L.H. Liu

On the discrete ordinates method for radiative heat transfer in anisotropically scattering media

Abstract In the discrete ordinates method (DOM), the normalized condition for the numerical quadrature of some complex scattering phase functions may not be satisfied. In this paper, a revised discrete ordinates method (RDOM) is developed to overcome this problem, in which a renormalizing factor is added into the numerical quadrature of in-scattering term. The RDOM is used to solve the radiative transfer problem in one-dimensional anisotropically scattering media with complex phase function. The radiative heat fluxes obtained by the RDOM are compared with those obtained by the conventional discrete ordinates method (CDOM) and Monte Carlo method. The results show the RDOM can overcome the false scattering resulted from the numerical quadrature of in-scattering term and improve largely the accuracy of solution of the radiative transfer equation by comparison with the CDOM.

Inverse radiation problem of sources and emissivities in one-dimensional semitransparent media

Abstract An inverse analysis is presented for simultaneous estimation of the source term distribution and the boundary emissivity for an absorbing, emitting, anisotropic scattering, and gray plane–parallel medium with opaque and diffuse bounding surfaces from the knowledge of the exit radiation intensities and temperature at boundary surfaces. The inverse problem is formulated as an optimization problem that minimizes the errors between the exit radiation intensities calculated and the experimental data. The conjugate gradient method and the two-dimensional network searching method are used to solve the inverse problem. The effects of the measurement errors, anisotropic scattering, single-scattering albedo, optical thickness, and boundary emissivity on the accuracy of the inverse analysis are investigated. The results show that the source term and the boundary emissivity can be simultaneously estimated accurately for exact and noisy data, and the estimation of boundary emissivity is more sensitive to the measurement errors.

Temperature distributions in an absorbing-emitting-scattering semitransparent slab with variable spatial refractive index

Abstract A Monte Carlo curved ray-tracing method is used to analyze the radiative heat transfer in one-dimensional absorbing–emitting–scattering semitransparent slab with variable spatial refractive index. A problem of radiative equilibrium with linear variable spatial refractive index is taken as an example in this paper. The predicted temperature distributions are determined by the proposed method and compared with the data in references. The results show that influences of refractive index gradient are important and the influences increase with the refractive index gradient, the temperature distribution approaches to the one obtained for a constant refractive index when the slab optical thickness is far greater than 1.0, and the effect of the scattering phase function is similar to that in the medium with constant refractive index.

Non-Fourier effects on transient temperature response in semitransparent medium caused by laser pulse

Abstract The non-Fourier effects on transient temperature response in semitransparent medium with black boundary surfaces caused by laser pulse are studied. The processes of the coupled conduction and radiation heat transfer in a one-dimensional semitransparent slab with black boundaries are analyzed numerically. The hyperbolic heat conduction equation is solved by the flux-splitting method, and the radiative transfer equation is solved by the discrete ordinate method. The transient temperature response obtained from hyperbolic heat conduction equation is compared with those obtained from the classical parabolic heat conduction equation. The results show that the non-Fourier effect can be important when the conduction-to-radiation parameter and the thermal relaxation time of heat conduction are larger. Under this condition, for the laser-flash measurement of the thermal diffusivity in semitransparent materials, omitting the non-Fourier effect can result in significant errors.

Internal distribution of radiation absorption in one-dimensional semitransparent medium

Abstract The internal distribution of spectral radiation absorption in one-dimensional semitransparent slab, sphere and cylinder irradiated uniformly and isotropically is determined by the ray tracing method, and the detailed theory for computation are deduced. The computed results show that the internal distribution of spectral radiation absorption in slab differs from that in spherical and cylindrical particles. The peak of internal volumetric spectral radiation absorption may locate at interior shell of the semitransparent sphere and cylinder. The dimensionless volumetric spectral radiation absorption is higher near the center for weakly absorbing or small size parameter, but is higher near the surface for strongly absorbing or large spheres and cylinders. Refraction focuses the rays close to the sphere and cylinder center. With the increases of the refractive index, the dimensionless volumetric spectral radiation absorption increases near the sphere and cylinder centers and decreases near the sphere and cylinder surfaces.

Internal distribution of radiation absorption in a semitransparent spherical particle

The internal distribution of spectral radiation absorption in a semitransparent spherical particle irradiated uniformly and isotropically is determined by the ray tracing method, and the detailed computation formulae for the internal spectral radiation absorption are deduced. The computed results show that the peak of internal volumetric spectral radiation absorption may locate at the interior shell of the particle. The dimensionless volumetric spectral radiation absorption is higher near the center for weakly absorbing or small spheres, but the dimensionless volumetric spectral radiation absorption is higher near the surface for strongly absorbing or large spheres. The corresponding physical interpretations of the internal spectral absorption distribution are given.

Reconstruction of time-averaged temperature of non-axisymmetric turbulent unconfined sooting flame by inverse radiation analysis

Abstract A multi-wavelength inversion method is extended to reconstruct the time-averaged temperature distribution in non-axisymmetric turbulent unconfined sooting flame by the multi-wavelength measured data of low time-resolution outgoing emission and transmission radiation intensities. Gaussian, β and uniform distribution probability density functions (PDF) are used to simulate the turbulent fluctuation of temperature, respectively. The reconstruction of time-averaged temperature consists of three steps. First, the time-averaged spectral absorption coefficient is retrieved from the time-averaged transmissivity data by an algebraic reconstruction technique. Then, the time-averaged blackbody spectral radiation intensity is estimated from the outgoing spectral emission radiation intensities. Finally, the time-averaged temperature is approximately reconstructed from the multi-wavelength time-averaged spectral emission radiation data by the least-squares method. Noisy input data have been used to test the performance of the proposed inversion method. The results show that the time-averaged temperature distribution can be estimated with good accuracy, even with noisy input data. The accuracy of the estimation decreases with the increase of turbulent fluctuation intensity of temperature and the effects of assumed PDF on the reconstruction of temperature are small.

Inverse radiation problem of axisymmetric turbulent sooting free flame

An iteration method is extended to reconstruct the time-averaged temperature distribution in turbulent axisymmetric sooting free flame by the multi-wavelength measured data of low time-resolution outgoing emission and transmission radiation intensities. A Gaussian probability density function is used to simulate the turbulent fluctuation of temperature. The reconstruction of time-averaged temperature profile consists of three steps. First, the time-averaged spectral absorption coefficient is retrieved from the time-averaged transmissivity data by iteration method. Then the time-averaged blackbody spectral radiation intensity is estimated from the low time-resolution outgoing spectral emission radiation intensities. Finally, the time-averaged temperature and its standard deviation are approximately reconstructed from the multi-wavelength time-averaged spectral emission radiation data by the least-square method. Both exact and noisy input data have been used to test the performance of the proposed inversion method. The results show that the time-averaged temperature profiles can be estimated with good accuracy by the presented inversion method, even with noisy input data, and the standard deviation of temperature is more sensitive to the measurement errors. In the case of large temperature fluctuation, the errors of estimation for time-averaged temperature profile are large if the turbulent fluctuation is not taken into account.

Transient coupled radiation–conduction in infinite semitransparent cylinders

Abstract A method is developed to analyze the transient coupled radiation–conduction in infinite semitransparent cylinders surrounded by isothermal black walls. The radiative heat source term is calculated by the radiative transfer coefficients and the transient energy equation is solved by an implicit finite difference method. The radiative transfer coefficients are deduced by use of the ray tracing method in combination with the Hottel and Sarofim zonal method. The effects of the related parameters on the transient radiative heat source and temperature distribution are analyzed. It is found that the peak of the dimensionless radial radiative heat source can be located at the interior shell of the cylinder with small optical thickness when heated by the surrounding irradiation. Treating the volume radiation as a surface radiation will result in large errors of transient temperature distribution for the cylinder with small optical thickness.

Emissive power of semitransparent spherical particle with nonuniform temperature

The ray tracing method and the zonal method is extended to study the emissive power of semitransparent spherical particle with nonuniform temperature. The particle emissive power is calculated by the radiative transfer coefficients. The effects of the related parameters on the particle emissive power and the errors resulted from omitting the nonuniformity of particle temperature are analyzed and discussed. The results show that omitting the nonuniformity of particle temperature will result in large errors of particle emissive power, and the errors increase with the nonuniformity of particle temperature. The particle emissive power based on the average temperature may deviate from its real value, and the size parameter and absorption index strongly affects the ratio of particle emissive power based on the average temperature and its real value. 2002 Elsevier Science Ltd. All rights reserved.