Transient Heat Conduction Equation Research Articles

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.

New thermal solver for mitigating surface temperature instability in laser-induced heating

An accepted approach to computing laser-induced peak surface temperature is to employ the enthalpy formulation of the transient heat conduction equation [Grigoropoulos et al., Adv. Heat Transfer 28, 75–144 (1996); Sawyer et al., J. Laser Appl. 29, 022212 (2017)]. This approach is generally implemented using an explicit numerical scheme to solve the thermal transport equation. While it offers the advantage of modeling the solid-melt phase transition automatically, the approach results in instability-like behavior in the computed surface temperature. When laser-induced ablation becomes significant, the heating rate in the surface cell becomes unrealistically large. This results in spikes in the computed peak surface temperature due to large errors in calculating the heating rate. In this paper, we present a new approach, which we refer to as the Moving Frame Solver, that employs a moving-coordinate frame of reference, located at the receding evaporating surface. We also use an analytical representation for the phase transition region of the enthalpy-temperature relationship. The Moving Frame Solver combined with an implicit scheme leads to a stable solution without surface temperature, pressure, or velocity spikes. In other words, any instability in these computed parameters due to use of an explicit scheme (such as Dufort–Frankel) has been eliminated. Details of the new thermal solver and example calculations are presented. Numerical experiments suggest that the surface cell size needs to be small, ∼0.1 μm, to obtain a highly accurate solution with a typical metal such as aluminum. Using the Moving Frame Solver with a refined grid near the surface, but coarse elsewhere, enables accurate and stable surface temperature computation.

Heat transport during drop impact onto a heated wall covered with an electrospun nanofiber mat: The influence of wall superheat, impact velocity, and mat thickness

Nanofiber surface coating is a promising method for the enhancement of heat transfer during spray cooling. In the present work, the drop dynamics as well as local and overall heat transfer during single drop impact onto a heated wall covered with a polyacrylonitrile (PAN) nanofiber mat are investigated to obtain insight into the mechanisms governing the heat transport enhancement. The influence of wall superheat, drop impact velocity, and mat thickness on the hydrodynamics and heat transfer from the heated wall to the fluid is studied. The experiments were conducted inside a temperature-controlled test cell with a pure vapor atmosphere maintained with refrigerant FC−72 (perfluorohexane). The temperature field at the solid-fluid interface was observed with a high-speed infrared camera, and the heat flux field was derived by solving a three-dimensional transient heat conduction equation within the substrate. The presence of the nanofiber mat on the heater surface suppresses the drop receding phase due to the pinning of the contact line at the end of the spreading phase. Two different heat transfer scenarios are observed depending on the wall superheat and drop impact velocity: scenario (I), in which the liquid drop completely penetrated through the porous nanofiber mat and made contact with the heater surface; and scenario (II) in which the vapor produced inside the pores of the nanofiber mat prevented the liquid drop from touching the heater surface. In scenario (I), an up 450% heat flow enhancement during the sessile drop evaporation stage has been observed. In scenario (II), a 80% heat flow enhancement at this stage has been registered.

Fast explicit time integration schemes for parabolic problems in mechanics

AbstractWe present a family of fast explicit time integration schemes of first, second and third order accuracy for parabolic problems in mechanics solved via standard numerical methods that have considerable higher computational efficiency versus existing explicit methods of the same order. The derivation of the new explicit schemes is inspired on the finite increment calculus (FIC) procedure used for obtaining stabilized numerical schemes in fluid and solid mechanics. The new (so‐called) explicit FIC‐Time (EFT) schemes allow considerable larger time steps than the standard first order forward Euler (FE) scheme and the second and third order Adams–Bashforth schemes. The comparison with Runge–Kutta schemes also favors the FIC‐Time schemes in terms of the limit time step size (for second order schemes) and the total number of matrix‐vector multiplications per time step (for second and third order schemes). The new first order explicit schemes have a faster convergence to steady‐state than the FE scheme. The accuracy and efficiency of the new EFT schemes are verified in examples of application to the transient heat conduction equation using the finite element method.

Nonlinear dynamic response of a temperature-dependent FGM spherical shell under various boundary conditions and thermal shocks: Examination of dynamic snap-through

In this study, a comprehensive exploration of the nonlinear dynamic snap-through phenomenon observed in shallow spherical shells composed of functionally graded materials (FGMs) is examined. The analysis encompasses the interplay of thermomechanical characteristics within metal and ceramic phases, taking into account their temperature-dependent behaviors. To address this problem, the transient heat conduction equation in a nonlinear form, accounting for the temperature dependence factor is derived. To solve this complexity, the Generalized Differential Quadrature (GDQ) method is employed, coupled with the Crank-Nicolson time integrating scheme, leveraging an iterative approach for precision. The formulation of nonlinear dynamic motion equations is based on Hamilton's principle and incorporates the utilization of nonlinear strain-displacement equations and uncoupled thermoelasticity. In the spatial domain, GDQ proves invaluable in solving these nonlinear, coupled equations. In the temporal domain, the Newmark procedure in conjunction with the Newton-Raphson iterative method are utilized to navigate through dynamic equations, yielding insights into shell's behavior over time. In the initial stages of the investigation, we validate our spherical shell formulation and methodologies by comparing shell's response with existing, simpler works. This step ensures the reliability and accuracy of our model. The primary objective is to assess the presence and characteristics of the thermal snap-through phenomenon within the shell under these diverse conditions.

Assessment of Axisymmetric Dynamic Snap-Through and Thermally Induced Vibrations in FGM Cylindrical Shells Under Instantaneous Heating

This paper addresses the investigation of axisymmetric thermally induced vibrations in Functionally Graded Material (FGM) cylindrical shells. It considers temperature-dependent (TD) properties and geometric non-linearity (the von Karman effect). The study systematically solves a transient heat conduction equation using finite differences method and the Crank–Nicolson method. During the heating stages, the evaluation of thermal forces and moments takes place. Equations of motion are derived through the application of Hamilton’s principle. Spatial dependencies are discretized using the generalized Ritz method, while temporal dependencies are approximated using the [Formula: see text]-Newmark method with Newton–Raphson linearization. A comparative analysis validates the procedure’s efficiency and precision. Parametric studies explore the influence of parameters, including the temperature-dependency material properties, geometric nonlinearity, and shell’s power-law index, providing valuable insights into FGM shell behavior under thermal shock.

Numerical analysis of thermal differential distribution in Bobbin tool friction stir welding of 7075-AZ31B dissimilar alloy

The theoretical model of bobbin tool friction stir welding (BT-FSW) structure and dynamic process is established. Because of the great differences in properties of different materials, the mathematical relationship between material stress and deformation will be expressed based on 3D transient heat conduction equation and J-C constitutive model. The BT-FSW process of 7075 and AZ31B was analyzed by finite element method, and the field distribution, thermal cycle curve and stress distribution characteristics of different welding areas of the welded material were simulated. To predict and control the non-equilibrium distribution of welding temperature and stress field, the influences of different Rotating speed, welding speed, offset and other variables on the BT-FSW process of 7075/AZ31B aluminum magnesium dissimilar alloy were studied comprehensively. Finally, the reliability of the model is proved by fitting the experimental and numerical analysis results. The research results show that the temperature of BT-FSW presents an “hourglass” symmetrical distribution, and the stress distribution on both sides of the plate is uniform when the aluminum alloy is placed on the advancing side, the stirring tool is offset to the aluminum alloy by 2mm, and the rotating speed and welding speed are 700r/min and 180mm/min, respectively.

Thermally Induced Vibrations of Functionally Graded Shallow Spherical Shells Under Cooling Shock

Based on the first-order shear deformation theory (FSDT), the large-amplitude vibration behavior of shallow spherical shells made of functionally graded materials (FGMs) due to the rapid decrease of temperature (cooling shock) is investigated. FGM is assumed to be a mixture of stainless steel and low-carbon steel whose properties are considered temperature dependent and are estimated based on the power-law model. According to FSDT and von Kármán assumptions, the governing equations of motion in conjunction with corresponding boundary conditions are obtained using Hamilton’s principle. The temperature distribution is also achieved by means of the 1D Fourier transient heat conduction equation, considering two time-dependent thermal loading scenarios. The considered thermal boundary conditions allow that a temperature difference is created with a time delay for the structure under cooling shock. Also, the results of recent experiments are employed to estimate the temperature-dependent properties of structures. In the solution approach, the generalized differential quadrature method and the Newmark-beta integration scheme are utilized. Selected numerical results are presented to study the effects of thermal load rapidity time, geometry, magnitude of thermal load, and material gradient index on the thermally induced vibrations of spherical shells. Stresses generated in the shell due to the thermally induced vibrations are also investigated.

Geometric imperfection sensitivity of nonlinear vibration responses of laminated beams under thermal shock

Initial geometric imperfections are bound to arise during a structure's service life or could emerge during the fabrication process. The exact influence of these geometric imperfections on the thermally-induced vibrations of the structure remains uncertain. This study represents the first endeavor to conduct a sensitivity analysis of geometric imperfections in the nonlinear vibration responses of laminated beams subjected to thermal shock. Considering uncoupled thermoelasticity theory, the temperature distribution is obtained using a one-dimensional Fourier-type transient heat conduction equation, and a nonlinear dynamic beam model accounting for initial geometric imperfections is developed based on the first-order shear deformation theory, Lagrange method, and Ritz method. An exact solution is implemented to capture the temporal temperature profile along the thickness of a beam subjected to a thermal boundary condition of the first kind (constant wall temperature). The sensitivity of various geometric imperfections to the nonlinear vibrations of laminated beams under thermal shocks is obtained using the Newmark scheme and Newton–Raphson iterative technique. A series of parametric studies is conducted to reveal the effects of different parameters on the nonlinear thermally induced vibration responses of the beams. Thermally induced vibration is shown to be sensitive to the initial geometric imperfections.

Development of Electro-Thermal Model for Air Cooling Study of Electric Vehicle Lithium-Ion Battery Module Operating in Indian Conditions and Drive Cycle

Lithium-ion cells have higher energy density at a specific power, making them a primary choice for battery packs in energy storage of modern technology, powering from smartphones to Electric vehicle (EV) and grid-scale energy storage systems. However, their performance, safety and lifespan are temperature sensitive and developing a fail-safe mechanism is considered as a pressing priority. In the present study, a 2D electro-thermal model is developed in MATLAB to characterize the thermal behaviour of cylindrical cells (individual and inline) during discharge. The thermal model solves the transient heat conduction equation with internal heat generation. The model is validated with the available experimental data for a single cell at a constant discharge rate and found to be in good agreement within 0.5°C. The analysis is extended to a battery module with 6 cell-inline configuration and validated. This validated model is used to study the impact of multiple cells (1-10) in inline configuration. For Indian drive cycle analysis, the current profile is estimated based on the drive cycle and its impacts on cell temperature rise in battery module is studied when subjected to variation in monthly average ambient temperature. Battery module maximum temperature rises to 60 °C and Δ T (Tmax - Tmin) across cells is 8.8 °C during summer conditions under natural convection due to multiple drive cycle loading. The battery module temperature can be kept under safe operating limits (Tmax≤ 45 °C, Δ T ≤ 4 °C) with forced convection of 2 m/s and 25 °C. The study aims to provide insights into the thermal behaviour of Lithium-ion cells that can be used to design safe and cost-effective air-cooled thermal management system for battery module.

Large-amplitude vibration analysis of FG rectangular plates under rapid cooling

In the present research, the large-amplitude vibration behavior of rectangular plates subjected to cooling shock is investigated based on a numerical solution approach. It is considered that the plates are made of functionally graded materials as a mixture of stainless steel and low-carbon steel. The thermoelastic properties are also considered temperature-dependent which are estimated based on available experimental data. According to the Mindlin plate theory, the nonlinear governing equations of motion together with corresponding boundary conditions are derived. The temperature profile is also obtained based on a one-dimensional Fourier-type transient heat conduction equation. Moreover, two time-dependent thermal loading scenarios for cooling shock on the plate’s top face are considered. To solve the problem numerically, the well-known generalized differential quadrature technique is used for discretization considering the Chebyshev–Gauss–Lobatto grid. In addition, the governing equations are traced in time by Newmark’s time integration scheme. After showing the validity of developed approach, a parametric study is presented to investigate the stress changes along the thickness and the effects of thermal load rapidity time, geometry, magnitude of thermal load, and material properties on the nonlinear vibrations of plates with various boundary conditions subjected to sudden decrease of temperature. It is concluded that the thermal load rapidity time has a significant role in the vibrational behavior and the stress distribution of the plate.

Finite integral transform with homogenized boundary conditions solution of transient heat conduction applied to thermal plug and abandonment of oil wells

After the end of productive life of a geological oil reservoir, a set of operations for well Plug and Abandonment (P&A) is performed to recover the soil layers to its natural state. As traditional P&A techniques are expensive activities, alternative technologies are gaining renewed interest for capital expenditure reduction and leak risks mitigation. Hence, the Thermal Plug and Abandonment (TP&A) procedure is here investigated by focusing on the fulfillment of two intermediate milestones, combining new technology with conventional P&A practices. The first milestone consists in applying a thermite exothermic reaction to generate enough heat for melting the entire thickness of the production tube. The second milestone aims to determine if the resulting molten section has sufficient dimensions to allow for the passage of the cement pumped downhole, thus ensuring the complete sealing of the oil well's cross-section. The thermal investigation was conducted by applying a homogenized form of the Finite Integral Transform (FIT) analytical framework to solve the transient heat conduction equation in 2-D polar coordinates. The well assembly was geometrically discretized as a multilayered circular domain. The thermite reaction was modeled as a theoretical volumetric heat source profile dependent on both time and space, placed in the innermost layer. While a pure FIT approach may only be used to solve Neumann boundary conditions, the combined scheme applied here copes with any type of boundary condition. Hence, the integration of FIT with homogenization satisfies the Dirichlet condition requirement of the TP&A procedure. The methodology was verified through an equivalent Finite Volume Method (FVM) solution obtained using a commercial code. The results evidenced that within the 5 min duration of thermite reaction, the entire circumferential section of the production tube exceeds the steel melting temperature by at least 227 °C, thus fulfilling both milestones set. The research outcomes are part of a series of investigation steps to thoroughly analyze the new TP&A technology with regard to its compliance with the regulatory norms established to P&A operations.

Nonlinear two-dimensional transient thermoelastic analysis of functionally graded composite plates subjected to localised cooling loads

In this study, for the first time, localized thermal cooling is considered for the transient thermoelastic response of an axisymmetric plate, addressing the underexplored area of localized low-temperature thermoelasticity. Functionally graded composite plates are analyzed which consist of a mixture of stainless steel (SUS304) and low-carbon steel (AISI1020). A two-dimensional transient heat conduction equation is employed to capture localized cooling effects, featuring various adaptive time-dependent thermal boundary conditions which mirror both loading and unloading scenarios, effectively simulating real-world phenomena. Novel parameters are introduced to facilitate a comprehensive understanding of various aspects of thermal load handling and their impact on thermoelastic responses. The heat conduction equations are solved using the Generalized Differential Quadrature Method (GDQM) and the Crank-Nicolson scheme. The nonlinear governing equations, incorporating geometrical nonlinearity through the von Kármán assumption within the First Shear Deformation Theory (FSDT) framework, are solved using the GDQM and the Picard's technique. The model is validated using Finite Element Method (FEM) through Abaqus. A comprehensive analysis is provided that considers influence of the ratio of thermally affected and unaffected plate surface, thermal load magnitude, rapidity of thermal loading and unloading, duration of the cooling load, geometrical nonlinearity, and temperature dependence of material. The study shows that the maximum von Mises stress within the structure remains consistent regardless of the duration of the cooling load, as long as the rate at which the cooling load is applied stays the same. Additionally, the findings reveal that even minor localized thermal loads can produce substantial stress, especially at the intersection of thermally affected and non-affected zones on the exposed surface in some situations.

Application of a soil temperature model for simulating the effect of dis-trict heating pipes on drinking water pipelines

According to Dutch legislation the drinking water temperature is required to be below 25°C at the tap to meet legionella prevention [1]. This requirement possibly limits the storage of heat in the subsoil and the transport of heat through pipelines. In order to asses the impact of heat transport pipelines on the drinking water temperature in adjacent drinking water pipelines, a soil temperature model was developed. This coupled micro-climate [2] and heat transport model relates atmospheric conditions to the temperature distribution in the sub-surface [3]. The subsurface part of the model is based on a finite element discretization of the transient heat conduction equation in a single phase material. The coupling of the subsurface part of the model with atmospherical part of the model requires that the sensible heat flux, the latent heat flux, net radiation and the subsurface heat flux are in equilibrium at the land surface. Simulation periods of a number of years are discritized in time steps of one hour. The model is validated by comparing the model results with a large number of field measurements in the Dutch city Rotterdam-Kralingen [3].
 The validated model is used to predict the drinking water temperature in transport, primary and secondary pipeline networks. The transport networks supply heat and water to the primary networks. Primary networks distribute their heat and water to secondary networks that transport it to buildings.
 with and without heating pipelines parallel to drinking water pipelines, with a limited flow of drinking water. The inner diameter of the heat supply and return pipelines, which are usually located close to each other, is 160.3 mm. The heat supply temperature is 90°C and the return temperature is 70°C. The inner diameter of the drinking water pipelines is 152 mm. The heat transport pipelines consist of an inner steel pipe with a wall thickness of 4 mm, a PUR isolation of 56 mm and a PE sleeve of 4 mm. The finite element discretization captures the geometry of the insulated pipeline and associated effective thermal properties. Drinking water pipelines have a PVC wall with a thickness of 4 mm. The drinking water pipeline cover depth is 1.00 m, the distance in between the drinking water pipeline and heat return pipeline is 0.75 m and the soil cover of the heat supply pipelines is 1.00 m. Tiles cover a dry sandy soil for which the specific heat capacity is 800 J/kgK, the thermal conductivity is 1.6 W/mK and the density of the soil is 1600 kg/m3. The figure shows a seasonal variation of the drinking water temperature of 13.8°C and an increase of the drinking water temperature due to the presence of the heat transport pipelines of 2.0°C, which is almost constant throughout the year. The maximum drinking water temperature is found in August and amounts to 23.8°C if the heat pipelines are absent and 25.8°C if they are present.
 Table 1 presents the results of a study with variation of the drinking water pipeline soil cover, the distance in between drinking water pipeline and heat return pipeline and the soil cover of the heat supply pipelines. Based on this table decisions can be made on the type of construction and the effect of the heat transport pipelines on adjacent drinking water pipelines.
 The table shows that the cover depth of the drinking water pipeline strongly influences the increase of the drinking water temperature due to atmospheric conditions. A decrease of the soil cover from 2.00 m to 0.75 m raises the maximal temperature by 4.2°C. The presence of the heat transport pipelines increases the temperature of the drinking water by 1.7°C for the case of a small cover depth and 2.1°C for a large cover depth. A decrease of the distance in between the drinking water pipeline and the heat return pipeline from 2.5 m to 0.25 m raises the maximal drinking water temperature by 2.6 °C. The calculation results show that the drinking water temperature is not sensitive to the proposed cover depth variation of the heat supply pipelines. The computational results for all networks considered, are collected in an expert tool that can be accessed via: https://www.tkiwatertechnologie.nl/projecten/opwarming-van-drinkwater/.

Nonlinear large-amplitude vibration analysis of annular sector plates made of FGMs subjected to cooling shock

A numerical approach is developed in the present article to study the geometrically nonlinear vibrations of annular sector plates made of functionally graded materials (FGMs) due to being exposed to cooling shock. The first-order shear deformation plate theory (FSDT) is used in the modeling of plate, and the governing equations of motion are derived based on Hamilton's principle considering von Kármán nonlinear kinematic relations. It is also considered that the properties of FGMs are dependent on temperature and distribution of materials. According to the uncoupled thermoelasticity theory, the temperature distribution is obtained via 1D Fourier-type transient heat conduction equation. To numerically solve the problem, the generalized differential quadrature (GDQ) method and Newmark-beta integration scheme are utilized. Considering two different types of thermal loading as cooling shock, the effects of various parameters including thermal load rapidity time, geometry, magnitude of thermal load and material properties on the large-amplitude vibrations of annular sector plates are investigated.

Mathematical and thermo-economic analysis of thermal insulation for thermal energy storage applications

Thermal energy storage (TES) is vital for achieving carbon neutrality in the energy sector. To achieve high storage efficiency, insulation with satisfactory performance is required. However, in the field of TES, limited attention has been paid to thermal insulation wherein the exergy loss under periodic operation conditions must be considered. In this study, we derived an analytical solution for the transient heat conduction equation and obtained a concise and explicit expression of the average exergy loss rate under a typical operation mode, where the variation of the internal temperature is a rectangular wave function. The effects of parameters such as the operation temperature, Biot number, Fourier number, insulation thickness, thermal conductivity, and volumetric heat capacity were discussed, and the insulation thickness was optimized by using the life-cycle cost analysis method. The transient effect caused by temperature fluctuations is expected to increase exergy loss. In addition to thermal conductivity, volumetric heat capacity has a considerable influence on insulation performance. In a real-world case, if the transient effect is ignored, the estimated exergy loss is 15.4%–62.5% smaller when the insulation thickness varies from 0.1 to 0.4 m, with the optimal thickness being 27.5% larger, resulting in an additional economic cost of 1.86%.

Analytical Solution of Transient Heat Conduction through a Hollow Cylindrical Thermal Insulation Material of a Temperature Dependant Thermal Conductivity

The one-dimensional, cylindrical coordinate, non-linear partial differential equation of transient heat conduction through a hollow cylindrical thermal insulation material of a thermal conductivity temperature dependent property proposed by an available empirical function is solved analytically using Kirchhoff’s transformation. It is assumed that this insulating material is initially at a uniform temperature. Then, it is suddenly subjected at its inner radius with a step change in temperature. Four thermal insulation materials were selected. An identical analytical solution was achieved when comparing the results of temperature distribution with available analytical solution for the same four case studies that assume a constant thermal conductivity. It is found that the characteristics of the thermal insulation material and the pressure value between its particles have a major effect on the rate of heat transfer and temperature profile.

Thermomechanical Modeling of Aircraft Tire–Runway Contact for Transient Maneuvers

ABSTRACT The aircraft tire is the link between the aircraft and the runway and transmits forces and moments within the contact area. During ground maneuvers, the tire is exposed to a wide range of operating conditions. The tires support the weight of the aircraft, they ensure safe rolling during taxiing, and they transfer the required forces to the runway during takeoff and landing. Temperature development during these maneuvers is of great importance because temperature can affect both material stiffness and friction behavior as well as wear characteristics. Heat is generated inside the tire due to energy dissipation caused by the cyclic loading of the tire during rolling. In addition, heat is generated due to friction in the tire–runway contact. Experimental measurements of the temperature distribution in the entire tire are not yet possible. The tire temperatures can only be determined at selected critical spots. Simulation models can help to obtain a better understanding of the overall tire temperature distribution during transient maneuvers. In this work, the transient thermomechanical processes of a rolling aircraft tire, e.g., directly following the touchdown process or during taxiing, are modeled based on a simple physical tire model. An extended brush model is used to simulate the contact forces. The tire temperature is determined via the transient heat conduction equation in radial and circumferential directions. The mechanical and thermal models are coupled via the coefficient of friction and the sliding velocities in contact. The free model parameters are parameterized using experimental data, and the overall model is validated by measurements on the whole tire. The validated thermomechanical tire model is used for simulations or the analysis of different driving maneuvers to get a better understanding of the temperature development in the aircraft tire.

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 simulation of thermal effects in laser diode end-pumped solid-state lasers utilizing finite element method

Based on the finite element numerical analysis method, the temperature field distribution in the crystal with time is simulated by using the PDE-tool in MATLAB. The critical factors affecting the crystal transient temperature distribution are simulated and analyzed. According to the simulation parameters and results, a simplified method for solving the transient heat conduction equation is proposed. The transient thermal focal length is solved to obtain its distribution expression and an approximate analytical solution. The results provide a theoretical basis for the thermal analysis process of suppressing or utilizing the transient thermal effect in high-energy solid-state lasers.