Estimation of time-dependent heat flux using temperature distribution at a point in a two layer system

AbstractIn this paper, the conjugate gradient method, coupled with the adjoint problem, is used in order to solve the inverse heat conduction problem and estimation of the time-dependent heat flux, using temperature distribution at a point in a two layer system. Also, the effect of noisy data on the final solution is studied. The numerical solution of the governing equations is obtained by employing a finite-difference technique. For solving this problem, the general coordinate method is used. The irregular region in the physical domain (r,z) is transformed into a rectangle in the computational domain (ξ,η). The present formulation is general and can be applied to the solution of boundary inverse heat conduction problems over any region that can be mapped into a rectangle. The obtained results for few selected examples show the good accuracy of the presented method. Also, the solutions have good stability even if the input data includes noise. The problem is solved in an axisymmetric case. Applications of this model are in the thermal protect systems (t.p.s.) and heat shield systems

Scalable explicit implementation of anisotropic diffusion with Runge-Kutta-Legendre super-time-stepping

An important ingredient in numerical modelling of high temperature magnetised astrophysical plasmas is the anisotropic transport of heat along magnetic field lines from higher to lower temperatures.Magnetohydrodynamics (MHD) typically involves solving the hyperbolic set of conservation equations along with the induction equation. Incorporating anisotropic thermal conduction requires to also treat parabolic terms arising from the diffusion operator. An explicit treatment of parabolic terms will considerably reduce the simulation time step due to its dependence on the square of the grid resolution ($\Delta x$) for stability. Although an implicit scheme relaxes the constraint on stability, it is difficult to distribute efficiently on a parallel architecture. Treating parabolic terms with accelerated super-time stepping (STS) methods has been discussed in literature but these methods suffer from poor accuracy (first order in time) and also have difficult-to-choose tuneable stability parameters. In this work we highlight a second order (in time) Runge Kutta Legendre (RKL) scheme (first described by Meyer et. al. 2012) that is robust, fast and accurate in treating parabolic terms alongside the hyperbolic conversation laws. We demonstrate its superiority over the first order super time stepping schemes with standard tests and astrophysical applications. We also show that explicit conduction is particularly robust in handling saturated thermal conduction. Parallel scaling of explicit conduction using RKL scheme is demonstrated up to more than $10^4$ processors.Comment: 15 pages, 9 figures, incorporated comments from the referee. This version is now accepted for publication in MNRA

Inverse solution to the heat transfer coeffcient for the oxidized ARMCO steel plate cooling by the air nozzle from high temperature

[EN] The inverse solution tests have been performed to the experimental data obtained during the oxidised Armco steel plate cooling by the air nozzle. The three-dimensional numerical model of heat transfer during the plate cooling has been considered. Steel products cooled in the air from high temperatures are covered with the oxide layer having significantly lower conductivity and a different surface structure comparing to the non-oxidised metal surface. The Armco steel has been selected as the experimental material because it oxidised in a similar way to carbon steels but there is no microstructure evolution process in Armco steel below 900oC. It eliminates in the inverse solutions serious problems caused by a latent heat of microstructure evolutions encountered during carbon steel cooling. In the present study the steel plate has been heated to about 900℃ and cooled by the circular air jet. The plate temperature has been measured by 36 thermocouples. The test of the selected inverse solution models involving a different number of degrees of freedom have been performed. The influence of the scale layer on the results of the inverse solution to the heat flux and heat transfer coefficient has been investigated.Scientific study financed from the regular activity of the Faculty of Metals Engineering and Industrial Computer Science of AGH University of Science and Technology.Jasiewicz, K.; Malinowski, Z.; Cebo-Rudnicka, A. (2022). Inverse solution to the heat transfer coeffcient for the oxidized ARMCO steel plate cooling by the air nozzle from high temperature. En Proceedings of the YIC 2021 - VI ECCOMAS Young Investigators Conference. Editorial Universitat Politècnica de València. 233-244. https://doi.org/10.4995/YIC2021.2021.12344OCS23324

Reconstruction of the thermal properties in a wave-type model of bio-heat transfer

Purpose: This study aims to at numerically retrieve five constant dimensional thermo-physical properties of a biological tissue from dimensionless boundary temperature measurements. Design/methodology/approach: The thermal-wave model of bio-heat transfer is used as an appropriate model because of its realism in situations in which the heat flux is extremely high or low and imposed over a short duration of time. For the numerical discretization, an unconditionally stable finite difference scheme used as a direct solver is developed. The sensitivity coefficients of the dimensionless boundary temperature measurements with respect to five constant dimensionless parameters appearing in a non-dimensionalised version of the governing hyperbolic model are computed. The retrieval of those dimensionless parameters, from both exact and noisy measurements, is successfully achieved by using a minimization procedure based on the MATLAB optimization toolbox routine lsqnonlin. The values of the five-dimensional parameters are recovered by inverting a nonlinear system of algebraic equations connecting those parameters to the dimensionless parameters whose values have already been recovered. Findings: Accurate and stable numerical solutions for the unknown thermo-physical properties of a biological tissue from dimensionless boundary temperature measurements are obtained using the proposed numerical procedure. Research limitations/implications: The current investigation is limited to the retrieval of constant physical properties, but future work will investigate the reconstruction of the space-dependent blood perfusion coefficient. Practical implications: As noise inherently present in practical measurements is inverted, the paper is of practical significance and models a real-world situation. Social implications: The findings of the present paper are of considerable significance and interest to practitioners in the biomedical engineering and medical physics sectors. Originality/value: In comparison to Alkhwaji et al. (2012), the novelty and contribution of this work are as follows: considering the more general and realistic thermal-wave model of bio-heat transfer, accounting for a relaxation time; allowing for the tissue to have a finite size; and reconstructing five thermally significant dimensional parameters

Effect of Heat Source and Imperfect Contact on Simultaneous Estimation of Thermal Properties of High-Conductivity Materials

In the current paper a novel methodology accounting for both the heater heat capacity and the imperfect thermal contact between a thin heater and a specimen is proposed. In particular, the volumetric heat capacity of the heater is considered by modelling it as a lumped capacitance body, while the imperfect thermal contact is considered by means of a contact resistance. Thus, the experimental apparatus consisting of three layers (specimen-heater-specimen) is reduced to a single finite layer (sample) subject to a "nonclassical" boundary condition at the heated surface, known as sixth kind. Once the temperature solution is derived analytically using the Laplace transform method, the scaled sensitivity coefficients are computed analytically at the interface between the heater and the sample (heater side and sample side) and at the sample backside. By applying the proposed methodology to a lab-controlled experiment available in the specialized literature, a reduction of the thermal properties values of about 1.4% is observed for a high-conductivity material (Armco iron)

Data-driven inverse modelling through neural network (deep learning) and computational heat transfer

In this work, the potential of carrying out inverse problems with linear and non-linear behaviour is investigated using deep learning methods. In inverse problems, the boundary conditions are determined using sparse measurement of a variable such as velocity or temperature. Although this is mathematically tractable for simple problems, it can be extremely challenging for complex problems. To overcome the non-linear and complex effects, a brute force approach was used on a trial and error basis to find an approximate solution. With the advent of machine learning algorithms it may now be possible to model inverse problems faster and more accurately. In order to demonstrate that machine learning can be used in solving inverse problems, we propose a fusion between computational mechanics and machine learning. The forward problems are solved first to create a database. This database is then used to train the machine learning algorithms. The trained algorithm is then used to determine the boundary conditions of a problem from assumed measurements. The proposed method is tested for the linear/non-linear heat conduction, convection–conduction, and natural convection problems in which the boundary conditions are determined by providing three, four, and five temperature measurements. This study demonstrates that the proposed fusion of computational mechanics and machine learning is an effective way of tackling complex inverse problems

Identification of conductivity in inhomogeneous orthotropic media

Purpose - The purpose of this paper is to solve numerically the identification of the thermal conductivity of an inhomogeneous and possibly anisotropic medium from interior/internal temperature measurements. Design/methodology/approach - The formulated coefficient identification problem is inverse and ill-posed and therefore, in order to obtain a stable solution, a nonlinear regularized least-squares approach is employed. For the numerical discretisation of the orthotropic heat equation, the finite-difference method is applied, whilst the nonlinear minimization is performed using the MATLAB toolbox routine lsqnonlin. Findings - Numerical results show the accuracy and stability of solution even in the presence of noise (modelling inexact measurements) in the input temperature data. Research limitations/implications - The mathematical formulation uses temporal tem- perature measurements taken at many points inside the sample and this may be too much information that is provided to identify a spacewise dependent only conductivity tensor. Practical implications - Since noisy data are inverted, the study models real situations in which practical temperature measurements recorded using thermocouples are inherently contaminated with random noise. Social implications - The identification of the conductivity of inhomogeneous and orthotropic media will be of great interest to the inverse problems community with applications in geophysics, groundwater flow and heat transfer. Originality/value - The current investigation advances the field of coefficient identification problems by generalising the conductivity to be orthotropic in addition of being heterogeneous. The originality lies in performing, for the first time, numerical simulations of inver- sion to find the anisotropic and inhomogeneous thermal conductivity form noisy temperature measurements. Further value and physical significance is brought in by determining the degree of cure in a resin transfer molding process, in addition to obtaining the inhomogeneous thermal conductivity of the tested material

ERROR ANALYSIS IN THE NUMERICAL SOLUTION OF 3D CONVECTION-DIFFUSION EQUATION BY FINITE DIFFERENCE METHODS

In this work an error analysis for numerical solution of 3D convectiondiffusionequation by finite difference methods has been done. The backward, the forward and the central difference schemes are applied for three applications: a case with diffusion dominant corresponding to high diffusion coefficients and two cases with convection dominant or with low diffusion coefficients. In the second application the convective coefficients are function only of the diffusion coefficient that in dimensionless form is named Reynolds numbers. In the third application the convective coefficients are function of both the Reynolds number and of the space. The three applications have analytical solutions to facilitate numerical comparisons of the solutions