Transient Heat Conduction Equation Research Articles

A boundary collocation method for anomalous heat conduction analysis in functionally graded materials

This paper applies a semi-analytical boundary collocation method, the singular boundary method (SBM), in conjunction with the dual reciprocity method (DRM) and Laplace transformation technique to solve anomalous heat conduction problems under functionally graded materials (FGMs). In this study, transient heat conduction equation with Caputo time fractional derivative is considered to describe anomalous heat conduction phenomena. In the present numerical implementation, Laplace transformation and numerical inverse Laplace transformation are used to deal with the Caputo time fractional derivative, which avoid the effect of time step on the computational efficiency of the time fractional derivation approximation. The SBM in conjunction with the DRM is employed to obtain the high accurate results in the solution of Laplace-transformed time-independent nonhomogeneous problems. To demonstrate the effectiveness of the proposed method for anomalous heat conduction analysis under functionally graded materials, three numerical examples are considered and the present results are compared with known analytical solutions and COMSOL simulation.

Finite Difference Approximation for Solving Transient Heat Conduction Equation of Copper

In this paper, a numerical technique is proposed to obtain the solution for transient heat conduction equation of Copper. The copper element is characterized by many characteristics; the most important of which is its high ability to conduct heat and electrical conductivity, in addition to being a flexible and malleable metal that is easy to form without being broken, making it one of the basic minerals that humans have benefited from for thousands of years, it is one of the first minerals. That has been discovered and extracted, and still plays a major role in the development of societies. The obtained solutions are compared with the available exact solutions and the obtained solutions using the finite difference method. The results indicate that the finite difference method is a highly effective method for obtaining approximate solutions for the thermal conductivity equation for copper. It is also clear from the numerical results from copper in the high conductivity of heat and electricity.

Analytical solution to thermal stresses of high speed PM rotor considering thermal load and heat convection

Aiming at the high speed permanent magnet (PM) rotor with the heat source, this work investigates the analytic solution to the transient temperature field and thermal stress field of the rotor, considering the influence of the forced air cooling of rotor surface on the stress field. Firstly, dimensionless formulation of the transient heat conduction equation including interior heat source is derived, where the axially non-uniform heat convection coefficient and the temperature of main flow region are equivalent to their mean values. Secondly, the Fourier integral transform method is used to solve the dimensionless heat conduction equation. Then, the obtained temperature field is loaded into the analytical solution of strength, in which three types of stress sources such as interference fit, centrifugal force and temperature gradient are included. Finally, examples are carried out to verify the analytical solutions and relative results are discussed.

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.

Dynamic snap-through of shallow spherical shells subjected to thermal shock

Present study deals with the dynamic snap-through phenomenon of an isotropic shallow spherical cap under transient type of thermal loading. The inner surface of the shell is subjected to sudden temperature elevation, whereas the outer surface is kept at reference temperature. Transient thermal shock is applied uniformly and shell thickness is assumed to be thin enough. Therefore, transient heat conduction equation is analytically solved across the thickness direction (one dimensional). Immovable simply-supported boundary conditions are assumed for the shell. Since the boundary conditions and the applied thermal shock are axisymmetric, the governing motion equations of the shell are restricted to the case of axisymmetric. First order shear deformation theory of shells is utilized to approximate the displacement field. The von Kármán type of geometrical non-linearity is used in strain-displacement relations. With the establishment of the associated Hamilton principle, differential equations of motion are extracted. Motion equations are discretized within the shell domain by means of the harmonic differential quadrature method (HDQM). The Newmark time marching scheme based on the constant average acceleration method is applied to turn the motion differential equations into a system of algebraic equations at each time step. Highly coupled equations are solved implementing the well-known Newton-Raphson iterative technique. By means of the Budiansky criterion, critical thermal shock parameters are distinguished. Critical dynamic snap-through temperatures are also verified using the phase-plane presentation. Comparison study is performed to validate the formulation and solution method of the present research with simple cases. Also, parametric investigations are performed to demonstrate the geometrical effects on the dynamic critical buckling temperatures of the shallow spherical shells subjected to thermal shock.

Coupled fluid-solid thermal interaction modeling for efficient transient simulation of biphasic water-steam energy systems

This paper proposes a fluid-solid coupled finite element formulation for the transient simulation of water-steam energy systems with phase change due to boiling and condensation. As it is commonly assumed in the study of thermal systems, the transient effects considered are exclusively originated by heat transfer processes. A homogeneous mixture model is adopted for the analysis of biphasic flow, resulting in a nonlinear transient advection-diffusion-reaction energy equation and an integral form for mass conservation in the fluid, coupled to the linear transient heat conduction equation for the solid. The conservation equations are approximated applying a stabilized Petrov-Galerkin FEM formulation, providing a set of coupled nonlinear equations for mass and energy conservation. This numerical model, combined with experimental heat transfer coefficients, provides a comprehensive simulation tool for the coupled analysis of boiling and condensation processes. For the treatment of enthalpy discontinuities traveling with the flow, a novel explicit-implicit time integration method based on Crank-Nicolson scheme is proposed, analyzing its accuracy and stability properties. To reduce problem size and enhance numerical efficiency, a modal superposition method with balanced truncation is applied to the solid equations. Finally, different example problems are solved to demonstrate the capabilities, flexibility and accuracy of the proposed formulation.

Energy Performance of Earthen Walls In A Hot Climate of Morocco

The building sector consumes more than 25% of energy consumption in Morocco. So, the reducing of the energy cost of building has become a necessity. In this context, the objectives of this work are to investigate the thermophysical properties and the energy performance of earthen walls made from three types of unfired clay bricks. This work is broken into two steps. In the first step the apparent thermal conductivity was obtained experimentally using the hot plate method. In the second step, the thermophysical properties obtained were used to calculate the decrement factor and the time lag of complete walls mad from these materials by solving the transient heat conduction equation with periodic loadings using the eigenfunction expansion and the integrating factor methods. The effect of the outer and inner combined convection and radiation heat transfer coefficient on these dynamic thermal characteristics was studied. The optimum thickness of the walls was also calculated. Finally, the effect of thermal insulation on energy performance of the walls was studied

Experimental design for estimation of the distribution of the convective heat transfer coefficient for a bubbly impinging jet

In this study, a two-dimensional inverse algorithm is developed to determine the heat transfer coefficient distribution of a two-phase air–water bubbly jet impinging on a steel cylindrical thermal mass. All procedures of thermal mass heating and cooling are simulated by solving two-dimensional, transient heat conduction equations using finite difference method. Afterward, the nonlinear inverse heat conduction problem is implemented to directly predict the local convective heat transfer coefficient of the bubbly jet. The sum of squared differences between calculated and measured temperature data is the objective function. Conjugate gradient method is employed sequentially in every time step to optimize the objective function at four gas Reynolds numbers which represent four different two-phase jets. The inverse scheme is validated using exact temperature data without noise. Local heat transfer coefficients are then estimated by inverse technique at five data acquisition times and four initial temperatures of thermal mass in the presence of noise. Furthermore, the effects of uncertainties due to indefinite lateral boundary conditions, temperature dependency of thermal conductivity, and the non-uniformity of the initial temperature distribution are investigated. A satisfactory agreement between exact and estimated heat transfer coefficients is achieved. However, the results show a greater sensitivity to the highest value of initial temperature, the shortest data acquisition time, and the lowest gas Reynolds number allowing a better estimation of heat transfer distribution for the bubbly jet.

Optimum position and distribution of insulation layers for exterior walls of a building conditioned by earth-air heat exchanger

In this paper, the optimum thermal insulation configuration in buildings with earth-air heat exchanger has been calculated based on time lag and decrement factor theory. In previous studies which all focused on buildings with conventional air-conditioning systems, the indoor air temperature has been assumed constant, while it is not valid for buildings conditioned by earth-air heat exchanger. Thereby, optimal insulation configuration in these buildings seems to be different from what previously reported. In this study, transient heat conduction equation is solved numerically for different multilayered wall configurations. Results showed that three-layer insulation, which is the optimal layout for buildings with conventional air-conditioning systems, is not desirable for buildings with earth-air heat exchanger. In such buildings, a wall with single-layer insulation at the outside and a wall with two layers of insulation (middle and outside) have the best performance, respectively. In these configurations, the decrement factor is 35% smaller, and the time lag is an hour longer than those of three-layer layout. Furthermore, results indicated that the optimum insulation configuration is independent of the system design characteristics and building wall materials.

Theoretical and experimental study on the thermally dependent transient response of the high power continuous wave Yb-doped fiber laser.

In this paper, transient thermal effects of the Yb-doped fiber and the laser diode (LD) in high power Yb-doped fiber lasers (YDFLs) are studied theoretically to analyze the transient response of the fiber laser. Based on the transient heat conduction equation, we simulated the temperature variation of the YDF and LD. It is found that the transient response of the laser is mainly affected by the thermally induced wavelength shifting of LDs. We modified the rate equations of the fiber laser according to the temperature variation of the LD. To improve the transient response speed of the fiber laser, we present three methods: (a) raising the cooling temperature, (b) increasing the length of gain fiber, and (c) using the YDF with high doping. Finally, experiments were carried out to verify the theory, and the results are in good agreement.

Nonlinear axisymmetric response of temperature-dependent FGM conical shells under rapid heating

Considering the uncoupled thermoelasticity assumptions, a thermally induced vibration analysis for functionally graded material (FGM) conical shells is performed in this research. Thermo-mechanical properties of the conical shell are assumed to be temperature and position dependent. The conical shell is under rapid heating with various cases of thermal loads on the outer surface, whereas the opposite surface is kept at reference temperature or thermally insulated. Since the ratio of thickness to radius is much smaller than one, the transient heat conduction equation, for simplicity, may be established and solved for one-dimensional condition. Assuming temperature-dependent material properties, the heat conduction equation is nonlinear and should be solved using a numerical method. A hybrid generalized differential quadrature (GDQ) and Crank–Nicolson method is used to obtain the temperature distribution in thickness direction, respectively. Based on the first-order shear deformation theory and geometrically nonlinear assumptions, the equations of motion are obtained applying the Hamilton principle. Discretization of the equations of motion in the space domain and boundary conditions is performed by applying the GDQ method, and then, the system of highly nonlinear coupled ordinary differential equations is solved by the iterative Newmark time-marching scheme and well-known Newton–Raphson method. Since the thermally induced vibration of the conical shells is not reported in the literature, the results are compared with the case of a circular plate. Also, studies of the FGM conical shells for various types of boundary conditions, functionally graded patterns, and thermal loads are provided. The effects of temperature dependency, geometrical nonlinearity, semi-vertex angle, shell length, and shell thickness upon the deflections of the conical shells are investigated.

Three-dimensional transient heat conduction analysis by boundary knot method

This paper makes the first attempt to apply the boundary knot method (BKM), in conjunction with dual reciprocity technique, for the solution of three-dimensional transient heat conduction problems. The BKM is a meshless, integration-free, easy-to-program boundary-only numerical technique for high-dimensional problems. The first step of our strategy is to use the finite difference method for temporal derivative to convert the transient heat conduction equation into a nonhomogeneous modified Helmholtz equation. And then the corresponding nonhomogeneous problem is solved using the proposed BKM strategy in conjunction with dual reciprocity technique. Four benchmark numerical examples are investigated in detail, and the numerical results show that the present scheme has the merits of high accuracy, wide applicability, good stability, and rapid convergence and is appealing to solve 3D transient heat conduction problems.

Geometrically nonlinear rapid surface heating of temperature-dependent FGM arches

Based on the nonlinear dynamic analysis, thermally induced vibrations of the FGM shallow arches subjected to different sudden thermal loads are studied. Temperature and position dependence of the material properties are taken into account. Based on the uncoupled thermoelasticity assumptions, The non-linear one-dimensional transient heat conduction equation is solved numerically by a hybrid iterative GDQ method and Crank-Nicolson time marching scheme. A first order shear deformation arch theory (FSDT) is also combined with the von Kármán type of geometrical non-linearity and the Donnell kinematic assumption to obtain the equations of motion employing the Hamilton principle. Discretization of the highly coupled non-linear equations of motion is done by using the GDQ method in the arch domain. The solution of the system of the ordinary differential equations is established by means of a hybrid iterative Picard-Newmark scheme. Comparison is also made with the existing results for the case of isotropic homogeneous shallow arches, where good agreement is obtained. Also, parametric studies are proposed to show the effects of temperature dependency, geometrical non-linearity, arch thickness, power law index, and the type of thermal-mechanical boundary conditions upon the arch deflection.

Simultaneous estimation of unknown parameters using a-priori knowledge for the estimation of interfacial heat transfer coefficient during solidification of Sn–5wt%Pb alloy—an ANN-driven Bayesian approach

The present methodology focuses on model reduction in which the prevalent one-dimensional transient heat conduction equation for a horizontal solidification of Sn–5wt%Pb alloy is replaced with Artificial Neural Network (ANN) in order to estimate the unknown constants present in the interfacial heat transfer coefficient correlation. As a novel approach, ANN-driven forward model is synergistically combined with Bayesian framework and Genetic algorithm to simultaneously estimate the unknown parameters and modelling error. Gaussian noise is then added to the temperature distribution obtained using the forward approach to represent real-time experiments. The hallmark of the present work is to reduce the computational time of both the forward and the inverse methods and to simultaneously estimate the unknown parameters using a-priori engineering knowledge. The results of the present methodology prove that the simultaneous estimation of unknown parameters can be effectively obtained only with the use of Bayesian framework.

Derivation and application of the adjoint method for estimation of both spatially and temporally varying convective heat transfer coefficient

In this study, a numerical algorithm is developed to determine the heat transfer coefficient distribution of mixed convection on a vertical plate with a two dimensional inverse method. Cooling procedures of hot plate include forced convection with buoyancy effect is simulated by solving two dimensional, transient heat conduction equation using finite difference method while the Grashof number is varied through natural convection in combination with forced convection resulting in a variation of the Richardson number from 0.1 to 1. The nonlinear inverse heat conduction problem is then implemented to directly predict the time-space-varying convective heat transfer coefficient of cooling in the mixed convection regime. The sum of squared differences between calculated and measured temperature data at thermocouples’ locations is the objective functional. The adjoint method is employed to optimize the functional using conjugate gradient method via the solutions of the direct, adjoint and sensitivity sub-problems in a whole time-domain optimization process. The inverse scheme is validated using exact temperature data without noise. Local heat transfer coefficients are then estimated by the adjoint method at four cooling strategies for ten minutes of data acquisition in the presence of noise with standard deviations of σ=0.01 °C and σ=0.1 °C. Although results are affected by noisy simulated temperatures, a satisfactory agreement between exact and estimated heat transfer coefficients is achieved. Furthermore, increasing data acquisition time to sixty minutes reveals that the inverse scheme is able to predict 1200 convective heat transfer coefficient components efficiently employing the rapidly convergent adjoint method.

Effect of Process Parameters and Material Properties on Laser Micromachining of Microchannels.

Laser micromachining has emerged as a promising technique for mass production of microfluidic devices. However, control and optimization of process parameters, and design of substrate materials are still ongoing challenges for the widespread application of laser micromachining. This article reports a systematic study on the effect of laser system parameters and thermo-physical properties of substrate materials on laser micromachining. Three dimensional transient heat conduction equation with a Gaussian laser heat source was solved using finite element based Multiphysics software COMSOL 5.2a. Large heat convection coefficients were used to consider the rapid phase transition of the material during the laser treatment. The depth of the laser cut was measured by removing material at a pre-set temperature. The grid independent analysis was performed for ensuring the accuracy of the model. The results show that laser power and scanning speed have a strong effect on the channel depth, while the level of focus of the laser beam contributes in determining both the depth and width of the channel. Higher thermal conductivity results deeper in cuts, in contrast the higher specific heat produces shallower channels for a given condition. These findings can help in designing and optimizing process parameters for laser micromachining of microfluidic devices.

Numerical manifold method for thermal–hydraulic coupling in fractured enhance geothermal system

The geothermal production from enhance geothermal system (EGS) involves complex thermal–hydraulic (TH) coupling process, which may affect the production efficiency and performance of EGS reservoir. In the study, a numerical manifold method (NMM) for TH coupling in fractured EGS is proposed. The EGS reservoir is considered to be formed from intact rock matrix blocks and discrete fractures networks. TH coupling control equations are firstly established using discrete fracture networks model. Then, the weight residual method was used to establish the discretized governing equation for transient heat conduction. Furthermore, the element sub-matrixes for heat conduction were presented. NMM approach simulating the fluid flow, heat conduction in fracture and rock matrix and TH coupling process was established. Finally, a single fracture heat extraction example was used to validate the effectiveness of the proposed method for solving a simplified TH coupling problem. Furthermore, a 2D fracture networks case was conducted to predict the performance of an EGS reservoir. The results in here validate the capability and accuracy of the proposed method and extend the application of NMM.

Analysis of temperature drops on the wall thickness of a supercritical boiler water-wall tube

The paper presents selected results of numerical computations related to simulations of a supercritical power boiler evaporator operation. A detailed analysis is carried out of temperature drops on the thickness of the water-wall tube and the tube fin. A comparison is also made between the results obtained at the fin division into one and two control volumes. The tube cross-sections analysed along the tube length and including the fins are divided into 24 control volumes, for which 2-D transient heat conduction equations are formulated. The 1-D equations describing the principles of mass, momentum, and energy conservation are solved on the side of the working fluid. A convective condition, described by an empirical heat transfer coefficient, is set on the water-wall tube inner surface. The developed mathematical model is a distributed parameter model. The numerical computations are performed for a boiler operating in a power plant in Poland. The analysis takes account of the non-uniformity of the furnace chamber thermal load along its height.

Transient Radiator Room Heating—Mathematical Model and Solution Algorithm

A mathematical model and robust numerical solution algorithm for radiator heating of an arbitrary room is presented in this paper. Three separate and coupled transient thermal energy equations are solved. A modified transient heat conduction equation is used for solving the heat transfer at multi-layer outer walls and room assembly. Heat exchange between the inner walls and the observed room are represented with their own transport equation and the transient thermal energy equation is solved for radiators as well. Explicit coupling of equations and linearization of source terms result in a simple, accurate, and stabile solution algorithm. Verification of the developed methodology is demonstrated on three carefully selected test cases for which an analytical solution can be found. The obtained results show that even for the small temperature differences between inner walls and room air, the corresponding heat flux can be larger than the transmission heat flux through outer walls or windows. The benefits of the current approach are stressed, while the plans for the further development and application of the methodology are highlighted at the end.

Investigation of thermal responses during metallic additive manufacturing using a “Tri-Prism” finite element method

Temperature distribution and thermal history have been demonstrated to greatly affect the microstructure of additive manufacturing (AM)-fabricated part, and further dedicate its mechanical properties. This paper, thus, develops an accurate and high-efficiency linear six-node “Tri-Prism” element for transient nonlinear heat transfer analyses of the selective electron beam melting (SEBM) of Ti-6Al-4V alloy. A stepwise activated approach is used to model the material layer-by-layer deposition process for the SEBM simulation. The physics of interaction between the electron beam and the material is modeled by a Goldak double-ellipsoid volumetric heat source. A backward difference technique featured with the unconditional stability is employed to address the transient heat conduction equations. In order to handle the material nonlinear and radiation issues, an accelerated Newton-Raphson method (ANRM) is utilized herein. Two examples of transient heat transfer are first given to demonstrate the accuracy and efficiency of “Tri-Prism” element. Finally, the thermal responses of an aircraft engine blade fabricated by SEBM are comprehensively investigated.