He-Ping Tan

Temperature field of radiative equilibrium in a semitransparent slab with a linear refractive index and gray walls

Abstract The temperature field in a semitransparent slab of absorbing–emitting gray medium at radiative equilibrium is solved in this paper. The medium has a linear refractive index and the two boundaries are diffuse gray walls. A curved ray tracing technique is combined with a pseudo-source adding method to deduce the radiative intensities on the gray walls. And on the basis of the previous work done by Ben Abdallah and Le Dez, the discrete temperature field in the slab is deduced. The influences of refractive index distribution, boundary wall emissivities and optical thickness on the radiative equilibrium temperature field are examined. The results display the significant influences of the refractive index distribution and the boundary wall emissivities.

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.

Radiative intensity solution and thermal emission analysis of a semitransparent medium layer with a sinusoidal refractive index

The radiative intensity in a sinusoidal refractive index semitransparent medium layer is solved by the curved ray-tracing method in combination with the pseudo-source adding method. One boundary of the medium layer is an opaque diffuse substrate wall. The other boundary is a semitransparent specular or diffuse surface, from which the medium thermal emission emerges. With considering a linear temperature distribution, the radiative intensity formulae are, respectively, deduced under the two boundary conditions. On the basis of the radiative intensity solutions, the directional and hemispherical emission of the medium layer with a specular surface as well as the hemispherical emission of that with a diffuse surface are calculated. The influences of the optical thickness, sinusoidal refractive index distribution and linear temperature distribution on the thermal emission are investigated. The results show that the effects of refractive index and temperature distribution are significant and are different under the two reflecting modes of the surface.

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.

Transient coupled radiative and conductive heat transfer in an absorbing, emitting and scattering medium

Abstract On the basis of our previous papers, the redistribution of radiative energy in the case of isotropic scattering is investigated and the radiative transfer coefficient (RTC) under specular reflection in an absorbing, emitting and isotropic scattering parallel slab is derived. Considering both multi-reflection and multi-scattering in the derivation, the RTC can accommodate various boundary conditions under specular reflection. By accumulating the RTC for specular reflection boundary and that for diffuse reflection boundary linearly, the RTC are calculated. The validity and high precision of the formula for the RTC are confirmed by comparing with references. The effects of single-scattering albedo ω, Planck number Np and refractive index of STM nm on the transient coupled heat transfer in a one-dimensional isotropic scattering medium are reviewed for: (a) two semi-transparent boundaries; and (b) one semi-transparent boundary and one opaque boundary. The presented calculation and formula for the redistribution of the scattering energy can also be applied to other radiative calculations, such as total radiative exchange area or total radiative transfer coefficient in multi-dimensional isotropic scattering media.

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.

Simultaneous identification of temperature profile and wall emissivities in one-dimensional semitransparent medium by inverse radiation analysis

An inverse analysis is presented for simultaneous estimation of the temperature distribution and the boundary emissivities for an emitting, nonscattering, gray, plane-parallel medium from the knowledge of the exit radiation intensities at boundary surfaces. The inverse problem is solved by using a least squares method. In this approach a zeroth-order regularization method is used to stabilize the solution. The effects of the measurement errors on the accuracy of the inverse analysis are investigated. The study shows that the temperature profile can be estimated accurately, even with noisy data, and the estimation of boundary emissivities is more sensitive to increases in measurement errors.

Inverse radiation problem in three-dimensional complicated geometric systems with opaque boundaries

Abstract An inverse radiation analysis is presented to identify the three-dimensional source term distribution in complicated geometric systems of known radiative properties from the knowledge of the exit radiative intensities. The inverse radiation problem is formulated as an optimization problem, and solved by the conjugate gradient method that minimizes the errors between the calculated exit radiative intensities and the experimental data. The measured data are simulated by adding random errors to the exact solution of the direct problem. The analysis consists of the direct problem, the gradient equation, and the sensitivity problem. In this approach, the discrete ordinates method is employed to solve the direct and the sensitivity problems in general body-fitted coordinates. The effects of the measurement errors, single scattering albedo, and scattering asymmetry parameter on the accuracy of the inverse analysis are investigated. The study shows that the three-dimensional source term distribution in complicated geometric systems with opaque and diffuse-gray boundaries can be estimated accurately for the exact and noisy data.

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.

Transient Coupled Radiation and Conduction in an Absorbing and Scattering Composite Layer

A method is developed for obtaining transient temperatures and heat flux densities in a two-layer isotropic scattering semitransparent composite with spectrally dependent radiative properties. The radiative transfer coefficients of the absorbing and isotropic scattering composite are deduced by the ray tracing method in combination with Hottel and Sarofims zonal method. The boundary surfaces and the Internal interface are semitransparent, and the reflections are assumed to be diffuse or specular. The radiative heat source term is calculated by the radiative transfer coefficients. The transient energy equation is solved by the full implicit control-volume method in combination with the spectral band model. An advantage of the method is that needs only to disperse the space position, but not to disperse the solid angle. A comparison of the results, obtained under diffuse reflection, with previous results shows that the equations are correct, and the results are more accurate. The analysis includes the influences of the scattering albedo, the dimensionless thickness, the conduction-radiation parameter, the convection-radiation parameter, the spectral properties, and the reflective mode on the transient temperature field and the heat flux density