Two-dimensional Transient Heat Research Articles

Simulation of Heat Storage and Heat Regeneration in Phase Change Material

The present study explores numerically the energy storage and energy regeneration during Melting and Solidification processes in Phase Change Materials (PCM) used in Latent Heat Thermal Energy Storage (LHTES) systems. Transient two-dimensional (2-D) conduction heat transfer equations with phase change have been solved utilizing the Explicit Finite Difference Method (FDM) and Grid Generation technique. A Fortran computer program was built to solve the problem. The study included four different Paraffin's. The effects of container geometrical shape, which included cylindrical and square sections of the same volume and heat transfer area, the container volume or mass of PCM, variation of mass flow rate of heat transfer fluid (HTF), and temperatures difference between PCM and HTF were all investigated. Results showed that the PCMs in a cylindrical container melt and solidify quicker than the square container. The increase in mass flow rate and/or temperature difference decreases the time required for complete phase change. Paraffin's solidify quicker than they melt and store more energy than they release

A self-scaling sequential quasi-Newton method for estimating the heat transfer coefficient distribution in the air jet impingement

Accurate knowledge of the heat transfer coefficient (HTC) distribution is of great significance to the control and optimization of the heat transfer system in the air jet impingement. For this purpose, the estimation of the temporally-spatially varying HTC in the air jet impingement is studied based on the inverse heat conduction problem. The transient two-dimensional heat conduction model of the target disc under a single cooling air jet is established and utilized as the direct problem. A self-scaling sequential quasi-Newton method (SS-SQNM) is developed for accurately and efficiently solving the inverse problem. As a modification of the sequential quasi-Newton method (SQNM), the SS-SQNM adopts the self-scaling updating equation to accelerate the single-step convergence rate during iteration, and thus attains higher inversion efficiency than the SQNM. A series of numerical tests are carried out to investigate the performance of the SS-SQNM and the effects of the regularization parameters on inverse results. The results reveal that the SS-SQNM can give stable and accurate estimates of the HTC when regularization parameters are appropriately selected through the discrepancy principle. The mean absolute percentage error of the estimated HTC distribution is generally less than 10.2% in the presence of measurement noises. Additionally, the iteration number required by the SS-SQNM for inversion is only about 7.81%∼27.78% of that of the SQNM under the same conditions, which saves lots of computational time for the SS-SQNM. Therefore, the SS-SQNM developed in this study is a promising inverse algorithm, which can be employed to estimate the HTC distribution in the air jet impingement with reasonable accuracy and efficiency.

Incorporating two-dimensional conduction modeling techniques into an energy simulation program: The EnergyPlus radiant system example

Due to the inherent complexity of multi-dimensional solutions for transient conduction in a multilayered building element, most heat balance-based algorithms assume that conduction occurs only in one dimension—through the building element between the inside and outside environments. Usually, since the length and width of the elements are much greater than the thickness of the element, this is a valid assumption. However, there are some cases such as hydronic radiant slabs where this assumption may not lead to an accurate estimate of the energy consumption of the system. In a hydronic radiant slab, water tubes are embedded in a slab, and hot or chilled water is circulated to meet any heating or cooling loads. The energy consumption of such a system will depend highly on the temperature of the water being sent to the slab (i.e., produced by the boiler or chiller). The temperature of the water that must be sent to the slab will depend on the spacing of the tubes, the multilayer construction of the building element, and the boundary conditions to which the construction is exposed. Thus, to accurately predict the energy consumption of such a system, a transient two-dimensional heat conduction model must be integrated with a heat balance-based simulation. This paper demonstrates the potential loss of accuracy associated with the one-dimensional radiant system models by providing a comparison of one- and two-dimensional solutions using EnergyPlus.

An improved sequential function specification coupled with Broyden combined method for determination of transient temperature field of the steel billet

Accurate knowledge of the steel billet temperature field in the reheating furnace is of crucial importance to improvement of product quality and energy conservation. For this purpose, the determination of the transient temperature field of the steel billet during reheating is investigated based on the inverse heat conduction problem. The transient two-dimensional heat transfer model of the billet in the furnace is established and used as the direct problem. The sequential function specification coupled with the Broyden combined method (BC-SFSM) is developed to obtain the temperature field of steel billet. A real measurement experiment is conducted for validation of the inverse model, which indicates that the BC-SFSM can accurately determine the billet temperature field from the temperatures measured at three points inside the billet. A series of numerical experiments are implemented to analyze the performance of the BC-SFSM and study the effects of the measurement point locations and of the number of future time steps on inverse results. The results illustrate that the BC-SFSM can take only 49.0% of the computing time of the sequential function specification method to estimate the billet temperatures, and the relative root-mean-square error of estimated temperatures is generally less than 1.5% when the reasonable number of future time steps and suitable measurement point locations are adopted. Therefore, the BC-SFSM in this work is a promising algorithm to determine the billet temperature field in the reheating furnace with high accuracy and efficiency.

Explicit Sensitivity Coefficients for Estimation of Temperature-Dependent Thermophysical Properties in Inverse Transient Heat Conduction Problems

Explicit expressions are obtained for sensitivity coefficients to separately estimate temperature-dependent thermophysical properties, such as specific heat and thermal conductivity, in two-dimensional inverse transient heat conduction problems for bodies with irregular shape from temperature measurement readings of a single sensor inside the body. The proposed sensitivity analysis scheme allows for the computation of all sensitivity coefficients in only one direct problem solution at each iteration with no need to solve the sensitivity and adjoint problems. In this method, a boundary-fitted grid generation (elliptic) method is used to mesh the irregular shape of the heat conducting body. Explicit expressions are obtained to calculate the sensitivity coefficients efficiently and the conjugate gradient method as an iterative gradient-based optimization method is used to minimize the objective function and reach the solution. A test case with different initial guesses and sensor locations is presented to investigate the proposed inverse analysis.

ANALYTICAL SOLUTION OF A 2D TRANSIENT HEAT CONDUCTION PROBLEM USING GREEN´S FUNCTIONS

Through the present work the authors determined the analytical solution of a transient two-dimensional heat conduction problem using Green’s Functions (GF). This method is very useful for solving cases where heat conduction is transient and whose boundary conditions vary with time. Boundary conditions of the problem in question, with rectangular geometry, are of the prescribed temperature type - prescribed flow in the direction x and prescribed flow - prescribed flow in the direction y, implying in the corresponding GF given by GX21Y22. The initial temperature of the space domain is assumed to be different from the prescribed temperature occurring at one of the boundaries along x. The temperature field solution of the two-dimensional problem was determined. The intrinsic verification of this solution was made by comparing the solution of a 1D problem. This was to consider the incident heat fluxes at y = 0 and y = 2b tending to zero, thus making the problem one-dimensional, with corresponding GF given by GX21. When comparing the results obtained in both cases, for a time of t = 1 s, it was seen that the temperature field of both was very similar, which validates the solution obtained for the 2D problem.

Thermal model for surface grinding application

Due to the characteristics of the grinding process, thermal damage may occur in the workpiece surface, resulting in the rejection of a component and considerably increasing the production costs. This study aims to analyze the heat fluxes, energy partition, and temperatures during surface grinding process with both conventional and MQL lubrication. Through the proposed analysis, the heat fluxes and maximum temperature can be predicted, enabling the avoidance of thermal damages and increasing the efficiency of the process. A comparison between the calculated and experimental value has shown that the difference is acceptable for various situations, in the order of 4.72% for the conventional method and 7.38% for the MQL method. A thermal model was developed. The transient two-dimensional heat diffusion equation was discretized by finite volume method in space and explicit discretized in time. The heat fluxes were estimated using inverse problem technique of heat transfer aiming the obtainment of the temperature of certain workpiece points. A comparison of the methods of lubrication showed that the conventional method was way more efficient than MQL, presenting considerably lower total heat flux and maximum reached temperature and any kind of thermal damage wasn’t observed. On the other hand, thermal damage occurred in the workpieces. Also, clogging phenomenon in the grinding wheel surface after the process in MQL condition was observed.

Transient temperature fields and thermal stress fields in glazing of different thicknesses exposed to heat radiation

Thermal performance analysis and cracking prediction of glazing elements are essential for fire safety evaluation of a building. This work investigated the transient temperature fields and the corresponding stress fields in glazing of different thicknesses with one side exposed to heat radiation. Based on the results of investigation, the criterion of critical temperature difference in cracking prediction was formulated. Glass samples of different thicknesses were tested under the same radiation exposure. Besides, a transient two-dimensional heat transfer model was presented to obtain the transient temperature fields in the glass samples. In addition, the temperature fields were loaded to ANSYS to obtain the thermal stress fields. It is found that in the initial heating period, the maximum stress was mainly caused by the temperature gradient component across thickness and the stress value was larger for thicker glazing; as time evolved, the maximum stress was gradually dominated by the temperature gradient component along the planar direction. Thus, for thick glazing under strong heat, thermal bending stress caused by temperature gradient across thickness should be calculated first to evaluate the cracking time, and the criterion of critical temperature difference is appropriate only if the thermal bending stress cannot reach the breaking value. For other conditions, cracks will result from the stress caused by the temperature gradient along the planar direction, and the criterion of critical temperature difference is applicable for cracking prediction of glazing in any thickness.

Numerical and experimental investigation of heat transfer in a solar receiver with a variable aperture

Variable aperture can assist in maintaining semi-constant temperatures within a receiver’s cavity under transient solar loading. An in-house code has been developed to model a receiver and effectively control its components to achieve semi-constant temperatures under transients. The code consists of a full optical analysis performed via the Monte Carlo ray tracing method in addition to a transient two-dimensional heat transfer analysis. The system studied consists of a cavity type solar receiver with 60 mm radius fixed aperture on the cavity body, a variable aperture mechanism mounted on the receiver’s flange, and a 7 kW Xenon arc solar simulator. A composite shape consisting of a hemisphere attached to a cylinder is proposed to model the Xenon arc. The in-house code has been experimentally validated through experimental tests for different input currents to the solar simulator, volumetric flow rates, and aperture’s radii. The optical analysis was validated based on heat flux measurements, where it had percentage errors of 0.8, 0.5, 1.1, and 3.2% for the peak power, total power, half width, and half power. For the heat transfer model, percentage errors of 3.2, 2.9, and 5.3% at the inlet, center, and outlet sections of the receiver were determined for different flow rates using maximum input current and opening radius. The aperture mechanism was capable of maintaining an exhaust temperature of 250 °C based on actual Direct Normal Irradiance data. Results showed that the variable aperture is a promising apparatus even in applications where the maximum temperatures are desired based on an observed optimum radius of 57.5 mm.

Analytical study of the heat transfer coefficient of the impinging air jet during cold spraying

A semi-empirical analytical model was developed to determine the convective heat transfer coefficient of an impinging air jet generated by a cold spraying nozzle. A low-pressure cold spraying unit was utilized to produce hot air jets that impinged upon a flat substrate surface. An infrared camera was used to measure the surface temperature of the substrate at different time intervals. A method involving Green's functions was employed to solve a transient two-dimensional heat conduction problem to obtain an expression for the temperature distribution within the substrate. By coupling the analytical results of temperature distribution and experimental surface temperature data, the radial variation of the non-dimensional heat transfer coefficient of the impinging air jets upon the substrate was estimated. The results showed that the maximum values of the heat transfer coefficient were present close to the stagnation point of the air jets. It was found that the heat transfer coefficient was independent of the time that the cold spray nozzle remained stationary over the substrate surface. It was found further that by increasing the stand-off distance of the nozzle, the radial variations of the heat transfer coefficient became negligible, compared to those for small stand-off distances. The close agreement between the experimental results and the predictions of the model suggested that the estimated heat transfer coefficient of the cold spray gas jet can be used to estimate the surface temperature of the substrate at any time.

Transient Two-Dimensional Heat Conduction Problem With Partial Heating Near Corners

This paper provides a solution for two-dimensional (2D) heating over a rectangular region on a homogeneous plate. It has application to verification of numerical conduction codes as well as direct application for heating and cooling of electronic equipment. Additionally, it can be applied as a direct solution for the inverse heat conduction problem, most notably used in thermal protection systems for re-entry vehicles. The solutions used in this work are generated using Green's functions (GFs). Two approaches are used, which provide solutions for either semi-infinite plates or finite plates with isothermal conditions, which are located a long distance from the heating. The methods are both efficient numerically and have extreme accuracy, which can be used to provide additional solution verification. The solutions have components that are shown to have physical significance. The procedure involves the derivation of previously unknown simpler forms for the summations, in some cases by virtue of the use of algebraic components. Also, a mathematical identity given in this paper can be used for a variety of related problems.

Figure of merit for the thermal performance of cementitious composites containing phase change materials

This paper presents a novel method to quantitatively characterize the thermal performance of composite materials containing phase change materials (PCM) based on a figure of merit we termed the energy indicator. The method features (i) commonly used specimen geometry, (ii) straightforward experimental implementation, and (iii) sensitivity to relevant design parameters including PCM volume fraction, enthalpy of phase change, composite effective thermal conductivity, and specimen dimensions. The experimental method and the concept of energy indicator were demonstrated on PCM-mortar composites using various volume fractions of two commercial microencapsulated PCMs. This was supported by transient two-dimensional heat transfer simulations. The energy indicator was shown to increase linearly with increasing microencapsulated PCM volume fraction and latent heat of fusion and quadratically with the specimen radius. This figure of merit can be used to rapidly screen and select microencapsulated PCM composite materials for energy efficient buildings or crack-resistant concretes.

Optimization of Friction Stir Welding Tool Advance Speed via Monte-Carlo Simulation of the Friction Stir Welding Process.

Recognition of the friction stir welding process is growing in the aeronautical and aero-space industries. To make the process more available to the structural fabrication industry (buildings and bridges), being able to model the process to determine the highest speed of advance possible that will not cause unwanted welding defects is desirable. A numerical solution to the transient two-dimensional heat diffusion equation for the friction stir welding process is presented. A non-linear heat generation term based on an arbitrary piecewise linear model of friction as a function of temperature is used. The solution is used to solve for the temperature distribution in the Al 6061-T6 work pieces. The finite difference solution of the non-linear problem is used to perform a Monte-Carlo simulation (MCS). A polynomial response surface (maximum welding temperature as a function of advancing and rotational speed) is constructed from the MCS results. The response surface is used to determine the optimum tool speed of advance and rotational speed. The exterior penalty method is used to find the highest speed of advance and the associated rotational speed of the tool for the FSW process considered. We show that good agreement with experimental optimization work is possible with this simplified model. Using our approach an optimal weld pitch of 0.52 mm/rev is obtained for 3.18 mm thick AA6061-T6 plate. Our method provides an estimate of the optimal welding parameters in less than 30 min of calculation time.

Numerical simulation of joining during the two-step resistance brazing

To improve the quality of a joint composed of metallic microstructural parts, a novel resistance brazing method called the ‘two-step resistance brazing method’ has been developed. In this method, filler metals plated on the surface of base metals are butted and alloyed due to the Joule's heat by performing two-step energizing. In this study, the numerical simulation of the joining process for the two-step resistance brazing method has been carried out. A mathematical formulation and corresponding calculation scheme, which is derived from using the control-volume method, are developed for a model based on contact resistance between the filler metals plated on the surface of the base metals, and transient two-dimensional heat conduction with the Joule's heat generation. The calculated result, based on a base metal of type 304 stainless steel and plated filler metals of gold and copper, is discussed and compared with experimental data.

Sustainable heat extraction from abandoned mine tunnels: A numerical model

Abandoned mines are often associated with enduring liabilities, which involve significant costs for decades after the decommissioning of the mine. Using a decommissioned mine as a geothermal resource can offset the environmental costs by supplying green heat to the communities living in and around the mine area. In this paper, a numerical assessment of geothermal heat extraction from underground mine workings using an open loop geothermal system is carried out. In this study, our focus is on fully flooded mines where the heat flow from the rock mass to the mine cavities is dominantly controlled by conduction in the rock mass. The sustainable heat flux into the mine workings is assessed using a transient two-dimensional axisymmetric heat transfer model. Finite volume method is applied to solve the model and simulate the transient temperature fields in the rock mass and within the water (flowing through cavities). The model is capable of controlling the rate of heat extraction through continuous adjustment of the rate of water flow through the mine. Sustainable rate of heat extraction is calculated for seasonally varied heat loads and for different project life cycles. It is shown that, with proper resource management, each kilometre of a typical deep underground mine tunnel, can produce about 150 kW of usable heat in a sustainable manner. The model is validated by comparing its results with other published models and realistic data available from Springhill mine, Nova Scotia, Canada. It is found that the sustainable heat extraction is controlled dominantly by virgin rock temperature, thermal conductivity of the rock mass, and seasonal heat load variations.

Two-Dimensional Inverse Transient Heat Conduction Analysis of Laminated Functionally Graded Circular Plates

A two-dimensional inverse transient heat conduction algorithm is developed to estimate the heat flux and convection heat transfer coefficient on the bottom (top) surface of the multilayered functionally graded (FG) circular plates using the measured temperature on their top (bottom) surface. The non-Fourier heat transfer, in conjunction with the layerwise-incremental differential quadrature method (LW-IDQM), is employed to accurately model the problem. The conjugate gradient method (CGM), coupled with the sensitivity and adjoint problems, are employed for the optimization procedure. The reliability and robustness of the presented approach is demonstrated by simulating the exact and noisy data through different examples.

An Investigation Into the Effects of Highly Transient Flight Maneuvers With Heat and Mass Transfer on the T-38 Air Force Trainer Inlet

The T-38 talon currently serves as the primary United States Air Force trainer for fighter aircraft. This supersonic trainer was developed in the 1960s but continues to be used today as the result of various modernization programs throughout its service life. The latest propulsion modernization program focused on improved takeoff performance of the T-38’s inlets, improved reliability of the twin J85 afterburning turbojet engines, and reduced drag with an improved exhaust nozzle design. The T-38’s inlet includes bleed holes upstream of the engine face to provide cooling airflow from the inlet to the engine bay. However, at various flight conditions, the bay air is pressurized relative to the inlet, resulting in reverse flow of hot engine bay air into the inlet. This reverse flow causes total temperature distortion that may reduce the engine stability margin. Partial inlet instrumentation of the left engine was used to estimate the total temperature distortion associated with reverse flow, however, flight testing of highly transient maneuvers revealed levels of total temperature distortion greater than that predicted for reverse flow alone. This discovery led to the hypothesis that thermal energy storage of the aluminum inlet during transient flight maneuvers resulted in increased temperature distortion at the engine face. Flight data analysis demonstrated the need for a near-real-time thermal inlet distortion analysis capability. A two-dimensional (2D) transient axisymmetric heat and mass transfer model was developed through the use of a lumped-parameter boundary-layer model to simulate the inlet flow and determine the time-dependent inlet duct heat transfer. This model was validated with transient 2D computational fluid dynamics and two flight maneuvers. The analysis of flight maneuvers revealed that in the absence of engine bay air re-ingestion, the time lag associated with the heating and cooling of the inlet walls generates radial temperature distortion, which has the effect of reducing engine stability margin up to 5.44% for the maneuvers analyzed.

A simplified 2-D model for the linear GTA welding of stainless steel plates

A simplified heat and fluid flow model has been used to solve the Gas Tungsten Arc (GTA) linear welding problem for austenitic stainless steel. In this study, an equivalent 2-D (two-dimensional) transient heat and fluid flow model has been used to study the effect of controlled boundary conditions with free bottom surface and with bottom insulation. The governing equations have been solved by a finite volume method. The peak temperature, maximum velocity in the weld pool, weld bead width and depth get reduced when the bottom surface of the stainless steel plate is insulated.

2段抵抗ろう付法における接合過程の数値シミュレーション

To improve the quality of a joint composed of metallic micro-structural parts, a novel resistance brazing method called the "two-step resistance brazing method" has been developed. In this method, filler metals plated on the surface of base metals are butted and alloyed due to the Joule's heat by performing two-step energizing. In this study, the numerical simulation of the joining process for the two-step resistance brazing method has been performed.A mathematical formulation and corresponding calculation scheme, which is derived from using the control-volume method, are developed for a model based on contact resistance between the filler metals plated on the surface of the base metals, and transient two-dimensional heat conduction with the Joule's heat generation.The calculated result, based on a base metal of type 304 stainless steel and plated filler metals of gold and copper, is discussed and compared with experimental data.

Different finite element approaches for inverse heat conduction problems

In this study, three different versions of the finite element method (FEM) based on Trefftz functions are presented for solving transient two-dimensional inverse heat conduction problems. The following cases are considered: (a) FEM with the condition of continuity of temperature in the common nodes of elements, (b) no temperature continuity at any point between elements and (c) nodeless FEM. Instead, in each finite element the temperature is approximated by a linear combination of Trefftz functions. These investigations are carried out using temperature data obtained from numerical simulations, exact and disturbed. For the inaccurate data some efficient method of smoothing is presented.