Time-dependent Heat Flux Research Articles

Unsteady magnetohydrodynamic stagnation point flow—closed-form analytical solutions

This paper investigates the unsteady stagnation point flow and heat transfer of magnetohydrodynamic (MHD) fluids over a moving permeable flat surface. The unsteady Navier-Stokes (NS) equations are transformed into a similarity nonlinear ordinary differential equation, and a closed form solution is obtained for the unsteadiness parameter of 2. The boundary layer energy equation is transformed into a similarity equation, and is solved for a constant wall temperature and a time-dependent uniform wall heat flux case. The solution domain, velocity, and temperature profiles are calculated for different combinations of parameters including the Prandtl number, mass transfer parameter, wall moving parameter, and magnetic parameter. Two solution branches are obtained for certain combinations of the controlling parameters, and a stability analysis demonstrates that the lower solution branch is not stable. The present solutions provide an exact solution to the entire unsteady MHD NS equations, which can be used for validating the numerical code of computational fluid dynamics.

Real-time estimation of time-dependent imposed heat flux in graded index media by KF-RLSE algorithm

Due to the large computational time caused by complex computational process of the existing inversion algorithm, real-time reconstruction of high-magnitude time-dependent heat flux in graded index media is quite challenging. In this study, based on hybrid technology of the Kalman filter and recursive least-square estimator (KF-RLSE), the real time reconstructed high-magnitude time-varying heat flux on graded index media surface, and the measurement information comes from the opposite side of the media. The ideal participating media, which is assumed to be isotropic scattering, constant thermophysical properties, and opaque and diffuse gray boundary, is employed to verify the reliability and validity of the proposed. All the reconstruction results show that the KF-RLSE algorithm can effectively reconstruct the boundary heat flux regardless of the positive or negative gradient of the refractive index. When the refractive index of each position increases or reduces, the transient heat flux on the surface can still be predicted effectively and acceptably. Furthermore, effects of different parameters on the accuracy and stability of the estimated results are also investigated. The reconstructed results show that the time-dependent heat flux can still be effectively reconstructed even when the measurement noise does not match its covariance. Meanwhile, the accuracy of the reconstruction results improves with the decrease of measurement noise covariance when the measurement noise distribution is fixed in a curtain range.

Solution of the Phase Change Stefan Problem With Time-Dependent Heat Flux Using Perturbation Method

Theoretical understanding of phase change heat transfer problems is of much interest for multiple engineering applications. Exact solutions for phase change heat transfer problems are often not available, and approximate analytical methods are needed to be used. This paper presents a solution for a one-dimensional (1D) phase change problem with time-dependent heat flux boundary condition using the perturbation method. Two different expressions for propagation of the phase change front are derived. For the special case of constant heat flux, the present solution is shown to offer key advantages over past papers. Specifically, the present solution results in greater accuracy and does not diverge at large times unlike past results. The theoretical result is used for understanding the nature of phase change propagation for linear and periodic heat flux boundary conditions. In addition to improving the theoretical understanding of phase change heat transfer problems, these results may contribute toward design of phase change based thermal management for a variety of engineering applications, such as cooling of Li-ion batteries.

Real-time retrieval of transient heat flux on the surface of participating medium by using the EKF-RLSE technique

A hybrid algorithm of extended Kalman filtering and recursive least-square estimator (EKF-RLSE) is developed to solve the nonlinear inverse heat conduction-radiation problems in participating medium. By measuring the time-resolved temperature signals from a thermocouple installed on the other boundary of the participating medium, the proposed EKF-RLSE algorithm can realize real-time retrieval of the time-varying heat flux on the medium surface. The influence factors of the sampling interval, measurement and process covariance, forgetting factor, and medium thickness on the reconstruction accuracy and stability of the EKF-RLSE are investigated by numerical experiments. All the estimated results show that the present algorithm is robust and accurate in real-time estimating transient heat flux. If the nonlinear coupled conduction and radiation heat transfer problems is regard as the linear problems, the unknown time-dependent heat flux predicted by the KF-RLSE are also illustrated in this work.

Wave Heat Transfer in the Orthotropic Half-Space Under the Action of a Nonstationary Point Source of Thermal Energy

A new analytical solution to the problem of wave heat transfer in the orthotropic half-space under the action of a time-dependent point heat flux is obtained and studied. The heat transfer is described by a hyperbolic heat conduction wave equation, in which the directions of the thermal conductivity coincide with the Cartesian coordinate system axes (the orthotropic solid). The obtained analytical solution has allowed us to trace the behavior of the point temperature profile in the vicinity of the initial time moment during a number of relaxation times, which is impossible to do when the classical parabolic heat conduction equation is used.

Analytical prediction of pyrolysis and ignition time of translucent fuel considering both time-dependent heat flux and in-depth absorption

This contribution reports an approximate analytical model to predict transient mass flux and ignition time of translucent fuel, black poly(methyl methacrylate) (PMMA), subjected to a time-dependent incident heat flux, atb, where t is time and a and b are constants. The model can be easily extended to other non-charring translucent solids. The model takes into account in-depth absorption of thermal radiation in the condensed phase, which is typically ignored in the analytical formulations. Both critical temperature and critical mass flux were employed as the ignition criteria to examine their effects on the predictions. The model was validated using exact numerical solutions and experimental data, and compared with earlier analytical models based on the assumption of surface absorption. Linear and quadratic heat fluxes were considered for validation and discussion. The results show that surface absorption accelerates the pyrolysis process and leads to higher mass flux and shorter ignition time with respect to the in-depth absorption case. The discrepancy between the predicted transient mass fluxes of these two absorption modes increases with increasing a. The ignition heat flux increases with increasing a and decreases with increasing b for both surface and in-depth absorption cases. However, the critical energy is independent of heat flux in in-depth absorption scenario. Furthermore, parametric studies of in-depth absorption coefficient and critical mass flux were conducted to investigate their effects on the quality of the model predictions. Also, the equivalent ignition temperature was calculated and compared with the experimental values. It is expected that the developed model will find its use in performance-based design applications.

Estimation of heat flux by using reduced model and the adjoint method. Application to a brake disc rotating

In this study, an approach for inverse problem is proposed to estimate a time dependent heat flux on a braking system. This approach combines modal reduction and conjugate gradient method with adjoint method. A brake disc with variable velocity and thermal solicitations is treated. In a first step, Branch Eigenmodes Reduction Method is used to build a reduced model which allows reduction of computing time while conserving a satisfying precision on temperature field. In a second step, an inverse algorithm based on adjoint method is applied to recover time dependent heat flux from temperature calculation. Compared to Beck method, adjoint method gives accurate results without any regularization. The proposed approach enables to perform a quasi on-line estimation.

Effect of radiation absorption modes on ignition time of translucent polymers subjected to time-dependent heat flux

Majority of previous solid ignition models, including numerical and analytical ones, considered only surface absorption of incident heat flux for simplification. However, the influence of in-depth absorption on pyrolysis and subsequent ignition cannot be ignored for infrared translucent polymers. This work addresses this problem and focuses on time-dependent heat flux to establish an analytical model for ignition behaviors prediction by means of theoretical analysis. Ignition temperature was utilized as the ignition criterion, and both surface and in-depth absorption scenarios were considered. Thermally thick polymethyl methacrylate and polyamide 6 were selected as reference materials to verify the reliability and applicability of the proposed model by comparing the analysis results with experimental data as well as numerical simulations. A method for determining the approximation parameters of the theoretical analysis was presented to derive the relationship between ignition time and the coefficients in heat flux expressions. The results show that the higher surface temperature owing to surface absorption accelerates the pyrolysis rate and results in a shorter ignition time, while in-depth absorption affects the ignition time inversely. The effect of surface heat loss was also evaluated quantitatively through both analytical and numerical models. The uncertainty of the proposed model is mainly caused by the selection of the approximation parameters. Nevertheless, it provides an alternative approach to estimate the ignition time of translucent polymers besides numerical simulation.

Application of KF-RLSE algorithm for on-line estimating the time-dependent melting thickness and input heat flux in participating media

This study proposes an inverse heat transfer process to estimate the time-dependent melting thickness and input heat flux in participating media. The inversion algorithm comprises Kalman filter technique and recursive least squares estimator (KF-RLSE) for on-line and real time predict the input heat flux and position of the solid–liquid phase front from the temperature history by using a sensor mounted on the surface. The KF technique is employed to generate a regression equation between the biased residual innovation and the unknown thermal coefficient. The recursive least squares estimator is utilized in the regression equation to predict time-varying phase front position and input heat flux. Based on simulation results, the proposed algorithm exhibits good reconstruction capability. The influences of forgetting factor, process noise, measurement noise, and absorption coefficient on the stability and precision of the retrieval results are also analyzed. The accuracy of the estimation increases with decrease of the distance between the acting surface of input heat flux and the sensor location. Furthermore, the present algorithm cannot accurately determine the abrupt heat flux with large time step.

An improved model for cold metal transfer welding of aluminium alloys

An improved model is proposed for automated cold metal transfer (CMT) welding based on a time-dependent double-ellipsoidal volumetric heat flux distribution. Equations are evaluated numerically within COMSOL Multiphysics for CMT welding of a 3-mm-thick AA5754 Al–Mg alloy plate. The simulation calculates transient and steady-state temperature distributions within the weld seam and heat-affected zone (HAZ). Validation of the model is achieved by comparing simulated temperatures with measured values from thermocouples in the HAZ during welding experiments, as well as through comparison of the calculated fusion zone and microscope images of the weld seam. Under steady-state conditions, large differences between the peak and average calculated temperatures in the weld pool highlight the underlying phenomenon responsible for improvements in weld quality for thin sheets with CMT compared to conventional joining processes. The developed simulation provides opportunities for process optimisation and sensitivity analysis in many applications.

Heat Transfer Analysis of Thermal Protection Structures for Hypersonic Vehicles

This research aims to develop an analytical approach to study the heat transfer problem of thermal protection systems (TPS) for hypersonic vehicles. Laplace transform and integral method are used to describe the temperature distribution through the TPS subject to aerodynamic heating during flight. Time-dependent incident heat flux is also taken into account. Two different cases with heat flux and radiation boundary conditions are studied and discussed. The results are compared with those obtained by finite element analyses and show a good agreement. Although temperature profiles of such problems can be readily accessed via numerical simulations, analytical solutions give a greater insight into the physical essence of the heat transfer problem. Furthermore, with the analytical approach, rapid thermal analyses and even thermal optimization can be achieved during the preliminary TPS design.

Simultaneous reconstruction of the time-dependent Robin coefficient and heat flux in heat conduction problems

This paper aims to solve an inverse heat conduction problem in two-dimensional space under transient regime, which consists of the estimation of multiple time-dependent heat sources placed at the boundaries. Robin boundary condition (third type boundary condition) is considered at the working domain boundary. The simultaneous identification problem is formulated as a constrained minimization problem using the output least squares method with Tikhonov regularization. The properties of the continuous and discrete optimization problem are studied. Differentiability results and the adjoint problems are established. The numerical estimation is investigated using a modified conjugate gradient method. Furthermore, to verify the performance of the proposed algorithm, obtained results are compared with results obtained from the well-known finite-element software COMSOL Multiphysics under the same conditions. The numerical results show that the proposed algorithm is accurate, robust and capable of simultaneously representing the time effects on reconstructing the time-dependent Robin coefficient and heat flux.

Heat-Flux Estimation in a Nonlinear Heat Conduction Problem with a Dual-Phase-Lag Model

In this study, an inverse algorithm based on the conjugate-gradient method and the discrepancy principle is used to solve the inverse nonlinear dual-phase-lag heat conduction problem in estimating the time-dependent boundary heat flux based on the measured back surface temperature. The accuracy of the presented method for solving the inverse problem is examined by utilizing two examples. The effects of measurement error, initial guess, measurement location, phase lag of heat flux, and nonlinearity on the accuracy of estimation are evaluated. The results demonstrate that the proposed method is an accurate and robust method to determine inversely the boundary heat flux in the hyperbolic heat conduction with the nonlinear behavior, even when the input data contain measurement errors. In addition, when the sensor is placed far from the heated surface, it is more difficult to conduct the reconstruction of the heat flux by using the non-Fourier model than the Fourier model. The comparison results obtained from the linear and nonlinear solutions demonstrate that, when the thermal conductivity is an increasing function of temperature, the accuracy of inverse solution is higher.

Pyrolysis and spontaneous ignition of wood under time-dependent heat flux

This work investigates experimentally and theoretically the effect of time-dependent incident heat flux (HF), which is more reasonable in fire-like environment, on thermal degradation process of wet pine wood. A feedback method was utilized to generate a time-dependent heat flux by controlling the output power of radiative heater. Both quadratic and linear heat fluxes were studied in this study. Comparison between measured heat fluxes and designed values indicates that the method provides high accuracy. Measurements of temperature distribution at different depths of material, ignition time and mass loss rate were implemented in the tests to examine the effects of time-dependent heat fluxes. Additionally, analytical model and numerical model were developed to predict the pyrolysis behaviors, and good agreement exists between the experimental and simulational results. Results showed that the heat penetration layer is restricted to a thinner depth for HF with higher increasing rate. A linear relationship was found between ignition time and HF parameters, which is also validated by experimental data and reexamined by constant heat flux circumstance. Mass loss rate was affected significantly by the changed heat flux compared with constant scenario. Furthermore, critical mass flux, which keeps almost unchanged, can be employed as ignition criterion due to the fact that the ignition temperature increases with increasing heat flux, which also certifies the conclusions of other researchers.

An analytical solution for temperature field around a cylindrical surface subjected to a time dependent heat flux: An alternative approach

An analytical solution for the temperature field in an infinite solid medium around a cylindrical surface subject to a time dependent heat flux proposed by Zanchini and Pulvirenti (2013) has been revisited. The well-known Laplace transform technique is applied to solve the time-dependent governing equation exactly in Laplace domain, while the method of Riemann-sum approximation approach is employed to invert the Laplace domain to the time domain in order to obtain the dimensionless temperature. For the case of sinusoidally varying heat flux, numerical values of the dimensionless temperature of the surface are obtained and compared with the benchmark values reported by Zanchini and Pulvirenti (2013), and an excellent agreement is found. An advantage of the proposed method could be a reduction in computation effort.

Estimation of time-dependent wall heat flux from single thermocouple data

Surface mounted sensors disturb the phenomena they are expected to record and are prone to damage in harsh environments, leading to incorrect measurements. The present study examines a robust alternative via the inverse estimation of time-dependent wall heat flux using single thermocouple data, as against an array of embedded thermocouples. The Levenberg-Marquardt algorithm for parameter estimation is employed with an even extension of truncated Fourier series as the trial function for inversion. The inversion error decreases with the length of the signal, number of harmonics in the even extension of truncated Fourier series, and proximity of the thermocouple relative to the boundary of interest. The method is numerically validated as well as experimentally benchmarked, against constant, periodic, and transient wall heat flux data in one-dimensional heat conduction problems and is seen to match the exact solution quite well. The developed method is very well suited for sensor development, particularly for application involving high heat flux.

Estimation of surface heat flux in a one-dimensional hyperbolic bio-heat conduction problem with temperature-dependent properties during thermal therapy

In this study, an inverse analysis is presented to determine the time-dependent surface heat flux of a living tissue in a one-dimensional non-Fourier bio-heat conduction problem by using temperature responses measured within the medium. The properties of tissue are assumed to be function of temperature. Therefore, this problem is classified as a nonlinear inverse problem of hyperbolic bio-heat conduction. The inverse problem is solved through the minimization of an objective function by the conjugate gradient method. Three examples are presented to show that the proposed method can be implemented to estimate the unknown surface heat flux. In addition, the effects of measurement error, measurement location, initial guess, nonlinearity and thermal relation time on the precision of the estimation are investigated. The results show that the proposed method is an accurate and robust method to determine inversely the surface heat flux in hyperbolic bio-heat conduction with temperature-dependent properties.

Estimation of Heat Flux in Variable Thickness Functionally Graded Annular Fin

An inverse heat conduction procedure for estimating the unknown time-dependent heat flux at the base of functionally graded variable thickness annular fins using only the measured temperature at the tip of fin is presented. The heat conduction formulation is based on the non-Fourier heat transfer equation. The conjugate gradient method is employed for optimization process, and the incremental differential quadrature method is applied to solve the governing differential equations. The accuracy of the presented approach is examined by simulating the exact and noisy data through different examples. Good accuracy of the obtained results validates the presented approach.

Determination of Radial Thermal Conductivity and Specific Heat in Externally-Heated LiCoO2 cells

In this presentation, we will investigate the thermophysical properties of LiCoO2 18650 battery cells. Due to their versatility and high energy density, lithium-ion batteries have become a major power source in a number of consumer and industrial products over the last decade. As their popularity and number of applications increase, so do instances of catastrophic battery failure. Excessive external heating, which in turn leads to thermal runaway, is a principle cause of cell failure and failure events continue to occur despite the installation of safety features, such as vents and thermal fuses, within the cells. Models of cell failure do exist and they have been used to improve upon battery safety features; however, failure models require prior knowledge of the cell’s thermophysical properties. For many common, commercial lithium-ion batteries, LiCoO2 18650 cells included, knowledge of their thermophysical properties is often either incomplete or absent entirely. To determine the radial thermal conductivity and specific heat of a LiCoO2 18650 cell, we conducted external heating experiments and created a numerical heat transfer model. In our experiments, a thermocouple and a heat flux sensor were mounted to the battery surface and, without damaging the cell, external heat was provided using NiCr wire, wrapped helically around the cell’s exterior. Using the time-dependent heat flux data measured at the battery’s surface, our computational model solved the heat conduction equation for the cell’s radial temperature profile. The radial thermal conductivity and specific heat of the LiCoO2 18650 cell were then found by minimizing the difference between the experimental surface temperature data and the model’s predicted surface temperature over time. Matching the model’s results with the experimental data was achieved with a parametric exploration of specific heat values and the Secant Method, which iterated over the radial thermal conductivity. Over the course of our presentation, the experimental setup, the numerical model, and the results we obtained will be explored in detail. We will also discuss how this method can be used to study the radial thermal conductivity and specific heat of any cylindrical cell.

Time-Dependent Heat Flux Estimation in Multi-Layer Systems by Inverse Method

In this paper, the conjugate gradient method, coupled with the adjoint problem, is used to solve the inverse heat conduction problem and to estimate the time-dependent heat flux, using temperature distribution at a point in a three-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 and body-fitted coordinate method are used. The irregular region in the physical domain (, ) is transformed into a rectangle in the computational domain (, ). 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 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 protection systems and heat shield systems.