Conduction Problem Research Articles

CNTs/Gr composite sandwich layered rare earth phthalocyanines MPcs (M = Yb, La) used as improved energy storage behaviors for lithium-ion batteries

To address the problem of aggregation and low electrical conductivity of rare earth phthalocyanines MPc (M = Yb, La), four composites are obtained by applying a physical method to combine two rare earth phthalocyanines with Gr/CNTs. These composites of rare earth phthalocyanines with Gr/CNTs effectively reduce the aggregation of phthalocyanine conjugated units and enhance the electrical conductivity. The electrochemical testing exhibits that the YbPc/CNTs electrode delivers a specific capacity of 1447 mAh/g after 275 cycles at a current density of 100 mA/g, demonstrating a capacity retention of 157.5 %. Similarly, the primitive specific capacity of LaPc/CNTs electrode is increasing from 515 mAh/g to 753 mAh/g at 100 mA/g after 400 cycles. Likewise, YbPc/Gr and LaPc/Gr electrodes have specific capacity of 912 mAh/g and 667 mAh/g after 450 cycles and 356 cycles, revealing capacity retention of 152 % and 74 %. It can be noted that compounding with Gr/CNTs can successfully solve the problems of a few exposures of redox-active sites caused by the aggregation of phthalocyanine conjugate units and poor rate performance owing to the sluggish ion/electrons transport of phthalocyanine, which furnishes a promising perspective for practical research and application of new organic anode materials in the field of superior electrochemical performance lithium-ion batteries.

A Moore-Gibson-Thompson heat conduction problem with second gradient

In this work, we study, from both analytical and numerical points of view, a heat conduction model which is based on the Moore-Gibson-Thompson equation. The second gradient effects are also included. First, the existence of a unique solution is proved by using the theory of linear semigroups, and the exponential energy decay is also shown when the constitutive tensors are homogeneous. The analyticity of the semigroup is also discussed in the isotropic case, and its spatial behavior is studied. The spatial exponential decay is also proved. Then, we provide the numerical analysis of a fully discrete approximation obtained by using the finite element method and an implicit Euler scheme. A discrete stability property is shown, and some a priori error estimates are derived, from which the linear convergence is concluded under suitable regularity conditions. Finally, some one-dimensional numerical simulations are presented to demonstrate the accuracy of the approximations and the behavior of the discrete energy.

Calculation of Green's functions for heat conduction in curved anisotropic media

AbstractSolving the inverse problems of heat conduction often requires performing a very large number of Green's function evaluations. This paper addresses the calculation of Green's functions in anisotropic curved media, such as composite lamina, where the principal axes of thermal conductivity follow the curved surface. We start with established exact series solutions for cylindrical convex and concave shapes. These solutions are extended to accommodate geometries of real‐world composite materials that do not form a closed surface like a cylinder. Unfortunately, the exact solutions have the form of an infinite series of sums or integrals and are computationally infeasible for the inverse problems of interest, especially for large radii of curvature. This motivates a perturbation solution that is accurate and computationally efficient at large radii of curvature. In addition, it motivates a phenomenological approximation that is extremely computationally efficient over a broad range of curvatures, but with some sacrifice in accuracy. These solutions are compared with exact solutions, with each other, and with numerical finite difference calculation. We identify regions in parameter space where the different approaches are preferable and where they lead to the same numerical result.

Constructing Metal-Organic Framework Films with Adjustable Electronic Properties on Hematite Photoanode for Boosting Photogenerated Carrier Transport.

Hematite (α-Fe2O3) has become a research hotspot in the field of photoelectrochemical water splitting (PEC-WS), but the low photogenerated carrier separation efficiency limits further application. The electronic structure regulation, such as element doping and organic functional groups with different electrical properties, is applied to alleviate the problems of poor electrical conductivity, interface defects, and band mismatch. Herein, α-Fe2O3 photoanodes are modified to regulate their electric structures and improve photogenerated carrier transport by the bimetallic metal-organic frameworks (MOFs), which are constructed with Fe/Ni and terephthalate (BDC) with 2-substitution of different organic functional groups (─H, ─Br, ─NO2 and ─NH2). The α-Fe2O3 photoanode loaded with FeNi-NH2BDC MOF catalyst exhibits the optimal photocurrent density (2mAcm-2) at 1.23VRHE, which is 2.33 times that of the pure α-Fe2O3 photoanode. The detailed PEC analyses demonstrate that the bimetallic synergistic effect between Fe and Ni can improve the conductivity and inhibit the photogenerated carrier recombination of α-Fe2O3 photoanodes. The ─NH2 group as an electron-donor group can effectively regulate the electron distribution and band structure of α-Fe2O3 photoanodes to prolong the lifetime of photogenerated holes, which facilitates photogenerated carrier transport and further enhances the PEC-WS performance of α-Fe2O3 photoanode.

The inverse problem of heat conduction in the case of non-uniqueness: A functional identification approach

Abstract A problem of identification of the set of a thermal-conductivity coefficients for the nonlinear heat equation in the case of non-uniqueness is considered. Classes of inverse heat conduction problems (IHCP) with a non-unique solution are defined. Explicit descriptions of sets of thermal-conductivity coefficients for these classes are obtained. For solving the identification problem the functional identification approach is used. Unlike traditional methods, the proposed algorithm does not utilize approximations of the coefficient with a finite system of basis functions. The results of computational experiments are presented. It is shown that the functional identification approach makes it possible to numerically identify the non-uniqueness of the solution of the inverse problem of heat conduction.

Hyper-reduced-order model for estimating convection heat transfer coefficients of turbine rotors

Forced convection heat transfer occurs inside the steam turbine, and it is crucial for thermal and strength analysis to accurately estimate the convection heat transfer coefficients of the turbine rotor. This paper presents a hyper-reduced-order model for efficiently estimating the forced convection heat transfer coefficients of a steam turbine rotor. Unlike the full-order finite element method, the hyper-reduced-order model uses the discrete empirical interpolation method to approximate the domain integration of transient heat conduction simulations. This approach greatly reduces the degrees of freedom of numerical integration, thus substantially improving the computational efficiency of forward heat conduction problems. The hyper-reduced-order model and Levenberg–Marquardt algorithm are combined to minimize temperature errors between the numerical simulation results and remote sensor measurements and iteratively estimate the heat transfer coefficients of forced convection in the rotor. The accuracy and efficiency of the proposed model are verified through a transient heat conduction simulation. Moreover, the effects of the initial guess values and measurement errors are investigated to demonstrate the universality and robustness of the proposed approach in determining the convection heat transfer coefficient. Compared with the full-order finite element model, the proposed hyper-reduced-order model can increase the efficiency of forward heat conduction simulations of the turbine rotor by more than 10 times, and it only takes 0.87 s computer runtime in single transient step simulation, and then rapidly estimate the forced convection heat transfer coefficients. Furthermore, the performance of the proposed approach is insensitive to the initial guess values, and it exhibits great robustness under measurement error. The mean errors are 4.78 %, 2.67 % and 10.51 % when the initial values are 0.015 mW·mm−3·°C−1, 0.05 mW·mm−3·°C−1 and 0.15 mW·mm−3·°C−1 respectively. When there is no measurement error, the estimated error of the convection heat transfer coefficients is 2.67 %, and even if the measurement error reaches 5 %, and the accuracy loss is only 5.57 %. Consequently, this approach is highly suitable for estimating convection heat transfer coefficients during the dynamic operation of steam turbines, and this research provides reliable guidance for the optimization design and safe operation of steam turbines.

An analysis of the suppression of a pair of odd and even temperature fluctuation in an area occupied by a two-phase one-directional periodically layered composite

The study concerns a heat conduction problem in a two-phase, one-directional periodically layered composite (commonly referred to as a laminate). The model is strictly related to considerations regarding continuous media and was introduced by Professor Czesław Woźniak. The basis for the study is a re-formulation of the heat conduction model for composites, referred to in the literature as the extended tolerance heat conduction model. The model equations consider coefficients responsible for the intensity of exponential and rotational suppression of boundary fluctuations of a temperature field in a limited area or in a half-space occupied by the composite. The study investigated the dependence of the suppression intensity of boundary fluctuations on the geometrical and material properties of the composite. Three special cases of pairs of interacting boundary fluctuations were analysed.

Stable recovery of piecewise constant conductance on spider networks

ABSTRACT We address the discrete inverse conductance problem for well-connected spider networks, that is, to recover the conductance function on a well-connected spider network from the Dirichlet-to-Neumann map. It is well-known that this inverse problem is exponentially ill-posed, requiring the implementation of a regularization strategy for numerical solutions. Our focus lies in exploring whether prior knowledge of the conductance being piecewise constant within a partition of the edge set comprising a few subsets enables stable conductance recovery. To achieve this, we propose formulating the problem as a polynomial optimization problem, incorporating a regularization term that accounts for the piecewise constant hypothesis. We show several experimental examples in which the stable conductance recovery under the aforementioned hypothesis is feasible.

An accelerated sequential function specification method based on LM gradient for transient inverse heat conduction problem

The sequential function specification method (SFSM) and the Levenberg-Marquardt method (LM) are two typical methods for inverse heat conduction problems (IHCP). However, these two methods have poor convergence and low convergence speed. Therefore, the combination of LM with SFSM is depended on to propose a modified Sequential Levenberg-Marquardt gradient method (SLM). The computation results of three shapes of heat fluxes with noise show the SLM is averagely improved over 16.3% more than SFSM in computation accuracy, but equally reduced over 39.9% less than in computation time. On other hand, when the emphasis is on the effect of the Aitken acceleration method, Tikhonov regularization and polynomial order on SLM, a neural network prediction model of the root mean square error is trained by regularization factor, polynomial order and the number of future time steps. Subsequently, the genetic algorithm optimization method is proposed for the minimum root mean square error so that it can improve the computation accuracy and efficiency of IHCP to the maximum extent. Finally, Second-order polynomial order and small order-of-magnitude regularization factor are highly recommended for parameter selection with multiple choices. The ASLM only needs to determine the optimal number of future time steps.

Oxygen-deficient ZnVOH@CC as high-capacity and stable cathode for aqueous zinc-ion batteries

Vanadium oxides have become promising cathodes for aqueous zinc ion batteries (AZIBs) due to their low cost and high theoretical capacity. However, vanadium oxides suffer from the problems of low electrical conductivity and structural instability. To address the challenges, ZnxV2O5·nH2O with oxygen-rich vacancies was synthesized in situ on carbon cloth (ZnVOH@CC) via a one-step hydrothermal method. The introduction of Zn2+ and oxygen vacancies is beneficial to enhance the structural stability and improve the Zn2+ diffusion rate. The pre-intercalated Zn2+/H2O acts as “pillars” to increase the interlayer spacing of vanadium pentoxide (V2O5), weakening the electrostatic interactions between Zn2+ and the ZnVOH host lattice. Benefiting from the above advantages, the ZnVOH@CC composite was used as the cathode for AZIBs, which showed a high capacity of 464.9 mAh/g at 0.1 A/g, a large capacity of 208.1 mAh/g at 1.0 A/g after 1,000 cycles, and a good cycling stability at a high current density of 5.0 A/g after 10,000 cycles. In addition, the assembled ZnVOH@CC//Cu2Se full cell by matching ZnVOH@CC with Cu2Se also exhibits a high reversible capacity of 68 mAh/g at 1.0 A/g and 92.5 % capacity retention after 260 cycles.

Finite element solution of heat conduction in complex 3D geometries

This paper presents the use of the finite element method (FEM) to solve heat conduction problems in complex 3-dimensional geometries not amenable to analytical solutions. Heat conduction is important across engineering domains, but closed-form solutions only exist for basic shapes. For intricate real-world component geometries, numerical techniques like FEM must be applied. The paper outlines the mathematical formulation of FEM, starting from the heat conduction governing equations. The domain is discretized into a mesh of interconnected finite elements. Element equations are derived and assembled into a global matrix system relating nodal temperatures. Boundary conditions are imposed and the matrix equations solved to find the temperature distribution. An example problem analyzes steady state conduction in an L-shaped block with 90-degree corners and surface convection. Results show FEM can capture localized gradients and discontinuities difficult to model otherwise. Detailed temperature contours provide insight. FEM enables robust thermal simulation of complex 3D geometries with localized effects, expanding analysis capabilities beyond basic analytical shapes. Proper application of FEM is critical for accurate results.

Geometry-independent spline finite element method to analyze two-dimensional heat conduction and elasticity problems

Traditional isogeometric analysis involves redundant geometric calculations and is more complex than the finite element method when imposing boundary conditions. To address this, a geometry-independent spline finite element method is proposed, which integrates the geometric construction techniques of isogeometric analysis and the element construction techniques of finite element method. Firstly, the geometrical precision element is constructed by using non-uniform rational B-spline and B-splines to describe the geometry and the physical field, where a transformation matrix is introduced to construct shape functions with interpolation properties. Secondly, calculation formats for two-dimensional heat conduction and elasticity problems are derived by parameter mapping. Finally, the feasibility and accuracy of the method are verified through several numerical examples. Unlike in isogeometric analysis, in the proposed method, geometry and physics are separated, and the shape functions have interpolation properties, which reduce redundant geometry calculations and allow the boundary conditions to be applied directly to the nodes. The numerical results indicate that the method has a higher convergence rate compared to the original isogeometric analysis.

A hybrid shape functions based temporal finite element method to solve transient heat conduction problems

A novel temporal finite element method is presented to solve Fourier and non-Fourier heat conduction problems. Two kinds of temporal finite element models are developed using Gurtin variational principle and weighted residual technique, respectively. A kind of hybrid shape function with polynomial and trigonometric basis is stressed to give more flexible and appropriate description of temporal behavior of temperature. A recursive algorithm is developed by which the temperature solution at a specific time can be obtained only via a matrix power product with the initial condition, and a criterion of stability analysis is derived, which is can numerically be conducted when the constitution of shape function and time step size are prescribed. The proposed approach is available for Fourier and non-Fourier heat transfer problems, and can conveniently be combined with well-developed numerical algorithms for the boundary value problem, such as FE, SBFEM, etc. Various numerical examples, including those with the nonlinear thermal conductivity, radiative boundary condition, etc. are provided to illustrate the efficiency of proposed approaches, and impacts of the temporal FE model, constitution of shape functions, and step size, etc. are taken into account. Satisfactory results are achieved in comparison with the analytical and the ABAQUS based numerical solutions.

Regularized kernel function methods for the backward heat conduction problem

In this letter, we are interested in the regularized numerical techniques for backward heat conduction problems, which aim at the detection of the previous status of physical field from its present information. It is well known that backward heat conduction problems are ill-posed. The small perturbation in the final status may result in large change in the previous status of physical field. It is difficult to get the stable numerical solutions of backward heat conduction problems. Based on the reproducing kernel theory, the letter aims at finding a new regularized kernel function methods for backward heat conduction problems.

Physics-Informed Fully Convolutional Networks for Forward Prediction of Temperature Field and Inverse Estimation of Thermal Diffusivity

Abstract Physics-informed neural networks (PINNs) are a novel approach to solving partial differential equations (PDEs) through deep learning. They offer a unified manner for solving forward and inverse problems, which is beneficial for various engineering problems, including heat transfer analysis. However, traditional PINNs suffer from low accuracy and efficiency due to the fully-connected neural network framework and the method to incorporate physical laws. In this paper, a novel physics-informed learning architecture, named physics-informed fully convolutional networks (PIFCNs), is developed to simultaneously solve forward and inverse problems in thermal conduction. The use of fully convolutional networks (FCNs) significantly reduces the density of connections. Thus, the computational cost is reduced. With the advantage of the nodal-level match between inputs and outputs in FCNs, the output solution can be used directly to formulate discretized PDEs via a finite difference method, which is more accurate and efficient than the traditional approach in PINNs. The results demonstrate that PIFCNs can flexibly implement Dirichlet and Neumann boundary conditions to predict temperature distribution. Remarkably, PIFCNs can also estimate unknown thermal diffusivity with an accuracy exceeding 99%, even with incomplete boundaries and limited sampling data. The results obtained from PIFCNs outperform those obtained from PINNs.

Analytic Modelling of 2-D Transient Heat Conduction with Heat Source Under Mixed Boundary Constraints by Symplectic Superposition

Abstract Many studies have been conducted on 2-D transient heat conduction, but analytic modelling is still uncommon for the cases with complex boundary constraints due to the mathematical challenge. With an unusual symplectic superposition method, this paper reports new analytic solutions to 2-D isotropic transient heat conduction problems with heat source over a rectangular region under mixed boundary constraints at an edge. With the Laplace transform, the Hamiltonian governing equation is derived. The applicable mathematical treatments, e.g., the variable separation and the symplectic eigenvector expansion in the symplectic space, are implemented for the fundamental solutions whose superposition yields the ultimate solutions. Benchmark results obtained by the present method are tabulated, with verification by the finite element solutions. Instead of the conventional Euclidean space, the present symplectic-space solution framework has the superiority on rigorous derivations without pre-determining solution forms, which may be extended to more issues with the complexity caused by mixed boundary constraints.

Numerical Computation of 2D Domain Integrals in Boundary Element Method by (α, β) Distance Transformation for Transient Heat Conduction Problems

When the time-dependent boundary element method, also termed the pseudo-initial condition method, is employed for solving transient heat conduction problems, the numerical evaluation of domain integrals is necessitated. Consequently, the accurate calculation of the domain integrals is of crucial importance for analyzing transient heat conduction. However, as the time step decreases progressively and approaches zero, the integrand of the domain integrals is close to singular, resulting in large errors when employing standard Gaussian quadrature directly. To solve the problem and further improve the calculation accuracy of the domain integrals, an (α, β) distance transformation is presented. Distance transformation is a simple and efficient method for eliminating near-singularity, typically applied to nearly singular integrals. Firstly, the (α, β) coordinate transformation is introduced. Then, a new distance transformation for the domain integrals is constructed by replacing the shortest distance with the time step. With the new method, the integrand of the domain integrals is substantially smoothed, and the singularity arising from small time steps in the domain integrals is effectively eliminated. Thus, more accurate results can be obtained by the (α, β) distance transformation. Different sizes of time steps, positions of source point, and shapes of integration elements are considered in numerical examples. Comparative studies of the numerical results for the domain integrals using various methods demonstrate that higher accuracy and efficiency are achieved by the proposed method.

SINGULAR INTEGRAL EQUATIONS FOR A CRACK SUBJECTED NORMAL STRESS IN A HEATED PLATE

In this paper, a crack in a heated plate is investigated, subjected to normal stress. Employing the relationship between the uniform and perturbation fields, as well as complex potential functions and stresses, the problems of heat conduction and heat stress are modeled as singular integral equations. The derivatives of the crack opening displacement function and the temperature jump function serve as the unknown functions. Gauss integration rules are applied to solve the obtained equations numerically. Analysis of the stress intensity factors(SIFs) for some particular crack configurations is presented.

Noninvasive reconstruction of internal heat source in biological tissue using adaptive simulated annealing algorithm

The heat distribution information of human lesions is of great value for disease analysis, diagnosis, and treatment. It is a typical inverse problem of heat conduction that deriving the distribution of internal heat sources from the temperature distribution on the body surface. This paper transforms such an inverse problem of bio-heat transfer into a direct one, thereby avoiding complex boundary conditions and regularization processes. To noninvasively reconstruct the internal heat source and its corresponding 3D temperature field in biological tissue, the adaptive simulated annealing (ASA) algorithm is used in the simulation module, where the position P(x, y, z) of point heat source in biological tissue and its corresponding temperature T are set as the optimization variables. Under a certain optimized sample, one can obtain the simulated temperature distributing on the surface of the module, then subtract the simulated temperature from the measured temperature of the same surface which was measured using a thermal infrared imager. If the sum of absolute values of the difference is smaller, it indicates that the current sample is closer to the true location and temperature of the heat source. When the values of optimization variables are determined, the corresponding 3D temperature field is also confirmed. The simulation results show the simulated position and temperature of the heat source are very approximate with those of the real experimental module. The method presented in this paper has enormous potential and promising prospects in clinical research and application, such as tumor hyperthermia, disease thermal diagnosis technology, etc.

Analytical Solution for Transient Thermal Behavior of Two Semisolids With Contact Resistance and Interfacial Heat Generation

Abstract The problem of two dissimilar semi-infinite solids at different initial temperatures brought into contact has a well-known simple analytical solution. In this work, this problem is reexamined with the additional simultaneous complications of both contact resistance and surface heat generation. While contact resistance is always present to some degree due to surface asperities or oxidation layers, heat generation at the contact interface can also occur in certain situations. These situations can occur in applications such as ultrasonic welding or arise in situations involving electromagnetic radiation passing through an optically transparent medium, but dissipating as heat at an interface with an opaque material that is in contact with the transparent material. In this paper, an analytical solution to the unsteady conduction problem is developed that accounts for both contact resistance and interfacial heat generation. The solution confirms that the initially warmer object rapidly decreases in temperature in the vicinity of the interface as heat flows into the cooler object and the heat generated at the interface preferentially flows to the cooler material. After a short time, however, the temperatures of both materials at the interface increase in temperature above even the initial temperature of the initially hotter material. An experiment was performed that verified the analytical solution.