Hybrid integral transforms analysis of the bioheat equation with variable properties

Pennes’ equation is the most frequently employed model to describe heat transfer processes within living tissues, with numerous applications in clinical diagnostics and thermal treatments. A number of analytical solutions were provided in the literature that represent the temperature distribution across tissue structures, but considering simplifying assumptions such as uniform and linear thermophysical properties and blood perfusion rates. The present work thus advances such analysis path by considering a heterogeneous medium formulation that allows for spatially variable parameters across the tissue thickness. Besides, the eventual variation of blood perfusion rates with temperature is also accounted for in the proposed model. The Generalized Integral Transform Technique (GITT) is employed to yield a hybrid numerical–analytical solution of the bioheat model in heterogeneous media, which reduces to the exact solution obtained via the Classical Integral Transform Method for a linear formulation with uniform coefficients. The open source UNIT code (“Unified Integral Transforms”) is utilized to obtain numerical results for a set of typical values of the governing parameters, in order to illustrate the convergence behavior of the proposed eigenfunction expansions and inspect the importance of accounting for spatially variable properties in predicting the thermal response of living tissues to external stimulus.Indisponível

Heat Transfer in Microchannels with Upstream–Downstream Regions Coupling and Wall Conjugation Effects

Heat transfer in microchannels is analyzed, including the coupling between the regions upstream and downstream of the heat transfer section and taking into account the wall conjugation and axial diffusion effects which are often of relevance in microchannels. The methodology is based on a recently proposed single-domain formulation for modeling the heat transfer phenomena simultaneously at the fluid stream and the channel walls, and applying the generalized integral transform technique (GITT) to find a hybrid numerical–analytical solution to the unified partial differential energy equation. The proposed mathematical model involves coefficients represented as space-dependent functions, with abrupt transitions at the fluid–wall interfaces, which carry the information concerning the transition of the two domains, unifying the model into a single-domain formulation with variable coefficients. Convergence of the proposed eigenfunction expansions is thoroughly investigated and the physical analysis is focused on the effects of the coupling between the downstream and the upstream flow regions.Indisponível

Enhanced convergence of eigenfunction expansions in convection-diffusion with multiscale space variable coefficients

A convergence enhancement technique known as the integral balance approach is employed in combination with the Generalized Integral Transform Technique (GITT) for solving diffusion or convection-diffusion problems in physical domains with subregions of markedly different materials properties and/or spatial scales. GITT is employed in the solution of the differential eigenvalue problem with space variable coefficients, by adopting simpler auxiliary eigenproblems for the eigenfunction representation. The examples provided deal with heat conduction in heterogeneous media and forced convection in a microchannel embedded in a substrate. The convergence characteristics of the proposed novel solution are critically compared against the conventional approach through integral transforms without the integral balance enhancement, with the aid of fully converged results from the available exact solutions.Indisponível

Theoretical analysis of conjugated heat transfer with a single domain formulation and integral transforms

The present work advances an analytical approach for conjugated conduction-convection heat transfer problems, by proposing a single domain formulation for modeling both the fluid stream and the channel wall regions. Making use of coefficients represented as space variable functions with abrupt transitions occurring at the fluid-wall interface, the mathematical model is fed with the information concerning the transition of the two domains, unifying the model into a single domain formulation with space variable coefficients. The Generalized Integral Transform Technique (GITT) is then employed in the hybrid numerical-analytical solution of the resulting convection-diffusion problem with variable coefficients, and critically compared for two alternative solution paths. A test problem is chosen that offers an exact solution for validation purposes, based on the extended Graetz problem including transversal conduction across the channel walls. The excellent agreement between approximate and exact solutions demonstrates the feasibility of the approach in handling more involved conjugated problems.Indisponível

Nonlinear eigenvalue problem in the integral transforms solution of convection-diffusion with nonlinear boundary conditions

The purpose of this paper is to propose the generalized integral transform technique (GITT) to the solution of convection-diffusion problems with nonlinear boundary conditions by employing the corresponding nonlinear eigenvalue problem in the construction of the expansion basis. The original nonlinear boundary condition coefficients in the problem formulation are all incorporated into the adopted eigenvalue problem, which may be itself integral transformed through a representative linear auxiliary problem, yielding a nonlinear algebraic eigenvalue problem for the associated eigenvalues and eigenvectors, to be solved along with the transformed ordinary differential system. The nonlinear eigenvalues computation may also be accomplished by rewriting the corresponding transcendental equation as an ordinary differential system for the eigenvalues, which is then simultaneously solved with the transformed potentials. An application on one-dimensional transient diffusion with nonlinear boundary condition coefficients is selected for illustrating some important computational aspects and the convergence behavior of the proposed eigenfunction expansions. For comparison purposes, an alternative solution with a linear eigenvalue problem basis is also presented and implemented. This novel approach can be further extended to various classes of nonlinear convection-diffusion problems, either already solved by the GITT with a linear coefficients basis, or new challenging applications with more involved nonlinearities.Indisponível

Fluid flow and conjugated heat transfer in arbitrarily shaped channels via single domain formulation and integral transforms

The present work advances a recently introduced approach based on combining the Generalized Integral Transform Technique (GITT) and a single domain reformulation strategy, aimed at providing hybrid numerical–analytical solutions to convection–diffusion problems in complex physical configurations and irregular geometries. The methodology has been previously considered in the analysis of conjugated conduction–convection heat transfer problems, simultaneously modeling the heat transfer phenomena at both the fluid streams and the channels walls, by making use of coefficients represented as space variable functions with abrupt transitions occurring at the fluid–wall interfaces. The present work is aimed at extending this methodology to deal with both fluid flow and conjugated heat transfer within arbitrarily shaped channels and complex multichannel configurations, so that the solution of a cumbersome system of coupled partial differential equations defined for each individual sub-domain of the problem is avoided, with the proposition of the single-domain formulation. The reformulated problem is integral transformed through the adoption of eigenvalue problems containing the space variable coefficients, which provide the basis of the eigenfunction expansions and are responsible for recovering the transitional behavior among the different regions in the original formulation. For demonstration purposes, an application is first considered consisting of a microchannel with an irregular cross-section shape, representing a typical channel micro-fabricated through laser ablation, in which heat and fluid flow are investigated, taking into account the conjugation with the polymeric substrate. Then, a complex configuration consisting of multiple irregularly shaped channels is more closely analyzed, in order to illustrate the flexibility and robustness of the advanced hybrid approach. In both cases, the convergence behavior of the proposed expansions is presented and critical comparisons against purely numerical approaches are provided.Indisponível

Inverse analysis with integral transformed temperature fields: Identification of thermophysical properties in heterogeneous media

The objective of this work is to introduce the use of integral transformed temperature measured data for the solution of inverse heat transfer problems, instead of the common local transient temperature measurements. The proposed approach is capable of significantly compressing the measured data through the integral transformation, without losing the information contained in the measurements and required for the solution of the inverse problem. The data compression is of special interest for modern measurement techniques, such as the infrared thermography, that allows for fine spatial resolutions and large frequencies, possibly resulting on a very large amount of measured data. In order to critically address the use of integral transformed measurements, we examine in this paper the simultaneous estimation of spatially variable thermal conductivity and thermal diffusivity in one-dimensional heat conduction within heterogeneous media. The direct problem solution is analytically obtained via integral transforms and the related eigenvalue problem is solved by the Generalized Integral Transform Technique (GITT). The inverse problem is handled with Bayesian inference by employing a Markov Chain Monte Carlo (MCMC) method. The unknown functions appearing in the formulation are expanded in terms of eigenfunctions as well, so that the unknown parameters become the corresponding series coefficients. Such projection of the functions in an infinite dimensional space onto a parametric space of finite dimension also permits that several quantities appearing in the solution of the direct problem be analytically computed. Simulated measurements are used in the inverse analysis; they are assumed to be additive, uncorrelated, normally distributed, with zero means and known covariances. Both Gaussian and non-informative uniform distributions are used as priors for demonstrating the robustness of the estimation procedure.Indisponível

Inverse analysis of forced convection in micro-channels with slip flow via integral transforms and Bayesian inference

The present work addresses the direct and inverse problems for convective heat transfer with incompressible laminar gas flow in micro-channels, within the range of validity of the slip-flow regime. The direct problem analysis combines the classical integral transform method and the generalized integral transform technique (GITT), by analytically solving the two-dimensional steady-state convection problem and finding a hybrid numerical-analytical solution for the required eigenvalue problem. The inverse problem analysis makes use of the accuracy and robustness of the direct problem solution and focus on the simultaneous identification of the momentum and thermal accommodation coefficients, related to gas flow and heat transfer within micro-channels, besides the usually unknown boundary condition parameters, here represented by the external Biot number. The inverse analysis is based on the availability solely of temperature measurements at the channel external wall, along its length, as obtained for instance via infrared camera thermography. A Bayesian inference approach is adopted in the solution of the identification problem based on the Monte Carlo Markov Chain method (MCMC) and the Metropolis-Hastings sampling algorithm. A typical example of slip flow in parallel-plates micro-channel is selected to illustrate both the direct and inverse problems solution approaches.Indisponível

Integral Transforms and Bayesian Inference in the Identification of Variable Thermal Conductivity in Two-Phase Dispersed Systems

This work illustrates the use of Bayesian inference in the estimation of spatially variable thermal conductivity for one-dimensional heat conduction in heterogeneous media, such as particle-filled composites and other two-phase dispersed systems, by employing a Markov chain Monte Carlo (MCMC) method, through the implementation of the Metropolis-Hastings algorithm. The direct problem solution is obtained analytically via integral transforms, and the related eigenvalue problem is solved by the generalized integral transform technique (GITT), offering a fast, precise, and robust solution for the transient temperature field, which are desirable features for the implementation of the inverse analysis. Instead of seeking the function estimation in the form of a sequence of local values for the thermal conductivity, an alternative approach is proposed here, which is based on the eigenfunction expansion of the thermal conductivity itself. Then, the unknown parameters become the corresponding series coefficients. Simulated temperatures obtained via integral transforms are used in the inverse analysis. From the prescription of the concentration distribution of the dispersed phase, available correlations for the thermal conductivity are employed to produce the simulated results with high precision in the direct problem solution, while eigenfunction expansions with reduced number of terms are employed in the inverse analysis itself, in order to avoid the so-called inverse crime. Both Gaussian and noninformative uniform distributions were used as priors for comparison purposes. In addition, alternative correlations for the thermal conductivity that yield different predictions are also employed as Gaussian priors for the algorithm in order to test the inverse analysis robustness.Indisponível

Improved lumped model for thermal analysis of high burn-up nuclear fuel rods

High burn-up nuclear fuel elements have been intensively studied for prolonged lifetime of existing reactors and for next-generation advanced reactors. This paper presents an improved lumped-differential formulation for one-dimensional transient heat conduction in a heat generating cylinder with temperature-dependent thermo-physical properties typical of high burn-up nuclear fuel rods. Two-points Hermite approximations for integrals (H1,1/H1,1) were used to obtain the average temperature and heat flux in the radial direction. As a testing case, transient heat conduction in a nuclear fuel rod was computed with the thermo-physical properties represented by the MATPRO correlations. The problem was formulated and solved by using the symbolic/numerical computation software system MATHEMATICA. The solutions of the proposed improved lumped model were validated by comparing with numerical solutions of the original distributed parameter formulation of the transient heat conduction problem, as well as by comparing with simulation results of a cold leg SBLOCA with RELAP5/MOD3.Indisponível