One-dimensional Transient Heat Conduction Problem Research Articles

An Analytical Model for Transient Heat Transfer with a Time-Dependent Boundary in Solar- and Waste-Heat-Assisted Geothermal Borehole Systems: From Single to Multiple Boreholes

With the increasing engineering applications of geothermal borehole heat exchangers (BHEs), accurate and reliable mathematical models can help advance their thermal design and operations. In this study, an analytical model with a time-dependent heat flux boundary condition on the borehole wall is developed, capable of predicting the thermal performance of single, double, and multiple closed-loop BHEs, with an emphasis on solar- and waste-heat-assisted geothermal borehole systems (S-GBS and W-GBS) for energy storage. This analytical framework begins with a one-dimensional transient heat conduction problem subjected to a time-dependent heat flux for a single borehole. The single borehole scenario is then extended to multiple boreholes by exploiting lines of symmetry (or thermal superposition). A final expression of the temperature distribution along the center line is attained for single, double, and multiple boreholes, which is verified with a two-dimensional finite-element numerical model (less than 0.7% mean absolute deviation) and uses much lesser computational power and time. The analytical solution is also validated against a field-scale experiment from the literature regarding the borehole and ground temperatures at different time frames, with an absolute error below 6.3%. Further, the thermal performance of S-GBS and W-GBS is compared for a 3-by-3 borehole configuration using the analytical model to ensure its versatility in thermal energy storage. It is concluded that our proposed analytical framework can rapidly evaluate closed-loop geothermal BHEs, regardless of the numbers of boreholes and the type of the heat flux on the borehole wall.

Eagle Strategy Based on Modified Barnacles Mating Optimization and Differential Evolution Algorithms for Solving Transient Heat Conduction Problems

Solving time-dependent heat conduction problems through a conventional solution procedure of iterative root-finding method may sometimes cause difficulties in obtaining accurate temperature distribution across the heat transfer medium. Analytical root-finding methods require good initial estimates for finding exact solutions, however locating these promising regions is some kind of a black-box process. One possible answer to this problem is to convert the root-finding equation into an optimization problem, which eliminates the exhaustive process of determining the correct initial guess. This study proposes an Eagle Strategy optimization framework based on modified mutation equations of Barnacles Mating Optimizer and Differential Evolution algorithm for solving one-dimensional transient heat conduction problems. A test suite of forty optimization benchmark problems have been solved by the proposed algorithm and the respective solution outcomes have been compared with those found by the reputed literature optimizers. Moreover, five challenging real-world constrained optimization problems have been solved to further scrutinize the effectiveness of the proposed framework. Finally, two case studies associated with a transient heat conduction problem have been solved. Results show that Eagle strategy can provide efficient and feasible results for various types of solution domains.

Solving time-dependent heat conduction problems using metaheuristic algorithms extended with a novel local search strategy

This study proposes a novel and dexterous local search scheme for improving the exploitation phase of a generic metaheuristic algorithm. The proposed local search considers a twofold probing mechanism, which takes advantage of a chaotic number generated by the hybrid chaotic map composed of Logistic map and Kent map to move around the so-far-obtained global best solutions to reach feasible candidate solutions. Also, an iterative local search scheme inspired by a variant of the differential evolution algorithm is incorporated into the proposed manipulation scheme to enhance intensification on the promising regions. The proposed scheme is included in the well-reputed metaheuristics of differential evolution, crow search, whale optimization, and sine–cosine algorithms to assess the resulting improvements made on the optimization accuracy. Forty optimization benchmark functions composed of unimodal and multimodal test problems have been solved by the local search improved and basic forms of these optimizers to identify the amelioration in terms of solution accuracy and robustness. Two different real-world constrained optimization problems have been solved by these algorithms to analyze the improvement in solution qualities maintained by the utilization of the proposed local search method. Furthermore, these mentioned optimization algorithms along with their improved forms have been applied to one-dimensional transient heat conduction problems to obtain accurate temperature distribution across the heat transfer medium. Optimization results reveal that utilizing local search enhanced metaheuristic algorithms can be considered a favorable alternative to conventional solution methods for solving transient heat conduction problems.

Simplified Theoretical Model for Temperature Evaluation in Tissue–Implant–Bone Systems during Ultrasound Diathermy

Deep heating procedures are helpful in treating joint contractures that frequently occur with fractures and joint diseases involving surgical implants and artificial joint prostheses. This study uses a one-dimensional composite medium model consisting of parallel slabs as a simplified approach to shed light on the influences of implants during ultrasound diathermy. Analytical solutions for the one-dimensional transient heat generation and conduction problem were derived using the orthogonal expansion technique and a Green’s function approach. The analytical solutions provided deep insight into the temperature profile by therapeutic ultrasound heating in the composite system. The effects of the implant material type, tissue thickness, and ultrasound operation frequency on temperature distribution were studied for clinical application. In addition, sensitivity analyses were carried out to investigate the influences of material properties on the temperature distribution during ultrasound diathermy. Based on the derived analytical solutions, the numerical simulations indicate that materials with high density, high specific heat, and low thermal conductivity may be optimal implant materials. Among available implant materials, a tantalum implant, which can achieve a lower temperature rise within the tissue (hydrogel) and bone layers during ultrasound diathermy, is a better choice thanks to its thermodynamics.

A fast-solving particle model for thermochemical conversion of biomass

Computational fluid dynamics (CFD) simulations of large-scale furnaces or reactors for thermal conversion of solid fuels remains challenging partially due to the high computational cost related to the particle sub-models. Owing to the thermally thick nature, it is particularly expensive to simulate the conversion of large fuel particles such as biomass particles. To address this issue, a fast-solving particle model was developed in this work with special attention to the computational efficiency. The model spatially discretizes a fuel particle in one homogenized dimension. The conversion process of the fuel particle is treated as a reactive variable-volume one-dimensional transient heat conduction problem. The model also utilizes several features that are typically found in sharp interphase-based models to reduce the computational cost. Validation of the model was carried out by comparing with experimental results under both pyrolysis and combustion conditions. The accuracy and computational efficiency of the model was thoroughly examined by varying the degrees of temporal and spatial discretization. It was found that the model well predicted pyrolysis and combustion of a single biomass particle within a broad range of temporal and spatial discretization. The time used to simulate the conversion of a biomass particle using the developed model can be more than one order of magnitude smaller than the conversion process itself. It was also revealed that a well-predicted conductive heat transfer inside the particle is essential for a precise simulation of the drying and devolatilization process. The char conversion process, however, is less sensitive to the external heat transfer as it is mainly controlled by the mass diffusion process. Further studies showed that a time step of 1×10−3 s and a spatial discretization of 20 cells were sufficient for simulating the conversion of typical fuel particles in grate-fired and fluidized-bed furnaces. We also demonstrated that when the particle model was implemented in a CFD solver, only 2.2% of computational overhead was introduced by the model. As the model can efficiently employ fixed time stepping, optimal load balancing during parallel computing of many simultaneous conversion processes becomes trivial. This performance opens up new possibilities for treating fuel polydispersity in Eulerian CFD simulations of biomass conversion.

Investigation of the one-dimensional transient heat conduction problem of a coated high strength steel plate

In this paper, the one-dimensional transient heat conduction problem is investigated of a coated high strength steel (HSS) plate which is composed of two coating layers and a HSS layer. As the coating is extremely thin, non-Fourier heat conduction is applied to this part, while the steel part is analyzed by Fourier conduction. Then the temperature increment equations are obtained, which can be calculated by the Newmark method. The effects of thermal relaxation time, temperature boundary conditions, and coating parameters on temperature increment distribution of the coated HSS plate are also presented. Thus, the one-dimensional transient heat conduction problem of a coated HSS plate can be solved, which contributes to practical application and engineering design.

WITHDRAWN: A procedure for solving transient nonlinear thermal problems of high burn-up nuclear fuel rods in a light water reactor

Abstract To ensure the safety of the high burn-up nuclear fuel rods (NFRs) in existing reactors, thermal behaviors of NFRs have been analyzed extensively. In this paper, an accurate and efficient procedure, called half-boundary method (HBM), is applied to solve the one-dimensional transient heat conduction problems of high burn-up NFRs with temperature-dependent material properties. HBM saves computation cost since it only needs to calculate the second-order matrix in the simulation. Compared with the low burn-up condition, the conductivity drops significantly in the rim zone of the high burn-up NFR. The huge temperature gradient in the radial direction render the nonlinear material properties affecting the temperature distribution greatly. In addition, the model contains a helium layer, so the radiation heat transfers are considered in this paper. The heating up process of a low burn-up nuclear fuel is first simulated, and these results are compared with those from reference to verify the accuracy of HBM in transient thermal problems of NFR. The high burn-up NFRs with and without central hole in heat-up process, reactivity initiated accidents (RIAs) and loss of coolant accidents (LOCAs) are then simulated. The results show that the NFR with the central hole (NFRCH) is safer than the NFR without the central hole (NFRWCH), and the temperature distribution of high burn-up NFR is higher than that of low burn-up NFR. The pellet, which is near the central axis, is going to melt when a critical RIA happens, while the inner surface of the cladding will melt first when serious LOCA happens.

A procedure for solving transient nonlinear thermal problems of high burn-up nuclear fuel rods in a light water reactor

To ensure the safety of the high burn-up nuclear fuel rods (NFRs) in existing reactors, thermal behaviors of NFRs have been analyzed extensively. In this paper, an accurate and efficient procedure, called half-boundary method (HBM), is applied to solve the one-dimensional transient heat conduction problems of high burn-up NFRs with temperature-dependent material properties. HBM saves computation cost since it only needs to calculate the second-order matrix in the simulation. Compared with the low burn-up condition, the conductivity drops significantly in the rim zone of the high burn-up NFR. The huge temperature gradient in the radial direction render the nonlinear material properties affecting the temperature distribution greatly. In addition, the model contains a helium layer, so the radiation heat transfers are considered in this paper. The heating up process of a low burn-up nuclear fuel is first simulated, and these results are compared with those from reference to verify the accuracy of HBM in transient thermal problems of NFR. The high burn-up NFRs with and without central hole in heat-up process, reactivity initiated accidents (RIAs) and loss of coolant accidents (LOCAs) are then simulated. The results show that the NFR with the central hole (NFRCH) is safer than the NFR without the central hole (NFRWCH), and the temperature distribution of high burn-up NFR is higher than that of low burn-up NFR. The pellet, which is near the central axis, is going to melt when a critical RIA happens, while the inner surface of the cladding will melt first when serious LOCA happens.

Optimization of the one-dimensional transient heat conduction problems using extended entransy analyses

Heat transfer optimization principle is critically important for further explaining the underlying mechanisms and guiding practical designs of heat transfer processes. Recently, the entransy theory has been successfully used to optimize various steady-state heat transfer processes. Nevertheless, it is still an open question whether this theory can be utilized in transient cases. Here, we examined the applicability of the entransy analyses on the one-dimensional transient heat conduction process. It was found that the entransy dissipation rate can neither derive the transient governing equation nor correspond to the optimal result in the transient optimization problem. Therefore, an extended entransy dissipation rate was defined as the convolution integral of heat flux and negative temperature gradient. The total extended entransy dissipation rate over the time and space domain can correspond to the optimal result of the transient optimization problem. Additionally, Fourier transform was used to convert the transient problem from the time domain into the frequency domain, and the total entransy dissipation rate in the frequency domain will give a convenient optimization criterion that the temperature gradient field should be spatially uniform to reach the shortest characteristic time. Also, the inverse Fourier transform of the entransy dissipation rate in the frequency will be the extended entransy dissipation rate in the time domain. Finally, these findings were used to optimize a practical transient heat conduction problem for a solid thermal energy storage unit.

Estimating Surface Temperature of a Calibration Apparatus for Contact Surface Thermometers from Its Internal Temperature Profile

A calibration apparatus for contact surface thermometers was developed. Temperature of the upper surface of a copper cube of the calibration apparatus was used as reference surface temperature, which was estimated at around $$50\,{^{\circ }}\hbox {C}$$ , $$100\,{^{\circ }}\hbox {C}$$ , and $$150\,{^{\circ }}\hbox {C}$$ by not only two conventional industrial platinum resistance thermometers (IPRTs) but also five small-sized platinum resistance thermometers (SSPRTs) calibrated based on the International Temperature Scale of 1990 (ITS-90). These thermometers were inserted horizontally into the copper cube and aligned along the center axis of the copper cube. In the case of a no-load state without anything on the upper surface, the temperature profile inside the copper cube linearly decreased from the lower part to the upper surface, which suggests that the heat conduction inside the copper cube can be regarded as a one-dimensional steady state. On the other hand, in the case of a transient state just after the contact surface thermometer was applied to the upper surface, the temperature profile became a round shape. We obtained good agreement between the curvature of the temperature profiles and the results estimated by using an error function used for a one-dimensional transient heat conduction problem. The temperature difference between the estimated temperature by linear extrapolation using two IPRTs and that by extrapolation using the error function was within $$0.2\,{^{\circ }}\hbox {C}$$ in the transient state at around $$150\,{^{\circ }}\hbox {C}$$ . Over 10 min after the contact surface thermometer was applied, the temperature profile showed a linear shape again, which indicated that linear extrapolation using two IPRTs was well for the estimation of the reference surface temperature because the heat conduction state inside the copper cube came back to the one-dimensional steady state. Difference between the surface temperature and temperature detected by the contact surface thermometer was also observed after the contact surface thermometer touched on the upper surface. The difference was over $$0.1\,{^{\circ }}\hbox {C}$$ at several minutes after the contact surface thermometer touching on the reference surface and was suppressed with passing time in the transient state and became negligible over 10 min.

Prediction of the Thermal Contact Resistance at the Steel-Concrete Interface of CFST Columns with Circular Cross-Section

Methods to predict the temperature field of a steel-concrete composite structure under fire conditions have been developed using different mathematical techniques. But, one of the drawbacks of these approaches is ignoring the effect of thermal contact resistance at the steel-concrete interface. The main object of the research in this article is the thermal contact resistance at steel-concrete interface of concrete-filled steel tubular (CFST) columns with circular cross-section. In order to obtain the temperature field of the columns under fire conditions, two circular steel tubes filled with normal weight concrete were used to measure the temperatures at some selected points. To obtain the solution of the governing equation for one-dimensional transient heat conduction problems by applying Laplace transformation and inverse Laplace transformation, the selected points temperatures are described as polynomial series or piecewise polynomial series of time. According to Fourier's law and Newton's law of cooling, the thermal contact resistance is obtained. The temperatures calculated for some representative problems show good agreement with the experimental results. The proposed approach can be applied to predict the thermal contact resistance.

The contact heat transfer during frictional heating in a three-element tribosystem

An analytical solution of a one-dimensional transient heat conduction problem for a tribosystem consisting of a semi-space sliding on a surface of a strip deposited on a semi-infinite foundation, is obtained. It is assumed, that the thermal contact of a strip and top semi-space is imperfect. Sliding with a constant speed or linearly decreasing with time (as at braking) is considered. Influence of Biot number, describing thermal conductivity of contact, on the temperature of a system component is studied.

Mould cooling simulation for injection moulding using a fast boundary element method approach

The existing cooling simulations for injection moulding are mostly based on the boundary element method (BEM). In this paper, a fast BEM approach for mould cooling analysis is developed. The actual problem is decoupled into a one-dimensional transient heat conduction problem within the thin part and a cycle-averaged steady state three-dimensional heat conduction problem of the mould. The BEM is formulated for the solution of the mould heat transfer problem. A dynamic allocation strategy of integral points is proposed when using the Gaussian integral formula to generate the BEM matrix. Considering that the full and unsymmetrical influence matrix of the BEM may lead to great storage space and solution time, this matrix is transformed into a sparse matrix by two methods: the direct rounding method or the combination method. This approximated sparsification approach can reduce the storage memory and solution time significantly. For validation, six typical cases with different element numbers are presented. The results show that the error of the direct rounding method is too large while that of the combination method is acceptable.

Inverse reconstruction of boundary condition coefficients in one-dimensional transient heat conduction

This paper describes the restoration of boundary conditions in one-dimensional transient inverse heat conduction problems (IHCP). In the formulation, convective boundary conditions are represented by linear relations between the temperature and the heat flux, together with an initial condition as a function of space. The temperature inside the solution domain together with the space-dependent ambient temperature or heat transfer coefficient are found from additional boundary temperature or average boundary temperature measurements. The determination of the spacewise dependent ambient temperature is a linear IHCP, whilst the determination of the spacewise dependent heat transfer coefficient is a nonlinear IHCP. For both problems uniqueness theorems are available. Numerical results obtained using the boundary element method are presented and discussed.

Reconstruction of heat transfer coefficients using the boundary element method

This paper describes the reconstruction of the heat transfer coefficient (space, Problem I, or time dependent, Problem II) in one-dimensional transient inverse heat conduction problems from surface temperature or average temperature measurements. Since the inverse problem posed does not involve internal temperature measurements, this means that non-destructive testing of materials can be performed. In the formulation, convective boundary conditions relate the boundary temperature to the heat flux. Numerical results obtained using the boundary element method are presented and discussed.

Advantages of Nonlinear-Programming-Based Methodologies for Inequality Path-Constrained Optimal Control Problems—A Numerical Study

We address some advantages of nonlinear programming (NLP)-based methods for inequality path-constrained optimal control problems. The analysis is carried out on the differential algebraic equation (DAE)-constrained optimization problem presented in [J. T. Betts and S. L. Campbell, Discretize Then Optimize, Technical report M&CT-TECH-03-01, The Boeing Company, Chicago, IL, 2003], which relates to a one-dimensional transient heat conduction problem. This optimal control problem possesses the essential characteristics of a typical inequality path-constrained optimal control problem. In a DAE-constrained optimization setting, the index of the path constraint can be arbitrarily increased, and this facilitates the study of the effect of the index of a path constraint on the solution obtained by using an NLP-based methodology. We show that the direct transcription approach leads to a problem whose limiting behavior does not satisfy the linear independence constraint qualification but satisfies the weaker Mangasarian–Fromowitz constraint qualification. The direct transcription approach leads to a convex quadratic programming problem with the result that the control obtained is the unique global minimizer. We also contrast this with the classical indirect approach where we show that it is difficult to numerically integrate the ODEs over the constrained arc because of stability and error control reasons. These observations explain some of the results of Betts and Campbell. This supports the fact that NLP-based methodologies have additional flexibility with respect to constraint qualifications, and this can be put to use in the case of inequality path-constrained optimal control problems to obtain well-defined solutions.

523 周期的加熱を受ける不均質長方形板の熱誘起振動問題(下面完全冷却条件の場合)(OS-4 熱弾性・等温弾性(2))

Theoretical analysis for thermally induced vibration problem of an inhomogeneous rectangular plate is treated in this study. It is assumed that material properties of the plate are given by the power product form with coordinate z'. Firstly, the one-dimensional transient heat conduction problem of the plate is developed when it is subjected to a cyclic heat supply. Then, the thermally induced vibration problem for the plate is analyzed under the condition that all edges are to be simply supported in such a manner that the in-plane displacements are fully restrained. Next, numerical calculations are carried out, and we discussed about an effect of aspect ratio on thermal displacements, thermal stresses and resonance characteristic in the plate.

220 周期的加熱を受ける不均質片持ちはりの熱誘起振動問題

This study is concerned with theoretical development for thermally induced vibration problem of an inhomogeneous beam. We suppose that thermal and mechanical material properties in beam are given as power function in term of transverse coordinate z. Firstly, we have developed one-dimensional transient heat conduction problem of an inhomogeneous beam due to cyclic heat supply. Then, we have developed thermally induced vibration problem of an inhomogeneous beam clamped at one edge and free at opposite edge. Next, we have carried out numerical calculations and have discussed about effects of both cyclic temperature fluctuation and inhomogeneity of material properties on thermal displacements, thermal stresses and resonance characteristics.

917 周期的加熱を受ける不均質はりの動的応答問題

This study is concerned with theoretical development for dynamic thermoelastic response problems of a nohomogeneous beam. We suppose that thermal and mechanical material properties in beam are given as power function in term of transverse coordinate z. Firstly, we have developed one-dimensional transient heat conduction problem of nonhomogeneous beam due to cyclic heat supply. Then. we have developed dynamic thermoelastic response problem under simply supported edge condition. Next, we have carried out numerical calculations and have discussed about effects of both harmonic temperature change and nonhomogeneity of material properties on dynamic or quasi-static thermoelastic responses in beam.

K-0616 周期的加熱を受ける不均質はりの動的応答問題(S05-5 複合材料システムの弾性数理モデリング(5))(S05 複合材料システムの弾性数理モデリング)

This study is concerned with theoretical development for dynamic thermoelastic response problems of a nohomogeneous beam. We suppose that thermal and mechanical material properties in beam are given as power function in term of transverse coordinate z. Firstly, we have developed one-dimensional transient heat conduction problem of nonhomogeneous beam due to periodic heat supply. Then, we have developed dynamic thermoelastic response problem under simply supported edge condition. Next, we have carried out numerical calculations and have discussed about effects of both harmonic temperature change and nonhomogeneity of material properties on dynamic or quasi-static thermoelastic responses in beam.