One-dimensional Heat Transfer Problem Research Articles

The Pseudo-Direct Numerical Simulation Method considered as a Reduced Order Model

The multiscale method called Pseudo-Direct Numerical Simulation (P-DNS) is presented as a Reduced Order Model (ROM) aiming to solve problems obtaining similar accuracy to a solution with many degrees of freedom (DOF). The theoretical basis of P-DNS is other than any standard ROM. However, from a methodological point of view, P-DNS shares the idea of an offline computation, as ROM does, providing the set of coefficients, as a database or table, needed to solve the main problem. This work highlights the advantages and disadvantages of both methodologies. In particular, the drawback of the standard ROM concerning problems where space and time are not separated variables is discussed. The so-called Idelsohn’s benchmark is possibly the most elemental test that can be proposed to point out this drawback. This one-dimensional heat transfer problem with a moving heat source shows that, unlike ROMs, P-DNS can solve it by reducing the number of degrees of freedom as much as needed.

A regularization method for inverse heat transfer problems using dynamic Bayesian networks with variable structure

In this paper, dynamic Bayesian networks (DBNs) were employed to solve one-dimensional inverse heat transfer problems (IHTPs) with temperature history data. Temperature-dependent conductivities and surface heat flux histories were discretized into a finite number of parameters and used as parent nodes in the DBN. Parameter sensitivity analyses were adopted to identify the dominant parameters at each time step. Non-dominant parameters were removed from the corresponding frames for the sparsification of the DBN, which reduces the number of parameters to be identified simultaneously. Thus, the structure of the DBN was dynamically optimized for the regularization of inverse problems. Numerical and experimental tests for IHTPs were conducted to validate the proposed method. Both thermal conductivities and surface heat flux were identified with relatively small error, even if considering the measurement error. Sensitivities of the remaining parameters increased significantly with the parameter identification process, and insensitive parameters in the initial DBN frames were also effectively identified in the subsequent frames. The accuracy of parameter identification can be improved, and the cost of the parameter sampling combination can be reduced. The proposed method is not subject to heat transfer problems and can be used for a wide range of applications.

Assessing the wall energy efficiency design under climate change using POD reduced order model

Within the environmental context, numerical modeling is a promising approach to assess the energy efficiency of building. Resilient buildings need to be designed, capable of adapting to future extreme heat. Simulations are required assuming a one-dimensional heat transfer problem through walls and a simulation horizon of several years (nearly 30). The computational cost associated with such modeling is quite significant and model reduction methods are worth investigating. The objective is to propose a reliable reduced-order model for such long-term simulations. For this, an alternative model reduction approach is investigated, assuming a known Proper Orthogonal Decomposition reduced basis for time, and not for space as usually. The model enables computing parametric solutions using basis interpolation on the tangent space of the Grassmann manifold. Three study cases are considered to verify the efficiency of the reduced-order model. Results highlight that the model has a satisfying accuracy of 10-3 compared to reference solutions. The last case study focuses on the wall energy efficiency design under climate change according to a four-dimensional parameter space. The latter is composed of the load material emissivity, heat capacity, thermal conductivity and thickness insulation layer. Simulations are carried over 30 years considering climate change. The solution minimizing the wall work rate is determined with a computational ratio of 0.1% compared to standard approaches.

Alpert multi-wavelets for functional inverse problems: direct optimization and deep learning

Computational engineering models often contain unknown entities (e.g. parameters, initial and boundary conditions) that require estimation from other measured observable data. Estimating such unknown entities is challenging when they involve spatio-temporal fields because such functional variables often require an infinite-dimensional representation. We address this problem by transforming an unknown functional field using Alpert wavelet bases and truncating the resulting spectrum. Hence the problem reduces to the estimation of few coefficients that can be performed using common optimization methods. We apply this method on a one-dimensional heat transfer problem where we estimate the heat source field varying in both time and space. The observable data is comprised of temperature measured at several thermocouples in the domain. This latter is composed of either copper or stainless steel. The optimization using our method based on wavelets is able to estimate the heat source with an error between 5% and 7%. We analyze the effect of the domain material and number of thermocouples as well as the sensitivity to the initial guess of the heat source. Finally, we estimate the unknown heat source using a different approach based on deep learning techniques where we consider the input and output of a multi-layer perceptron in wavelet form. We find that this deep learning approach is more accurate than the optimization approach with errors below 4%.

Eigenfunction-based solution for solid-liquid phase change heat transfer problems with time-dependent boundary conditions

Solid-liquid phase change heat transfer problems appear in a broad variety of engineering systems, such as thermal management and latent energy storage. Boundary conditions in such systems often vary with time, for example, due to sinusoidal heating, or step changes in externally applied temperature or heat flux. Unfortunately, phase change problems with time-dependent boundary conditions do not generally admit an exact solution, and therefore, approximate analytical solutions are of much interest. This paper presents an eigenfunction expansion based technique for solving one-dimensional phase change heat transfer problems with time-dependent temperature or heat flux boundary conditions. The temperature field is expressed as a series solution of appropriate eigenfunctions, and with time-dependent coefficients, which are determined by deriving and solving an ordinary differential equation that accounts for the time-dependent nature of the boundary condition. Solutions for Cartesian, cylindrical and spherical problems are derived. Results are found to be in excellent agreement with past work and numerical simulations. The effect of Stefan problem on the nature of phase change propagation is studied. Two practical problems involving sinusoidal and step function boundary conditions are also solved and analyzed in detail. In addition to improving the fundamental understanding of phase change heat transfer, this work may also contribute towards design and optimization of practical engineering problems involving phase change heat transfer.

Eigenfunction-based solution for one-dimensional solid-liquid phase change heat transfer problems with advection

Heat transfer problems involving solid-liquid phase change occur in a wide variety of engineering applications. Most phase change heat transfer problems do not admit an exact analytical solution, and therefore, development of approximate analytical methods is of much interest. This work analyzes a one-dimensional phase change heat transfer problem in the presence of advection due to fluid flow. An approximate, eigenfunction-based solution for the temperature distribution and propagation of the phase change front with time is derived, which may be interpreted as a generalization of the classical quasistationary method. It is shown that even a single term of this series offers improved accuracy compared to the classical quasistationary solution. The method is shown to retain good accuracy even at large values of the Stefan number where approximate analytical methods usually lose accuracy. Results are shown to be in good agreement with numerical simulation results. As expected, in the absence of advection, results are shown to reduce to well-known Neumann and Stefan solutions. The impact of various problem parameters, including Stefan and Peclet numbers on the rate of phase change front propagation is investigated. The theoretical treatment presented here can also be used to solve similar mass transfer problems with a chemical reaction front where species advection may play a key role. This work improves the theoretical understanding of phase change heat transfer in the presence of advection, and may find applications in the design and optimization of engineering processes and systems involving phase change.

A new damping strategy of Levenberg-Marquardt algorithm with a fuzzy method for inverse heat transfer problem parameter estimation

In this paper for the first time, the Levenberg-Marquardt (LM) algorithm assisted by a fuzzy method is presented to solve an inverse heat transfer problem. Given the importance of the damping factor in the consistency and efficiency of the LM algorithm in solving ill-posed problems, a new method based on fuzzy logic theory is used to adjust the value of the damping factor during the solution iteration step. In order to investigate the convergence speed, stability and robustness of the fuzzy LM algorithm, a one-dimensional heat transfer problem is solved using three different experimental heat fluxes including quadratic polynomial, cubic polynomial, and sine function, and the results are compared with classical LM algorithm. According to the results, the speed and the convergence stability of the solution of an inverse problem for different noisy temperatures are significantly improved by combining fuzzy logic with classical LM algorithm. Also, the new method robustness is comparable with the LM algorithm.

Multilayer one-dimensional Convection-Diffusion-Reaction (CDR) problem: Analytical solution and imaginary eigenvalue analysis

This paper presents a theoretical analysis of a one-dimensional multilayer heat transfer problem with diffusion, advection and linear temperature-dependent heat generation occurring in each layer. A general solution of the problem is derived. Orthogonality of eigenfunctions is proved, and an explicit expression for the eigenequation is derived. The special case of a two-layer body is discussed. It is shown that, under specific conditions, this problem admits two types of imaginary eigenvalues, one of which is related to divergence of temperature at large times, corresponding to the thermal runaway phenomenon in batteries. The impact of various problem parameters related to diffusion, advection and heat generation on the appearance of imaginary eigenvalues is discussed. Specifically, due to the directional nature of fluid flow, advection in each layer of a two-layer body has opposing impact on the occurrence of imaginary eigenvalues. It is also shown that a balance between heat generation, diffusion and advection determines whether an imaginary eigenvalue is encountered, and consequently, whether thermal runaway occurs. Results presented here expand the theoretical understanding of multilayer heat transfer, and may also contribute towards improved thermal design of multilayer engineering systems such as flow batteries.

Generalized source term multiflux method coupled with Runge-Kutta ray tracing technique for arbitrary radiative intensity of graded-index media.

The generalized source term multiflux method (GSMFM) combined with Runge-Kutta ray tracing technique is developed to calculate arbitrary directional radiative intensity of graded-index media. In this method, the finite volume method is employed to solve source terms along the curved ray path determined by the Fermat principle. Runge-Kutta ray tracing technique is adopted to obtain the ray trajectory numerical solution in graded-index media. And the GSMFM is used to solve radiative intensity to be expected. One-dimensional and two-dimensional radiative heat transfer problems are investigated to verify the performance of this method. The numerical results show that the accuracy of the GSMFM is close to that of backward Monte Carlo (BMC) method, while the efficiency of GSMFM is much higher than that of the BMC. Therefore, the GSMFM developed can be considered as a promising method to solve arbitrary radiative intensity in graded-index media.

Automatic differentiation approach for solving one-dimensional flow and heat transfer problems

Traditional reactor system analysis tools are confronted with challenging difficulties in model development, low accuracy, and poor convergence. To solve these problems, the automatic differentiation (AD) method that allows the automatic numerical calculation of derivatives of functions was adopted to develop the reactor system code in this paper. For the simulation of single-phase models, the steady and transient responses were presented to investigate the effects of the spatial and temporal discretization schemes on modeling accuracy and efficiency. Meanwhile, the comparison of convergence performance between the automatic differentiation using operator overloading (ADOO) and the traditional hand-coded method was completed. Further, in the case of two-phase flow problems, the high-order discrete schemes were applied in this code. It was demonstrated that the reactor system code with single-phase and five-equation two-phase flow models, which adopted the high-order discretization and the ADOO method, performed very well for one-dimensional flow and heat transfer problems.

Experimental investigation of wall heat transfer due to spray combustion in a high-pressure/high-temperature vessel

Combustion chamber wall heat transfer is a major contributor to efficiency losses in diesel engines. In this context, thermal swing materials (adapting to the surrounding gas temperature) have been pinpointed as a promising mitigative solution. In this study, experiments are carried out in a high-pressure/high-temperature vessel to (a) characterise the wall heat transfer process ensuing from wall impingement of a combusting fuel spray, and (b) evaluate insulative improvements provided by a coating that promotes thermal swing. The baseline experimental condition resembles that of Spray A from the Engine Combustion Network, while additional variations are generated by modifying the ambient temperature as well as the injection pressure and duration. Wall heat transfer and wall temperature measurements are time-resolved and accompanied by concurrent high-speed imaging of natural luminosity. An investigation with an uncoated wall is carried out with several sensor locations around the stagnation point, elucidating sensor-to-sensor variability and setup symmetry. Surface heat flux follows three phases: (i) an initial peak, (ii) a slightly lower plateau dependent on the injection duration, and (iii) a slow decline. In addition to the uncoated reference case, the investigation involves a coating made of porous zirconia, an established thermal swing material. With a coated setup, the projection of surface quantities (heat flux and temperature) from the immersed measurement location requires additional numerical analysis of conjugate heat transfer. Starting from the traces measured beneath the coating, the surface quantities are obtained by solving a one-dimensional inverse heat transfer problem. The present measurements are complemented by CFD simulations supplemented with recent rough-wall models. The surface roughness of the coated specimen is indicated to have a significant impact on the wall heat flux, offsetting the expected benefit from the thermal swing material.

A general procedure for infrared thermography heat transfer measurements in hypersonic wind tunnels

Heat transfer measurements in hypersonic wind tunnels are particularly challenging when dealing with fully three-dimensional geometries and/or flows with high spatial temperature gradients. In the present work, a full and consistent description of a general procedure to perform infrared thermography heat transfer measurements is proposed. Radiometric and optical camera calibration, spectral directional emissivity characterization, model oscillations correction and 3D surface temperature reconstruction problems are theoretically and practically analyzed along with new solutions. A novel hybrid heat transfer problem is proposed. This technique is based on the solution of the direct problem during the wind tunnel unsteady start-up to accurately estimate the initial condition for the inverse problem solution during the quasi-steady test phase. The data reduction procedure was validated through an experimental campaign conducted in the Boeing/AFOSR Mach-6 Quiet Tunnel on a sharp cone at 0 deg angle of attack. The experimental Stanton number obtained by solving a one-dimensional heat transfer problem is in agreement with the theoretical laminar solution for a range of free-stream conditions with an average error between −1.1÷0.64% and a standard deviation of 1.55%. Finally, a two-dimensional heat transfer problem has been solved when the cone is tested at 6 deg angle of attack demonstrating the IR procedure capability to analyze multi-dimensional flows and to take into account tangential conduction effects caused by surface temperature gradients due to the presence of crossflow vortices.

Direct methods for solving the Boltzmann equations: Comparisons with direct simulation Monte Carlo and possibilities

The possibilities of direct methods for solving the Boltzmann equation in comparison with direct simulation Monte Carlo are discussed. The general features of these different methods are considered, in particular, from the point of view of application of different variants of discretization in phase space. The advantages and disadvantages of both approaches are clarified. Comparative solutions of some simple problems are given. An important issue concerns anomalous heat transfer and validation of the effect by calculations based on these two methods. The solutions of the stationary one-dimensional heat transfer problem between two infinite plates with nonclassical nonequilibrium reflection from the surface are obtained; the anomalous heat transfer with a temperature gradient and a heat flux having the same sign is observed. One-dimensional and two-dimensional (in the square domain) problems with nonequilibrium “membranelike” boundary conditions are solved numerically; the anomalous heat transfer for all the considered cases is demonstrated.

On the basic concepts of the direct simulation Monte Carlo method

In this paper, the basic ideas underlying the Direct Simulation Monte Carlo (DSMC) method are examined and a novel nonhomogeneous N-particle kinetic equation describing the randomized mathematical model of DSMC is derived. It is shown that different collision-partner selection schemes, including No-Time-Counter (NTC) and Bernoulli-trials schemes, are approximations of the general transition operator of the randomized model. The popular collision-partner selection schemes, represented by the standard NTC and Bernoulli-trials approximations of the general transition operator, represented by Simplified Bernoulli-trials and Generalized Bernoulli-trials schemes, are tested on the one-dimensional rarefied gas heat transfer problem against conditions of two approximation limits: first, leading to the Boltzmann equation and, second, leading to the novel N-particle kinetic one.

Study on Heterogeneous Characteristics of Vertical Subsurface Flow Exchange in River Channel Based on Temperature Tracing Method

Through the monitoring of the riverbed temperature in the Xi’an subsurface flow in the Weihe River, Shaanxi Province in the spring and autumn, based on the analytical solution of the one-dimensional steady-state vertical heat transfer problem, the water exchange flux and the surface water temperature are given by the temperature tracer method. Parameters, and explain the distribution of the depth of the submerged zone by the relationship between the groundwater temperature and the depth of the submerged zone. The results show that the distribution of latent flux is 99.61∼356.25L/(m2·d), which obeys the normal distribution. There are large differences in the values of different positions in the same section, there is a concentrated discharge area, showing significant spatial heterogeneity, and the river channel. The heterogeneity of the bank is more significant than that of the bank; the depth of the submerged zone is inversely related to the amount of latent flow. The greater the latent flow, the smaller the depth of the submerged zone.

Modeling the one-dimensional inverse heat transfer problem using a Haar wavelet collocation approach

In this paper, a numerical method to solving the one-dimensional inverse heat transfer problem in Cartesian and Cylindrical coordinates, which is combination of the Haar wavelet collocation and Tikhonov regularization approach, has been used. The audit consisted in comparing the heat transfer in reality with numerical result as a noisy data, in order to identify errors of 1%–5% has been presented. In this study, the Haar functions, in addition to estimating the unknown functions, are also used to reduce output noises. Based on the obtained results, two main advantages of the repeated method have been proven, first, the precision of this method in estimating the unknown boundary condition and the second, processing speed due to the lack of need for wavelet functions to be collocated at low intervals. This suggested that this method, also has a high speed. According to the obtained results, it can be asserted that the present method by applying a small error in the input data, is preserved its stability.

Optimal Adaptive Designs with Inverse Ordinary Differential Equations.

Many industrial and engineering applications are built on the basis of differential equations. In some cases, parameters of these equations are not known and are estimated from measurements leading to an inverse problem. Unlike many other papers, we suggest to construct new designs in the adaptive fashion 'on the go' using the A-optimality criterion. This approach is demonstrated on determination of optimal locations of measurements and temperature sensors in several engineering applications: (1) determination of the optimal location to measure the height of a hanging wire in order to estimate the sagging parameter with minimum variance (toy example), (2) adaptive determination of optimal locations of temperature sensors in a one-dimensional inverse heat transfer problem and (3) adaptive design in the framework of a one-dimensional diffusion problem when the solution is found numerically using the finite difference approach. In all these problems, statistical criteria for parameter identification and optimal design of experiments are applied. Statistical simulations confirm that estimates derived from the adaptive optimal design converge to the true parameter values with minimum sum of variances when the number of measurements increases. We deliberately chose technically uncomplicated industrial problems to transparently introduce principal ideas of statistical adaptive design.

A thermal model for predicting the drilling temperature in deep lunar regolith exploration

Lunar drilling will work in an extreme environment and may lead to a high drilling temperature due to the poor heat dissipation approaches, with potentially devastating effects on the drill tool. This paper presents a mathematical thermal model for predicting the drilling temperature in lunar regolith exploration. The proposed model combines a unique heat balance equation for the system of the drill tool and the drilled cuttings. Modeling of the drill tool-cuttings system assumes an axial temperature distribution along the drill tool and one-dimensional transient heat transfer problem in a semi-infinite plane is employed to calculate the temperature. Geometric features of the drill tool, thermal and mechanical properties of the drill tool and lunar regolith, drilling parameters, and penetration force are taken into consideration in the model. The theoretical investigation addresses the significance of heat generation by the penetration force and heat dissipation by the drilled cuttings. Convolution is utilized to establish the heat transfer processes along the drill tool. Using the presented thermal model, a thermal property test bed was employed to simulate drilling in an evacuated lunar environment, and experimental and computational temperatures were compared in penetrating a lunar regolith simulant with different drilling parameters. Experimental results indicated that the theoretical model was effective in predicting the drilling temperature in sampling the lunar regolith simulant.

Error estimation and adaptive moment hierarchies for goal-oriented approximations of the Boltzmann equation

This paper presents an a-posteriori goal-oriented error analysis for a numerical approximation of the steady Boltzmann equation based on a moment-system approximation in velocity dependence and a discontinuous Galerkin finite-element (DGFE) approximation in position dependence. We derive computable error estimates and bounds for general target functionals of solutions of the steady Boltzmann equation based on the DGFE moment approximation. The a-posteriori error estimates and bounds are used to guide a model adaptive algorithm for optimal approximations of the goal functional in question. We present results for one-dimensional heat transfer and shock structure problems where the moment model order is refined locally in space for optimal approximation of the heat flux.

Mold Simulator Study of Heat Transfer Phenomenon During the Initial Solidification in Continuous Casting Mold

In this paper, mold simulator trials were firstly carried out to study the phenomena of the initial shell solidification of molten steel and the heat transfer across the initial shell to the infiltrated mold/shell slag film and mold. Second, a one-dimensional inverse heat transfer problem for solidification (1DITPS) was built to determine the temperature distribution and the heat transfer behavior through the solidifying shell from the measured shell thickness. Third, the mold wall temperature field was recovered by a 2DIHCP mathematical model from the measured in-mold wall temperatures. Finally, coupled with the measured slag film thickness and the calculations of 1DITPS and 2DIHCP, the thermal resistance and the thickness of liquid slag film in the vicinity of the meniscus were evaluated. The experiment results show that: the total mold/shell thermal resistance, the mold/slag interfacial thermal resistance, the liquid film thermal resistance, and the solid film thermal resistance is 8.0 to 14.9 × 10−4, 2.7 to 4.8 × 10−4, 1.5 to 4.6 × 10−4, and 3.9 to 6.8 × 10−4 m2 K/W, respectively. The percentage of mold/slag interfacial thermal resistance, liquid film thermal resistance, and solid film thermal resistance over the total mold/shell thermal resistance is 27.5 to 34.4, 17.2 to 34.0, and 38.5 to 48.8 pct, respectively. The ratio of radiation heat flux is around 14.1 to 51.9 pct in the liquid slag film.