Quadrupole Formalism Research Articles

A fast quadrupole-based analytical model for solving transient heat transfer in ventilated double-skin walls and assessing their energy saving potential

This research proposes a fast and accurate method for modeling transient heat transfer in ventilated double-skin façades (VDFs) with forced ventilation to predict their energy-saving potential. Using the thermal quadrupole formalism, the method solves the VDF problem in the Laplace domain while considering the full transient nature of the heat transfer; the only approximation is in space. The solution involves obtaining a general transfer function that predicts heat transfer rates or air temperatures at ventilated cavity's exit. The quadrupole method is validated against computational fluid dynamic numerical simulations conducted under similar meteorological, design and boundary conditions. A very good agreement was found (less than 3 % average deviation) with a significant reduction in computation time (less than 3s against 6h for CFD calculations). The model is computationally efficient and can consider important factors such as the opacity of the walls, construction materials, and air cavity design parameters. Finally, the model allows the assessment of the energy-saving potential of VDFs under various scenarios compared to conventional systems, which helps contributing to more sustainable building design practices. The energy efficiency of a VDF configuration was compared against a conventional wall system. Energy savings of 8.4 and 5.2 % were obtained, in cold and hot climates respectively.

A numerical investigation of longitudinal wave for thermo-acoustoelastic effects on fluid-embedded in two viscoelastic layer plates at higher temperatures

Ultrasound has become indispensable for inspecting complex structures due to its efficient propagation in both liquids and solid materials. Extensive research has been conducted across various domains, encompassing wave behavior, structural interactions, material dispersion due to heterogeneity, propagation characteristics near boundaries, defect detection, and signal processing. In this context, longitudinal waves are particularly favored for examining thick components exhibiting surface defects or thermal gradients across their depth. In our current work, we employed a numerical approach based on the Transfer Matrix Method (TMM) to model ultrasound signals backscattered under normal incidence by a multilayer structure immersed in water. This method involves representing each layer using a quadrupole formalism that combines stresses and velocities. Simultaneously, we harnessed the thermodynamic equation for aqueous solutions and Fourier’s law to derive the expression for the thermo-acoustic coefficient of longitudinal waves within a fluid embedded between two parallel immersed plates, enabling precise control of the fluid temperature over a range from 0 °C to 700 °C. In the time domain, we evaluated the ultrasonic parameters of a fluid exposed to a temperature range from 0 to 700 °C. This evaluation was supported by a variety of multilayer structures designed to ensure the accurate processing of backscattered signals. These structures were excited by longitudinal waves, with a central frequency fixed at 5 MHz. The results obtained emphasize the sensitivity of longitudinal waves to temperature variations and reveal the presence of wave dispersion induced by temperature fluctuations. In order to investigate this dispersion, we conducted an extensive study of ultrasonic longitudinal wave propagation across a range of excitation frequencies from 2 to 12 MHz. This study encompassed the calculation of phase and group velocities, as well as the corresponding attenuation. The results confirm the relevance of our research to various applications and highlight the complex interaction between longitudinal wave dispersion and temperature variations.

Online system identification of global lung heat transfers

In cardiac surgery where extracorporeal circulation is used, the lungs are temporarily disconnected from the body and are connected to a device that provides air and blood. To minimize the risk of tissue damage, the lungs are subjected to mild hypothermia. Heat transfer modeling offers the potential to enhance temperature regulation through a more advanced approach. From a thermal quadrupole formalism combined with the heat equation or the bio-heat equation when considering blood perfusion according to the branch level, a more compact model is used for system identification. The feasibility of online system identification is proposed for coefficient estimation by using an extended Prediction Error Method algorithm more suited for estimating fractional order systems: the Long Memory Prediction Error Method (LMRPEM).

High power density laser estimation using quantitative thermal imaging method

ABSTRACT The knowledge of the amplitude and the spatial distribution of an excitation flux is of great interest for the quantification of heat sources. In this work, the development of a non-contact imaging powermeter based on the association of a bolometer with an infrared camera is described. This powermeter allows, thanks to infrared thermographic measurements and image processing methods, the quantitative estimation of the spatial distribution of the power of the flux delivered by a high-power laser. First, the experimental setup used is described. Then, the complete modelling of the heat transfer within the bolometer using the 3D thermal quadrupole formalism is presented. After that, an inverse method based on the Wiener filter in Fourier-Laplace transform spaces to estimate the spatial distribution of the power flux is described. Finally, power estimation results using two metallic plates as a bolometer are presented and discussed.

Characterization of semi-infinite delaminations using lock-in thermography: Theory and numerical experiments

Delaminations are buried defects parallel to the sample surface. In the last decades infrared thermography with optical excitation has been used to detect and size the depth of this kind of defects. However, sizing the delamination thickness has been usually disregarded. In a recent paper we proposed a method to size both depth and thickness of ideal delaminations (infinite area) using modulated excitation. Here, we extend the previous work to approach more realistic situations, tackling the case of semi-infinite delaminations. First, we calculate analytically the surface temperature oscillation of a sample containing a semi-infinite delamination using the thermal quadrupoles formalism. Then, we corroborate the analytical results by solving the same problem numerically. Finally, we perform an inverse parametric estimation of synthetic temperature amplitude and phase data with added Gaussian noise to retrieve the three geometrical parameters characterizing the delamination: length, depth and thickness.

A novel approach for modeling of the ultrasonic signal backscattered in immersed multilayer structures at normal incidence: Time, Frequency, and Velocity dispersion representation

This study proposes a theoretical procedure for modeling ultrasonic signal backscattered in normal incidence by a multilayer structure (MS) immersed in water. The propagation of a longitudinal ultrasonic wave in a MS is studied analytically by using the transfer matrix method (TMM), in which each layer is established as a quadrupole formalism combining the stresses and velocities. The transfer matrices allow us to determine the reflection coefficient of the multilayer structure. The MS solid/liquid/solid case is studied in both insonation configurations (direct and reverse). The modeled ultrasonic response is represented in both time and frequency domains and validated by comparison with experimental data published previously. The time–frequency methods such as spectrogram (SP) and the smoothed pseudo-Wigner–Ville (SPWV) are also used to investigate the character and the value of the non-dispersive of the longitudinal wave. Moreover, the TMM method is used under different conditions (different layer thicknesses, non-uniform distribution of layers, and any frequency range), which are among the causes of MS dispersion. TMM method is able to show this behavior and is exploited to produce (Distance versus Time) and (Distance versus Frequency) representations of wave propagation for each studied MS. The perfect agreement of the comparison of TMM dispersion results with the results is achieved by using time–frequency methods in the case of a 5 MHz central frequency.

Time-frequency simulation of ultrasonic longitudinal wave propagation in rocks

Ultrasound numerical simulation can be used as a powerful tool to investigate the generation, propagation, and interaction of elastic waves in solid materials for non-destructive evaluation (NDE). It has been widely applied in several domains such as medical, mechanical, construction, and geotechnical engineering, especially for the detection of cracks and defects in structures and for the identification of the characteristics of rock materials. The comprehension of the propagation of the ultrasonic wave in homogeneous and isotropic elastic materials is very important. This paper presents a method for simulating normal incidence ultrasonic waves (Longitudinal waves) propagating in rock structures. A mathematical model of the response of an ultrasonic wave in a submerged rock structure is introduced analytically based on the transfer matrix method (TMM), in which the layer is established in a quadrupole formalism. The simulation results of the ultrasonic response of rock structures are represented in a time–frequency representation. The results have been compared with the experimental data available in the literature and show good agreement. The numerical simulation method established in this work can be proposed as an effective complement to experiments and used for rock characterization by performing the simulation of ultrasonic responses for different thicknesses of rock structures.

Transient linear analytical heat transfer model for a building, validation with a non-linear coupled finite volume code

This work aims to propose a fast and accurate method, based on the thermal quadrupole formalism, to model transient heat transfer and predict the thermal behavior in buildings. The core principles and concepts of the analytical method are presented where no approximation is made to model the transient heat conduction through the building’s envelope. A general equation is obtained for each zone heat balance that can either be used to obtain heat rates or temperatures. Furthermore, this method can account for thermal masses inside a room that are at different temperatures than the surrounding air. In four different cases, the analytical quadruple method is compared to numerical simulations under similar conditions and the results demonstrate a very good agreement. Finally, the quadrupole method is straight forward, requires very low computational time, and with no timescale constraints which makes it relatively easy to use for computer programming or embedded in existing optimization codes.

Unconventional flash technique for the identification of multilayer thermal diffusivity tensors

This paper presents a novel non-intrusive technique for the thermal characterization of opaque multilayer materials that consists in the simultaneous estimation of the thermal diffusivities tensors of each constituting layers that may be isotropic or orthotropic. The present identification method is based on the resolution of an inverse heat conduction problem consisting in the minimization of the least square objective function constructed with the temperature measurements and the prediction of a pseudo-analytical model. The pseudo-analytical model relies on the quadrupoles formalism that proves its capability to treat multi-layered materials and to mimic the experimental method investigated in this work. This latter, inspired from the flash method, relies on a unique and non-intrusive experiment, in terms of excitation and measurements. One of the sample surfaces is locally and shortly heated by means of a CO2 laser, while the resulting transient temperature field is continuously recorded during the cooling period by an IR camera. The minimization procedure invokes a PSO algorithm that has been proved to be convenient for such complex problems that deal with a non-linearity and a large number of unknown parameters, including those related to the excitation. The overall identification method is validated using an isotropic opaque monolayer material of known properties. The identification method is then performed on an actual two layers sample. The accuracy and robustness is discussed depending on the face of excitation and measurement leading to two distinct experimental configurations. A strong emphasis is placed on the sensitivity analysis in order to test the feasibility of the estimation for both configurations.

Finite element modeling of the Hot Disc method

The Hot Disc method, also known as the transient plane source (TPS) technique, is an experimental approach to determine the thermal transport properties of materials. The core of the method is the Hot Disc sensor, an electrically conducting metallic strip, shaped as a double spiral clad with a protective polymer film. The mean temperature increase in the sensor has been approximated from various analytical approaches such as: the concentric ring sources model, the thermal quadrupoles formalism, and concentric circular strips structure approach. However, full numerical simulation of the sensor has not been addressed so far. Here we develop a 3D model of Hot Disc sensors and compare simulated mean temperature increase to experimental recordings. Joule heating coupled with heat transfer of solids (of COMSOL Multiphysics software) is used to simulate the working principle of the sensor. The volume mean temperature increase in the sensor from the simulations proves to be in a good agreement with the corresponding experimental recordings. The temperature distributions of the metallic strip are also evaluated and discussed with respect to the previous experimental findings. Furthermore, the current distribution across the strip is obtained. Such simulation can potentially be used in further optimizing geometry and parameter estimation.

Estimation of radiative and conductive properties of a semitransparent medium using genetic algorithms

The aim of this work is to simultaneously identify the conductive and radiative parameters of a semitransparent sample using a photothermal method associated with an inverse problem.The identification of the conductive and radiative proprieties is performed by the minimization of an objective function that represents the errors between calculated temperature and measured signal.The calculated temperature is obtained from a theoretical model built with the thermal quadrupole formalism. Measurement is obtained in the rear face of the sample whose front face is excited by a crenel of heat flux. For identification procedure, a genetic algorithm is developed and used. The genetic algorithm is a useful tool in the simultaneous estimation of correlated or nearly correlated parameters, which can be a limiting factor for the gradient-based methods. The results of the identification procedure show the efficiency and the stability of the genetic algorithm to simultaneously estimate the conductive and radiative properties of clear glass.

Analytical theories for near coplanar and polar circumbinary orbits

The recently discovered Kepler circumbinary systems and the circumbinary polar-ring debris disc in the 99 Herculis system call for the study of circumbinary orbits. As an approximation, it is advisable to investigate the problem with the assumption that the outer body is massless. In such a scheme, we briefly summarize the main characteristics of the evolution of the orbits and give the general solutions to some secular orbital elements under the quadrupole formalism and then, using the solution, explore the octupole Hamiltonian, which allows us to reveal the evolution of the eccentricity of the outer orbit. By doing so, we have discovered that the evolution of slightly inclined orbits does not deviate much from coplanar ones. For the near polar configurations, the inner binary can significantly excite the eccentricity of such orbits (in some cases to about 0.3); this may influence the stability of orbits near the binary.

Simultaneous estimation of the thermophysical properties of liquids and of the boundary conditions

The determination of thermophysical properties from an inverse method is an attractive technique both from the experimental and methodological point of view. The aim of this paper is to identify simultaneously the thermal conductivity and the thermal diffusivity of liquids and the heat transfer coefficient by the utilization of the well-known flash method which is coupled to an inverse parameter identification method. To be characterized, a thin layer of the liquid is sandwiched between two metal plates and a cylindrical sample holder. Then it is subjected on its front face to a heat flux density during a short time. The temperature evolution of the rear face of the container is measured during and after the thermal excitation. The identification procedure based on the inverse method consists in the minimization of an objective function that represents the errors between the measured temperature at the rear face and a theoretical model that describes the temperature evolution of the system (liquid and container). The theoretical model, which is a function of the unknown thermophysical properties of the three layer system, is calculated by the use of the thermal quadrupoles formalism. To minimize the objective function, a genetic algorithm is developed and used. Genetic algorithm is categorized as global search heuristics. It is an iterative procedure which borrows the ideas of natural selection and survival of the fittest from natural evolution which were first described by Darwin.The identification procedure based on the use the flash method and the developed genetic algorithm is used in this work to identify thermophysical properties of distilled water, glycerol, acetone, and ethylene glycol. A good agreement is found between our experimental results and the thermophysical properties indicated in the literature of the studied liquids.

Comprehensive characterization of thermophysical properties in solids using thermal impedance

Thermal impedance Zth(iω) is a way of defining the thermophysical characteristics and behavior of thermal systems. Existing photoacoustic and photothermal approaches based on thermal impedance formalism merely allows a partial thermal characterization of the materials (generally, either thermal diffusivity or thermal effusivity). In this work, a new approach based on the thermal impedance concept in terms of its characteristic thermal time constant is developed from thermal quadrupoles formalism. The approach outlined in this contribution presents a set of analytical equations in which through a single measurement of thermal impedance is sufficient to obtain a comprehensive characterization of the thermophysical properties of solid materials in a simple way.

An inverse problem based on genetic algorithm to estimate thermophysical properties of fouling

In the past few decades, the study on how to apply genetic algorithms to problems in the industrial engineering world has aroused a great deal of curiosity of many researchers in the area of management science, industrial operations and engineering systems. This paper shows an experimental process of thermophysical properties estimation of fouling deposited on internal surface of a heat exchanger tube using genetic algorithms. In brief, the deposits on heat exchanger tubes are caused by the presence of inorganic salts, of small quantities of organic materials and products of corrosion in the water. From thermophysical point of view, the deposited fouling has harmful effects on the heat exchanger efficiency. For these reasons the determination of its thermophysical properties became very important. The experimental bench using a photothermal method with a finite width pulse heat excitation is used. The genetic algorithm is used to minimize an objective function containing a calculated and a measured temperature. This last is measured on the rear face of a bi-layer system composed of a section of a heat exchanger and the fouling deposited on during and after a finite width pulse heat excitation on its front face. The calculated temperature, that is a function of the unknown thermophysical properties of the bi-layer system, is calculated by the resolution of the one-dimensional linear inverse conduction problem, and by the use of the thermal quadrupoles formalism. The motivation in using genetic algorithms was their potential to overcome the restriction to the estimation of non-correlated parameters of gradient-based methods, and their powerful ability to work well for many complex problems which are very difficult to solve by conventional techniques. The results of the estimation procedure show on the one hand the efficiency and the stability of the developed genetic algorithm to estimate the thermophysical properties of fouling and the high accuracy of the obtained results on the other hand.

Thermal Depth Profiling Reconstruction by Multilayer Thermal Quadrupole Modeling and Particle Swarm Optimization

A new hybrid inversion method for depth profiling reconstruction of thermal conductivities of inhomogeneous solids is proposed based on multilayer quadrupole formalism of thermal waves, particle swarm optimization and sequential quadratic programming. The reconstruction simulations for several thermal conductivity profiles are performed to evaluate the applicability of the method. The numerical simulations demonstrate that the precision and insensitivity to noise of the inversion method are very satisfactory.

Estimation of thermophysical properties of fouling using inverse problem and its impact on heat transfer efficiency

At high temperature, the circulation of fluid in heat exchangers provides a tendency for fouling accumulation to take place on the internal surface of tubes. In brief, the deposits on heat exchanger tubes are caused by the presence of inorganic salts, of small quantities of organic materials and products of corrosion in the water. From thermophysical point of view, the deposited fouling has harmful effects on the heat exchanger efficiency. Indeed, it increases the thermal resistance which can raise the energy consumption. This study shows an experimental and a theoretical process of estimation of thermophysical properties of the fouling deposited on a section of a heat exchanger and its effects on the heat transfer efficiency. The estimation method is based on the Gauss–Newton algorithm that minimizes the ordinary least squares function comparing a measured temperature and a theoretical one. The temperature response is measured on the rear face of a bi-layer system composed of a section of a heat exchanger and the fouling deposited on during and after a finite width pulse heat flux on its front face. The theoretical temperature, that is a function of the unknown thermophysical properties of the bi-layer system, is calculated by the resolution of the one-dimensional linear inverse conduction problem, and by the use of the quadrupole formalism. The results of the estimation procedure show, on the one hand the efficiency and the stability of the optimization algorithm to estimate the thermophysical properties of the fouling. On the other hand they underline the necessity of the maintenance of fluid circulating tubes at high temperature.

Periodic heat conduction in a solid homogeneous finite cylinder

Analytic solution of the steady periodic, non-necessarily harmonic, heat conduction in a homogeneous cylinder of finite length and radius is given in term of Fourier transform of the fluctuating temperature field. The solutions are found for quite general boundary conditions (first, second and third kind on each surface) with the sole restriction of uniformity on the lateral surface and radial symmetry on the bases. The thermal quadrupole formalism is used to obtain a compact form of the solution that can be, with some exception, straightforwardly extended to multi-slab composite cylinders. The limiting cases of infinite thickness and infinite radius are also considered and solved.

Measurement of thermal radiative and conductive properties of semitransparent materials using a photothermal crenel method

This paper deals with a theoretical and an experimental study allowing the measurement of the radiative and the conductive properties of semitransparent materials. The method consists of applying a crenel heat flux on the front face of a semitransparent sample and recording the temperature at the rear face using an open thermocouple junction. Parameter identification is performed by the minimization of the ordinary least-squares function comparing the measured and the calculated temperatures. This later is obtained from the thermal model describing the heat transfer by conduction and radiation in the medium. This model is built by the thermal quadrupole formalism. Measurements are reported on commercial glasses and plexiglass samples, and the used iterative algorithm is based on the Gauss–Newton method.

An analytical two-temperature model for convection–diffusion in multilayered systems: Application to the thermal characterization of microchannel reactors

The main purpose of this paper is to implement some convenient analytical solutions of the two-dimensional convection–diffusion equations in a multilayered system, in the form of some relationships between average temperature fields, based on the thermal quadrupole formalism. Some equivalent analogical networks are proposed in order to implement the model in a more convenient form, based on the electrical analogy. The important advantage of such approach is to connect different layers through simple network connections between the respective interface variables. Special emphasis is laid on the case where the lateral boundary conditions correspond to insulated walls. The transient case is also presented, for non-insulated lateral boundary conditions, coupled with a third layer. Some examples are given in order to illustrate the suitability of the proposed model in the case of temperature field image processing in a microfluidic chip.