Time-dependent Surface Heat Flux Research Articles

Real-time reconstruction of the time-dependent heat flux and temperature distribution in participating media by using the Kalman filtering technique

The Kalman filtering (KF) algorithm is introduced to solve the inverse coupled radiation-conduction heat transfer problem in the participating medium for the first time. The time-dependent surface heat flux and internal temperature distribution of the participating medium is reconstructed simultaneously from the (non-intrusive transient temperature) measurements made on the other surface. The governing energy equation and radiative transfer equation are calculated by using the finite volume method (FVM). Different forms of transient heat flux are employed to test the performance of the KF method. The influence of initial state, initial state error covariance, measurement noise, process noise, measurement noise covariance, process noise covariance, sampling step, refractive index, medium thickness, and absorption coefficient on the accuracy and stability are discussed thoroughly. All the retrieval results show that the surface heat flux and internal temperature distribution can be reconstructed simultaneously in real time. Compared with the hybrid algorithm of KF and recursive least-square estimator (RLSE), the KF algorithm can obtain better reconstruction results and a noticeable decrease of the sensitivity to measurement noise, initial temperature distribution, and absorption coefficient is observed from the retrieval results.

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.

Dual-Phase-Lag Heat Conduction in Finite Slabs With Arbitrary Time-Dependent Boundary Heat Flux

Applying a constant or transient heat flux on a plane slab is a common technique in microelectronics technology and material processing, including laser patterning, micromachining, and laser surface treatment processes. Although Fourier's law is typically very precise for evaluating temperatures in solids, a number of experimental observations suggest the existence of non-Fourier transient conduction in these applications. Since the dual-phase-lag (DPL) model of heat conduction can be compatible with the hypothesis of local equilibrium thermodynamics (as shown here), the effects of temperature gradient relaxation time on the non-Fourier hyperbolic conduction in a finite slab subjected to an arbitrary time-dependent surface heat flux is examined by this model. The combination of diffusion- and wave-like features in heat conduction process is properly monitored by the DPL model for two types of heat flow regimes, namely, gradient precedence and flux precedence. The results indicate considerable deviations between the predictions of these regimes.

Estimation of surface heat flux and temperature distributions in a multilayer tissue based on the hyperbolic model of heat conduction

In this study, an inverse algorithm based on the conjugate gradient method and the discrepancy principle is applied to solve the inverse hyperbolic heat conduction problem in estimating the unknown time-dependent surface heat flux in a skin tissue, which is stratified into epidermis, dermis, and subcutaneous layers, from the temperature measurements taken within the medium. Subsequently, the temperature distributions in the tissue can be calculated as well. The concept of finite heat propagation velocity is applied to the modeling of the bioheat transfer problem. The inverse solutions will be justified based on the numerical experiments in which two different heat flux distributions are to be determined. The temperature data obtained from the direct problem are used to simulate the temperature measurements. The influence of measurement errors on the precision of the estimated results is also investigated. Results show that an excellent estimation on the time-dependent surface heat flux can be obtained for the test cases considered in this study.

Simultaneous determination of time-varying strength and location of a heating source in a three-dimensional domain

In this article, identification of heating source location and time-dependent surface heat flux is investigated considering point temperature measurements on a boundary of the studied three-dimensional geometry. Such Inverse Heat Conduction Problems are ill-posed, since solution stability is not satisfied when observations are noisy/disturbed. We propose a robust algorithm for a simultaneous estimation of location and time-varying strength of a plane heat source. This iterative regularization method based on the conjugate gradient method is tested in several numerical configurations.

An inverse hyperbolic heat conduction problem in estimating surface heat flux of a living skin tissue

In this study, an inverse algorithm based on the conjugate gradient method and the discrepancy principle is applied to solve the inverse hyperbolic heat conduction problem in estimating the unknown time-dependent surface heat flux in a living skin tissue from the temperature measurements taken within the tissue. The inverse solutions will be justified based on the numerical experiments in which three different heat flux distributions are to be determined. The temperature data obtained from the direct problem are used to simulate the temperature measurements. The influence of measurement errors upon the precision of the estimated results is also investigated. Results show that an excellent estimation on the time-dependent surface heat flux can be obtained for the test cases considered in this study.

Comparison of Large Eddy Simulations of a convective boundary layer with wind LIDAR measurements

Abstract. Vertical profiles of the horizontal wind speed and of the standard deviation of vertical wind speed from Large Eddy Simulations of a convective atmospheric boundary layer are compared to wind LIDAR measurements up to 1400 m. Fair agreement regarding both types of profiles is observed only when the simulated flow is driven by a both time- and height-dependent geostrophic wind and a time-dependent surface heat flux. This underlines the importance of mesoscale effects when the flow above the atmospheric surface layer is simulated with a computational fluid dynamics model.

Particle Swarm Optimization-based algorithms for solving inverse heat conduction problems of estimating surface heat flux

An inverse analysis of estimating a time-dependent surface heat flux for a three-dimensional heat conduction problem is presented. A global optimization method known as Particle Swarm Optimization (PSO) is employed to estimate the unknown heat flux at the inner surface of a crystal tube from the knowledge of temperature measurements obtained at the external surface. Three modifications of the PSO-based algorithm, PSO with constriction factor, PSO with time-varying acceleration of the cognitive and social coefficients, and PSO with mutation are carried out to implement the optimization process of the inverse analysis. The results show that the PSO with mutation algorithm is significantly better than other PSO-based algorithms because it can overcome the drawback of trapping in the local optimum points and obtain better inverse solutions. The effects of measurement errors, number of dimensionalities, and number of generations on the inverse solutions are also investigated.

Inverse Identification of Thermal Properties of Charring Ablators

The modified Levenberg-Marquardt method is used for simultaneous estimation of decomposition kinetic coefficients and temperature-dependent thermophysical properties of charring ablators with a moving boundary over a wide temperature range. No prior information is used for the functional forms of the unknown thermal conductivity and specific heat. The procedure used differs from the traditional one in that it does not require prescribed time-dependent surface heat flux, recession rate, and pyrolysis gas mass flow rate. These time-dependent quantities may recover during an iterative procedure. The measured temperatures are simulated numerically by the Charring material ablation code, which accounts for unsteady ablation. The method can determine unknown parameters in an efficient manner with reasonable accuracy, without exact advance knowledge about the net surface heat flux, surface recession, and gas flux through the material.

Two-Dimensional Inverse Heat Transfer Analysis of Functionally Graded Materials in Estimating Time-Dependent Surface Heat Flux

Two-dimensional transient inverse heat conduction problem (IHCP) of functionally graded materials (FGMs) is studied herein. A combination of the finite element (FE) and differential quadrature (DQ) methods as a simple, accurate, and efficient numerical method for FGMs transient heat transfer analysis is employed for solving the direct problem. In order to estimate the unknown boundary heat flux in solving the inverse problem, conjugate gradient method (CGM) in conjunction with adjoint problem is used. The results obtained show good accuracy for the estimation of boundary heat fluxes. The effects of measurement errors on the inverse solutions are also discussed.

Heat wave phenomena in solids subjected to time dependent surface heat flux

Hyperbolic heat conduction in a plane slab, infinitely long solid cylinder and solid sphere with a time dependent boundary heat flux is analytically studied. The solution is based on the separation of variables method and Duhamel’s principle. The temperature distribution, the propagation and reflection of the temperature wave and the effect of geometry on the shape of the wave front are studied for the case of a rectangular pulsed boundary heat flux. Comparisons with the solution obtained for Fourier heat conduction are performed by considering the limit of a vanishing thermal relaxation time.

Methodology for estimation of time-dependent surface heat flux due to cryogen spray cooling.

Cryogen spray cooling (CSC) is an effective technique to protect the epidermis during cutaneous laser therapies. Spraying a cryogen onto the skin surface creates a time-varying heat flux, effectively cooling the skin during and following the cryogen spurt. In previous studies mathematical models were developed to predict the human skin temperature profiles during the cryogen spraying time. However, no studies have accounted for the additional cooling due to residual cryogen left on the skin surface following the spurt termination. We formulate and solve an inverse heat conduction (IHC) problem to predict the time-varying surface heat flux both during and following a cryogen spurt. The IHC formulation uses measured temperature profiles from within a medium to estimate the surface heat flux. We implement a one-dimensional sequential function specification method (SFSM) to estimate the surface heat flux from internal temperatures measured within an in vitro model in response to a cryogen spurt. Solution accuracy and experimental errors are examined using simulated temperature data. Heat flux following spurt termination appears substantial; however, it is less than that during the spraying time. The estimated time-varying heat flux can subsequently be used in forward heat conduction models to estimate temperature profiles in skin during and following a cryogen spurt and predict appropriate timing for onset of the laser pulse.

AN OPTIMAL SOLUTION TO INVERSE HEAT CONDUCTION PROBLEMS BASED ON FREQUENCY-DOMAIN INTERPRETATION AND OBSERVERS

Observers are proposed as a solution algorithm for the inverse heat conduction problem (IHCP) of reconstructing a time-dependent surface heat flux at the boundary of a linear heat conductor. The derivation of optimal observer equations follows directly from a novel interpretation of the IHCP in the frequency domain: Solving the IHCP is viewed as a filter design problem in which the reconstructed heat flux is obtained by low-pass filtering of the true heat flux. It is demonstrated that observers are easier to tune, yield a better trade-off between the deterministic bias and the sensitivity to measurement errors, and lead to smaller-size computational problems than other solution approaches.

KALMAN SMOOTHING TECHNIQUE APPLIED TO THE INVERSE HEAT CONDUCTION PROBLEM

The time-dependent surface heat flux at one boundary of a one-dimensional system is reconstructed by using the Kalman smoothing technique, given the initial temperature distribution and the time-temperature history at an interior location. The study makes a parametric investigation and analyzes the behavior of two finite-difference schemes. The numerical results show the very good performance of the proposed technique, which provides a comprehensive way for using future temperature measurements. Although in this study attention is devoted to the one-dimensional linear problem, the algorithm can be generalized for the stochastic nonlinear multidimensional case.

Unsteady Temperature Distribution in a Sphere Subjected to Time-Dependent Surface Heat Flux and Internal Heat Source

General expressions are derived for the three-dimensional unsteady heating of a solid sphere under the influence of a variable internal heat source and an arbitrary initial temperature distribution when the entire surface is subjected to a variable heat flux. Both the volume heat source and the surface heat flux are prescribed in terms of arbitrary functions of space and time. Various forms of solutions are noted. These expressions, not available hitherto, contain the solutions of many special problems of technological importance. The results presented in two recent studies on the arbitrary heating of spheres are shown to be special cases of the general sphere problem treated here and the errors appearing in these studies are corrected. Other special cases are also discussed.