One-dimensional Transient Heat Conduction Research Articles

Effects of heat wave propagation and initial stress on the thermo-elastic behavior of porous-auxetic FG circular plate with variable hybrid foundation

Engineering structures' heterogeneity and porous-auxetic features generate some prominent effects on their thermo-mechanical behavior. In addition, initial stress and heat wave propagation also play important roles in the mechanical behavior of FGM structures. Therefore, this paper investigates the effects of heat wave propagation and initial stress on the thermo-elastic behavior of porous-auxetic FG circular/annular plates with variable hybrid foundations. The one-dimensional transient hyperbolic heat conduction equation is modeled using the Cattaneo-Vernotte (C-V) theory and numerically solved using the Crank-sNicolson (C-N) finite difference scheme. Besides, a mathematical model of the other governing equations is extracted based on the classical thermoporoelasticity theory and semi-analytically solved by employing the state-space method and differential quadrature technique. Numerical examples are provided to illustrate the findings of this investigation graphically. The results reveal that the grading indices, and thermal relaxation time have a significant effect on heat wave propagation in the FG circular plate. Initial stress, porosity, and auxeticity also play an important role in its thermoelastic behavior.

Vibrations of skew cylindrical shells made of FG-GPLRCs under rapid surface heating

An analysis is performed in this research to examine the thermally induced vibrations in skew cylindrical shells. It is assumed that shell is made from a composite laminated material where each layer is reinforced with graphene platelets (GPLs). The amount of GPLs in the layers may be different which results in a piecewise functionally graded media. The one-dimensional transient heat conduction equation is established through the thickness of the shell. Heat conduction equation is solved using the finite difference method for each layer and continuity of temperature and heat flux is applied between the layers. Crank Nicolson algorithm is implemented to obtain the temperature profile at each time step and after that thermally induced force and moment are evaluated. The kinematics of the shell are estimated following the first order shear deformation shell theory. Also it should be noted that an oblique coordinates system in addition to orthogonal system is defined which make it easier to apply the boundary conditions. The definition of strain and kinetic energies are constructed and Following the Ritz method whose shape functions are estimated by Chebyshev polynomials, energies are provided in discrete form of the governing equations. Results of this study are provided to explore the effects of thermal and mechanical boundary conditions, geometry of the shell, weight fraction and distribution patterns of GPLs. It is highlighted that thermally induced vibrations indeed exist especially in thin class of shells.

Computational Analysis and Regression Laws for Nozzle Erosion Prediction in Hybrid Rockets

The erosion of the nozzle throat can represent one of the major limitations against the future widespread use of hybrid rocket engines (HREs) in the space industry. In fact, nozzle erosion in HREs can be more severe and harder to predict than in solid rockets due to the higher concentration of oxidizing species in the combustion products and to mixture ratio shifts and/or throttling. Therefore, an accurate understanding of the erosion phenomenon is of fundamental importance for the technological advancement of HREs. This work is focused on the investigation of graphite nozzle erosion in HREs burning high-density polyethylene with two different oxidizers, oxygen and nitrous oxide. First, the results of a computational fluid dynamics parametric analysis are used to derive closed-form regression laws for the rapid estimation of nozzle throat erosion and wall temperature depending on chamber pressure and mixture ratio. Then, a one-dimensional transient heat conduction solver is loosely coupled with the aforementioned regression laws, allowing to reconstruct the transient heating process within the solid. The obtained numerical results are validated against experimental data. Finally, the effect of gas-phase reactions on the heterogeneous reactions occurring at the nozzle surface is highlighted when moving from fuel-rich to oxidizer-rich conditions.

Numerical modeling of soil temperature variation under an extreme desert climate

The soil temperature profile is useful for different engineering, architecture, and agriculture areas. In recent years, the growth of direct uses of geothermal energy has increased the interest in soil temperature prediction models. This research paper numerically models the soil temperature variation under extreme desert climate conditions and compares it with experimental data measured on-site to evaluate the reliability of numerical models based on the one-dimensional transient state heat conduction equation as a tool for the evaluation of low enthalpy geothermal resource using meteorological data and soil thermal properties. From the study results, it can be concluded that the numerical models present temperature values close to those measured experimentally. The model that presents the more significant mismatches concerning the experimental data is the 1 m model, presenting the following statistical values; R2 of 0.97, mean absolute error of 0.61 °C, RMSE of 0.76 °C, NRMSE of 3.01%, and a MBIAS of 0.23. Although the numerical models present values are close to those measured, some coefficients, such as the soil surface water content greatly influence the model's accuracy regardless of depth.

Numerical Analysis of Airfoil Flow Control Using a Nanosecond Pulsed Plasma Actuator

The control mechanism of a nanosecond dielectric barrier discharge (NS-DBD) actuator for suppressing laminar separation on airfoils was investigated using two-dimensional scale-adaptive simulations. To model the control effect of the NS-DBD actuator, a surface heating approach based on the analytical solution for transient one-dimensional heat conduction has been used. The focus of the investigation lies in the study of vortex formation and evolution induced by the actuator. A NACA 0015 profile was simulated at 14° with 250,000 chord-based Reynolds number. Regarding energy deposition, a low-energy case and a high-energy case were simulated. Several actuation parameters were varied, including the electrode position and surface temperature. In addition, the influence of constant and temperature-dependent modeling of the kinematic viscosity was investigated. The results for the time-averaged pressure coefficients show excellent agreement with measurement results. High grid resolution in the boundary layer allowed a detailed investigation of the vortex formation process. The main finding is that vortices are generated by baroclinic torque and vortex dilatation and not by a Kelvin–Helmholtz instability. Initially, tiny vortices form at the beginning of the separation region. Subsequently, these vortices are carried downstream and grow in size, thus preventing full separation due to momentum transfer from the external flow.

Application of GDQ method to large amplitude response of FGM joined spherical-conical shells under rapid surface heating

Geometrically non-linear thermally induced vibrations of functionally graded material (FGM) joined spherical-conical shells are analyzed in the current research. Thermo-mechanical properties of the shells are assumed to be temperature and position dependent. The system of joined spherical-conical shells is subjected to thermal shock on the ceramic-rich surface, whereas the metal-rich one is kept at reference temperature. The one-dimensional transient heat conduction equation is established and solved via the generalized differential quadrature (GDQ) and Crank-Nicolson methods. This equation is non-linear since the thermo-mechanical properties of the shells are temperature dependent. The total functional of the shells is obtained under the assumptions of uncoupled thermoelasticity laws, first order shear deformation shell theory, and the von Kármán type of geometrical non-linearity. Non-linear coupled equations of motion are solved via the iterative Picard method accompanied with the β-Newmark time approximation technique. Numerical results are well-validated with the available data for the case of single FGM spherical shell. Parametric studies are conducted to examine the influences of conical and spherical shell geometries, material composition, temperature dependence, in-plane and out-of-plane mechanical boundary conditions, various configuration of conical/spherical shell system, and thermal boundary conditions. It is highlighted that thermally induced vibrations indeed exists.

Geometrically nonlinear response of FGM joined conical–conical shells subjected to thermal shock

Geometrically nonlinear thermally induced vibrations of functionally graded material (FGM) joined conical–conical shells is analyzed in the current research. Thermo-mechanical properties of the shell are assumed to be temperature and position dependent. The system of joined conical–conical shells is subjected to thermal shock on the ceramic-rich surface, whereas the metal-rich one is kept at reference temperature. The one-dimensional transient heat conduction equation is established and solved via the generalized differential quadrature (GDQ) and the Crank–Nicolson methods. This equation is nonlinear since the thermo-mechanical properties of the shell are temperature dependent. The total functional of the shell is obtained under the assumptions of uncoupled thermoelasticity laws, first order shear deformation shell theory, and the von Kármán type of geometrical non-linearity. Non-linear coupled equations of motion are solved via the iterative Picard method accompanied with the β-Newmark time approximation technique. Numerical results are well validated with the available results for the case of FGM single conical shell. Parametric studies are conducted to examine the influences of conical geometry, material composition, temperature dependence, in-plane and out-of-plane mechanical, various configuration of joined shell system, and thermal boundary conditions.

Determination of Film Cooling Effectiveness and Heat Transfer Coefficient Simultaneously on a Flat Plate

In this paper, flat plate film cooling with two rows of compound angle cylindrical film cooling holes was investigated. A data processing method was evaluated which could determine the film cooling effectiveness and heat transfer coefficient simultaneously from the transient wall temperature data. The method was based on solving an inverse problem of the one-dimensional transient heat conduction equation. To evaluate the performance of the method, wall temperature data were obtained using the known film cooling effectiveness and heat transfer coefficient data as the convection boundary condition. Then, the method was applied to calculate the film cooling effectiveness and heat transfer coefficient based on the wall temperature data. Different blowing ratios, heat transfer coefficients, mainstream temperatures, and material thermal conductivities were investigated. In general, the data and calculation were in good agreement. It was found that the error decreased when the heat transfer coefficient increased and the material thermal conductivity decreased. The percentage error of the span-wise averaged film cooling effectiveness was mainly between 0% and 10%, and the percentage error of the span-wise averaged heat transfer coefficient was mainly between 0% and 4%.

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.

Heat Transfer and Recovery Factor of Aerodynamic Heating on a Flared Cone

This paper investigates aerodynamic heating on a flared cone in a Mach 6 wind tunnel. Two data processing methods to calculate the Stanton number and recovery factor simultaneously are presented, using the experimentally measured wall temperature in a hypersonic wind-tunnel test. The first method is the linear fitting of the measured temperature and calculated heat flux. The second method is to solve an inverse problem of a one-dimensional transient heat conduction equation with the convection boundary condition. Distributions of heat flux, Stanton number, and recovery factor for the flared cone with a smooth wall or wavy wall are obtained. Compared with the smooth wall, the wavy wall can remove the local high Stanton number region before the boundary-layer transition. In addition, the wavy wall can reduce the growth rate of Stanton number in the boundary-layer transition and delay the transition process.

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.

Electrocaloric devices part I: Analytical solution of one-dimensional transient heat conduction in a multilayer electrocaloric system

The analytical solution is reported for one-dimensional (1D) dynamic conduction heat transfer within a multilayer system that is the typical structure of electrocaloric devices. Here, the multilayer structure of typical electrocaloric devices is simplified as four layers in which two layers of electrocaloric materials (ECMs) are sandwiched between two semi-infinite bodies representing the thermal sink and source. The temperature of electrocaloric layers can be instantaneously changed by external electric field to establish the initial temperature profile. The analytical solution includes the temperatures in four bodies as a function of both time and location and heat flux through each of the three interfaces as a function of time. Each of these analytical solutions includes five infinite series. It is proved that each of these series is convergent so that the sum of each series can be calculated using the first [Formula: see text] terms of the series. The formula for calculating the value of [Formula: see text] is presented so that the simulation of an electrocaloric device, such as the temperature distribution and heat transferred from one body to another can be performed. The value of [Formula: see text] is dependent on the thickness of electrocaloric material layers, the time of heat conduction, and thermal properties of the materials used. Based on a case study, it is concluded that the [Formula: see text] is mostly less than 20 and barely reaches more than 70. The application of the analytical solutions for the simulation of real electrocaloric devices is discussed.

A Hybrid Numerical-Experimental Method for Determining Thermal Conductivities

Abstract Determining the thermal conductivity of a material from temperature measurements in a cooling or heating process belongs to the class of inverse problems. In this article, we present a method for a simple experimental setup, consisting of a glass tube containing the material under investigation, two thermistors for temperature monitoring (one at the central axis and the other attached on the outer surface of the tube), and a water heat bath maintained at a desired temperature. We solve the direct problem, i.e., the transient heat conduction equation, treating the thermal conductivity as a parameter whose value is determined by minimizing the difference between the calculated and the experimentally measured temperatures. The method is based on the numerical solution of the one-dimensional transient heat conduction equation in cylindrical coordinates that accurately describes the temperature evolution of a material in a narrow, long glass tube. The technique has been validated by applying it to the lauric and capric acids, whose thermal conductivities are accurately known and therefore it could be a valuable tool for the determination of the thermal properties of phase change materials suitable for thermal storage applications.

Contact resistance analysis applied to simultaneous estimation of thermal properties of metals

This work aims to present a method for simultaneously estimating the thermal conductivity, k, and specific heat, cp, in samples of stainless steel 304 and cemented carbide by accounting the imperfect thermal contact at the heater–sample interface. A transient one-dimensional heat conduction model with constant thermal properties is used. The metallic sample is placed in the middle of a polyimide heater and a thermal insulator. A constant heat flux is applied on the specimen upside surface and a thermal insulation condition is maintained on the opposing surface, where a type T thermocouple measures the temperature response. Contact resistance is determined and accounted for as a reducing factor on heat flux. Instead of considering a lumped model, a microscopic approach of the contacting regions is used to describe both the surface roughness and the fluid gap. With the intention of guaranteeing simultaneous and reliable estimates for both thermal properties, sensitivity analysis is used to establish heat flux intensity, experiment duration, time step for data acquisition, and other characteristics of the experiments. To simultaneously achieve both thermal properties, the optimization method BFGS (Broyden–Fletcher–Goldfarb–Shanno) is employed to minimize a least-squares objective function comparing experientially measured and numerically calculated temperatures. The numerical solution for the transient conduction problem is determined with COMSOL, which solves the transient heat diffusion problem by discretizing it applying the finite element method (FEM). The linear system of equations is solved using the PARDISO solver and backward differentiation formula (BDF) method. Heat flux is estimated employing the sequential function specification method (SFSM) as a means of verifying the robustness of parameter estimation and experimental procedure. Moreover, uncertainty analysis is performed to confirm the quality of the results obtained. Finally, the uncertainty analysis outcomes and thermal properties estimated are compared with literature.

Analytical model for prediction of bend angle in laser forming of sheets

This paper covers an analytical approach for predicting the bend angle of a laser formed sheet with a single laser scan. The proposed approach is based on the strain energy stored under the laser-irradiated area that causes plastic deformation of sheet due to laser heating. With any particular intensity of laser, the temperature profile established across the sheet thickness is determined by employing a one-dimensional transient heat conduction equation and by considering the loss of heat to the surrounding material by thermal conduction. Based on the temperature gradient across the sheet thickness, thermal strain, strain-induced stress, and bending moment are evaluated. By equating the strain energy stored in the sheet material due to thermal stress and strain with the strain energy stored due to the bending moment, the angle of bend in the sheet is determined. The results obtained with the proposed model are validated with the experiments performed on AISI 304 stainless steel and AA 1100 aluminum alloy sheet material of different thickness using CO2 laser with varying power and scanning speed. A three-dimensional finite element model is developed to validate the temperature profile predicted by the proposed model. The effectiveness of the proposed approach is validated by comparing the predicted bend angle with those predicted by other analytical models. Apart from this, the effect of various process parameters on the laser formed sheet was studied using microstructure and microhardness analysis. Finally, with the application of the proposed model, a process map is generated to achieve pure bending during laser forming.

Cold-crystallizing erythritol-polyelectrolyte: Scaling up reliable long-term heat storage material

Renewable energy usage would benefit from efficient and high-capacity long-term heat storage material. However, these types of material solutions still lack reliable and durable operation on bulk level. Previously, we showed that cold-crystallizing material (CCM), which consists of erythritol in cross-linked polymer matrix, stored heat for a long-term period in a milligram scale by supercooling stably and preventing undesired crystallization during storage. Crystallization of CCM can be triggered efficiently by re-heating the material (i.e. cold-crystallization). Supercooling and cold-crystallization are stochastic phenomena which manifest in a way that the properties in bulk scale often deviate from the microscale. In this work, we scale up CCM to a bulk size of 160 g, and analyze its supercooling and crystallization characteristics for long-term heat storage. In order to identify the impact of the scale-up on the tested compositions and to discover optimal storage conditions, CCM samples are maintained in storage mode at constant temperature between 0 and 10 °C and up to 97 days. To this end, the thermal chamber measurement procedure estimates the heat release of CCM samples based on the measured temperature data and the one-dimensional transient heat conduction model. Results indicate that the heat release in cold-crystallization is over 70% of the melting heat. This heat can be stored without reduction for at least 97 days, demonstrating the reliable performance of long-term heat storage. Analysing the thermal properties of CCM compositions indicates a maximum volumetric storage capacity of 250 MJ/m3 and excellent properties for further heat storage applications.

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 model for puffing/microexplosions in water/fuel emulsion droplets

A new analytical solution to the one-dimensional transient heat conduction equation in a composite spherically symmetric water-fuel emulsion droplet, suspended in a hot gas, is obtained. The Robin boundary condition at the surface of a droplet and conditions at the fuel-water interface are used. A water sub-droplet is assumed to be located at the centre of a fuel droplet, the radius of which was fixed at each time step; it could change at the next time step. The Abramzon and Sirignano model is applied for the approximation of the droplet evaporation process. This solution and the evaporation model are incorporated into a numerical code in which droplet heating/evaporation and the variable thermophysical properties are accounted for. The time instant at which the temperature at the fuel-water interface became equal to the boiling temperature of water is identified with the initiation of puffing, giving rise to microexplosion. This allowed us to compute the minimal microexplosion delay time. The new solution is applied to a typical case of puffing/microexplosion of water/diesel emulsion droplets in high temperature gas. It is shown that the new model allows us to understand better the underlying physics of the processes leading to puffing/microexplosion. The experimental observations of the microexplosion delay time for various initial droplet sizes are shown to be compatible with the predicted values of this time. It is shown that puffing/microexplosions are expected well before the droplet surface temperature reaches the boiling temperature of n-dodecane. The numerical code can be potentially implemented into Computational Fluid Dynamics codes, which can be applied to the modelling of other fuel and water/fuel blends.

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.