Heat Conduction Problem Research Articles

Mathematical model of slab heating in a furnace with walking beams

Slab heating before hot rolling is necessary for obtaining required metal ductility. The most effective for this purpose are furnaces with walking beams that provide heat supply to all sides of the slab. However, the areas of slabs lower surfaces, contacting with water-cooled beams, are shielded from the radiation of the furnace lower heating zones and take the heat from the beams.To study the inhomogeneity of the slab temperature field and its dependence on the peculiarities of their transport system design, a mathematical model of slab heating in a furnace with walking beams was developed and programmatically implemented, based on numerical solution of a three-dimensional heat conduction problem with piecewise defined boundary conditions on the lower surface. For the open areas of the slab bottom surface, the same boundary conditions were set as on the top surface; and for the areas of contact with the beams, modified boundary conditions were set, taking into account the duration of this contact. For the numerical solution of the system of difference equations, the line-by-line method was applied, which allows us to obtain a system with a three-diagonal matrix of coefficients. The calculations carried out in the approximation of adiabaticity of the contact areas of the slab with the beams during the contact period allowed us to obtain temperature fields for different slab sections. As a result, a significant irregularity of temperature field of the slab lower surface was revealed, affecting the irregularity of temperature field of the entire slab. The developed program for calculating and visualizing the results can be used to study the temperature field of the slab under various heating modes if there is experimental information that allows one to clarify the tuning parameters of the model.

The Fading Memory Formalism with Mittag-Leffler-Type Kernels as A Generator of Non-Local Operators

Transient heat conduction problems are systematically applied to the fading memory formalism with different Mittag-Leffler-type memory kernels. With such an approach, using various memories naturally results in definitions of various fractional operators. Six examples are given and interpreted from a common perspective, covering the most well-liked versions of the Mittag-Leffler function. The fading memory approach was used as a template and demonstrated that, if the constitutive equations are correctly built, it is also possible to directly determine where the hereditary terms are located in the models.

A coupled RKPM and dynamic infinite element approach for solving static and transient heat conduction problems

A new accurate and efficient coupled method RKPM-DIEM is proposed. This is a stable and efficient meshfree nodally-integrated reproducing kernel particle method (RKPM) coupled with a dynamic infinite element method (DIEM) for solving half-space problems. The half-space domain is defined as the near field (bounded) and the far field (unbounded) analyzed by the RKPM and DIEM, respectively. Unlike the element-based methods, RKPM is constructed using only nodal data in the global Cartesian coordinates directly to avoid mesh issues such as mesh distortion and entanglement. Also, it provides flexible control of the local smoothness and order of basis, as well as easy construction for a higher-order gradient by changing the kernel function directly. DIEM is first used to show that this approach could solve not only dynamic but also static problems by setting the wave number and the decay coefficient properly. Furthermore, various meshfree integration methods, such as the Gaussian integration, the direct nodal integration, and the natural stabilized nodal integration, are tested to show accuracy and stability. Several benchmark problems are investigated to verify the effectiveness of the proposed method. It has been found that numerical results can achieve high accuracy and stability.

An internal cooling grinding wheel: From design to application

The cooling and lubrication conditions during the grinding process significantly impact the nickel-based superalloy's final service performance. The existing jet cooling and heat pipe technology can solve the heat conduction problem in the grinding process of superalloy. Still, managing cooling, lubrication, and chip removal are difficult. This paper describes the design and fabrication of a novel central fluid-through internal cooling slotted grinding wheel with an ordered grain pattern to improve the grinding machinability of a nickel-based superalloy. The pressurized grinding fluid was ejected into the grinding zone via the pipe and tool holder from the lower-end face of the inner cooling wheel. The structure of the grinding wheel was optimized using computational fluid dynamics (CFD). The flow field in the grinding area achieved the highest overall flow rate, distribution homogeneity, and effective exit flow when the internal flow channel had four through-holes. The exit for the inner runner is located at the abrasive edge and diamond staggered pattern. Single-layer brazing was used to create cubic boron nitride (CBN) abrasive rings with various abrasive patterns. The internal cooling wheel matrix and various components were prepared according to the optimized grinding wheel geometry model. A grinding test bench was built to conduct an experimental study of grinding the nickel-based alloy GH4169. The results show that, under the same conditions, a diamond-shaped staggered pattern obtains lower grinding temperature, lower surface roughness, better surface morphology, and more significant residual compressive stress distribution than an abrasive cluster diagonal circular staggered pattern or disordered pattern. The average effective flow rate calculated by CFD is increased by 42.3% when compared to the disordered pattern. In the experiment, compared to the disordered arrangement, with the increase of grinding wheel’s rotating speed and coolant pressure, the average grinding temperature of abrasive grain with diamond-interleaved arrangement decreases by 58.2% and 51.7% respectively, and its surface hardening degree decreases by 11.1% and 11.7% respectively.

Level set-based multiscale topology optimization for a thermal cloak design problem using the homogenization method

Artificially designed composite materials consist of microstructures, that exhibit various thermal properties depending on their shapes, such as anisotropic thermal conductivity. One of the representative applications of such composite materials for thermal control is the thermal cloak. This study proposed a topology optimization method based on a level set method for a heat conduction problem to optimally design composite materials that achieve a thermal cloak. The homogenization method was introduced to evaluate its effective thermal conductivity coefficient. Then, we formulated a multiscale topology optimization method for the composite materials in the framework of the homogenization method, where the microstructures were optimized to minimize objective functions defined using the macroscopic temperature field. We presented examples of optimal structures in a two-dimensional problem and discussed the validity of the obtained structures.

Inertial projected gradient method for large-scale topology optimization

We present an inertial projected gradient method for solving large-scale topology optimization problems. We consider the compliance minimization problem, the heat conduction problem and the compliant mechanism problem of continua. We use the projected gradient method to efficiently treat the linear constraints of these problems. Also, inertial techniques are used to accelerate the convergence of the method. We consider an adaptive step size policy to further reduce the computational cost. The proposed method has a global convergence property. By numerical experiments, we show that the proposed method converges fast to a point satisfying the first-order optimality condition with high accuracy compared with the existing methods. The proposed method has a low computational cost per iteration, and is thus effective in a large-scale problem.

Shannon-Taylor technique for solving one-dimensional inverse heat conduction problem

The Shannon-Taylor interpolation technique was introduced by Butzer and Engels in 1983. In this work, the sinc-function is replaced by a Taylor approximation polynomial. In this work, we implement the Shannon-Taylor approximations to solve a one-dimensional heat conduction problem. One of the major advantages of this approach is that the resulting linear system of equations of the approximation procedure has an explicit coefficient matrix. This is not the case of the classical sinc methods due to finite integrals involving e^{-x^2}. We establish rigorous error estimates involving an additional Taylor’s series tail. Numerical illustrations are depicted.

Enhanced surrogate modelling of heat conduction problems using physics-informed neural network framework

Solving partial differential equations (PDEs) using deep-learning techniques provides opportunities for surrogate models that require no labelled data, e.g., CFD results, from the domain interior other than the boundary and initial conditions. We propose a new ansatz of the solution incorporated with a physics-informed neural network (PINN) for solving PDEs to impose the boundary conditions (BCs) with hard constraints. This ansatz comprises three subnetworks: a boundary function, a distance function, and a deep neural network (DNN). The new model performance is assessed thoroughly in terms of convergence speed and accuracy. To this end, we apply the PINN models to conduction heat transfer problems with different geometries and BCs. The results of 1D, 2D and 3D problems are compared with conventional numerical methods and analytical results. The results reveal that the neural networks (NNs) model with the proposed ansatz outperforms counterpart PINN models in the literature and leads to faster convergence with better accuracy, especially for higher dimensions, i.e., three-dimensional case studies.

Numerical simulations for artillery barrel temperature variation considering mechanical friction heat under continuous shots

Accurate modeling of artillery barrel temperature distribution is important to evaluate thermochemical-mechanical erosion and thermal shock. For such heat conduction problems, the previous works ignore frictional heat, especially for continuous shots. To end this, this paper proposes a numerical simulation approach for the temperature field on the inner barrel wall considering the frictional temperature rise. In the novel scheme, the frictional temperature between the barrel and the rotating band is derived from the frictional heat theory, and the heat transfer model is computed coupling the interior ballistic codes with the finite difference method. A close-formed solution for the continuous shots with an incomplete cooling barrel is provided, and followed by a finite element numerical simulation. The good agreement between the aforementioned two simulations demonstrates a good accuracy of the developed solution form. The results show that there is a temperature peak at the initial part of the rifling, which is much higher than the other parts, and far exceeds the critical phase transition temperature of the barrel steel material.

Liquid film and heat transfer characteristics during superheated wall cooling via pulsed injection of a liquid jet

Pulse firing is one of the major operation modes of bipropellant thrusters for the attitude control of small-scale spacecrafts. In the pulse-firing mode, the operational range of the thruster depends on the deterioration of liquid film cooling performance. Because cooling performance is determined by the balance of heat removed by the liquid film and transferred to the manifold, the behavior and heat transfer characteristics of liquid films under the intermittent injection of liquid jets must be understood to enable a wider range of operational patterns. We conducted cooling tests on a superheated metal plate via the pulsed injection of a liquid jet for better understanding of the pulsed cooling process. Different injection patterns, including continuous injection, with the same injection quantity were examined on two types of metal plates (aluminum alloy and copper), because the thermal properties of metal plates affect both the heat transfer characteristics of the liquid film and temperature rise of the plate during the inter-pulse duration. The cooling process was evaluated based on the evolution of the liquid film on the metal plate and rear-side temperature of the metal plate. The evolution and temperature were simultaneously visualized using a high-speed camera and infrared camera, respectively. To link the liquid film state to the heat transfer process, the temperature and heat flux on the cooled surface were estimated by solving the inverse problem of three-dimensional transient heat conduction. The results indicate that the wetting front position corresponds to the position of the maximum temperature gradient. Additionally, the residual liquid film is consumed through the evaporation and the droplet dispersion related to the nucleate boiling. The effects of thermal inertia and diffusivity of the metal plate highlight the differences in the amount of heat removed by the liquid film. The heat removed by the liquid film during pulsed cooling exhibited a peak and was higher than that removed by continuous injection on the aluminum alloy plate. Less heat was removed under low-duty-cycle conditions, and the amount of heat removed by the liquid film was lower than that removed by continuous injection on the copper plate.

Heat Conduction Problem for a Nonstationary Point Heat Source in a Plane-Layered Medium

Heat Conduction Problem for a Nonstationary Point Heat Source in a Plane-Layered Medium

Estimation of the local instantaneous heat flux inside a pulsating heat pipe for space applications

Pulsating Heat Pipes (PHP) are promising and effective passive two-phase heat transfer devices in terms of high heat transfer capability, efficient thermal control, adaptability and low cost and therefore they have been extensively studied in the last years. Many authors have estimated the heat fluxes at the evaporator and at the condenser area only in terms of the average values based on first principle considerations. In the present study the application of an inverse analysis technique to experimental infrared temperature data is proposed to investigate the local convective heat flux for forced convection flow inside these devices along the adiabatic zone. A PHP specifically designed to be hosted on board the heat transfer host of the International Space Station was tested in microgravity during the 67th Parabolic Flight Campaign organized by the European Space Agency. The device consists of an aluminium tube closed in a loop with 14 turns in the evaporator section, 3 mm inner diameter, half filled with FC-72 fluid. The external temperatures of the device are measured in the adiabatic zone with a high-speed infrared camera (50 Hz, 1280x1024 pixels). The images are thereafter post-processed and adopted as input data for the solution of the inverse heat conduction problem in the pipe wall (Tikhonov regularisation method) to extrapolate time-varying local heat fluxes on the tube internal surface in contact with the fluid.

A novel defect identification design of gas pipeline based on inverse heat conduction problem

A novel defect identification design of gas pipeline based on inverse heat conduction problem

Algorithm for Processing Data on the Thermophysical Properties of Phosphorites by Solving the Inverse-Coefficient Heat-Conduction Problem

Algorithm for Processing Data on the Thermophysical Properties of Phosphorites by Solving the Inverse-Coefficient Heat-Conduction Problem

Toward the Design of a Representative Heater for Boiling Flow Characterization under PWR’s Prototypical Thermalhydraulic Conditions

In order to improve the understanding of the phenomena underlying the boiling occurrence, CEA Cadarache (France) is designing a new experimental setup, intended to operate for pressures ranging from atmospheric to PWR conditions. This will allow optical access to the convective boiling flow as well as thermal imaging (infrared thermography) of the heated surface. A two-step methodology for designing the heater (particularly its thickness since it directly influences the boiling mechanisms) was developed. This approach is based on solving the heat conduction problem within the heater, considering realistic time-dependent boundary conditions representative of the boiling process. Since those boundary conditions are measured on the external face of the heater, this heat transfer problem is known as an inverse problem that is difficult to solve because of its ill-posedness and high sensitivity to boundary condition uncertainties. In the first stage, we considered one-dimensional modeling to determine the order of magnitude of the heater’s thickness that guaranteed a correct reconstruction of the wet temperature from the measured dry temperature in terms of uncertainties. This value was confirmed in the second stage using a two-dimensional model that accounted for the presence of multiple bubbles on the wet side.

Temperature during Repetitive Short-Term Operation of a Brake with Functionally Graded Friction Element.

The object of study is the temperature of a braking system, operating in repetitive short-term (RST) mode. One element of the considered friction pair is made of a functionally gradient material (FGM), and the other of a homogeneous material. To determine the temperature on the friction surfaces of both elements, the previously obtained, exact solution of the boundary value problem of heat conduction was adopted, with account of the heat generation due to the friction. A calculation scheme was proposed that takes into consideration thermal sensitivity of materials and variations of the friction coefficient under the influence of temperature. Calculations were performed for two-component FGM (ZrO2-Ti-6Al-4V) in combination with gray cast iron (ChNMKh). It was found that for selected friction pair materials, consideration of their thermal sensitivity reduces the time of braking and the value of temperature achieved on the friction surfaces. At the same time, the whole process was characterized by a good stability of braking with a slight decrease in efficiency in each subsequent cycle.

Jeffreys heat conduction in coupled semispaces subjected to interfacial heating

A Jeffreys heat conduction problem for coupled semispaces subjected to the action of an interfacial heat source was defined. An analytical solution of the problem was derived for a polynomial specific power of the heat source using the Laplace transform approach. The asymptotic and parametric analysis was performed for different ratios of thermal conductivities K1,2, thermal diffusivities k1,2, thermal relaxation times τ1,2 and coefficients α1,2 indicating the relative contribution of Fourier heat conduction. It was found that Jeffreys heat conduction results in continuous variation of the contact temperature, whilst its particular case — Cattaneo heat conduction — is accompanied by a step change of the contact temperature at the initial time. Another finding is that the initial heat partition occurs due to the ratio of K1α1/k1 and K2α2/k2 under Jeffreys heat conduction and due to the different ratio of K1/k1τ1 and K2/k2τ2 under Cattaneo heat conduction. The solution applicability was demonstrated on the simulation example of ultrashort laser pulse welding. The type of heat conduction was revealed to have qualitative and quantitative impacts on the contact temperature and heat fluxes.

A multiscale scaled boundary finite element method solving steady-state heat conduction problem with heterogeneous materials

A new numerical model is proposed by bridging small and large scales via the numerical base functions constructed by Scaled Boundary Finite Element Method (SBFEM) for the multiscale heterogeneous analysis of steady-state heat conduction problems. Utilizing the advantages of polygonal Scaled Boundary Finite Element (SBFE) and image-based quadtree gridding, the construction of base functions can be conveniently and efficiently conducted particularly for the heterogeneous media with complex geometries or singularity of heat flux at the small-scale, and the solution accuracy at the large-scale can be improved by increasing nodes of coarse element only without increasing new nodes inside. Consequently, the SBFEM based solutions at the large-scale can be solved with a reduced order and sufficient accuracy, and the solutions at the small-scale can be achieved using SBFEM based base functions. Numerical examples are provided and the effectiveness of the proposed approach is stressed at both the large and small scales.

The efficient calculation methods for stochastic nonlinear transient heat conduction problems

This paper proposes an efficient model reduction strategy in conjunction with non-intrusive statistical moment methods for the stochastic nonlinear transient heat conduction problem. In the proposed strategy, the deterministic reduced basis vectors (RBVs) are adopted to approximate the random temperature response, and the thermal conductivity of the random nonlinear heat conduction matrix in the RBVs space is decoupled from the geometric characteristic matrix (GCM). The computational efficiency of solving the reduced random nonlinear heat conduction models can be enhanced by implementing these two technologies. The large coefficient of variation and the small coefficient of variation stochastic heat conduction problems, respectively, are addressed by the two non-intrusive statistical methods, i.e., the quasi-Monte Carlo method (QMCM) based on number theory and the modified stochastic perturbation method (MSPM). These two methods necessitate deterministic analysis and are easily integrated into the model reduction method. In four numerical instances, the selection of RBVs, the influence of the nonlinear term on the random temperature response, and the accuracy and efficiency of the two non-intrusive statistical moment methods are discussed. The numerical results highlight that the model reduction strategy based on the decoupling of thermal conductivity and GCM, in conjunction with the QMCM and the MSPM, enables the analysis of stochastic nonlinear transient heat conduction problems with a large coefficient of variation and a small coefficient of variation, respectively.

MODELING THE INFLUENCE OF THE SHAPE OF THE LOCAL HEAT FLOW INTENSITY DISTRIBUTION ON THE SURFACE OF A SEMI-INFINITE BODY ON THE STRESS STATE IN THE VICINITY OF A SUBSURFACE CRACK

Purpose. A mathematical model to determine the two-dimensional thermoelastic state in a semi-infinite solid weakened by an internal crack under conditions of local heating is examined. Heat flux due to frictional heating on the local area of the body causes changes in temperature and stresses in the body, which significantly affects its strength, as it can lead to crack growth and local destruction. Therefore, the study of the problem of frictional heat is of practical interest. This paper proposes to investigate the stress-deformed state in the vicinity of the crack tip, depending on the crack placement. Methodology. The methods for studying the two-dimensional thermoelastic state of a body with cracks as stress concentrators are based on the method of complex variable function. Reducing the problem of stationary heat conduction and thermoelasticity to singular integral equations (SIE) of the first kind, the numerical solution by the method of mechanical quadrature was obtained. Findings. In this paper, we present graphical dependencies of stress intensity factors (SIF) at the crack tip on the angle orientation of the crack as well as forms of the intensity distribution of the local heat flux. The obtained results will be used later to determine the critical value of the intensity of the local heat flux from equations of limit equilibrium at which crack growth and the local destruction of the body occur. Originality. The scientific novelty lies in the fact that the solutions to two-dimensional problems of heat conduction and thermoelasticity for a half-plane containing a crack due to local heating by a heat flux were obtained. This would make it possible to obtain a comparative analysis of the intensity of thermal stresses around the top of the crack, depending on the form of distribution of the intensity of the heat flow on the surface of the body. Practical value. The practical value is the ability to extend our knowledge of the real situation in the thermoelastic elements of engineering structures with the crack that operate under conditions of heat stress (frictional heat) in various industries, particularly in mechanical engineering. The results of specific values of SIF at the crack tip in graphs may be useful in the development of sustainable modes of structural elements in terms of preventing the growth of cracks.