Transient Heat Diffusion Research Articles

Exploring thermal dynamics and porosity of cold-sprayed Ti-6Al-4V coatings on Al6061-T6 substrates: A pulsed thermography and numerical modeling approach

In this investigation, we use pulsed thermography and a multilayer analytical model to critically evaluate the thermal diffusivity and inherent porosity in cold-sprayed Ti-6Al-4V coatings on Al6061-T6 substrates. These coatings are of importance in deploying cold spray as an additive manufacturing technique for the restoration of high-cost alloy components. Our study draws upon experimental data obtained at spray angles from 50°to 90°, illustrating the relationship among the spray angle, porosity within the coating, and the resultant thermal diffusivity. Our results reveal a decrease in porosity and an increase in thermal diffusivity as the spray angle increases from 50°to 90°. In conjunction with our experimental approach, a numerical model was developed, incorporating binary images derived from optical micrographs to simulate transient heat diffusion during pulsed thermography experiments. The congruence between our empirical findings and simulation data affirms the effectiveness of the applied methodology, shedding light on the complex interplay between processing parameters and the resulting thermal properties. This study explores the effects of porosity and thermal gradients introduced during pulsed thermography, providing a profound understanding of the thermal dynamics of cold-sprayed coatings.

A new approach for solving heat conduction under zero and non-zero initial conditions

This paper provides a new approach for the analysis of transient heat diffusion arising from conduction. The diffusion is subject to zero and non-zero initial conditions in a two-dimensional domain, containing confined heterogeneous subdomains. This system can be subjected to a heat source. A new formulation is presented, in the frequency domain. It allows coupling between the boundary element method (BEM) and other methods (such as the method of fundamental solutions (MFS), the finite element method (FEM), the finite difference method (FDM), the meshless Petrov-Galerkin (MLGP) method, the virtual element method (VEM), and analytical formulations). The BEM is used for the homogeneous exterior domain and the other method is used to evaluate the solutions for the non-homogeneous confined subdomains, to overcome the specific limitations of each of these methods.

Errors Incurred in Local Convective Heat Transfer Coefficients Obtained through Transient One-Dimensional Semi-Infinite Conduction Modeling: A Computational Heat Transfer Study

In typical turbulent flow problems, detailed heat transfer coefficient (h) maps obtained through short-duration experiments are based on inverse heat transfer methods that take the wall temperatures measured via liquid crystals or infrared thermography as input, and an error minimization routine is adopted to determine the best value of h that satisfies the wall temperature temporal evolution under a certain change in fluid temperature. A common practice involves modeling the solid as a one-dimensional semi-infinite medium by selecting the solid material that has low thermal conductivity and low thermal diffusivity. However, in certain flow scenarios, the neglection of the lateral heat diffusion may lead to significant errors in the deduced h values. It is imperative to understand the reasons behind large errors that may be incurred by using the 1D heat conduction assumption in order to accurately determine high-resolution h maps for better heat exchanger designs in a wide range of thermal management applications. This paper presents a computational heat transfer study on different jet impingement scenarios to demonstrate the errors incurred in the determination of h when calculated under the assumption of one-dimensional (1-d) heat conduction into a solid. To this end, three different cases are studied: (a) single jet, (b) array jet (theoretical distribution), (c) array jet (experimental distribution), along with three different mainstream temperature evolution profiles representing step change, moderately fast transient and slow transient nature of flow driving the heat transfer in the solid. A known distribution of heat transfer coefficient (“true h”) for each of the three cases is considered, and three-dimensional transient heat diffusion equations were solved to populate temperatures of each node in the solid at every time step. It is found that stagnation zones’ h1d calculations were lower than the “true h” while the low heat transfer zones exhibited significantly higher h1d compared to the “true h”. For the array jet (experimental distribution) case, it was observed that errors can be as high as 10% in certain low heat transfer zones. Different data reduction procedures, configurations, and conditions explored in this study indicate that a suitable balance can be achieved if shorter time durations in transient experiments are used as a reference for tracking in h1d calculations to keep the deviations from the “true h” low.

Effects of Charged Solute-Solvent Interaction on Reservoir Temperature during Subsurface CO2 Injection

A short-term side-effect of CO2 injection is a developing low-pH front that forms ahead of the bulk water injectant, due to differences in solute diffusivity. Observations of downhole well temperature show a reduction in aqueous-phase temperature with the arrival of a low-pH front, followed by a gradual rise in temperature upon the arrival of a high concentration of bicarbonate ion. In this work, we model aqueous-phase transient heat advection and diffusion, with the volumetric energy generation rate computed from solute-solvent interaction using the Helgeson–Kirkham–Flowers (HKF) model, which is based on the Born Solvation model, for computing specific molar heat capacity and the enthalpy of charged electrolytes. A computed injectant water temperature profile is shown to agree with the actual bottom hole sampled temperature acquired from sensors. The modeling of aqueous-phase temperature during subsurface injection simulation is important for the accurate modeling of mineral dissolution and precipitation because forward dissolution rates are governed by a temperature-dependent Arrhenius model.

Explosion of metastable droplets in immiscible liquids

Evaporation of metastable droplets in an immiscible liquid can occur in a wide range of practical applications and in particular in cryogenic systems due to the low saturation temperature of the cryogenic liquid. In this article, we show experimentally that the incomplete explosion of a metastable droplet can lead to a ‘film evaporation’ configuration, where a remaining droplet is eventually separated from the host immiscible liquid by an established vapour layer. Based on a transient heat diffusion model, we identified that the necessary but not sufficient condition for the complete explosion of a metastable droplet is that the droplet must absorb, before explosion, an amount of heat larger than the latent heat required for a full evaporation, i.e. the global Jacob number Ja≥1. In addition, the measured superheat limits of pentane and isohexane are 420.65 K and 444.95 K respectively, which are very close to the theoretical Spinodal points (≤ 1.5 K) and other reported values in existing literature. Therefore, the experimental setup reported here can be used as an alternative way to quantify the superheat limit. The Droplet relaxation time was also found scaled to the square root of the diffusion characteristic time.

A thermo-mechanical numerical approach for the analysis of a historical rockslide in the Italian Alps

In this work, a thermo-mechanical (TM) numerical approach is presented and applied to investigate the stress-strain evolution of an alpine rock-slope located in the Central Italian Alps (Sondrio Province). Along the “Cimaganda” slope a massive rockslide event occurred around 900 A.D. mobilizing an estimated volume of 7.5 Mm3 of material, and reaching the bottom of the valley. Interest in this historic event was raised again in recent times, as a new rockslide took place in 2012, mobilizing 20.000 m3 of rock material and blocking the SS36 National Road.To understand the general evolution of the Cimaganda rock slope, the recent geomorphological history of the Valley (post Last Glacial Maximum) was considered. In particular, to explore how glacial loading and unloading, associated with thermo-mechanical processes can promote rock mass damage, a 2D DEM numerical approach was adopted, calibrated upon the collected experimental and field data, and supported by a 2D FEM analysis to simulate transient heat diffusion over the Valley cross-section due to ice retreat and paleo-temperature evolution. Results show a clear relation between TM stresses and the occurrence of rock-mass damage and slip propagation along discontinuities. Simulated displacement and the development of a deep region of shear strain localization, allow to highlight the significance of temperature influence in preparing the rock slope to instability.

Effects of surface wettability on pool boiling process: Dynamic and thermal behaviors of dry spots and relevant critical heat flux triggering mechanism

In this paper, we investigate experimentally and numerically the influences of surface wettability on the boiling process from nucleate boiling to critical heat flux (CHF). The boiling surfaces with four different wettabilities (apparent contact angles) are examined. Using a simple optic setup, sequential phase distribution images on the boiling surfaces were acquired, and we analyzed the dynamics of microlayer, bulk liquid region, and dry spots with the processed images. The variation of surface wettability significantly affected the liquid and vapor phase behaviors on the boiling surface, such as the growth and shrinkage of the microlayer and dry spot, wetting velocity and area portion of the dry spot. As a result, the CHF increased with the surface wettability, and we found the CHF is triggered from a local dry spot where the liquid wets the spot no longer and is violently scattered by strong evaporation at the triple contact line. Our numerical simulation results, in which the three-dimensional transient heat diffusion equation was solved for the boiling substrate using the phase distribution images as boundary conditions, indicated that certain dry spots (critical dry spot) can be highly overheated under high heat flux condition, and they lead significant heat transfer degradation and, eventually, the CHF. With the numerical results and observation of the phase distribution images at the CHF conditions, we postulated the allowable wetting temperature and formulated it as a function of surface wettability through considering the balance between vapor recoil and surface tension forces. The wetting temperature increased with surface wettability, and it was confirmed by the numerical results. Finally, we proposed a qualitative relation to elucidate the influence of surface wettability on the CHF based on the heat conduction via the vapor on a critical dry spot and heat flux partitioning concept.

Convolved energy variational principle in heat diffusion

One of the major, long-standing challenges in analytical mechanics involves the inability to address systems with dissipation in a rigorous manner. In this paper, we overcome that difficulty by formulating a novel temperature-based stationary variational principle for transient heat diffusion based upon a temporal convolution operator and fractional derivatives. The associated Euler-Lagrange equations provide the governing heat equation, along with the initial conditions on temperature and specified heat flux boundary conditions. A further integration-by-parts then leads to a formulation that is somewhat less symmetric but can be written without introducing fractional calculus. Finally, the resulting principle is used to solve two basic one-dimensional problems, as an illustration of a Ritz-type approach.

Modelling and Simulation of Thermo-Physical Property of Composite Material

Thermo-physical property measurement is a significant methodology as these properties are the input for designing thermal systems. Properties of materials like thermal conductivity, thermal diffusivity and specific heat capacity has to be estimated in prior to estimate the thermal performance of any material. In the present work, numerical simulations were performed to estimate the thermo-physical properties of a polymer based composite material using porous medium modelling. The weight fraction of the composites and fibres are incorporated into the simulations through porosity. The transient heat diffusion equation are solved using ANSYS Fluent for a polymer based composite material using semi-infinite transient conduction approach. The thermal conductivity and thermal diffusivity are estimated simultaneously for various volume fractions of polyester and banana fibre. The temporal temperature distribution data predicted from ANSYS Fluent is curve fitted using MATLAB and the obtained values of thermal conductivity and thermal diffusivity are compared with the previously reported experimental results and are found to be within a deviation of 7 % for all cases.

Connections Between the Meshfree Peridynamics Discretization and Graph Laplacian for Transient Diffusion Problems

Formulations for diffusion processes based on graph Laplacian kernels have been recently used to solve linear transient heat transfer problems with insulated boundary conditions by way of a spectral-based semi-analytical approach. This has been called the “spectral graph” (SG) approach. In this paper, we show that the meshfree discretization for corresponding peridynamic (PD) models leads to a graph structure that allows us to introduce the “graph Laplacian PD” (GL-PD) method using the semi-analytical approach employed in the SG approach to solve the transient heat diffusion problems. The new method, GL-PD, can be seen as a “hybrid” between SG and PD with meshfree discretization. We discuss the similarities and differences between the GL-PD and the SG approaches. Main differences are related to calibration and discretization procedures. We use a 1D heat diffusion example to highlight some limitations the spectral-based semi-analytical method in the SG approach has compared with the direct time-integration normally used in computing solutions to transient diffusion problems. We then propose an extension of the semi-analytical approach to solve transient diffusion problems with Dirichlet boundary conditions, using a recent scaling and squaring algorithm that calculates the matrix exponential of non-symmetric matrices.

A simple method to predict transient diffusion processes in ellipsoids

This work is focused on transient diffusion processes inside semi-solid and solid agri-food products presenting ellipsoidal shape, i.e. prolate and oblate spheroids and general ellipsoid. Many typical industrial operations like drying, soaking, rehydration, heating, cooking, and cooling, can be effectively modelled by using Fick's law for mass diffusion and Fourier's law for heat conduction, with constant properties and neglecting volume/shape variations. To predict such processes, there are some methods available in the literature, although they are limited to certain shapes and/or operating conditions and their implementation can be quite complicated. Therefore, our objective was to develop a simple method to predict transient diffusion processes in ellipsoids and spheroids, considering both local and average values of industrial interest. The methodology to obtain such a simple method consisted in first generating numerical data by simulating a wide range of shapes and conditions, and then fitting a simple two-parameter expression, which is based on the analytical solution for regular geometries. The developed method can be easily implemented through the use of tabulated values provided in this article. Finally, the capability of the proposed method was verified by using previously reported data, showing very good results and demonstrating that it is easily implementable. It is expected that the developed simple method can be helpful for industrial applications and to improve calculations in related fields of research, which may not be familiar with transport phenomena modelling. • Many food processes involve transient mass and/or heat diffusion in ellipsoids. • A semi-analytical, user-friendly and accurate prediction method is developed. • Different prolate, oblate and general ellipsoids, and Biot values are covered. • Evolution of average, central and surface values of dependent variable is predicted. • The simple method can also estimate transport properties by inverse methodology.

An enriched finite element method for efficient solutions of transient heat diffusion problems with multiple heat sources

We propose an efficient formulation using the framework of Generalized Finite Element Method (GFEM) for the solutions of transient heat diffusion problems having multiple heat sources in the solution domain. The purpose here is to use minimal computational resources and rely on coarse mesh grids to capture the sharp variations of temperature field. We use Gaussian functions of global nature to enrich the GFEM approximation space which ensure efficient solution in the whole solution domain. To capture steep thermal gradients at multiple locations, a multiplicity of enrichment functions is used and the peaks of enrichment functions are centred at the cores of heat sources. The advantage of this approach is that no further degrees of freedom (DOFs) are added to capture the solution at multiple locations. Besides, the enrichment functions are time-independent; the temporal variation of temperature is embedded in the definition of the enrichment functions. This formulation requires the assembly of system matrix once and only the right-hand side of the system of equations is updated at subsequent time steps which results in a significant reduction in the overall computational time. We consider problems in two-dimensional (2D) and three-dimensional (3D) domains to show the effectiveness of the proposed approach. A reduction of more than 95% in DOFs and more than 80% in computational time is achieved as compared to the h-version of finite element method.

Evidence for a thermally driven charge-density-wave transition in 1T-TaS2 thin-film devices: Prospects for GHz switching speed

We report on the room-temperature switching of 1T-TaS2 thin-film charge-density-wave devices, using nanosecond-duration electrical pulsing to construct their time-resolved current–voltage characteristics. The switching action is based upon the nearly commensurate to incommensurate charge-density-wave phase transition in this material, which has a characteristic temperature of 350 K at thermal equilibrium. For sufficiently short pulses, with rise times in the nanosecond range, self-heating of the devices is suppressed, and their current–voltage characteristics are weakly nonlinear and free of hysteresis. This changes as the pulse duration is increased to ∼200 ns, where the current develops pronounced hysteresis that evolves nonmonotonically with the pulse duration. By combining the results of our experiments with a numerical analysis of transient heat diffusion in these devices, we clearly reveal the thermal origins of their switching. In spite of this thermal character, our modeling suggests that suitable reduction of the size of these devices should allow their operation at GHz frequencies.

An experimentally-verified temperature control simulation model for spray evaporative cooling of vibrating powertrain parts

A new simulation capability is presented to enable the performance of a hardware-based temperature control system to be assessed in thermally-managing heat-generating automotive vehicle powertrain parts. Temperature control is assumed to involve spray evaporative cooling of powertrain parts exposed to vibration. Two hypotheses are proposed to enable construction of a practical simulation that is both accurate and computationally efficient. The first is that a dynamic correlation model for single-nozzle spray evaporative cooling of a flat test-piece exposed to vibration, can be used as a reasonable model for multiple-nozzle spray evaporative cooling of component parts with curved cooling surfaces of non-horizontal orientation. The second is that the transient heat diffusion properties of a particular 3-dimensional component can be replaced by a 1-dimensional (1D) equivalence. To test this hypothesis, Finite Element models for two representative parts have been constructed and used to demonstrate the quality of the 1D heat diffusion equivalence, for which a fast Finite Difference solution can be exploited. To test the accuracy of test-piece surface temperature control simulation, an experimental test facility has been built in hardware, in which the temperature of two instrumented test-pieces exposed to vibration (from a shaker) are controlled by spray evaporative cooling. Each test piece is electrically-heated and the hardware control system is configured using PID control, for which appropriate gains are selected. Detailed comparisons of temperature control by hardware and simulation are given for the two test-pieces under static and dynamic conditions. Good agreement is generally obtained between simulated surface temperatures compared with measurements taken from both test-pieces. The paper shows that temperature control of a hardware-based control system using spray evaporative cooling of powertrain parts can be confidently simulated.

Influence of porosity on the thermal, electrical, and mechanical performance of selective laser melted stainless steel

This study verifies a novel approach to determine the thermal conductivity developed by the first two authors (Tomanek and Stutts) [1] as applied to additively manufactured selective laser melted stainless steel 304L specimens having a range of 1.4 to seven percent porosity. The selective laser melting technique is highly dependent on the process parameters used, unlike traditionally manufactured materials, and can cause the thermal, electrical, and mechanical properties to vary considerably from the bulk alloy. For this study, the thermal conductivity and several auxiliary parameters were estimated using a Levenberg-Marquardt nonlinear least squares algorithm. The parameter estimation used a model of a one-dimensional transient heat diffusion PDE with a closed-form solution of a slender rod under forced convection. In addition to the thermal conductivity’s dependency on porosity, the correlated porosity dependency on electrical conductivity was examined. The results were corroborated by mechanical tensile tests as well. The stainless steel 304L selective laser melted specimens saw a degradation of mechanical, thermal, and electrical performance with increasing porosity.

Coupling the BEM and analytical solutions for the numerical simulation of transient heat conduction in a heterogeneous solid medium

This paper simulates the three-dimensional transient heat diffusion conduction in a solid medium containing multi-layered cylindrical circular heterogeneities using a coupling formulation between the boundary element method (BEM) and analytical solutions. The analytical solutions are incorporated as Green's functions, thereby making the discretization of the interfaces of the multi-layered system unnecessary; this improves the efficiency of the algorithm. The coupling is enforced by imposing the continuity of temperature and heat fluxes at virtual interfaces created around each heterogeneity.The solution is first computed in the frequency domain, for a wide range of frequencies and axial wavenumbers, using complex frequencies to avoid aliasing phenomenon. Time series can then be obtained by means of a (fast) inverse Fourier transform.The accuracy, efficiency, and stability of the proposed algorithms are confirmed by comparing the results against reference solutions. The transient analysis of heat diffusion in a solid medium in the presence of multi-layered inclusions is used to illustrate the potential of the proposed coupling formulation.

Material thermal properties estimation via a one-dimensional transient convection model

This study investigates an approach to determine the thermal conductivity of a material from transient temperature data. For this study, a one-dimensional transient convection heat diffusion PDE with a closed-form solution is used to model slender test coupons. The inhomogeneous boundary condition is handled using eigenfunction expansion and Green’s second identity. A modified Levenberg–Marquardt (LM) nonlinear least squares (NLS) algorithm is used to determine the thermal conductivity from the experimental data using a flux boundary condition. The boundary conditions require an open-loop heater control and forced convection along the rod, which is simpler to implement than current methods. The estimated thermal conductivity from the experimental data was within ten percent of the nominal published values of the materials tested. The method described here promises to be easier to implement than current standards with similar accuracy and is applicable over a wide range of thermal conductivities – ranging from 15 to 400 W/mK. The method utilizes a small amount of material and a simple geometry, and is therefore suitable for the thermal characterization of additively-manufactured materials on a batch-by-batch basis.

Local adaptive [formula omitted]-enrichments and generalized finite elements for transient heat diffusion problems

A residual a-posteriori error estimate is used in conjunction with a q−adaptive procedure for selecting enrichment functions in the modelling of transient heat diffusion problems with the Generalized Finite Element Method (GFEM). The error estimate allows to assess local as well as global errors of the GFEM solutions and provides a tool to adaptively enrich the approximation space. With the proposed adaptive algorithm, the total number of degrees of freedom (DOFs) is significantly reduced in comparison to uniform enrichment. For a comparable accuracy, a reduction of up to 80% in DOFs is achieved. Moreover, the error estimate captures the deterioration of the conditioning of the system matrix typically encountered in enrichment based methods and enables the possibility of keeping the condition number within an acceptable limit.

Contact resistance analysis applied to simultaneous estimation of thermal properties of metals

This work aims to present a method for simultaneously estimating the thermal conductivity, k, and specific heat, cp, in samples of stainless steel 304 and cemented carbide by accounting the imperfect thermal contact at the heater–sample interface. A transient one-dimensional heat conduction model with constant thermal properties is used. The metallic sample is placed in the middle of a polyimide heater and a thermal insulator. A constant heat flux is applied on the specimen upside surface and a thermal insulation condition is maintained on the opposing surface, where a type T thermocouple measures the temperature response. Contact resistance is determined and accounted for as a reducing factor on heat flux. Instead of considering a lumped model, a microscopic approach of the contacting regions is used to describe both the surface roughness and the fluid gap. With the intention of guaranteeing simultaneous and reliable estimates for both thermal properties, sensitivity analysis is used to establish heat flux intensity, experiment duration, time step for data acquisition, and other characteristics of the experiments. To simultaneously achieve both thermal properties, the optimization method BFGS (Broyden–Fletcher–Goldfarb–Shanno) is employed to minimize a least-squares objective function comparing experientially measured and numerically calculated temperatures. The numerical solution for the transient conduction problem is determined with COMSOL, which solves the transient heat diffusion problem by discretizing it applying the finite element method (FEM). The linear system of equations is solved using the PARDISO solver and backward differentiation formula (BDF) method. Heat flux is estimated employing the sequential function specification method (SFSM) as a means of verifying the robustness of parameter estimation and experimental procedure. Moreover, uncertainty analysis is performed to confirm the quality of the results obtained. Finally, the uncertainty analysis outcomes and thermal properties estimated are compared with literature.

A residual a posteriori error estimate for partition of unity finite elements for three‐dimensional transient heat diffusion problems using multiple global enrichment functions

SUMMARYIn this article, a study of residual based a posteriori error estimation is presented for the partition of unity finite element method (PUFEM) for three‐dimensional (3D) transient heat diffusion problems. The proposed error estimate is independent of the heuristically selected enrichment functions and provides a useful and reliable upper bound for the discretization errors of the PUFEM solutions. Numerical results show that the presented error estimate efficiently captures the effect of h‐refinement and q‐refinement on the performance of PUFEM solutions. It also efficiently reflects the effect of ill‐conditioning of the stiffness matrix that is typically experienced in the partition of unity based finite element methods. For a problem with a known exact solution, the error estimate is shown to capture the same solution trends as obtained by the classical L2 norm error. For problems with no known analytical solutions, the proposed estimate is shown to be used as a reliable and efficient tool to predict the numerical errors in the PUFEM solutions of 3D transient heat diffusion problems.