Modeling nongray gas-phase and soot radiation in luminous turbulent nonpremixed jet flames

Much progress has been made in radiative heat transfer modeling with respect to treatment of nongray radiation from both gas-phase species and soot particles, while radiation modeling in turbulent flame simulations is still in its infancy. Aiming at reducing this gap, this paper introduces state-of-the-art models of gas-phase and soot radiation to turbulent flame simulations. The full-spectrum k-distribution method (Modest, M.F., 2003, Journal of Quantitative Spectroscopy & Radiative Transfer, 76, 69–83) is implemented into a three-dimensional unstructured CFD code for nongray radiation modeling. The mixture full-spectrum k-distributions including nongray absorbing soot particles are constructed from a narrow-band k-distribution database created for individual gas-phase species, and an efficient scheme is employed for their construction in CFD simulations. A detailed reaction mechanism including NO _x and soot kinetics is used to predict flame structure, and a detailed soot model using a method of moments is employed to determine soot particle size distributions. A spherical-harmonic P₁ approximation is invoked to solve the radiative transfer equation. An oxygen-enriched, turbulent, nonpremixed jet flame is simulated, which features large concentrations of gas-phase radiating species and soot particles. Nongray soot modeling is shown to be of greater importance than nongray gas modeling in sooty flame simulations, with gray soot models producing large errors. The nongray treatment of soot strongly influences flame temperatures in the upstream and the flame-tip region and is essential for accurate predictions of NO. The nongray treatment of gases, however, weakly influences upstream flame temperatures and, therefore, has only a small effect on NO predictions. The effect of nongray soot radiation on flame temperature is also substantial in downstream regions where the soot concentration is small. Limitations of the P₁ approximation are discussed for the jet flame configuration; the P₁ approximation yields large errors in the spatial distribution of the computed radiative heat flux for highly anisotropic radiation fields such as those in flames with localized, near-opaque soot regions.     

气粒混合物非灰辐射特性合并宽窄谱带K分布模型

气粒混合物辐射问题具有全场性、非灰性、耦合性等特点,准确预估高温燃气/粒子非灰辐射特性是非常重要的。本文将合并宽窄谱带K分布模础(CWNBCK)与离散坐标法(DOM)结合,开展了非灰气粒混合物辐射换热问题的模拟工作,分别验证了一维和三维情况下应用该模型的准确性,给出不同工况下的热流源项、壁面热流或辐射热流等。结果表明:该模型能够给出与SNB模型精度基本相同的结果,考虑其计算效率的提高,可以在工程实际中应用该模型计算非灰气粒混合物辐射换热。     

Non-coherent scattering in subordinate line—V: Solutions of the transfer problem

We present several examples of the numerical solution of the radiative transfer in subordinate lines. Using a simplified physical model that yields the line source function analogous to the usual two-level-atom form modified by the presence of the redistribution function R_v in the scattering integral, we have solved the transfer problem for isothermal, plane-parallel atmospheres, both finite and semi-infinite. For finite atmospheres, we have found substantial differences between the solutions with R_v and those with complete redistribution. On the other hand, for semi-infinite atmospheres the complete redistribution appears to be a good approximation, at least for a_l?a_u (damping parameters for the lower and upper levels, respectively). It is shown that the effect of R_v becomes more pronounced with increasing ratio a_u/a_l. Finally, it is demonstrated that an approximate form for R_v analogous to that of Kneer for R_II serves as a very good approximation for computing the line profiles, particularly in the line wings.     

The Yang-Lee distribution for a class of lattice gases

Classical lattice gases moving on a simple cubic lattice are considered. The lattice is assumed to grow only one-dimensionally. The gas particles have hard cores (of diameter greater than the lattice spacing) and are further subject to interactions of finite range and finite order. The interactions outside the hard cores may be represented as the components of av-dimensional vector, ?, which is initially allowed to be complex. Using a transfer matrix technique, an asymptotic expression is obtained for the grand canonical pressure (at complex values of the inverse absolute temperature β and the fugacityz). Let λ₁ ... λ _M denote the eigenvalues of the transfer matrix. Define ? to be aD*-interaction if and only if the quotients, λ _j /λ _k , 1≦j<k≦M, regarded as functions of β,z (with ? fixed) arenonconstant. In this paper it is assumed that there exists at least one allowable D*-interaction. With this assumption, the main result is that ifF denotes the set of interaction vectors for which the distribution, Ω, of limit points of zeros of the grand partition function in the complexz-plane at fixed β (res. complex β-plane at fixedz) contains a domain, thenF contains no product setA ₁×...×A _v ,A _k ??, 1≦k≦v unless one or more of theA _k consists of (at most) isolated points. This implies that the set of vectors for which Ω consists of arcs is dense in the set of all allowable vectors (in the usual topology for ?v).     

Evaluation method for radiative heat transfer in polydisperse water droplets

Simplifications of the model for nongray radiative heat transfer analysis in participating media comprised of polydisperse water droplets are presented. Databases of the radiative properties for a water droplet over a wide range of wavelengths and diameters are constructed using rigorous Mie theory. The accuracy of the radiative properties obtained from the database interpolation is validated by comparing them with those obtained from the Mie calculations. The radiative properties of polydisperse water droplets are compared with those of monodisperse water droplets with equivalent mean diameters. Nongray radiative heat transfer in the anisotropic scattering fog layer, including direct and diffuse solar irradiations and infrared sky flux, is analyzed using REM². The radiative heat fluxes within the fog layer containing polydisperse water droplets are compared with those in the layer containing monodisperse water droplets. Through numerical simulation of the radiative heat transfer, polydisperse water droplets can be approximated by using the Sauter diameter, a technique that can be useful in several research fields, such as engineering and atmospheric science. Although this approximation is valid in the case of pure radiative transfer problems, the Sauter diameter is reconfirmed to be the appropriate diameter for approximating problems in radiative heat transfer, although volume-length mean diameter shows better accordance in some cases. The CPU time for nongray radiative heat transfer analysis with a fog model is evaluated. It is proved that the CPU time is decreased by using the databases and the approximation method for polydisperse particulate media.     

On the energy transport velocity in a dissipative medium

An exact definition of the group velocity v _g is proposed for a wave process with arbitrary dispersion relation ω = ω′(k) + iω″(k). For the monochromatic approximation, a limit expression v _g (k) is obtained. A condition under which v _g (k) takes the form of the Kuzelev–Rukhadze expression [1] dω′(k)/dk is found. In the general case, it appears that v _g (k) is defined not only by the dispersion relation ω(k), but also by other elements of the initial problem. As applied to the dissipative medium, it is shown that v _g (k) defines the field energy transfer velocity, and this velocity does not exceed thee light speed in vacuum. An expression for the energy transfer velocity is also obtained for the case where the dispersion relation is given in the form k = k′(ω) + ik″(ω) which corresponds to the boundary problem.     

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.     

Resonant and surface polaritons in quantum wells

Summary We show how the breaking of the translational invariance in a quantum well modifies the concept of polariton with respect to that defined for bulk material. Polaritons in quantum wells result from the combination of the exciton states with the radiation field. They are here obtained as the solutions of Maxwell equations with retardation, provided an appropriate nonlocal response function is used for the electric susceptibility, and Maxwell boundary conditions are imposed. We find two types of polaritons depending on the values of the in-plane wavevectork _II: those atk _II<ω/v (wherev=c/n is the velocity of light in the sample) are resonant with the radiation field in the barrier and those atk _II>ω/v cannot be coupled to waves in the barrier. In both cases explicit expressions are given for radiative shifts and radiative broadenings as functions ofk _II. Numerical results are obtained for GaAs-Ga_1−x Al _x As and for CuCl quantum wells and new experiments are suggested. The existence of resonant and surface polaritons justifies an interpretation of the temperature dependence of the radiative lifetime suggested by the same authors. It also decreases the radiative efficiency in the direction perpendicular to the planes and increases the radiative efficiency parallel to the planes with increasing temperature. Due to the relevance of its scientific content, this paper has been given priority by the Journal Direction.     

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.     

A method for performing radiative transfer calculations in media where population inversion may occur

Numerical problems are associated with performing coupled radiative transfer—kinetic equilibrium calculations in media where population inversion (i.e. negative opacity) may occur. Firstly, the formal solution of the radiative transfer equation is inherently unstable if the Feautrier (u, v) variables are used. Secondly, it is not appropriate to use an analytic equivalent two level atom source function coupled to a provisional optical depth scale in such conditions, since the optical depth may change rapidly as a function of the radiation field.A non-Λ method for overcoming these problems is presented in this paper. Its application to a simple problem is described and its utility for more general problems is discussed. Results are compared with those from a Λ iteration method.     

Effect of line or band shape on the radiative flux of an isothermal spherical layer

Radiative transfer in a nongray, absorbing-emitting spherical layer is investigated. The absorption coefficient consists of an array of equal-intensity, nonoverlapping, narrow bands or lines. Specific attention is directed toward the evaluation of the effect of band or line shape on the local radiative flux of an isothermal layer. The integral equation for radiative equilibrium is formulated and functions associated with it are studied. Numerical and graphical results are presented for the rectangular, triangular, exponential, Doppler, and Lorentz profiles.     

Effective surface potential method for calculating surface states

A novel formalism (the effective surface potential method) is developed for calculating surface states. Like the Green function method of Kalkstein and Soven and the transfer matrix method of Falicov and Yndurain, the technique is exact for simple tight binding Hamiltonians. As well as offering an alternative viewpoint, the present method provides a simple analytic expression describing the surface states. At each point k_s in the surface Brillouin zone the semi-infinite solid is viewed as an effective linear chain where each element of the chain is a planar layer. The solution to the linear chain problem can be expressed in terms of an effective potential h(k_s,E) at each energy E. A number of examples are presented in detail; “sp^d” Hamiltonians for a linear chain (d = 1), the honeycomb lattice (d = 2), the 111 surface of silicon (d = 3), and a dissected Bethe lattice. Various exact results are given, e.g. the extremities of surface state bands and the surface density of states of p-like (delta function) bands. The results of Kalkstein and Soven for the 100 surface of a simple cubic solid with a perturbation on the surface layer are rederived.     

Inductive-resonant theory of nonradiative transitions in lanthanide and transition metal ions (review)

The review is devoted to the theory of nonradiative transitions in tricharged ions of lanthanides and transition metals in the condensed phase, which was proposed in 1971. The theory is based on the phenomenon of nonradiative energy transfer from an electronically excited ion to surrounding molecular groups with excitation of resonant vibrational states and makes it possible to calculate the nonradiative transition rate constant (k _nr) by a formula that is similar to the Förster formula. The primary emphasis is placed on recent experimental works that directly confirm the proposed theory. It is shown that the theory satisfactorily quantitatively accounts for (i) the effect of deuteration of molecular groups surrounding ions on k _nr, (ii) the energy gap law, and (iii) the dependence of k _nr on the distance between the ion and deactivating groups. Furthermore, it is shown that (iv) the theory makes it possible to satisfactorily quantitatively calculate in the dipole-dipole approximation the constant k _nr of the electronic transition based on the knowledge of the radiative rate constant and the vibrational absorption spectra of molecular groups in the range of overlap with the luminescence spectrum of the ion; (v) the temperature dependence of k _nr; and (vi) the anomalously low k _nr in the case where the corresponding radiative transition is caused by the magnetic rather than the electric dipole. Literature data are presented that directly experimentally support the proposed theory of nonradiative transitions. In addition, works where this approach is used to calculate k _nr of transitions in laser media are described.     

On spectral representations for the functions Wi(v, q2) (i = 1, 2) describing electroproduction process

It is shown that the Deser, Gilbert, Sudarshan representation (DGSR) does not follow from microcausality and spectrality only. Examples of the functions W_i(v, q²) satisfying the microcausality and spectrality conditions are given which cannot be written as the DGSR with spectral function h(a, α) that is a temperature distribution. Instead of the DGSR the spectral representation for W_i(v, q²) has been proved (eq. (3)) which follows only from microcausality and spectrality.     

Calculation of high-order rotational centrifugal-distortion matrix elements for a Hund's case (a) basis set

The derivation of explicit expressions for the Hund's case (a) matrix elements of R^2k is discussed, where R is the mechanical rotational angular momentum operator of the molecule. A recursion relation is developed that permits matrix elements of R^2k to be expressed in terms of those of R^2(k?1), thus affording a straightforward means of calculating the case (a) matrix elements of rotational centrifugal-distortion constants D_v, H_v, L_v, M_v, etc., to an arbitrarily high order. The explicit matrix elements of L_v are listed.     

Exploiting the linearity of radiative transfer

A standard problem in radiative transfer is finding the external and internal radiative fields produced by uniform, parallel rays illuminating the top of a one-dimensional, scattering and absorbing medium of finite optical thickness. This problem has been solved in several ways with various physical restrictions. One approach is by finding the source function that represents the rate of production of scattered radiation per unit volume per unit solid angle at each point in the medium. The present paper develops and uses the idea that the standard source function is an influence function for a given medium. The linearity of radiative transfer is then used to find certain general source functions in terms of the standard one. The usefulness of the above concept is demonstrated by the following four problems: (1) derivation of Chandrasekhar's four principles of invariance from the radiative transfer equation, (2) derivation of the equations governing Chandrasekhar's X- and Y- functions without using the invariance principles or resolvent kernels, (3) finding the source function for a medium with a Lambert's-law bottom, and (4) finding the source function for a medium with a bottom that is a perfect specular reflector.     

Lichtausbeute von α-Teilchen in kristallinem und dampfförmigem Anthrazen

The light output,S _v by α-particles stopped in anthracene vapour has been measured as a function of vapour pressure (10–700 mm Hg) and temperature (250°C–385°C). The comparison of the results for an idealised vapour neglecting collisions with the light output,S _c, from anthracene crystals by α-particles impinging parallel to thec′-axis yields the unexpected results: S_v(8.78 MeV)/S_c(8.78 MeV)=0.46±0.05 andS _v(6.05 MeV)/S _c(6.05 MeV)=0.57±0.08. A simple model assuming quenching by collisions of the vapour molecules could explain the observed dependence of the light output on the vapour pressure at fixed temperature. The path lengthsR _v of α-particles in anthracene vapour were determined to be R_v(8.78 MeV)=(9.0±0.6) mg/cm²,R _v(6.05 MeV)=(4.9±0.6) mg/cm² and the ratio of the light output by the two different α-energiesS _v(8.78 MeV)/S _v(6.05 MeV)=1.42±0.2.     

Field transformation in a multimode optical waveguide with random irregularities of the film surface

A self-consistent mathematical model for the transformation of the average intensity of the mode spectrum I(z) of a waveguide field in a multimode planar optical waveguide with a step profile and rough surface is developed. This model is based on the matrix model for multiple scattering of modes in an optical waveguide. The elements of the intermode scattering matrix are found, which describe the process of mutual transfer of the energy of modes along a waveguide and their transformation into radiation modes. The transformation of the I(z) modes in waveguides with large-and small-scale inhomogeneities is investigated. It is shown that the largest qualitative differences in the noted dependences manifest themselves only in the initial portions of the optical waveguide. The length z of these portions is much smaller than the characteristic scale length L _k at which the fundamental energy of the kth mode excited in the optical waveguide is renewed. The effect of self-filtration of the mode spectrum I(z) is described, as a result of which a stable (normalized), independent of distance z, distribution I* is formed. It is established that irregularities of the optical waveguide boundaries exert a depolarizing effect on a guided light beam. The specific features of the normalization of the radiative dissipation of a group of modes I_i(z) in an optical waveguide are investigated. It is ascertained that, in the case of small-scale irregularities, the attenuation coefficient is described by a nonlinear monotonic dependence α(z), which asymptotically converges to the value α*, characteristic of the normalized field I*. When the optical-waveguide film has large irregularities, the dependence α(z) is characterized by a pronounced maximum due to the formation of alternative channels of radiative dissipation of the energy of waveguide modes.     

Influence of atomic motion on the shape of two-photon resonance in gas

A gas of three-level atoms with Λ configuration of energy levels was taken as an example to demonstrate that the influence of particle motion on the two-photon resonances extends further than the residual Doppler shift (k ₁?k ₂)v. In particular, a narrow dip in the absorption spectrum (“dark” resonance) undergoes substantial narrowing, as compared to the atoms at rest. The width of this resonance is studied nonperturbatively as a function of the intensities of probe and strong fields.     

A semi-analytical numerical method for time-dependent radiative transfer problems in slab geometry with coherent isotropic scattering

We describe a semi-analytical numerical method for coherent isotropic scattering time-dependent radiative transfer problems in slab geometry. This numerical method is based on a combination of two classes of numerical methods: the spectral methods and the Laplace transform (LTS_N) methods applied to the radiative transfer equation in the discrete ordinates (S_N) formulation. The basic idea is to use the essence of the spectral methods and expand the intensity of radiation in a truncated series of Laguerre polynomials in the time variable and then solve recursively the resulting set of “time-independent” S_N problems by using the LTS_N method. We show some numerical experiments for a typical model problem.