more rapidly than the weak-line approximation.     

A database for the SLMB modeling of the full spectrum radiative properties of CO2

Development and application of a database for the Spectral-Line Moment-Based (SLMB) modeling of the full spectrum radiative properties of mixtures of carbon dioxide and nitrogen is presented. The critical issue of the definition of a reference thermophysical condition is addressed together with the suggestion of a coherent and precise methodology to derive parameters of the model for any other configuration. The database is built accordingly from the CDSD-1000 high temperature spectroscopic databank for gas and blackbody-weighting temperatures in the range [300; 2700 K]. Accuracy of both the modeling and the database is assessed through comparisons with LBL results in terms of full spectrum k-distributions and emission functions. Results obtained from the application of FSK correlations and the Leckner's formula are also provided for extended analyses.     

Evaluation of solution methods for radiative heat transfer in gaseous oxy-fuel combustion environments

The exact solution to radiative heat transfer in combusting flows is not possible analytically due to the complex nature of the integro-differential radiative transfer equation (RTE). Many different approximate solution methods for the solution of the RTE in multi-dimensional problems are available. In this paper, two of the principal methods, the spherical harmonics (P₁) and the discrete ordinates method (DOM) are used to calculate radiation. The radiative properties of the gases are calculated using a non-gray gas full spectrum k-distribution method and a gray method. Analysis of the effects of numerical quadrature in the DOM and its effect on computation time is performed. Results of different radiative property methods are compared with benchmark statistical narrow band (SNB) data for both cases that simulate air combustion and oxy-fuel combustion. For both cases, results of the non-gray full spectrum k-distribution method are in good agreement with the SNB data. In the case of oxy-fuel simulations with high partial pressures of carbon dioxide, use of gray method for the radiative properties may cause errors and should be avoided.     

Entropy maximum in a nonlinear system with the 1/<Emphasis Type="Italic">f</Emphasis> fluctuation spectrum

Analysis of the control and subordination is carried out for the system of nonlinear stochastic equations describing fluctuations with the 1/f spectrum and with the interaction of nonequilibrium phase transitions. It is shown that the control equation of the system has a distribution function that decreases upon an increase in the argument in the same way as the Gaussian distribution function. Therefore, this function can be used for determining the Gibbs-Shannon informational entropy. The local maximum of this entropy is determined, which corresponds to tuning of the stochastic equations to criticality and indicates the stability of fluctuations with the 1/f spectrum. The values of parameter q appearing in the definition of these entropies are determined from the condition that the coordinates of the Gibbs-Shannon entropy maximum coincide with the coordinates of the Tsallis entropy maximum and the Renyi entropy maximum for distribution functions with a power dependence.     

High fidelity radiative heat transfer models for high-pressure laminar hydrogen–air diffusion flames

Radiative heat transfer is studied numerically for high-pressure laminar H₂–air jet diffusion flames, with pressure ranging from 1 to 30 bar. Water vapour is assumed to be the only radiatively participating species. Two different radiation models are employed, the first being the full spectrum k-distribution model together with conventional Radiative Transfer Equation (RTE) solvers. Narrowband k-distributions of water vapour are calculated and databased from the HITEMP 2010 database, which claims to retain accuracy up to 4000 K. The full-spectrum k-distributions are assembled from their narrowband counterparts to yield high accuracy with little additional computational cost. The RTE is solved using various spherical harmonics methods, such as P₁, simplified P₃ (SP₃) and simplified P₅ (SP₅). The resulting partial differential equations as well as other transport equations in the laminar diffusion flames are discretized with the finite-volume method in OpenFOAM^®. The second radiation model is a Photon Monte Carlo (PMC) method coupled with a line-by-line spectral model. The PMC absorption coefficient database is derived from the same spectroscopy database as the k-distribution methods. A time blending scheme is used to reduce PMC calculations at each time step. Differential diffusion effects, which are important in laminar hydrogen flames, are also included in the scalar transport equations. It was found that the optically thin approximation overpredicts radiative heat loss at elevated pressures. Peak flame temperature is less affected by radiation because of faster chemical reactions at high pressures. Significant cooling effects are observed at downstream locations. As pressure increases, the performance of RTE models starts to deviate due to increased optical thickness. SP_N models perform only marginally better than P₁ because P₁ is adequate except at very high pressure.     

Formal equivalence between Tsallis and extended Boltzmann-Gibbs statistics

A formal correspondence between the q-distribution obtained from the Tsallis entropy and non-Maxwellian distributions obtained from the Boltzmann-Gibbs (BG) entropy is obtained. This formal correspondence is obtained by imposing an infinite number of constraints when one maximizes the BG entropy. Different from the approach of Tsallis, Prato and Plastino [C. Tsallis, D. Prato, A.R. Plastino, Astrophys. Space Sci., 290 (2004) 259-274], we relate the constraints to the central moments, providing a natural meaning to the q-parameter.     

Non-gray chemical composition based radiative property model of fly ash particles

Particle radiation has a spectral dependence and is closely related to the chemical composition of the material. Iron oxide, one of the main components of fly ash, observably affects the complex index of refraction of the particles. In this study, following the theory of the spectrum k-distribution based weighted sum of gray particles model (Guo et al. [4,13]), a non-gray fly ash radiative property model involving the chemical composition was developed. First, four typical fly ash particles with different iron oxide contents were selected, and the corresponding particle radiative parameters were obtained using the Mie theory. Then, the absorption efficiency and weighting factors of the non-gray model were directly obtained from the Gaussian integral points of the k-distribution. The scattering efficiency of the particles was obtained from the Planck mean. The accuracy of the newly developed model was evaluated in a one-dimensional plane-parallel slab system through comparison with the line-by-line (LBL) model and two commonly used gray radiative property models. The results show that the new non-gray model agrees well with the LBL solution and becomes more accurate as the iron oxide content increases. When the iron oxide content of the fly ash increased from 5.47% to 30.50%, the maximum relative error of the radiative heat flux and the radiative source term decreased from 12.50% to 5.68% and from 20.97% to 12.62%, respectively. The new model can improve the prediction accuracy of radiative heat transfer in pulverized coal-fired furnaces.     

A nonuniform narrow band correlated-k approximation using the k-moment method

Narrow band k-moment (NBKM) formulations of the transmission function for nonuniform gaseous paths are developed. One of them, called the scaled variance (SV) approximation, is based on the assumption of correlated absorption coefficients. Theoretical derivations are properly detailed and several test cases taken from the literature are provided to assess the approximate modeling results against LBL reference calculations. In most cases representative of combustion configurations, it is shown that the scaled variance approximation is more accurate than the Curtis–Godson one for CO₂ but not for H₂O.     

Pricing European options with a log Student’s t-distribution: A Gosset formula

The distributions of returns for stocks are not well described by a normal probability density function (pdf). Student’s t-distributions, which have fat tails, are known to fit the distributions of the returns. We present pricing of European call or put options using a log Student’s t-distribution, which we call a Gosset approach in honour of W.S. Gosset, the author behind the nom de plume Student. The approach that we present can be used to price European options using other distributions and yields the Black-Scholes formula for returns described by a normal pdf.     

Sustained turbulence in the three-dimensional Gross-Pitaevskii model

We study the three-dimensional forced-dissipated Gross-Pitaevskii equation. We force at relatively low wave numbers, expecting to observe a direct energy cascade and a consequent power-law spectrum of the form k^−α. Our numerical results show that the exponent α strongly depends on how the inverse particle cascade is attenuated at ks lower than the forcing wave-number. If the inverse cascade is arrested by a friction at low ks, we observe an exponent which is in good agreement with the weak wave turbulence prediction k⁻¹. For a hypo-viscosity, a k⁻² spectrum is observed which we explain using a critical balance argument. In simulations without any low k dissipation, a condensate at k=0 is growing and the system goes through a strongly turbulent transition from a 4-wave to a 3-wave weak turbulence acoustic regime with evidence of k^−3/2 Zakharov-Sagdeev spectrum. In this regime, we also observe a spectrum for the incompressible kinetic energy which formally resembles the Kolmogorov k^−5/3, but whose correct explanation should be in terms of the Kelvin wave turbulence. The probability density functions for the velocities and the densities are also discussed.     

The volume dependence of the Grüneisen parameter and crystal stability

Utilizing only the general features of the lattice structure and potential, the explicit volume dependence of the entire frequency spectrum is derived in terms of its zero pressure values. It is shown that even though the frequencies and the mode Grüneisen parameters depend on the various details of the interactions and lattice structure, their variation with volume can be relatively simple functions of the volume. A criterion for mode instability with pressure is established involving only the measured value of γ(kλ). The volume dependence of the high temperature Grüneisen parameter is derived utilizing the characteristics of the full frequency spectrum through the first two moments of the mode gammas and the results are applied to the alkali-halides.     

Gaussian Fluctuations of Eigenvalues in Wigner Random Matrices

We study the fluctuations of eigenvalues from a class of Wigner random matrices that generalize the Gaussian orthogonal ensemble. We begin by considering an n×n matrix from the Gaussian orthogonal ensemble (GOE) or Gaussian symplectic ensemble (GSE) and let x _k denote eigenvalue number k. Under the condition that both k and n?k tend to infinity as n→∞, we show that x _k is normally distributed in the limit. We also consider the joint limit distribution of eigenvalues $(x_{k_{1}},\ldots,x_{k_{m}})$ from the GOE or GSE where k ₁, n?k _m and k _i+1?k _i , 1≤i≤m?1, tend to infinity with n. The result in each case is an m-dimensional normal distribution. Using a recent universality result by Tao and Vu, we extend our results to a class of Wigner real symmetric matrices with non-Gaussian entries that have an exponentially decaying distribution and whose first four moments match the Gaussian moments.     

A new approach to solve correlated k-distribution function

The resorting profile of absorption coefficient is fitted with Voigt function by employing the similarity between it and a single mixed broadened Voigt line type. In this way, simple mathematical fitting formulae are obtained to solve k-distribution (k-D) function at any pressure and temperature. Thus, a new correlated k-D method is proposed on the basis of it. Finally, taking mid-latitude summer atmosphere as an example, longwave cooling rates are calculated for three major gases by the new method, and then compared with the corresponding results by line-by-line integration.     

The transverse momentum profile of a quark jet

The first and second moments of the transverse momentum distribution of pions ine ⁺ e ^?-annihilation at fixed longitudinal momentum with respect to the jet axis are discussed, treating the hadronic final state evolving from the 2-quark system in the framework of a fireball-model. Good agreement with the published data is found. A comparison with thep _T -distribution of electroproduction is made, showing qualitative agreement. This casts serious doubts on previous conclusions favoring a large partonk _T inside the proton.     

Inclusive production of non-strange resonances inK − p interactions at 32 GeV/c

New data on the inclusive production of the non-strange resonances ?⁰(770), ω(783), ?(1020) andf(1270) inK ^? p interactions at 32 GeV/c are presented. The inclusive production cross sections are equal to (4.32±0.72) mb, (3.7±1.4) mb, (0.65±0.10) mb and (0.91±0.35) mb respectively. Estimates of the topological cross sections are also obtained. The invariant and non-invariantx-distributions for the vector mesons ?⁰ and ? indicate the prevalence of forward resonance production in the c.m. system. For the tensorf-meson the rapidity andx-distributions are presented. Thet′-distributions for ?⁰, ?, andf have exponential slopes of 0.6±0.1 GeV^?2, 1.2±0.2 GeV^?2, and 0.8±0.5 GeV^?2 respectively. The exponential slope ofp _T ² -distribution of thef-meson is equal to (2.3±0.5) GeV^?2.     

Statistical properties of many-particle spectra: III. Ergodic behavior in random-matrix ensembles

The ergodic problem is defined for random-matrix ensembles and some conditions for ergodicity given. Ergodic properties are demonstrated for the orthogonal, unitary and symplectic cases of the Gaussian and circular ensembles, and also for the Poisson ensemble. The one-point measures, viz., the eigenvalue density, the number statistic and the k'thnearest-neighbor spacings are shown to be ergodic and the ensemble variances of the corresponding spectral averages are explicitly calculated. It is moreover shown, by using Dyson's cluster functions, that all the k-point correlation functions are themselves ergodic as are therefore the fluctuation measures which follow from them. It is proved also that the local fluctuation properties of the Gaussian ensembles are stationary over the spectrum.     

Investigation of the formula for the average of two S-matrix elements in compound nucleus reactions

Recently the problem of calculating the average of two S-matrix elements describing compound nucleus reactions was solved by using the technique of generating functions and grassmann integration. In this paper we analyze the solution, a three-dimensional integral, both analytically and numerically. We study the expansion of the integral in powers of the transmission coefficients and the expansion in inverse powers of the transmission coefficients. The former one, which also contains logarithmic terms, is shown to agree with previous work. The latter one is identical to the expansion obtained with the help of the replica trick. After a suitable change of integration variables the integrand becomes finite everywhere in its domain and numerical integration is without any problem. Earlier Monte Carlo calculations are confirmed. The average of two S-matrix elements at the same energy obtained by a Mexican group using general properties of the S-matrix and the maximum entropy principle is shown to coincide with the microscople result.