Transient Radiative Transfer Research Articles

Numerical Analysis of Non-Fourier Model-Based Bio-Heat Transfer in the Laser-irradiated Axisymmetric Living Tissue.

The current work is related to the numerical investigation of non-Fourier heat transfer inside the short-pulsed laser-irradiated axisymmetric soft tissue phantom. It utilizes the modified discrete ordinate method to solve the transient radiative transfer equation (TRTE) for determining the intensity field. The laser energy absorbed by the soft tissue phantom behaves like a source in the Fourier/non-Fourier heat conduction model based-bio-heat transfer equation (BHTE), which is solved by employing the finite volume method (FVM) to determine the temperature distribution. Despite the prevalent use of non-Fourier BHTE for this purpose, a second law analysis is considered crucial to detect any potential anomalies. Equilibrium entropy production rates (EPR) are initially computed based on classical irreversible thermodynamics (CIT), which may yield negative values, possibly contravening the second law. Consequently, the EPR based on CIT is adjusted using the extended irreversible thermodynamics (EIT) hypothesis to ensure positivity. After that, the current research findings are compared with the results from the literature, and found good agreement between them. Then, the independent study is performed to select the optimum grid size, control angle size, and time step. A comparative analysis of results between the traditional Fourier and non-Fourier models has been performed. The impact of different parameters on the temperature fields and EPRs, is discussed. The effect of the optical properties of the inhomogeneity on the temperature distribution has been investigated. This study may help to enhance the effectiveness of the laser-based photo-thermal therapy.

Thermal relaxation parameters estimation of non-Fourier heat transfer in laser-irradiated living tissue using the Levenberg-Marquardt algorithm

The present work is concerned with solving the inverse heat transfer problem (IHTP) to estimate the thermal relaxation parameters corresponding to the dual phase lag (DPL) model-based bio-heat transfer in laser-irradiated living tissue. This problem is divided into two parts: direct and inverse problems. The solution of the inverse problem has been obtained using the Levenberg–Marquardt algorithm. On the other hand, the solution to the direct problem is obtained by solving the DPL model-based bio-heat transfer equation (BHTE) with the transient radiative transfer equation (TRTE). So, the TRTE has been numerically solved by the modified discrete ordinate method (DOM) to find the intensity field in the living tissue subjected to short-pulsed laser radiation, and it acts as a heat source in the DPL model-based BHTE. So, the DPL model-based BHTE has been analytically solved by employing the finite integral transform technique to get the temperature variation inside the laser-irradiated living tissue. First, the computer code developed for estimating unknown thermal relaxation parameters using the Levenberg–Marquardt algorithm has been verified against the data given in the literature, and found good agreement between them. Then, the thermal response of laser-irradiated living tissue has been investigated. After that, the influence of sensor position on the sensitivity of the IHTP solution has been discussed. The impact of the total transient readings and the measurement error on estimating the unknown parameters have been investigated. The present study’s findings can significantly contribute to various parameter estimation problems where the conventional methods for direct measurements of parameters are impossible or very difficult. Furthermore, the current results may considerably impact the study of the thermic response of living tissue during laser-based photothermal therapy.

Lattice Boltzmann simulations on transient radiative transfer problems in irregular geometries with constant or graded refractive index

This work is concerned with the transient radiative transport in inhomogeneous two- or three-dimensional semitransparent graded-index media with irregular geometry. The lattice Boltzmann method is applied to the graded-index media mapped with non-orthogonal grids to solve the radiative transfer while the finite volume method expresses the angular processing. The solution of the transient radiation from the methodology developed is firstly validated with good accuracy by considering benchmark solutions available in the literature for comparisons for various forms of geometry including a cube, elliptic cylinder as well as irregular shapes. Afterwards, the solver is used to analyze the radiative transfer in graded-index media including regular and irregular geometries, and various governing parameters such as laser and optical properties, scattering laws, incident angle of the incoming beam, refractive index, and temperature distributions. Suitable calculations of the transient radiometric heat flux and incident energy help to evaluate the time-resolved reflectance and transmittance. The results from this study indicate that the present methodology and boundary treatments are simple and suitable to deal with multidimensional radiative transfer in irregular domains.

Thermodynamic stability analysis of non‐Fourier model‐based bio‐heat transfer in laser‐irradiated cylindrical‐shaped multilayered biological tissue

AbstractThe present research focuses on examining the thermic response of living tissue in the form of a triple‐layered cylindrical structure when subjected to laser light and the compatibility analysis of non‐Fourier heat transfer with thermodynamics second law. The temperature field in the triple‐layered cylindrical living tissue subjected to laser light is determined by numerically solving the transient radiative transfer equation in conjunction with the dual phase lag (DPL) based bioheat equation. Once the temperature field is known, the equilibrium and nonequilibrium entropy production rate (EPR) is calculated based on the hypothesis of classical irreversible thermodynamics and extended irreversible thermodynamics, respectively. The present results are verified against the data from the literature and found a good match between them. A comparative analysis of the Fourier and non‐Fourier models is accomplished. The equilibrium and nonequilibrium EPR values for the Fourier model are found to be positive. While the equilibrium EPR is negative for non‐Fourier heat conduction and does not satisfy the thermodynamics second law, nonequilibrium EPR is always a positive value for Fourier, DPL, and hyperbolic models and satisfies thermodynamics second law. It has been investigated how thermal relaxation times affect the temperature field and EPRs in tissue are subjected to laser light.

Transient radiative transfer in two graded – index slab adhered by an infinitely thin vacuum via the lattice Boltzmann method

The lattice Boltzmann method is presented to solve the radiative transport of both diffuse and collimated problems in inhomogeneous and semitransparent two – layered slab with graded refractive indices. The two layers are bounded by Fresnel's walls and separated by an infinitely thin vacuum interlayer with negligible thickness to represent the case with surface roughness or assembling of two layers using a transparent binder or intumescent resin. The high – order discrete ordinates is used to simulate the angular redistribution due to variable refractive indices and multiple reflections at the boundaries and interface. The accuracy and efficiency of the methodology is firstly tested for transient and steady state solution in two – layered slab with constant or variable refractive indices. Effects of the distributions of refractive indices and surrounding, scattering albedos, and the optical thicknesses on the transient radiative signals are examined. Calculation results show that, these parameters have a significant influence on radiative transfer and the angular intensities at different few times levels are presented to illustrate the effects of the interface characteristics, and the discontinuous nature of the intensity at the interface is discussed. The developed LBM is a promising, efficient and simple solver to perform the radiative transport in inhomogeneous layered media with graded – index functions.

Three-dimensional mesoscopic simulation of radiative transfer in graded index media

Light propagation in three-dimensional (3D) graded index (GRIN) media has a complex spatial path, making it difficult to solve thermal radiative transfer problems. In this paper, a mesoscopic lattice Boltzmann method (LBM) that implements simple steady-state and transient radiative transfer solutions in 3D GRIN media is proposed. During the construction of the lattice Boltzmann (LB) model, the concept of the “local radiative transfer direction”, which addresses complex curved trajectories in 3D GRIN media simply, is introduced. The radiative transfer equation of 3D GRIN media can be strictly recovered from the LB model by adopting the Chapman-Enskog analysis. Numerical results indicate that radiative transfer problems in 3D GRIN media can be solved effectively by the LBM. Additionally, the influences of different optical parameters on steady-state and transient radiative transfer in 3D GRIN media are discussed. The LBM is a promising mathematical tool for radiative transfer in 3D GRIN media.

Predicting radiative intensity in transient radiative transfer with an inhomogeneous medium by spectral element discontinuous Galerkin method

Accurate prediction of the transient radiative intensity within an inhomogeneous medium is very useful in laser pulse technology. In the present work, a spectral element discontinuous Galerkin method (SEDGM) is proposed to solve transient radiative transfer with two kinds of boundary conditions, i.e., intense diffuse irradiation and short square laser irradiation. In the solving processing, the discontinuous elements are adopted to capture the large variety of radiative intensity on the element boundary. Then, the implicit temporal differencing scheme is introduced to deal with the time discretization. Several test cases are chosen to test the accuracy of the SEDGM. Results show that: (1) the SEDGM can provide good accuracy and efficiency for transient radiative transfer in an inhomogeneous medium; (2) the SEDGM has h-p convergence characteristics. Otherwise, the angle-time distributions of radiative intensities at boundaries are exhibited, and the influences of pulse duration and scattering phase function are investigated.

Conjugate heat transfer analysis of laser-irradiated cylindrical-shaped biological tissue embedded with the optical inhomogeneity

The current study is associated with the numerical investigation of the thermal behavior of cylindrical-shaped biological tissue during laser-based photo-thermal therapy. The light-tissue interaction has been modeled using the transient radiative transfer equation (TRTE). The TRTE in a cylindrical coordinate system is solved using the modified discrete ordinate method to obtain the intensity distribution inside the biological tissue. The solution of TRTE is coupled with Pennes bio-heat transfer equation (BHTE) to understand the thermic response of biological tissue. The Pennes BHTE is solved using the finite volume method (FVM). The two different types of optical inhomogeneity (absorption and scattering inhomogeneity) are considered in the current study. The inhomogeneity's optical and thermophysical properties may be the same/different from the homogeneous biological tissue. This conjugate heat transfer problem (CHTP) is solved using the harmonic mean technique. First, the present result is verified with the data available in the literature. Subsequently, the effect of inhomogeneity's optical properties on the temperature distribution is investigated. A comparative study between the same and different thermophysical properties of the biological tissue and inhomogeneity is performed. This study will help to accurately model the thermal characteristics of the laser-irradiated biological tissue having different thermophysical properties than the inhomogeneity.

Transient three-dimensional radiative heat transfer analysis of a turbulent diffusion flame-with a focus on the effect of angular discretization scheme

Radiation modeling is a challenging and computational expensive task for turbulent combustion modeling of various fuels. Radiative intensity varies greatly in the angular direction due to turbulent fluctuations of temperature and species fields. A large number of rays are needed in the discrete ordinates method (DOM) or the finite volume method (FVM) to describe the angular variations. Determining the adequate number of rays is essential for the efficient modeling of radiation heat transfer. For this purpose, the effect of angular scheme on predicting the transient three-dimensional radiation heat transfer of a turbulent diffusion flame is investigated. Results show that a smaller number of rays produce strong ray effects and may generate significant errors for radiative heat flux prediction on surrounding walls, while the angular scheme has negligible influence on the radiative heat source. Transient radiative heat fluxes on different walls are also analyzed in detail, and they show completely different contours with unimodal frequency behavior. Recommendation for radiation modeling strategies is proposed for turbulent combustion modeling.

Optical Tomography in Variable Index Media Using the Transient Discrete Transfer Method

This paper presents the optical tomography based on the reconstruction of optical properties in one- and two-dimensional variable index media. A transient discrete transfer method is used to solve the forward problem of transient radiative transfer. A multistart conjugate gradient method is applied to solve the inverse problem, and a straightforward approach is used to obtain the sensitivity coefficients. Different numerical examples are presented to show the efficiency of this method. The results show that the proposed algorithm can effectively estimate the distribution of absorption and scattering coefficients, even with noisy measured data.

Numerical investigation on transient third-grade magnetized nanofluid flow and radiative convection heat transfer from a stationary/moving cylinder: nanomaterial and nanoparticle shape effects

In this study, a mathematical model is developed for analyzing the time-dependent magneto-convective flow and heat transfer characteristics of an electrically conducting (functional) third-grade Reiner-Rivlin non-Newtonian nanofluid from a moving or stationary hot cylinder in the presence of magnetic field and thermal radiation. A well-tested convergent Crank–Nicolson type finite difference algorithm is employed to solve the transformed, nonlinear boundary value problem. The Tiwari-Das nanofluid volume fraction model is adopted for nanoscale effects and the Rosseland algebraic flux model for radiative heat flux effects. It has been shown that the shape of nanoparticles remarkably contributes to the enhancement of heat transfer. Several metallic nanoparticle types such as Al2O3, Cu, and TiO2 are examined. It is found from the investigation that the viscoelastic nanofluid with TiO2 nanoparticles results in more heat transfer than the other nanoparticles. Lower velocity and higher temperature values are computed at transient conditions with a higher third-grade fluid parameter for the flow of nanofluid (Al2O3-SA). The plots of transient friction and heat transfer coefficients are visualized at the surface of a hot cylinder. The tabulated heat transfer coefficient is comparatively more for the moving cylinder than the stationary cylinder. Detailed validation of results of the numerical scheme with previous studies is included. The simulations find applications in coating deposition (enrobing) of magnetic nanomaterial at high temperatures, functional nanomaterial synthesis, etc.

On the performance of a MRT lattice Boltzmann algorithm for transient radiative transfer problems

An exact multiple-relaxation-time lattice Boltzmann (MRT-LB) algorithm without any assumptions or approximations is raised for solving time-dependent radiative transfer problems (RTPs). The macroscopic radiative transfer equation (RTE) is exactly recovered from the mesoscopic lattice Boltzmann equation (LBE). The proposed MRT-LB algorithm is utilized to obtain the time-dependent radiation distribution with strong variations. Analysis of its numerical performance, including the accuracy and convergence order, is carried out. Results demonstrate that the present MRT-LB is accurate for solving the RTE, with a convergence order of two. The MRT-LB has a good ability to capture the time-dependent properties, including the radiation intensity, radiative heat flux, as well as wavefront, for the radiation propagation problems.

Discontinuous finite element method applied to transient pure and coupled radiative heat transfer

The discontinuous finite element method (DFEM) is applied to transient pure and coupled radiative heat transfer problems in participating media. The DFEM discretization is presented for the time-dependent governing equations that take the full time-dependent term into account. Transient pure radiative transfer problems in a slab medium illuminated by a collimated Gaussian pulse and in a vacuum gapped medium subjected to diffuse irradiation is solved via the DFEM. The DFEM is further extended to transient radiation-conduction problems in two-dimensional media. The DFEM solutions of time-dependent radiation-intensity-based quantities are verified to be highly accurate and they are presented in the form of tabulated numbers with four decimal digits.

A multiple-relaxation-time lattice Boltzmann model for radiative transfer equation

A novel multiple-relaxation-time (MRT) lattice Boltzmann model is proposed for the radiative transfer equation (RTE). In this paper, the discussion and implementation are restricted to the grey (frequency-independent) radiative transfer equation. We establish this model by regarding the RTE as a particular convection–diffusion equation without the diffusion term. The equilibrium distribution function for the RTE is constructed, and the RTE is rigorously derived from the lattice Boltzmann equation via the Maxwell iteration under the diffusive scaling. Several numerical cases are conducted, including one-dimensional, two-dimensional and three-dimensional transient and steady cases. Results demonstrate that this MRT lattice Boltzmann model is accurate and stable for solving transient and steady radiative transfer problems in participating media. Besides, the second-order accuracy and convergence rate are observed for this model. This model may be further extended to various radiative transfer problems in participating media.

FTn Finite Volume Analysis of Ultrafast Laser Radiation Transport through Human Skin Cancer

In this work, we develop a numerical tool for the early detection of skin cancer using a 3D numerical transient radiative heat transfer study of ultrafast-laser transport through normal and malignant human skins. The curved-line advection method (CLAM) spatial scheme and the FTn angular scheme of the finite volume method (FVM) are investigated to analyze the above-cited physical phenomena. Both Fresnel specular and diffuse boundary conditions are analyzed. Human skin is considered based on optical properties available from other sources. The temporal radiative signals of skin with malignancies were compared to those of normal skin. Malignancies in the basal layer and epidermis were simulated. Further, the effects of laser light wavelength and the volume of the cancerous region and its scattering coefficient on these signals were studied. The results show that (1) the effect of the Fresnel boundary in the modeling was pronounced; (2) the peak magnitude for human skin with the cutaneous melanoma (CM) had the maximum value in comparison with those obtained for the two other malignancies; and (3) when cancer fully affected the living epidermis with any of these malignancies, the reflectance was more than its predicted value when affected by the basal only.

Transient radiative transfer in variable index media using the discrete transfer method

The discrete transfer method is adopted to solve the transient radiative transfer with collimated or diffuse irradiation in variable index media with arbitrary distribution of refractive index. The advantage of this approach is its ability to simulate the transient radiation in semi-transparent media with non-continuous refractive index distributions. The solution method is verified by comparison the results with those reported in literature for one, two and three-dimensional cases. The reflectance and transmittance for some locations over the boundaries are obtained, and the effects of optical thickness, scattering albedo, and scattering phase function are examined.

Transient radiative transfer in semi-transparent slab with arbitrary refractive index and collimated irradiation

The variable discrete ordinates method is adopted to solve the transient radiation heat transfer in a semi-transparent slab with arbitrary refractive index distribution, exposed to collimated or diffuse irradiation. Influences of the optical thickness, scattering albedo, and anisotropically scattering for different refractive index profiles are considered. The proficiency of the present method is tested by results of other validated methods. The privilege of this approach is to handle the semi-transparent media with non-continuous and non-monotonic refractive index distributions.

Solving steady and transient radiative transfer problems with strong inhomogeneity via a lattice Boltzmann method

A complete lattice Boltzmann method (LBM) is developed to solve the steady and transient radiative transfer problems with strong inhomogeneity. The transient radiative transfer equation (RTE) is regarded as a hyperbolic conservation law equation and then solved by the LBM. A forward difference scheme is applied to the time derivative of the source term in the evolution equation, which makes the LBM accurate and stable for radiative transfer problems with strong inhomogeneity. The correctness and accuracy of the proposed LBM is verified first. Afterward, the LBM is applied to solve the one-dimensional radiative transfer problems involving strong inhomogeneous radiative intensity and discontinuous medium system, including a planar slab subjected to Gaussian-shaped source, an infinite slab with an inclusion layer, and a vacuum gapped medium system. The time-resolved radiation is presented and analyzed as well as the steady results. Finally, the present LBM is extended to multidimensional radiative transfer problems.

A matrix model of particle-scale radiative heat transfer in structured and randomly packed pebble bed

Thermal radiation in pebble bed is very hard to compute accurately and efficiently since it occurs between the surfaces no matter how far they are. Herein, radiative heat flux and effective thermal conductivity were derived mathematically by a matrix model to present the thermal radiation between particles in pebble beds. For structured packing, the transient radiative heat transfer is proven to be similar to thermal diffusion. This study proposed a correlation to predict the radiation exchange factor under different porosity. Radiative effective thermal conductivity was found to grow slightly with void fractions. For random pebble beds, a regression model of the multi-layer neural network was trained by large datasets to compute the view factor matrix efficiently. With the trained model and GPU acceleration, it is feasible to perform real-time-simulation of full-range radiations inside large-scale pebble beds, even for real reactors (e.g. HTR-PM). A thermal radiation simulation of the decay heat removal of a real reactor adopted this model as a demonstrative application and showed that the highest temperature is still within the design limitation of the packed bed.

Three-dimensional FTn finite volume solution of short-pulse laser propagation through heterogeneous medium

In this paper, 3-D heterogeneous medium, containing small inhomogeneous zones, subjected to a short-pulse laser has been examined by solving the transient radiative transfer equation. Both curved-line advection method and STEP schemes of the FTn finite volume method have been applied. The curved-line advection method predictions proved that a decrease of the false scattering and ray effects are obtained. In fact, there was a good agreement between the curved-line advection method and the Monte Carlo method. However, the STEP results are slightly mismatching the predictions of the aforementioned reference method. Then, the effects of the absorption coefficient, the size, the number and the position of inhomogeneous zone on the transmittance and reflectance signals have been analyzed. The predictions showed that the increase of the size of the inhomogeneity reduces the intensity of radiation. For both homogenous and heterogonous medium, the change of the detector position varies both the broadening of the signal pulse-width and the time with peak reflectance and/or transmittance. That is can be explained by the effects of the distance and the medium property between the laser-incident source and the detector position. Thus, these both parameters are the main factors for determining the peak position and the pulse broadening. Finally, the effects of the absorption coefficient in the inhomogeneity zone on the absolute values of logarithmic slope has been discussed. The results proved that the absolute values of logarithmic slope may be a perfect indicator for detecting any abnormal absorbing zones in the medium.