Transient Three-dimensional Heat Conduction Equation Research Articles

Experimental investigation of hydrodynamics and heat transport during horizontal coalescence of two drops impinging a hot wall

This paper presents an experimental study on hydrodynamics and heat transport during the horizontal coalescence of two drops impinging a hot wall. The study addresses the influence of distance between impact locations, the time interval between drop impact, and wall superheat on the transport processes. The experiments were conducted under a pure vapor atmosphere with the refrigerant FC-72 at a saturation temperature of 54°C, corresponding to a system pressure of 0.94 bar. The drops were generated with a constant diameter and a constant impact velocity. The temperature field at the surface of the heater was measured by an infrared camera with a high spatial and temporal resolution. The local heat flux distribution was derived from the temperature field by solving the transient three-dimensional heat conduction equation within the substrate. The total heat flow was evaluated by integrating the local heat fluxes over the footprint of the drop. The impact parameters (drop size and impact velocity), the time interval between drop impact, and the distance between impact locations were evaluated through post-processing of the black-and-white images captured employing a high-speed camera. The results for simultaneous drop impingement reveal that the heat transported from the wall to the fluid is not affected by the presence of neighboring drop as long as the drops do not contact each other. In the case of horizontal coalescence of drops, the heat flow is reduced both during the spreading and receding phases and during the sessile drop evaporation phase. This can be explained by the reduction of solid–liquid interface and of three-phase contact line length after the coalescence. An increase in wall superheat leads to reducing the footprint of the drop and increasing of heat flow. Asymmetric behavior of the drop hydrodynamics and heat flux is observed during the non-simultaneous drop impingement.

Experimental investigation of hydrodynamics and heat transport during vertical coalescence of multiple successive drops impacting a hot wall under saturated vapor atmosphere

In this study, hydrodynamics and heat transport during the vertical coalescence of multiple successive drops impacting a hot wall are analyzed experimentally. This study addresses the influence of wall superheat and the frequency of drop generation on the hydrodynamics and heat transport. The experiments are conducted under a pure vapor atmosphere with the refrigerant FC-72 at a saturation temperature of 54.5°C, corresponding to a system pressure of 0.94 bar. The drops are generated with a constant diameter of 1.14 mm and a constant impact velocity of 0.54 m s−1. An infrared camera with a high spatial and temporal resolution is used to capture the temperature field on the surface of the heater. The local heat flux distribution is derived from the temperature field by solving the transient three-dimensional heat conduction equation within the substrate. The total heat flow is evaluated by integrating the local heat flux over the footprint of the drop. The impact parameters (drop size and impact velocity) are evaluated through post-processing of the black/white images captured using a high-speed camera. The maximum spreading radius and maximum heat flow observed after the impact of each successive drop are higher than those observed after the impact of the previous drop. For instance, in the case of a wall superheat of 12.4 K and an impact frequency of 10 Hz, the maximum spreading radius and maximum heat flow observed after the impact of the fourth drop increased by approximately 35% compared with those observed after the impact of the first drop on a dry wall. The distribution of wall heat flux during the spreading phase of the second impacting drop is characterized by the appearance of two thin ring-shaped zones of high heat flux. The time evolutions of the drop shape and heat flow depend on the wall superheat and are independent of the frequency of drop generation within the studied range of parameters.

Unsteady Conjugate Natural Convection in a Three-Dimensional Porous Enclosure

Transient-free convection in a porous enclosure having heat-conducting solid walls of finite thickness under conditions of convective heat exchange with an environment was studied numerically. A heat source of constant temperature was located at the bottom of the cavity. The governing equations in porous volume formulated in dimensionless variables such as the temperature and vector potential functions within the Darcy–Boussinesq approach and the transient three-dimensional heat conduction equation based on the Fourier hypothesis for solid walls with corresponding initial and boundary conditions were solved using an iterative implicit finite-difference method. The main objective was to investigate the influence of the Rayleigh number 103 ≤ Ra ≤ 106, the Darcy number 10−5 ≤ Da ≤ 10−3, the thermal conductivity ratio 1 ≤ k1,2 ≤ 20, the solid wall thickness ratio 0.1 ≤ l/L ≤ 0.3, and the dimensionless time 0 ≤ τ ≤ 200 on the fluid flow and heat transfer. Comprehensive analysis of the effects of these key parameters on the average Nusselt number at the heat source surface was conducted.

Prediction of HAZ Width of Submerged Arc Welded Plates

This paper is focused on deriving an analytical solution to predict the transient temperature distribution on the plate during the process of Submerged Arc Welding (SAW). An analytical solution is derived from the transient three-dimensional heat conduction equation. The energy input that is applied on the plate is taken as the volume of heat lost from the electric arc. The electric arc is assumed to be a moving double-ellipsoidal heat source with a close proximity to a Gaussian distribution. It is observed that the predicted values are in good agreement with the experimental results. HAZ width calculation is also done with the help of the analytical solution of the transient-three dimensional heat conduction equation. Very good agreement between predicted and experimental values has been archived.

Effect of Heat input on Submerged Arc Welded Plates

This paper is focused on deriving an analytical solution to predict the transient temperature distribution on the plate during the process of Submerged Arc Welding (SAW). An analytical solution is derived from the transient three-dimensional heat conduction equation. The energy input that is applied on the plate is taken as the volume of heat lost from the electric arc and the kinetic energy of filler droplets specifically driven by gravity, electromagnetic force, arc drag force, carring mass, momentum and thermal energy. These driving forces periodically impinge onto the base metal, leading to a liquid weld puddle. The electric arc is assumed to be a moving double central conicoidal heat source with a close proximity to a Gaussian distribution. It is observed that the predicted values are in good agreement with the experimental results. HAZ width calculation is also done with the help of the analytical solution of the transient-three dimensional heat conduction equation. Analyses of micro-structural changes are critically investigated to comprehend the HAZ softening and phenomena.

(156) Serologic profile of HBS antigen carriers

This paper is focused on deriving an analytical solution to predict the transient temperature distribution on the plate during the process of Submerged Arc Welding (SAW). An analytical solution is derived from the transient three-dimensional heat conduction equation. The energy input that is applied on the plate is taken as the volume of heat lost from the electric arc and the kinetic energy of filler droplets specifically driven by gravity, electromagnetic force, arc drag force, carring mass, momentum and thermal energy. These driving forces periodically impinge onto the base metal, leading to a liquid weld puddle. The electric arc is assumed to be a moving double central conicoidal heat source with a close proximity to a Gaussian distribution. It is observed that the predicted values are in good agreement with the experimental results. HAZ width calculation is also done with the help of the analytical solution of the transient-three dimensional heat conduction equation. Analyses of micro-structural changes are critically investigated to comprehend the HAZ softening and phenomena.

Thermal fatigue on pistons induced by shaped high power laser. Part II: Design of spatial intensity distribution via numerical simulation

In the laser induced thermal fatigue simulation test on pistons, the high power laser was transformed from the incident Gaussian beam into a concentric multi-circular pattern with specific intensity ratio. The spatial intensity distribution of the shaped beam, which determines the temperature field in the piston, must be designed before a diffractive optical element (DOE) can be manufactured. In this paper, a reverse method based on finite element model (FEM) was proposed to design the intensity distribution in order to simulate the thermal loadings on pistons. Temperature fields were obtained by solving a transient three-dimensional heat conduction equation with convective boundary conditions at the surfaces of the piston workpiece. The numerical model then was validated by approaching the computational results to the experimental data. During the process, some important parameters including laser absorptivity, convective heat transfer coefficient, thermal conductivity and Biot number were also validated. Then, optimization procedure was processed to find favorable spatial intensity distribution for the shaped beam, with the aid of the validated FEM. The analysis shows that the reverse method incorporated with numerical simulation can reduce design cycle and design expense efficiently. This method can serve as a kind of virtual experimental vehicle as well, which makes the thermal fatigue simulation test more controllable and predictable.

Analytical solution for a transient, three-dimensional temperature distribution due to a moving laser beam

This paper presents an analytical solution of the transient temperature distribution in a finite solid when heated by a moving heat source. The analytical solution is obtained by solving the transient three-dimensional heat conduction equation in a finite domain by the method of separation of variables (SOV). Meanwhile previous studies focus on analytical solutions for semi-infinite domains, here an analytical solution is provided for a finite domain. The non-homogeneous equation is solved by using the Laplace transform for a unit impulse and then convoluted with the actual heat source. Two different distributions are used: a Gaussian distribution and a spatially uniform plane heat source.

Analytical solution for transient temperature distribution in gas tungsten arc welding with consideration of filler wire

This paper is focused on deriving an analytical solution to predict the transient temperature distribution of the plate during the process of gas tungsten arc (GTA) welding in which a filler wire is fed. An analytical solution is derived from the transient three-dimensional heat conduction equation by applying a convection boundary condition to the top and bottom surface of an infinite plate of finite thickness. The heat input that is applied on the plate is exactly the same as the amount of heat lost from the electric arc while fusing the filler wire. The electric arc is assumed to be a moving heat source with a Gaussian distribution. The point heat sink is located in the spot where the filler wire is fed to the weld pool. However, the feed position of the filler wire is ever-evolving with time because the size of the weld pool changes in accordance with the welding conditions. Nevertheless, the algorithm is capable of determining the feed position. Also, the influence of the filler wire on the temperature distribution of the plate is established by removing the filler wire from the analytical solution and comparing these two different situations. In order to verify the validity of this solution, a GTA welding experiment is conducted for various welding conditions. After this stage, the resulting isotherms of the cross-sections of the plate and the analytical results are compared. The analytical solution, which is in good agreement with experimental results, is the very first to take the filler wire feed into consideration. It can be used as a model to control the GTA welding process with account taken of the filler wire, and it can also be used as an optimization tool for welding process parameters.

Three-dimensional temperature distribution in laser surface hardening processes

This paper introduces a new analytical solution to predict transient temperature distributions in a finite thickness plate during laser surface hardening. This analytical solution was obtained by solving a transient three-dimensional heat conduction equation with convection boundary conditions at the surfaces of the workpiece. To calculate the temperature field numerically in laser surface hardening processes, laser beam absorptivity, one of the most important parameters, should first be determined. It was extremely difficult to find an accurate value for laser beam absorptivity owing to the use of coating material on the workpiece surfaces, essentially to improve the efficiency of laser beam power. Therefore, in this paper, absorptivities were measured experimentally under various hardening conditions, including variations in coating thickness, laser beam power and beam travel velocity. Thereafter, to prove the validity of the model, a series of laser surface hardening experiments was performed on medium-carbon steel under various hardening conditions. The hardened layer of the etched cross-sectional view was examined and compared with the isothermal lines predicted by the proposed analytical model, in which the values of the experimentally measured absorptivity were employed. The results showed that the solution predicted well the transient temperature distribution of finite plates with satisfactory accuracy. Also, owing to the simplicity of the solution method, the analytical model developed may be easily implemented for simulation work for analysis and prediction of laser surface hardening processes under various hardening conditions.

An analytical solution for transient temperature distribution in fillet arc welding including the effect of molten metal

This paper presents an analytical solution to predict the transient temperature distribution in fillet arc welding, including the effect of the molten metal generated from the electrode. The analytical solution is obtained by solving a transient three-dimensional heat conduction equation with convection boundary conditions on the surfaces of an infinite plate with finite thickness, and mapping an infinite plate on to the fillet weld geometry, including the molten metal with energy equation. The electric arc heat input on the fillet weld and on the infinite plate is assumed to have a travelling bivariate Gaussian distribution. To check the validity of the solution, flux cored arc (FCA) welding experiments were performed under various welding conditions. The actual isotherms of the weldment cross-sections at various distances from the arc start point are compared with those of the simulation result. As the result shows a good accuracy, this analytical solution can be used to predict the transient temperature distribution in the fillet weld of finite thickness under a moving bivariate Gaussian distributed heat source. The simplicity and short calculation time of the analytical solution provides the rationale for using the analytical solution to model the welding control systems or to obtain an optimization tool for welding process parameters.

A study of a multidimensional transient heat conduction process with a moving heat source Part 2: Numerical and experimental

A theoretical and experimental study of heat flow distribution during the autogenous GTA welding of metal plates is made. The theoretical part of this study involves the development of a numerical model based on a finite-difference implicit-solution method for the transient three-dimensional heat conduction equation with a moving heat source/sink, which simulates the welding process. The experimental part of this study involves the measurement of temperature distribution in the welding workpieces. These experimental results are compared with the numerical ones and the agreement is fairly good. The numerical solution developed is easier and more accurate for studying the effect of many welding parameters on heat flow distribution, the welding speed, the input-heat energy and the type and geometry of welding workpieces.

Transient Temperature Distribution in Arc Welding of Finite Thickness Plates

This paper introduces a new analytical solution to predict the transient temperature distributions in a finite thickness plate during arc welding. This analytical solution is obtained by solving a transient three-dimensional heat conduction equation with convection boundary conditions at the surfaces of the weldment. The heat source due to electric arc is assumed to take a travelling Gaussian distribution. To prove the validity of the model, a series of GTA bead-on-plate welding were performed on a medium carbon steel under various welding conditions. The isothermal lines of the cross-sectional view at various distances from the arc start were examined and compared with those predicted by the proposed analytical model. The results show that the finite thickness solution well predicts the transient temperature distribution of the finite thickness or thin plates with satisfactory accuracy. Due to the simplicity of the solution method, the developed analytical model may be implemented to the feedback control of the arc welding process or to the optimization of the process parameters.