Steady-state Heat Research Articles

Surface Damage and Microstructure Evolution of Yttria Particle-Reinforced Tungsten Plate during Transient Laser Thermal Shock

Tungsten and its alloys are considered to be the most nominated plasma-facing materials in fusion reactors, which will be exposed to enormously rigorous conditions such as thermal load, plasma exposure, and neutron radiation. At present, the research on the behavior of oxide particle-reinforced tungsten-based materials under long-term steady-state heat load and transient thermal shock is insufficient. The purpose of this study is to investigate the performance of yttria particle-reinforced tungsten plates prepared by the wet chemical method under heat loads by means of indirect coupling experiments. An Nd:YAG laser device is used to perform thermal shock events. The surface damage and microstructure evolution of rolled and fully recrystallized samples exposed to laser thermal shock are observed and analyzed. The cracking threshold of the rolled and fully recrystallized samples is about 0.40~0.48 GW/m2; the degree of surface damage of them aggravates with the increased laser power density. What is more, cracks or even melting damage could be observed on the surface and be accelerated by the process of recrystallization, resulting in the degradation of the ability to withstand the thermal shock of the material.

Steady State Heat Transport by Microbubble Dispersions Mediating Convection With Phase Change Dynamics

A new theory for additional heat transfer convected by a dispersed phase of microbubbles was posited recently. An additional convection term in the heat transport equation reflects the latent heat of vapor of the liquid carried by the microbubbles from hot zones that vaporize more liquid to cold zones where condensation releases the latent heat. This theory was shown to be consistent with analysis of observations of freezing times measured by in the original Mpemba effect study, by inferring heat transfer coefficients fitted by Newton’s law of cooling. In this paper, the scaling analysis, leading to the proposition that the additional heat flux is proportional to the phase fraction of microbubbles, is tested by steady state solutions of the canonical hot wall / cold wall buoyant convection problem. For phase fractions 0.02 and 0.1, the maximum ratio of additional Nusselt number emergent is five, occurring in the microfluidic regime. Increasing the characteristic length of the domain maintains the monotonicity of the increase in additional Nusselt number ratio over the case of no microbubbles present. The additional heat transfer due to the microbubble dispersion, ranging from 5-50%, is found to be nearly proportional to the microbubble phase fraction for the range of 0.02 to 0.2. However, larger characteristic lengths introduce insufficient heat flux from the hot wall to maintain a “driven cavity” flow structure, so that the steady state structure that emerges is a stable stratification with thin boundary layers near the hot and cold walls, with weak shear flow convection. The stable stratification resultant at higher characteristic lengths suppresses the additional heat flux due to microbubble mediation, but only moderately deviating from proportionality.

Rotor Transverse Slots’ Influence on Negative Sequence Loss and Heat Distribution Prediction of Large Generators Based on Improved Radial Basis Function Process Neural Network

Owing to the advantages of scientific computation and the data feature support provided by artificial intelligence technology, the theoretical exploration and application research of new computing methods for large generators, one of the most expensive energy equipment in a power system, has become a research hotspot toward solving the bearing limit and operation capacity under abnormal working conditions. Because of many factors affecting the distribution of negative sequence loss and temperature rise that have an extremely complex nonlinear relationship, the traditional calculation and prediction methods of negative sequence conditions cannot suitably reflect the time accumulation effect. Therefore, a prediction method of rotor steady-state negative sequence heating based on radial basis function process neural network is proposed in this paper, and a negative sequence working condition prediction model is established. Accordingly, this study focuses on the study of the relationship among negative sequence heating characteristics, negative sequence component proportion, and transverse slot; additionally, their influence degree, variation relationship, and main principles are further explored that provide a theoretical basis for the design and operation of large generators. As observed from the test results, the steady-state negative sequence condition prediction method based on the improved genetic algorithm radial basis function process neural network features high accuracy; it is a feasible prediction method, specifically for negative sequence conditions of large generators.

Temperature response and thermal performance analysis of a super-long flexible thermosyphon for shallow geothermal utilization: Field test and numerical simulation

• Super-long flexible thermosyphon (SFTS) fabricated by metal corrugated pipe, 32m in length. • Field tests of SFTS for shallow geothermal utilization. • Start-up and steady-state thermal performance of SFTS. • CFD simulation of long-term underground temperature variation near SFTS. • A method to avoid the heat transfer problem due to super-high liquid column of working fluid in super-long thermosyphon. A super-long flexible thermosyphon (SFTS) fabricated by the corrugated pipe, with a total length up to 32m, was field-tested in a drilling, in order to evaluate its potential for use in shallow geothermal utilization. The seasonable dependence of local underground temperature and air temperature, the corresponding temperature response of SFTS, and the its heat transfer characterists were investigated. The underground temperature recovery characteristics under the operation of uni-directional heat extraction were also conducted by a combination analysis of field tests and CFD simulation. The temperature response characteristics of SFTS in start-up of field tests show that the SFTS could operate effectively as the vaporization of working fluid occurs all over the whole evaporator. This indicates that employing a corrugated pipe prevents the formation of a super-high liquid pool at evaporator, which would inhibit the boiling of working fluid at the bottom. The maximum heat transfer power of SFTS obtained in field tests of this work is approximately 305W under the conditions with the inlet temperature of cooling water controlled at 5.5°C and the flow rate at 1500mL/min. The analysis of steady-state heat transfer results reveal that the underground temperature plays the most important role in the overall thermal performance of SFTS, followed by the flow rate and the temperature of cooling water. The long-term uni-directional heat extraction of SFTS from underground for heating is feasible and however the heat extraction rate is recommended to be maintained below 30W/m at the drilling site aiming to make sure a good sustainability. This work reports field tests of a super-long flexible thermosyphon in geothermal utilization for the first time.

Steady-State Heat Transport and Work With a Single Artificial Atom Coupled to a Waveguide: Emission Without External Driving

We observe the continuous emission of photons into a waveguide from a superconducting qubit without the application of an external drive. To explain this observation, we build a two-bath model where the qubit couples simultaneously to a cold bath (the waveguide) and a hot bath (a secondary environment). Our results show that the thermal-photon occupation of the hot bath is up to 0.14 photons, 35 times larger than the cold waveguide, leading to nonequilibrium heat transport with a power of up to 132 zW, as estimated from the qubit emission spectrum. By adding more isolation between the sample output and the first cold amplifier in the output line, the heat transport is strongly suppressed. Our interpretation is that the hot bath may arise from active two-level systems being excited by noise from the output line. We also apply a coherent drive, and use the waveguide to measure thermodynamic work and heat, suggesting waveguide spectroscopy is a useful means to study quantum heat engines and refrigerators. Finally, based on the theoretical model, we propose how a similar setup can be used as a noise spectrometer which provides a new solution for calibrating the background noise of hybrid quantum systems.

Thermodynamic Relationships for Perfectly Elastic Solids Undergoing Steady-State Heat Flow.

Available data on insulating, semiconducting, and metallic solids verify our new model that incorporates steady-state heat flow into a macroscopic, thermodynamic description of solids, with agreement being best for isotropic examples. Our model is based on: (1) mass and energy conservation; (2) Fourier’s law; (3) Stefan–Boltzmann’s law; and (4) rigidity, which is a large, yet heretofore neglected, energy reservoir with no counterpart in gases. To account for rigidity while neglecting dissipation, we consider the ideal, limiting case of a perfectly frictionless elastic solid (PFES) which does not generate heat from stress. Its equation-of-state is independent of the energetics, as in the historic model. We show that pressure-volume work (PdV) in a PFES arises from internal interatomic forces, which are linked to Young’s modulus (Ξ) and a constant (n) accounting for cation coordination. Steady-state conditions are adiabatic since heat content (Q) is constant. Because average temperature is also constant and the thermal gradient is fixed in space, conditions are simultaneously isothermal: Under these dual restrictions, thermal transport properties do not enter into our analysis. We find that adiabatic and isothermal bulk moduli (B) are equal. Moreover, Q/V depends on temperature only. Distinguishing deformation from volume changes elucidates how solids thermally expand. These findings lead to simple descriptions of the two specific heats in solids: ∂ln(cP)/∂P = −1/B; cP = nΞ times thermal expansivity divided by density; cP = cVnΞ/B. Implications of our validated formulae are briefly covered.

Transient and steady-state heat transfer for forced convection of helium gas in minichannels with various inner diameters

An experimental research was carried out on steady-state and transient heat transfer characteristics for forced convection of helium gas in minichannels. Platinum minichannels with different inner diameters were heated by an exponential power source with a wide range of e-folding time of heat generation rate and cooled by forced convection of helium gas with different flow velocities, inlet temperatures, and inlet pressures. The diameter effect of minichannels on the transient and steady-state heat transfer was investigated. According to the results, the heat transfer coefficients were improved with a decrease in the inner diameters of minichannels under both transient and quasi-steady states. A significant enhancement was obtained by using a 0.8 mm diameter minichannel. For the steady-state heat transfer, a new correlation was acquired for the 0.8 mm diameter minichannel and the correlation of a 1.8 mm diameter minichannel was appropriate for a 2.8 mm diameter minichannel. The influence of Reynolds number on steady-state Nusselt number for the 0.8 mm diameter minichannel was more significant compared with the 2.8 and 1.8 mm diameter minichannels. The transient heat transfer correlations for the 0.8 and 2.8 mm diameter minichannels were obtained by introducing Fourier number, respectively.

Thermal performance of a microchannel primary surface recuperator for portable microturbine generators: Design and experimental study

The recuperator with high compactness and power/weight ratio is a mandatory component of portable microturbine generators (PMTG) to achieve the required energy density and power density. In order to explore the application prospects of cross-wavy primary surface recuperators (CW-PSR), an annular microchannel CW-PSR with heat transfer matrix compactness up to 2428 m2/m3 was designed by an in-house one-dimensional steady-state heat transfer model and successfully manufactured. Furthermore, an experimental system consisting of gas and air flow paths was built to validate the developed model and investigate the thermal performance of the new CW-PSR. On this basis, heat transfer experiments were performed under steady-state by changing the mass flow rate and inlet temperature of the gas. The experimental results show a good agreement between modeled results and experimental data of the power/weight ratio and effectiveness within an error of ±2%. There exists a significant heat transfer non-uniformity in the radial direction of the CW-PSR, but it was found to have little impact on the design accuracy of overall thermal performance. Due to corrugated microchannels, the heat transfer performance of the developed CW-PSR has been enhanced by up to 113.3% compared with straight channels and the experimental power/weight ratio of the CW-PSR is in the range between 9.6 kW/kg and 16.1 kW/kg, which means the microchannel CW-PSR is a promising heat exchanger for the PMTG.

Studies on heat transfer characteristics of vapor chamber under periodic-pulsed heat source

In a high-power microwave device, the intense electron beam causes a high-heat-flux on the collector’s inner surface which results in an instant temperature increase which eventually causes the output power to drop. The vapor chamber is one of the passive cooling devices effective for high-heat-flux heat dissipation. To study the heat dissipation capacity and the surface temperature control effect of the vapor chamber, a numerical heat transfer model of the vapor chamber is proposed based on the effective thermal conductivity method. The accuracy of the steady-state heat transfer analysis and the applicability of the transient simulation are verified by comparing with experimental data, and the heat transfer are calculated under the condition of periodic pulsed heat sources. Lastly, the transient performance of a vapor chamber relative to a copper heat spreader of the same external dimensions is investigated as a function of the wick effective thermal conductivity, pulsed heat flux, and pulse duration. The transient behavior of the vapor chambers are examined to find a performance threshold to determine if the performance is superior to that of a copper heat spreader. The present work provides a basis to understand the advantages and limitations of metal heat spreader in pulse mode operation of vapor chamber and plays a vital role in the design of improving transient performance.

Systematic approach to estimate non-uniform heat generation rate in heat transfer problems using liquid crystal thermography and inverse methodology

ABSTRACT This paper presents an inverse methodology to estimate the parameters of the non-uniform heat generation (function estimation) within a flat plate assembly using steady-state conjugate heat transfer experiments on the flat plate assembly. Steady-state laminar conjugate forced convection experiments on a flat plate assembly are conducted on a horizontal wind tunnel to estimate the parameters of the non-uniform heat generation within flat plate assembly using the inverse methodology. Bayesian inference based Metropolis Hastings–Markov Chain Monte Carlo (MH–MCMC) algorithm and experimental temperatures are employed in the inverse methodology. The experimental temperatures are measured at convenient locations of the flat plate assembly using liquid crystal thermography. In order to accomplish the retrieval, first, steady-state experiments on only the cork material are conducted to estimate the thermal conductivity of the cork material accurately for use in the estimation of the heat generation rate so that the additional error due to uncertainty in the thermal conductivity of the cork material does not affect our final goal of estimating heat generation rate. Following this, steady-state experiments on the cork setup (consisting of a non-uniform heat generation heater and two symmetric cork plates) are conducted to ascertain the nature of heat generation of the heater using measured temperatures and fundamental rate laws. The priors are generated using coupled artificial neural network (ANN) and Levenberg–Marquardt (LM) algorithm for Bayesian inference. Using the Bayesian inference with priors, the parameters of non-uniform heat generation are then estimated in terms of the mean, maximum a posteriori with standard deviation. Finally, the simulated heat powers and temperatures are estimated with retrieved parameters of the non-uniform heat generation. These compared very well with the measured heat powers and temperatures. Finally, a recipe for solving a practical problem, in which only measured temperatures are available, is provided.

Design of Novel Cooling Systems Based on Metal Plates with Channels of Shapes Inspired by Nature

The effect of the channel shape of aluminum plates on cooling capacity was evaluated by studying different configurations. Common shapes of the channel, such as square and fork shapes, were compared with novel configurations inspired by shapes found in nature, specifically the shape of the outline of flowers, inspired these new configurations, consisting of channels with crateriform, salverform, and cruciform shapes. The aim of the study is to evaluate the effect of the channel shape on the cooling capacity of the metal plate. To that end, all the configurations were analyzed from a geometrical point of view, determining the minimum distance of each point across the plate to the channel. A finite difference method was implemented to study both transient and steady state heat dissipation across the plates for each configuration. Even though the effect of the channel shape on the average temperature of the plate is slight, the maximum temperature, the size and location of hot spots, and the temperature homogeneity of the plate are strongly affected by the shape of the channel through which the cooling fluid is circulated. A reduction in the maximum temperature of the plate during transient cooling of around 2 °C for the crateriform and salverform channels and approximately 4.5 °C for the cruciform channel can be attained, compared to the standard configurations. The steady state heat dissipation analysis concluded that the crateriform and salverform configurations reduced the maximum variation in temperature of the common configurations by roughly 15%, whereas a reduction of approximately 28% could be reached by the cruciform configuration. Regarding the homogeneity of temperature across the plate, a reduction up to 34.5% of the index of uniform temperature can be attained using the novel configurations during the steady state refrigeration of the plate. The cruciform channel is the optimal configuration for both transient and steady state cooling processes, reducing the size and temperature of hot spots and improving the temperature homogeneity of the plate, a result already anticipated by the geometrical analysis. In fact, the main conclusions attained from the cooling study are in good agreement with the results of the geometrical analysis. Therefore, the geometrical analysis was found to be a simple and reliable method to design the shape of channels of a cooling system.

Fully compressible convection for planetary mantles

SUMMARY The numerical simulations of convection inside the mantle of the Earth or of terrestrial planets have been based on approximate equations of fluid dynamics. A common approximation is the neglect of the inertia term which is certainly reasonable as the Reynolds number of silicate mantles, or their inverse Prandtl number, are infinitesimally small. However various other simplifications are made which we discuss in this paper. The crudest approximation that can be done is the Boussinesq approximation (BA) where the various parameters are constant and the variations of density are only included in the buoyancy term and assumed to be proportional to temperature with a constant thermal expansivity. The variations of density with pressure and the related physical consequences (mostly the presence of an adiabatic temperature gradient and of dissipation) are usually accounted for by using an anelastic approximation (AA) initially developed for astrophysical and atmospheric situations. The BA and AA cases provide simplified but self-consistent systems of differential equations. Intermediate approximations are also common in the geophysical literature although they are invariably associated with theoretical inconsistencies (non-conservation of total energy, non-conservation of statistically steady state heat flow with depth, momentum and entropy equations implying inconsistent dissipations). We show that, in the infinite Prandtl number case, solving the fully compressible (FC) equations of convection with a realistic equation of state (EoS) is however not much more difficult or numerically challenging than solving the approximate cases. We compare various statistical properties of the Boussinesq, AA and FC simulations in 2-D simulations. We point to an inconsistency of the AA approximation when the two heat capacities are assumed constant. We suggest that at high Rayleigh number, the profile of dissipation in a convective mantle can be directly related to the surface heat flux. Our results are mostly discussed in the framework of mantle convection but the EoS we used is flexible enough to be applied for convection in icy planets or in the inner core.

Development and Processing of Thermal-Hydraulic Model for GFR 2400 Fast Reactor Design and NESTLE Coupled Transient Code

Abstract The paper investigates the influence of the used thermal-hydraulic approximations on the coupled calculations of gas-cooled fast reactor design (hereby GFR 2400). The NESTLE code is used as coupled simulation tool and solves the multigroup neutron diffusion equation by the finite difference method that is internally coupled with a thermal-hydraulic subchannel code. The in-house developed tempin code and the computational fluid dynamics (CFD) code fluent (from ANSYS code system, Canonsburg, PA) are used to prepare the thermal-hydraulic data for the GFR 2400 calculations. The tempin code solves the steady-state heat balance equation with flowing coolant in triangular lattice cell together with temperature dependent thermal-hydraulic properties of the fuel, cladding, and coolant. Based on the calculated fuel bundle temperature distributions by the tempin code, the thermal-hydraulic material properties (approximations) suitable for the NESTLE coupled code are processed for the GFR 2400 design. The influence of the constant and radial heat generation term within the fuel pin is studied within the paper. The performance of the NESTLE code with thermal-hydraulic approximations processed by both (tempin and fluent) methods is compared with the findings of the GoFastR project. Moreover, both the thermal-hydraulic approximations were compared for one steady-state and one transient state, related to the rapid withdrawal of one control rod assembly from the core. Changes in thermal-hydraulic distributions are described and visualized in the paper.

Simulation of Heat Transfer in Single-Crystal Lithium Niobate in Interaction with Continuous-Wave Laser Radiation

The paper presents the simulation results of heat transfer in single-crystal lithium niobate (LiNbO3) in the form of cylinder of diameter mm and height mm in interaction with continuous-wave laser radiation with the output power of W and the wavelength of nm. The density of the LiNbO3 crystal is kg/m3; the thermal conductivity along the [001] direction is W/(m×K); the thermal conductivity in the (001) plane is W/(m×K); the specific heat at constant pressure is J/(kg×K); the absorption coefficient is %/cm @ 1064 nm. The laser beam propagates along the optical axis of the crystal. The laser beam intensity profile is represented as a Gaussian function, and the absorption of laser radiation of the single-crystal lithium niobate is described by Beer-Lambert’s law. The numerical solution of the non-stationary heat conduction problem is obtained by meshless scheme using anisotropic radial basis functions. The time interval of the non-stationary boundary-value problem is 2 h 30 min. The results of numerical calculations of the temperature distribution inside and on the surface of the single-crystal lithium niobate at times s are presented. The time required to achieve the steady-state heating mode of the LiNbO3 crystal, as well as its temperature range over the entire time interval, have been determined. The accuracy of the approximate solution of the boundary-value problem at the n-th iteration is estimated by the value of the norm of relative residual . The results of the numerical solution of the non-stationary heat conduction problem obtained by meshless method show its high efficiency even at a small number of interpolation nodes.

Cross-code comparison of the edge codes SOLPS-ITER, SOLEDGE2D and UEDGE in modelling a low-power scenario in the DTT

As reactor-level nuclear fusion experiments are approaching, a solution to the power exhaust issue in future fusion reactors is still missing. The maximum steady-state heat load that can be exhausted by the present technology is around 10 MW m−2. Different promising strategies aiming at successfully managing the power exhaust in reactor-relevant conditions such that the limit is not exceeded are under investigation, and will be tested in the Divertor Tokamak Test (DTT) experiment. Meanwhile, the design of tokamaks beyond the DTT, e.g. EU-DEMO/ARC, is progressing at a high pace. A strategy to work around the present lack of reactor-relevant data consists of exploiting modelling to reduce the uncertainty in the extrapolation in the design phase. Different simulation tools, with their own capabilities and limitations, can be employed for this purpose. In this work, we compare SOLPS-ITER, SOLEDGE2D and UEDGE, three state-of-the-art edge codes heavily used in power exhaust studies, in modelling the same DTT low-power, pure-deuterium, narrow heat-flux-width scenario. This simplified, although still reactor-relevant, testbed eases the cross-comparison and the interpretation of the code predictions, to identify areas where results differ and develop understanding of the underlying causes. Under the conditions investigated, the codes show encouraging agreement in terms of key parameters at both targets, including peak parallel heat flux (1%–45%), ion temperature (2%–19%), and inner target plasma density (1%–23%) when run with similar input. However, strong disagreement is observed for the remaining quantities, from 30% at outer mid-plane up to a factor 4–5 at the targets. The results primarily reflect limitations of the codes: the SOLPS-ITER plasma mesh not reaching the first wall, SOLEDGE2D not including ion-neutral temperature equilibration, and UEDGE enforcing a common ion-neutral temperature. Potential improvements that could help enhance the accuracy of the code models for future applications are also discussed.

The effect of the altitude on the performance of a solar chimney

This study aimed to show the relationship between the altitude and the performance of a solar chimney based on simulation analysis and numerical calculation. Steady-state heat transfer from isothermal vertical walls of a chimney is modelled analytically. The paper introduces the basic concepts and governing equations for modelling heat transfer and air flow rates within the solar chimney at various altitudes. The relationship between altitude, temperature, and mass flow rates for the buoyancy-driven chimney flow is complex. A multidimensional equation comprising the inverse tangent and inverse hyperbolic tangent functions is developed. A Matlab program was developed to overcome the problem of the non-linearity of the equation. The results obtained showed that the altitude has modest influence but not substantial on the performance of the solar chimney if we assumed a constant ambient air temperature for all altitudes. But in real climatic conditions, the ambient temperature is altitude dependent. In these circumstances, the performance of the solar chimney is significantly influenced by the altitude. Therefore, microclimatic conditions should be used rather than large-scale area climatic data for solar chimney design.

Boundary Element and Sensitivity Analysis of Anisotropic Thermoelastic Metal and Alloy Discs with Holes.

The main aim of this paper was to develop an advanced processing method for analyzing of anisotropic thermoelastic metal and alloy discs with holes. In the boundary element method (BEM), the heat impact is expressed as an additional volume integral in the corresponding boundary integral equation. Any attempt to integrate it directly will necessitate domain discretization, which will eliminate the BEM’s most distinguishing feature of boundary discretization. This additional volume integral can be transformed into the boundary by using branch-cut redefinitions to avoid the use of additional line integrals. The numerical results obtained are presented graphically to show the effects of the transient and steady-state heat conduction on the quasi-static thermal stresses of isotropic, orthotropic, and anisotropic metal and alloy discs with holes. The validity of the proposed technique is examined for one-dimensional sensitivity, and excellent agreement with finite element method and experimental results is obtained.

Integrated design strategy for EU-DEMO first wall protection from plasma transients

This work presents an overview of the integrated strategy developed, as part of the DEMO Key Design Integration Issue 1 (KDII1), to protect the EU-DEMO first wall (FW) from planned and unplanned plasma transients by employing discrete limiters. The present Breeding Blanket (BB) FW design, which aims at minimizing the loss of neutrons while travelling to the breeding zone, is able to withstand steady state heat fluxes up to ≈1-1.5 MW/m² [1], which is not sufficient to guarantee its integrity for most plasma-FW direct contact. This is different from ITER, which has a FW designed for peak heat loads up to 4.6 MW/m2 [2], and it does not have the DEMO BB breeding related requirement. A series of documents was compiled in the DEMO Pre-Conceptual Design Phase, in support of the KDII1. The work presented here was presented at the 2020 DEMO Gate 1 (G1) review, and collects also the comments of the panel and the relative additional studies triggered by them. The design process, presented in this paper was adopted to systematically evaluate the impact of design changes, or new physics inputs, on the FW protection strategy and integration issues. It includes compiling the list of transients, and performing the relative plasma simulations, the design of discrete limiters and the evaluation of their capability to reduce the heat flux density on the FW, and finally a preliminary analysis of the heat loads effects on the Plasma Facing Components (PFC). All these aspects, together with preliminary limiter design, where considered since the beginning, in an integrated way.

Comparative Analysis of Two-Dimensional Steady State Heat Flow Problem of a Rectangular Plate by Analytical and Numerical Approach

Heat transfer problems play a very vital role in many fields of engineering. This present analysis work focuses on analytical and numerical approaches to resolve a two-dimensional steady-state heat conduction problem of a rectangular plate. The temperatures at the inside points of the rectangular plate are calculated analytically and the results are verified numerically. For numerical solution of the problems programming is done in using METLAB software. The material of the rectangular plate used is steel. It will be seen from the literature survey that rigorous analytical solutions are available only for a very few simple boundary conditions and these conditions does not seem to be favourable for complex boundaries. However, majority of engineering problems with simple as well as complex boundary conditions will be solved with the help of numerical approaches. In this analysis work, an attempt is made to obtain solutions using numerical method for a two-dimensional steady state heat conduction problems of a steel rectangular plate. The mathematical formulation of problems is done using Laplace method. The results obtained from the two different methods that is analytical and numerical are then compared with each other. The agreement between the analytical and numerical results is a sign of the accuracy of solution method.

Schmidt method to study the disturbance of steady-state heat flows by an arbitrary oriented crack in bonded functionally graded strips

This article aims to investigate the impact of a partially insulated crack on vertical and horizontal components of heat flow. The crack is arbitrary oriented and its center is located at the interface of bonded finitely thick functionally graded strips. Employing the integral transformation, the thermoelastic equations are reduced to a set of singular integral equations. The unknown variables of these equations are the jump in temperature and displacements across the crack surfaces. Representing the unknowns as a series of Jacobi polynomials, the solutions of the singular integral equations are obtained by employing the Schmidt method which further helps to determine the analytical forms of crack tip heat flux intensity factors (HFIFs) and thermal stress intensity factors (TSIFs). The results of the present study are also validated for a particular case.To quantify the strength of heat flow components, the crack tip HFIFs and TSIFs as a function of crack angle, crack insulation parameter, strips’ width, and non-homogeneity parameters’ ratio are demonstrated pictorially.