Heat Conduction Problem Research Articles

A novel numerical modeling of microsecond laser beam percussion micro-drilling of Hastelloy X: experimental validation and multi-objective optimization

The paper investigates the characteristics of the laser beam percussion micro-drilling (LBPMD) process in aerospace nickel-based superalloy Hastelloy X using microsecond pulses. The quality of the drilled hole is crucial in laser beam micromachining, and selecting appropriate process parameters significantly impacts the hole’s quality. The objective is to achieve predefined hole dimensions with minimal taper angles. Additionally, the study focuses on the alteration of pulse width, which is a combination of laser pulse frequency and duty cycle. Laser power (P), duty cycle % (D), focal plane position (FPP), and laser frequency (f) are considered input parameters, while geometric features such as inlet and outlet diameters, hole taper angle, and inlet circularity are examined as process responses. ANOVA is employed to establish significant relationships between process parameters and response variations based on experimental tests. Creating a precise simulation model that accurately accounts for the moving boundary of the target material’s receding surface is a crucial and challenging task in formulating the laser heat conduction problem. It is necessary to simultaneously capture the material’s dynamic front movement and update the boundary conditions of the laser source. To model the micro-drilled hole with LBPMD, the UMESHMOTION and DFLUX subroutines, along with the arbitrary Lagrangian-Eulerian (ALE) adaptive remesh algorithm in the Abaqus™ software, are utilized. Notably, no previous numerical study has predicted the geometry of micro-drilled holes using this technique. The proposed procedure is validated through the predictions of inlet and outlet hole diameters. Special emphasis is placed on the validation of models. Consequently, the numerical model and statistical model are compared as well as the need to define model applicability. The study demonstrates that all input parameters significantly influence the inlet hole diameter, while the pulse width notably affects the taper angle and circularity. The interaction between high laser frequency and low duty cycle results in reduced pulse duration. Multi-objective optimization is performed to determine the optimal process parameter settings for desired quality characteristics, considering minimum hole taper angle, precise inlet diameter, and maximum inlet circularity of the hole as optimization criteria. The findings show that with the optimized predicted results obtained from the optimal input variables, a composite desirability of 92% can be achieved.

Conjugate gradient method with regularization in estimating mold surface heat flux during continuous casting

The reconstruction of the boundary heat flux between solidifying steel and water-cooled mold wall using measured temperature data is recognized as an inverse heat conduction problem (IHCP). A Tikhonov spatial regularization approach to enhance the smoothness of the Conjugate Gradient Method (CGMr) was proposed which incorporates three different orders of spatial regularization: zeroth, first, and second order. The optimal regularization parameter was selected using a modified L-curve method. The accuracy of the CGMr is investigated with Case 1: a triangular time-spatial variation heat flux, and Case 2: a step change in the form of rectangular variation heat flux. The effects of measurement noise, grid points and time-step size are investigated. The results show that the minimum relative error (eRMS) of the predicted Case 1 heat flux is 1.98%, 7.64%, and 8.02% for zeroth-, first-, and second-order spatial regularization, respectively. The corresponding values for the predicted Case 2 heat flux are 3.82%, 13.42%, and 14.91%. Subsequently, CGMr algorithm is applied to calculate the heat flux in a mold simulator experiment. By comparing the relationship between heat fluxes reconstructed by CGMr after different iteration numbers, it is observed that the recovered heat flux of 2.14 MW/m2 with zeroth regularization remains highly stable.

Thermo-Mechanical analysis of laminated Doubly-Curved Shells: Higher order Equivalent Layer-Wise formulation

The paper presents a refined two-dimensional formulation for the thermo-mechanical analysis of laminated doubly-curved shell structures under thermodynamic equilibrium conditions. Both the kinematic configuration variables and the temperature variation with respect to the natural equilibrium state are described with a generalized formulation, following the Equivalent Layer-Wise (ELW) approach employing higher order polynomials, along with some proper zigzag functions. The governing equations are derived from the stationary configuration of the Helmholtz free energy of the system, and a semi-analytical solution is found. In the post-processing phase, the Fourier-based Generalized Differential Quadrature (F-GDQ) is applied to recover the actual three-dimensional response of the doubly-curved shell solid, and very accurate results are obtained for the quantities of both mechanical and heat conduction problems. In addition, the integrals occurring in the theory are performed numerically with the Taylor-based Generalized Integral Quadrature (GTIQ), showing a high level of accuracy with a reduced number of sample points. The model is validated in some case studies, where the accuracy of the model is shown, and the numerical predictions are successfully compared with those ones from refined three-dimensional numerical simulations. After that, an extensive set of parametric investigations is reported, pointing out the effects of the panel curvature, lamination schemes and different levels of coupling on the thermo-mechanical structural behavior of the investigated panels.

Determination of interfacial heat transfer coefficient at the frozen sand mold casting process of ZL101 alloy

Compared to the resin sand mold casting process, frozen casting is more environmentally friendly, providing a better working environment and enhanced supercooling degree. The interfacial heat transfer coefficient (IHTC) between frozen sand mold and metal is an important parameter that significantly influences the final mechanical properties and microstructure of the castings. This paper solved the inverse heat conduction problem using the finite difference method (FDM). In addition, the conjugate gradient method (CGM) was adopted to calculate the temperature distribution and heat flux in the molten metal. At the same time, the particle swarm optimization algorithm (PSO) was used in temperature distribution determination in frozen sand mold. The interfacial heat transfer coefficient (IHTC) was estimated during the solidification of ZL101. The results showed a good agreement between calculated and experimental data, obtaining accurate casting interface temperature Tm, frozen sand mold interface temperature Ts, heat flux q, and IHTC. The analysis of the IHTC variation revealed a water content value within the range of 4 wt.% to 5 wt.% resulted in IHTC in two types of interpretation, called ‘fluctuation type’ and ‘turning type’.

Monitoring internal heat fluxes on Pulsating Heat Pipes using Kalman filter – Numerical and experimental results

The present study proposes an approach to monitor the wall-to-fluid internal heat flux on Pulsating Heat Pipes (PHP), which provides meaningful information about the efficiency and limiting operation conditions of such devices. In this sense, the current work adopts the Kalman Filter (KF) to estimate the heat flux between the working fluid and the internal tube wall along the adiabatic section of a PHP device by solving an inverse heat conduction problem. The KF is a well-known tracking technique, and therefore the current study is attractive due to the possibility of real-time evaluation of internal heat fluxes, a prospect that is extremely difficult to realize by standard whole domain inverse problem techniques. A high-speed and high-resolution infrared camera performs the temperature acquisition on the external wall of the PHP device. Two mathematical heat conduction models are considered: complete and thin-wall models. The complete model determines the temperature and heat flux evolution over time, and the thin-wall model is used to correct the random walk parameter on the KF formulation. Since the time step needed to the KF evolution model is smaller than the measurements acquisition interval, a matrix manipulation is used during the prediction step, reducing the number of mathematical operations. Hence, instead of predicting the states until reaching the measurement times, a proper mathematical manipulation reduces several mathematical operations to a single one. Then, the KF estimation capability is evaluated by considering synthetic data with different noise levels and different heat flux values. Subsequently, real experimental data are employed, and the results are compared with the ones obtained via a standard whole domain technique. The results obtained with the KF presented good visual agreement with the exact solution. Once the KF presents a delayed response, which is a characteristic of some filtering techniques, the errors with synthetic data were calculated offsetting the delay of the estimation in time in order to make a fair comparison, showing a maximum estimation error lower than 36 % considering the highest noise level. The synthetic data considered 20 s of time experiment, and the KF was able to estimate the PHP internal heat flux in less than 12 s, including the smoothing temperature step, showing fast and accurate results. For the real experimental data, the agreement between the KF and a whole domain technique is highly commendable, demonstrating a deviation of 25.5 % between both techniques. Moreover, the KF required significantly less computational time. The proposed methodology for heat flux estimation on PHPs is promising to evaluate their thermal behavior in real-time due to a low CPU time and computational memory required, as well as its simplicity to be implemented, possibly on embarked devices. Such information may therefore provide a thermal indicator of the correct functioning of PHPs in real time applications.

Fundamental Solutions of the Equations of Classical and Generalized Heat Conduction Models

This article presents the mathematical formulations of transient heat conduction problems corresponding to the models of classical heat conduction using the Fourier law and generalized heat conduction based on the Cattaneo–Vernotta–Lykov law (Maxwell–Cattaneo model), as well as the generalized Green–Nagdy type II and III models. The Fourier transforms in spatial coordinates and the Laplace transforms in time were used to obtain the fundamental solutions of the equations of the Maxwell–Cattaneo and Green–Nagdy type II and III models of classical and generalized heat conduction. The results were displayed graphically and analyzed. Differences between the considered heat conduction models were shown, and suggestions for their practical application were given.

Nonlinear Estimators for Hypersonic Heat Flux Reconstruction

A novel class of nonlinear estimators that minimizes the projection of the error on a multiresolution basis is developed to improve the accuracy and efficiency of the multidimensional reconstruction of the heat flux from discrete temperature sensors in nonplanar hypersonic boundary layers. This approach uses the local temperature time derivative as the pre-estimator of the heat flux estimation risk in the inverse heat conduction problem. This new approach leads to improved results both in terms of computational efficiency and signal-to-noise ratio when compared to both regularization of the linear response operator and filtering of the temperature measurements. The approach is verified against an analytical verification test case, which shows significant accuracy improvement compared to optimal linear estimators and L2 regularization. It is then applied to experimental data to resolve the heat flux from noisy temperature data collected using fast-response sensors in the Virginia Tech hypersonic wind tunnel at Mach 6. The heat flux agrees very well with the Fay–Riddell approximation. Moreover, the heat transfer coefficient variation over the surface of the test is consistent with computational fluid dynamics simulations of the Reynolds-averaged Navier–Stokes equations.

Numerical Model of Simultaneous Multi-Regime Boiling Quenching of Metals

This work presents a heat transfer and boiling model that computes the evolution of the temperature field in a representative steel workpiece quenched from 850 or 930 °C by immersion in water flowing at average velocities of 0.2 or 0.6 m/s, respectively. Under these conditions, all three boiling regimes were present during cooling: stable vapor film, nucleate boiling, and single-phase convection. The model was based on the numerical solution of the heat conduction equation coupled to the solution of the energy and momentum equations for water. The mixture phase approach was adopted using the Lee model to compute the rates of water evaporation–condensation. Heat flux at the wall was calculated for all regimes using a single semi-mechanistic model. Therefore, the evolution of boiling regimes at every position on the wall surface was automatically determined. Predictions were validated using laboratory results, namely: (a) videorecording the upward motion of the wetting front along the workpiece wall surface; and (b) cooling curves obtained with embedded thermocouples in the steel probe. Wall heat flux calculations were used to determine the importance of the simultaneous presence of all three boiling regimes on the heat flux distribution. It was found that this simultaneous presence leads to high heat flux variations that should be avoided in production lines. In addition, it was determined that the corresponding inverse heat conduction problem to estimate the active heat transfer boundary condition must be set-up for 2D heat flow.

Registration of melting temperature at phase boundaries in melting powder mixtures

An integral constituent of the induction surfacing process for parts hardening pertains to the thermal treatment of rigid alloy particulates and the flux contained within the surfacing amalgam. This scholarly exposition delineates the outcomes of a comprehensive investigation, oriented towards the quantification and simulation of thermal gradients at interphase boundaries within intricate amalgamations of melting and thermosetting powdery substrates. In order to monitor the thermal dynamics during induction surfacing, the application of the CA-microthermocouple methodology and the thermal indication technique utilizing self-propagating SHS-process compositions is posited. The study encompasses both computational simulations and the resolution of the unsteady-state heat conduction problem within the chosen model of composite contacting materials. The devised methodologies for the intricate tracking of thermal profiles during induction surfacing offer the capability to ascertain the temperatures at which the surfacing amalgam and its individual constituents undergo liquefaction, in conjunction with the temperature gradient exhibited by the surface of the targeted component. This facet holds marked significance in the context of optimizing the hardening regimen and ensuring the requisite attributes of the amalgamated materials post-fusion. The outcomes of this research bear practical relevance in industrial applications, wherein enhancements to the caliber and dependability of hardened components are sought, simultaneously facilitating the curtailment of production overheads.

Peridynamic modeling of transient heat conduction in solids using a highly efficient algorithm

This article presents a novel peridynamic algorithm to model the transient heat conduction in two-dimensional (2-D) problems. The peridynamic method is used to solve the heat transfer equation, and a new algorithm is developed to simulate heat conduction efficiently. The new peridynamic algorithm can simulate the heat conduction for the whole geometry within only 2 s when using up to 5,000 particles and 5,000 simulation steps, where the mesh-free nature of peridynamics is well utilized. Moreover, the thermal results obtained from the new peridynamic algorithm demonstrate high accuracy by solving several heat conduction problems under different boundary conditions. The results are validated against exact and analytical solutions and compared with other numerical models. The developed algorithm can be easily implemented in software packages and industrial applications.

Effects of flow direction and thermal short-circuiting on the performance of coaxial ground heat exchangers

The effects of flow direction and thermal short-circuiting on the performance of coaxial ground heat exchangers are studied by finite-element simulations, performed through COMSOL Multiphysics 3.4 (©Comsol, Inc.). The real 2-D axisymmetric unsteady heat conduction and convection problem is considered. The distribution of the fluid bulk temperature in the inner circular tube is determined by means of the “weak boundary form” boundary condition available in COMSOL Multiphysics; the laminar convective heat transfer in the outer annular passage is simulated directly. Two Coaxial Ground Heat Exchangers (CGHEs) with the same length but different cross sections are examined; moreover, two values of the ground thermal conductivity, as well as two materials for the inner tube wall are considered. The results point out that the annulus-in flow direction (fluid inlet in the outer annular passage) is more efficient than the center-in flow direction (fluid inlet in the inner circular tube) and that the effect of short circuiting is not very important, if the annulus-in flow direction is employed. Indeed, with this inlet configuration the energy loss for thermal short-circuiting is less than 1% for time intervals longer than 1 hour.

Identifying the Heat Source in Radially Symmetry and Axis-Symmetry Problems

This paper solves the inverse source problem of heat conduction in which the source term only varies with time. The application of the discrete regularization method, a kind of effective radial symmetry and axisymmetric heat conduction problem source identification that does not involve the grid integral numerical method, is put forward. Taking the fundamental solution as the fundamental function, the classical Tikhonov regularization method combined with the L-curve criterion is used to select the appropriate regularization parameters, so the problem is transformed into a class of ill-conditioned linear algebraic equations to solve with an optimal solution. Several numerical examples of inverse source problems are given. Simultaneously, a few numerical examples of inverse source problems are given, and the effectiveness and superiority of the method is shown by the results.

Modeling of the thermophysical properties of molding materials by solving the inverse heat conductivity problem

Modeling of the thermophysical properties of molding materials by solving the inverse heat conductivity problem

A new MIB-based time integration method for transient heat conduction analysis of discrete and continuous systems

By incorporating the matched interface and boundary (MIB) method into time integration computation, a new MIB-based time integration (MIBTI) approach is proposed for solving transient heat conduction problems in this work. Indeed, this is a truly effective time-domain MIB method that can directly discretize space and time derivatives, and the transient heat conduction problem (an initial-value problem) is transformed into a pseudo-space boundary-value problem. The initial and final time points are regarded as boundary points in the transformed space domain. Two algorithms are developed to deal with the final time point. Unlike conventional recursive schemes for time integration, all unknown variables can be determined by solving the transformed boundary-value problem in the present approach. Various discrete and continuous systems without and with internal heat sources under different boundary conditions are considered to examine the working performance of the proposed method. The applicability of the time stability criterion in the conventional recursive schemes to the proposed method is also discussed. It is found that excellent numerical stability and convergence can be achieved by using the present MIBTI method for solving transient heat conduction problems.

Vat photopolymerization additive manufacturing process modeling: a thermal-chemical coupling approach informed by in-situ and ex-situ characterization data

Vat photopolymerization (VPP) is one of the most widely used additive manufacturing methods. The VPP process temperature and material curing reaction interplay with each other to critically determine the final product quality. Insights about the time-varying process temperature and degree of conversion (DoC) is desired for VPP process control but difficult to attain due to lacking effective operando characterization technologies. This work reports a new method to create a thermal-chemical model of the VPP process by solving an inverse heat conduction problem (IHCP) based on in-situ observable temperature measurement to estimate the chemistry reaction-induced heat source that is a function of DoC. Ex-situ photo differential scanning calorimetry (Photo-DSC) characterization is used to initialize the chemistry reaction model parameters so that DoC can be calculated. Specifically, vat substrate temperature is measured using an in-situ infrared thermal camera and used as input to solve an IHCP for estimating exothermic heat generation rate for the internal heat generation component at the curing part. Overall, the newly developed VPP modeling framework combines an IHCP that is optimized by in-situ thermal monitoring with a chemical reaction heat generation and conduction model that is educated by Photo-DSC characterization. The model predictions of temperature and DoC are experimentally validated by comparing against in-situ temperature measurement and ex-situ spectroscopy measurement of prints at different exposure times.

Simulation of thermal processes in a half-space under heating by a moving source with a uniformly distributed heat flow

Using the method of applying instantaneous point sources, a solution was obtained to the problem of heat conduction during surface heating of a body in the form of a half-space by a uniformly distributed highly concentrated heat flux moving at a constant speed along a rectilinear trajectory with a different shape of the heating spot at constant thermophysical characteristics of the material. The effect of temperature loading modes and the shape of the heating spot on thermal processes in the heat-affected zone is studied. The surfaces and lines of the temperature level are constructed for different moments of time and speed loading modes in different planes of the heating zone. Time dependences of temperatures, heating and cooling rates for body points are given. The shortcomings of the methods used for linear thermal conductivity, the lack of direct consideration in the design scheme of the surface melt zone of the material do not allow one to reliably assess the effect of heat treatment modes on changes in material properties, focusing only on the level of the maximum design temperature. In this regard, the structure formation of metal in the zone of thermal action is proposed to be associated with a thermal impulse, i.e. the total thermal energy perceived by the material at a given point of the body, as well as with the effective structurization impulse introduced into consideration, which characterizes the energy spent on the process of structural transformations of the material, and the structurization time at a point and some volume of the body. The dependencies of these values on the speed of movement and the shape of the heating spot are presented. The considered approaches can be applied to various metals and alloys. The research results can be used to develop more effective methods for determining the optimal modes of surface hardening of metal products with a high-energy source.

A Detailed Study of Boiling Phenomena in Forced Convective Quenching Experiments

A detailed study at the microscopic level of the transient behavior of boiling phenomena during quenching of an AISI 304 stainless steel, conical-end, cylindrical probe in flowing water at 60 °C was conducted using high-speed video recordings and cooling curve data acquisition. Two free-stream velocities (0.2 and 0.6 m/s) and two initial probe temperatures (850 and 950 °C) were investigated. It was complemented by video recordings at 60 fps to calculate the wetting front velocity. Surface heat flux histories at the thermocouple positions were estimated by solving the corresponding Inverse Heat Conduction Problem. As the water velocity increases the duration of the vapor film at each thermocouple position shortens and the difference among cooling curves at the thermocouple positions decreases. Increasing the initial temperature increases the duration of the vapor film stage. The wetting front velocity increased from 4 to 6 m/s. The time to reach the maximum surface heat flux at the position of the thermocouple closest to the probe tip ranged from 9.7 to 18 s. Undulations at the vapor-liquid interface that appear periodically and propagate in the direction of quenchant flow were observed early during the vapor film stage. Later, before the collapse of the vapor film, a kind of a shock wave was generated at the probe tip which modified the appearance of the film. Once the Leidenfrost temperature is reached, a wetting front (which consists of many small bubbles that coalesce rapidly in a very small area) is formed and travels in the direction of quenchant flow while fewer and larger bubbles nucleate and grow upstream. The nucleation, growth and detachment of the larger bubbles was studied; as the free-stream velocity decreases larger values of bubble maximum diameter and half-life time were observed, while the initial temperature has a marginal effect on these quantities.

A new distance transformation method of estimating domain integrals directly in boundary integral equation for transient heat conduction problems

Accurate estimation of domain integrals is of crucial importance for the pseudo-initial condition method applied in transient heat conduction analysis. When small time step is used, the integrand of the domain integral changes dramatically and is close to singular. A straightforward evaluation of the domain integrals using Gaussian quadrature may produce large errors. To improve the computational accuracy of two-dimensional domain integrals in boundary integral equation, a new distance transformation method is presented in this paper. The proposed method takes the time step as the transformation parameter, implementing the new transformation in the radial direction directly. With the new distance transformation method, the integrand of the domain integral can be smoother, thus more accurate results can be obtained. Numerical examples have demonstrated the efficiency and accuracy of the proposed method.

Termostressed state of a three-layer rectangular plate under non-stationary convective heating conditions

The study considers a rectangular isotropic plate with a layered irregular structure. It is convectively non-stationarily heated by an external environment. The initial relationships of the non-stationary heat conduction and thermoelasticity problem are formulated using a five-mode mathematical model based on the shear deformation theory of thermoelasticity. Using the methods of Fourier and Laplace integral transforms, general solutions have been obtained for the non-stationary heat conduction problem and the quasi-static thermoelasticity problem for a hinge-supported plate along its edges. A numerical analysis of the temperature field, radial deflections, normal forces, bending moments, and normal stresses, depending on geometric parameters and the Bi criterion, has been performed for a three-layer plate. The materials of its layers are made of ceramics and metal. The temperature and mechanical parameters have been analyzed for the layering configuration of the plate: metal-ceramic-metal.

Development of Methods for Solving Problems of Transient Heat Conduction of Plane Layered Media

Development of Methods for Solving Problems of Transient Heat Conduction of Plane Layered Media