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

ABSTRACT

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.     

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.     

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.     

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.     

A review of hybrid integral transform solutions in fluid flow problems with heat or mass transfer and under Navier–Stokes equations formulation

Abstract

The Generalized Integral Transform Technique (GITT) is reviewed as a hybrid numerical–analytical approach for fluid flow problems, with or without heat and mass transfer, here with emphasis on the literature related to flow problems formulated through the full Navier–Stokes equations. A brief overview of the integral transform methodology is first provided for a general nonlinear convection–diffusion problem. Then, different alternatives of eigenfunction expansion strategies are discussed in the integral transformation of problems for which the fluid flow model is either based on the primitive variables or the streamfunction-only formulations, as applied to both steady and transient states. Representative test cases are selected to illustrate the different eigenfunction expansion approaches, with convergence being analyzed for each situation. In addition, fully converged integral transform results are critically compared to previously reported simulations obtained from traditional purely discrete methods.     

Improved lumped-differential formulations and hybrid solution methods for drying in porous media

The present paper illustrates the construction of hybrid tools for the problem formulation and solution methodology towards the simulation of heat and mass transfer during drying in capillary porous media. First, a problem reformulation strategy is discussed, known as the coupled integral equations approach (CIEA), which offers improved lumped-differential formulations in different classes of problems, in comparison against classical lumping schemes, allowing for a reduction on the number of independent variables to be considered in specific formulations. Second, the generalized integral transform technique (GITT) is employed, as a hybrid numerical–analytical solution methodology for convection–diffusion problems. An example is provided related to drying in capillary porous cylindrical media as formulated by the two-dimensional Luikov's system of equations.     

Analytical solution of the advection–diffusion transport equation using a change-of-variable and integral transform technique

This paper presents a formal exact solution of the linear advection–diffusion transport equation with constant coefficients for both transient and steady-state regimes. A classical mathematical substitution transforms the original advection–diffusion equation into an exclusively diffusive equation. The new diffusive problem is solved analytically using the classic version of Generalized Integral Transform Technique (GITT), resulting in an explicit formal solution. The new solution is shown to converge faster than a hybrid analytical–numerical solution previously obtained by applying the GITT directly to the advection–diffusion transport equation.     

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.     

Laminar counterflow parallel-plate heat exchangers: Exact and approximate solutions

Multilayered, counterflow, parallel-plate heat exchangers are analyzed numerically and theoretically. The analysis, carried out for constant property fluids, considers a hydrodynamically developed laminar flow and neglects longitudinal conduction both in the fluid and in the plates. The solution for the temperature field involves eigenfunction expansions that can be solved in terms of Whittaker functions using standard symbolic algebra packages, leading to analytical expressions that provide the eigenvalues numerically. It is seen that the approximate solution obtained by retaining the first two modes in the eigenfunction expansion provides an accurate representation for the temperature away from the entrance regions, specially for long heat exchangers, thereby enabling simplified expressions for the wall and bulk temperatures, local heat-transfer rate, overall heat-transfer coefficient, and outlet bulk temperatures. The agreement between the numerical and theoretical results suggests the possibility of using the analytical solutions presented herein as benchmark problems for computational heat-transfer codes.