Finite Difference-Peridynamic Differential Operator for Solving Transient Heat Conduction Problems

Transient heat conduction problems widely exist in engineering.In previous work on the peridynamic differential operator(PDDO)method for solving such problems,both time and spatial derivatives were discretized using the PDDO method,resulting in increased complexity and programming difficulty.In this work,the forward difference formula,the backward difference formula,and the centered difference formula are used to discretize the time derivative,while the PDDO method is used to discretize the spatial derivative.Three new schemes for solving transient heat conduction equations have been developed,namely,the forward-in-time and PDDO in space(FT-PDDO)scheme,the backward-in-time and PDDO in space(BT-PDDO)scheme,and the central-in-time and PDDO in space(CT-PDDO)scheme.The stability and convergence of these schemes are analyzed using the Fourier method and Taylor’s theorem.Results show that the FT-PDDO scheme is conditionally stable,whereas the BT-PDDO and CT-PDDO schemes are unconditionally stable.The stability conditions for the FT-PDDO scheme are less stringent than those of the explicit finite element method and explicit finite difference method.The convergence rate in space for these three methods is two.These constructed schemes are applied to solve one-dimensional and two-dimensional transient heat conduction problems.The accuracy and validity of the schemes are verified by comparison with analytical solutions.

A Novel Localized Meshless Method for Solving Transient Heat Conduction Problems in Complicated Domains

This paper first attempts to solve the transient heat conduction problem by combining the recently proposed local knot method(LKM)with the dual reciprocity method(DRM).Firstly,the temporal derivative is discretized by a finite difference scheme,and thus the governing equation of transient heat transfer is transformed into a non-homogeneous modified Helmholtz equation.Secondly,the solution of the non-homogeneous modified Helmholtz equation is decomposed into a particular solution and a homogeneous solution.And then,the DRM and LKM are used to solve the particular solution of the non-homogeneous equation and the homogeneous solution of the modified Helmholtz equation,respectively.The LKM is a recently proposed local radial basis function collocationmethod with themerits of being simple,accurate,and free ofmesh and integration.Compared with the traditional domain-type and boundary-type schemes,the present coupling algorithm could be treated as a really good alternative for the analysis of transient heat conduction on high-dimensional and complicated domains.Numerical experiments,including two-and three-dimensional heat transfer models,demonstrated the effectiveness and accuracy of the new methodology.

Evaluation of ultra-fine grained tungsten under transient high heat flux by high-intensity pulsed ion beam

Pure tungsten, oxide dispersion strengthened tungsten and carbide dispersion strengthened tungsten were fabricated by high-energy ball milling and spark plasma sintering process. In order to evaluate the properties of the tungsten alloys under transient high heat flues, four tungsten samples with different grain sizes were tested by high-intensity pulsed ion beam with a heat flux as high as 160 MW/（m^2·s^-1/2）. Compared with the commercial tungsten, the surface modification of the oxide dispersion strengthened tungsten by high-intensity pulsed ion beam is completely different. The oxide dispersion strengthened tungsten shows inferior thermal shock response due to the low melting point second phase of Ti and Y2O3, which results in the surface melting, boiling bubbles and cracking. While the carbide dispersion strengthened tungsten shows better thermal shock response than the commercial tungsten.

A new complex variable meshless method for transient heat conduction problems

In this paper, based on the improved complex variable moving least-square （ICVMLS） approximation, a new complex variable meshless method （CVMM） for two-dimensional （2D） transient heat conduction problems is presented. The variational method is employed to obtain the discrete equations, and the essential boundary conditions are imposed by the penalty method. As the transient heat conduction problems are related to time, the Crank-Nicolson difference scheme for two-point boundary value problems is selected for the time discretization. Then the corresponding formulae of the CVMM for 2D heat conduction problems are obtained. In order to demonstrate the applicability of the proposed method, numerical examples are given to show the high convergence rate, good accuracy, and high efficiency of the CVMM presented in this paper.

Transient Heat Transfer Study of Direct Contact Condensation of Steam in Spray Cooling Water

We conducted a transient experimental investigation of steam–water direct contact condensation in the absence of noncondensible gas in a laboratory-scale column with the inner diameter of 325 mm and the height of 1045 mm. We applied a new analysis method for the steam state equation to analyze the molar quantity change in steam over the course of the experiment and determined the transient steam variation. We also investigated the influence of flow rates and temperatures ofcooling water on the efficiency ofsteam condensation. Our experimental results show that appropriate increasing of the cooling water flow rate can significantly accelerate the steam condensation. We achieved a rapid increase in the total volumetric heat transfer coefficient by increasing the flow rate of cooling water, which indicated a higher thermal convection between the steam and the cooling water with higher flow rates. We found that the temperature ofcooling water did not play an important role on steam condensation. This method was confirmed to be effective for rapid recovering ofsteam.

Transient analysis and optimization of passive residual heat removal heat exchanger in advanced nuclear power plant

The transient performance and optimization of a passive residual heat removal heat exchanger(PRHR HX)were investigated.First,a calculation method was developed for predicting the heat transfer of the PRHR HX.The calculation results were validated through comparisons with ROSA experimental data.The heat-transfer performance of the AP1000 PRHR HX in the initial period was predicted,and it satisfied the design requirements.Second,the distributions of the heat flux,tube-inside/outside heattransfer coefficients,and heat load for the AP1000 PRHR HX over 2000 s were examined.Third,an optimization study was conducted by adjusting the horizontal length and tube diameter.Their effects on the four main heat-transfer parameters and the heat-transfer area were analyzed.Furthermore,the influence of the initial in-containment refueling water storage tank(IRWST)temperature was investigated using an established simulation procedure.The results indicated that it significantly affected the trends of the IRWST temperature and reactor outlet temperature.Finally,the minimum required flow rates over time to maintain the reactor outlet temperature at the safety line were determined for different start-up times.The trends of the minimum required flow rate and the peak flow rate were analyzed.

Thermo-structural analysis on evaluating effects of friction and transient heat transfer on performance of gears in high-precision assemblies

The high precision assemblies with considerable radial interference should be accompanied by heating and cooling processes.However,the mechanical properties of metals are greatly affected by thermal operations.So,for evaluating the stress distribution and distortion of teeth profiles in a gear/shaft assembly,a transient thermal analysis is necessary for finding the change in mechanical properties.The friction on the contact surface is another important parameter in interaction of the gear with the shaft.Evaluating the gear stress and deformation fields for several modes of heat transfer and friction coefficients showed that the maximum radial or tangential stresses on contact surface of the joint may have more than 8%increase by increasing friction coefficient;while the intensity of heat transfer at cooling stage has lower effect on stress distribution.

Combining the complex variable reproducing kernel particle method and the finite element method for solving transient heat conduction problems

In this paper, the complex variable reproducing kernel particle （CVRKP） method and the finite element （FE） method are combined as the CVRKP-FE method to solve transient heat conduction problems. The CVRKP-FE method not only conveniently imposes the essential boundary conditions, but also exploits the advantages of the individual methods while avoiding their disadvantages, then the computational efficiency is higher. A hybrid approximation function is applied to combine the CVRKP method with the FE method, and the traditional difference method for two-point boundary value problems is selected as the time discretization scheme. The corresponding formulations of the CVRKP-FE method are presented in detail. Several selected numerical examples of the transient heat conduction problems are presented to illustrate the performance of the CVRKP-FE method.

Transient heat transfer characteristics of air-array-jet impingement with different nozzle arrangements on high-tempera-ture flat plate in small jet-to-plate distance

Numerical studies on transient heat transfer characteristics of air-array-jet impingement with a small jet-to-plate distance and a large temperature difference between nozzles and plate were presented.The dimensionless jet-to-plate distance(H/D)was 0.2,and non-dimensional nozzle-to-nozzle spacing(S/D)was 3,4,5 and 6,respectively.It is found that the quenching time is shortened at a constant total mass flow at air jet inlet m·(m·=218.21 kg/h),and the heat transfer uniformity is deterio-rated as S/D increases.However,the adding reversed-flow nozzles can shorten the quenching time of the glass plate considerably with a modest change in the heat transfer uniformity.The results at variable m·are the same as those at a fixed m·.Furthermore,the parity and arrangement of nozzles are also discussed,It is found that an odd number of nozzles is more beneficial for transient heat transfer.Based on these results,an appropriate proposal for ultra-thin glass tempering process is presented.

Hybrid graded element model for transient heat conduction in functionally graded materials

This paper presents a hybrid graded element model for the transient heat conduction problem in functionally graded materials （FGMs）. First, a Laplace transform approach is used to handle the time variable. Then, a fundamental solution in Laplace space for FGMs is constructed. Next, a hybrid graded element is formulated based on the obtained fundamental solution and a frame field. As a result, the graded properties of FGMs are naturally reflected by using the fundamental solution to interpolate the intra-element field. Further, Stefest＇s algorithm is employed to convert the results in Laplace space back into the time-space domain. Finally, the performance of the proposed method is assessed by several benchmark examples. The results demonstrate well the efficiency and accuracy of the proposed method.

Theoretical solution of transient heat conduction problem in one-dimensional double-layer composite medium

To make heat conduction equation embody the essence of physical phenomenon under study, dimensionless factors were introduced and the transient heat conduction equation and its boundary conditions were transformed to dimensionless forms. Then, a theoretical solution model of transient heat conduction problem in one-dimensional double-layer composite medium was built utilizing the natural eigenfunction expansion method. In order to verify the validity of the model, the results of the above theoretical solution were compared with those of finite element method. The results by the two methods are in a good agreement. The maximum errors by the two methods appear when τ（τ is nondimensional time） equals 0.1 near the boundaries of ζ =1 （ζ is nondimensional space coordinate） and ζ =4. As τ increases, the error decreases gradually, and when τ =5 the results of both solutions have almost no change with the variation of coordinate 4.

A meshless model for transient heat conduction analyses of 3D axisymmetric functionally graded solids

A meshless numerical model is developed for analyzing transient heat conductions in three-dimensional （3D） axisymmetric continuously nonhomogeneous functionally graded materials （FGMs）. Axial symmetry of geometry and boundary conditions reduces the original 3D initial-boundary value problem into a two-dimensional （2D） problem. Local weak forms are derived for small polygonal sub-domains which surround nodal points distributed over the cross section. In order to simplify the treatment of the essential boundary conditions, spatial variations of the temperature and heat flux at discrete time instants are interpolated by the natural neighbor interpolation. Moreover, the using of three-node triangular finite element method （FEM） shape functions as test functions reduces the orders of integrands involved in domain integrals. The semi-discrete heat conduction equation is solved numerically with the traditional two-point difference technique in the time domain. Two numerical examples are investigated and excellent results are obtained, demonstrating the potential application of the proposed approach.

Finite Difference Approximation for Solving Transient Heat Conduction Equation of Copper

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

Transient analysis on surface heated piezoelectric semiconductor plate lying on rigid substrate

Based on the thermo-electro-elastic coupling theory,the mathematical model for a surface heated piezoelectric semiconductor(PS)plate is developed in the time domain.Applying the direct and inverse Laplace transformations to the established model,the mechanical and electrical responses are investigated.The comparison between the analytical solution and the finite element method(FEM)is conducted,which illustrates the validity of the derivation.The calculated results show that the maximum values of the mechanical and electrical fields appear at the heating surface.Importantly,the perturbation carriers tend to concentrate in the zone near the heating surface under the given boundary conditions.It can also be observed that the heating induced elastic wave leads to jumps for the electric potential and perturbation carrier density at the wavefront.When the thermal relaxation time is introduced,all the field quantities become smaller because of the thermal lagging effect.Meanwhile,it can be found that the thermal relaxation time can describe the smooth variation at the jump position.Besides,for a plate with P-N junction,the effect of the interface position on the electrical response is studied.The effects of the initial carrier density on the electrical properties are discussed in detail.The conclusions in this article can be the guidance for the design of PS devices serving in thermal environment.

CoNi nanoparticles anchored inside carbon nanotube networks by transient heating:Low loading and high activity for oxygen reduction and evolution

Transitional metal alloy and compounds have been developed as the low cost and efficient bifunctional electrocatalysts for oxygen reduction reaction(ORR)and oxygen evolution reaction(OER).However,a high mass loading of these catalysts is commonly needed to achieve acceptable catalytic performance,which could cause such problems as battery weight gain,mass transport blocking,and catalyst loss.We report herein the preparation of fine CoNi nanoparticles(5-6 nm)anchored inside a nitrogendoped defective carbon nanotube network(CoNi@N-DCNT)by a transient Joule heating method.When utilized as an electrocatalyst for oxygen reduction and evolution in alkaline media,the CoNi@N-DCNT film catalyst with a very low mass loading of 0.06 mg cm^(-2) showed excellent bifunctional catalytic performance.For ORR,the onset potential(Eonset)and the half-wave potential(E_(1/2))were 0.92 V versus reversible hydrogen electrode(vs.RHE)and 0.83 V(vs.RHE),respectively.For OER,the potential at the current density(J)of 10 mA cm^(-2)(E_(10))was 1.53 V,resulting in an overpotential of 300 mV much lower than that of the commercial RuO_(2) catalyst(320 mV).The potential gap between E_(1/2) and E_(10) was as small as 0.7 V.Considering the low mass loading,the mass activity at E_(10) reached at 123.2 A g^(-1),much larger than that of the RuO_(2) catalyst and literature results of transitional metal-based bifunctional catalysts.Moreover,the CoNi@N-DCNT film catalyst showed very good long-term stability during the ORR and OER test.The excellent bifunctional catalytic performance could be attributed to the synergistic effect of the bimetal alloy.

EXPERIMENTAL RESEARCH ON TRANSIENT HEAT TRANSFER IN SAND

In order to examine if the heat conduction in porous material behaves like a wave as claimed by other researchers, experiments on transient heat transfer is conducted in casting sand. The results show that the heat propagation can be described neither by wave nor by diffusion model. The CV-wave concept is discussed and challenged according to the experimental results in the entire transient stage, including the transient response of temperature and the response curve of penetration-depth versus penetration-time.

Consideration of transient heat conduction in a semi-infinite medium using homotopy analysis method

In the current work, transient heat conduction in a semi-infinite medium is considered for its many applications in various heat fields. Here, the homotopy analysis method （HAM） is applied to solve this problem and analytical results are compared with those of the exact and integral methods results. The results show that the HAM can give much better approximations than the other approximate methods： Changes in heat fluxes and profiles of temperature are obtained at different times and positions for copper, iron and aluminum.

An improved local radial point interpolation method for transient heat conduction analysis

The smoothing thin plate spline （STPS） interpolation using the penalty function method according to the optimization theory is presented to deal with transient heat conduction problems. The smooth conditions of the shape functions and derivatives can be satisfied so that the distortions hardly occur. Local weak forms are developed using the weighted residual method locally from the partial differential equations of the transient heat conduction. Here the Heaviside step function is used as the test function in each sub-domain to avoid the need for a domain integral. Essential boundary conditions can be implemented like the finite element method （FEM） as the shape functions possess the Kronecker delta property. The traditional two-point difference method is selected for the time discretization scheme. Three selected numerical examples are presented in this paper to demonstrate the availability and accuracy of the present approach comparing with the traditional thin plate spline （TPS） radial basis functions.

A complex variable meshless local Petrov-Galerkin method for transient heat conduction problems

On the basis of the complex variable moving least-square （CVMLS） approximation, a complex variable meshless local Petrov-Galerkin （CVMLPG） method is presented for transient heat conduction problems. The method is developed based on the CVMLS approximation for constructing shape functions at scattered points, and the Heaviside step function is used as a test function in each sub-domain to avoid the need for a domain integral in symmetric weak form. In the construction of the well-performed shape function, the trial function of a two-dimensional （2D） problem is formed with a one-dimensional （1D） basis function, thus improving computational efficiency. The numerical results are compared with the exact solutions of the problems and the finite element method （FEM）. This comparison illustrates the accuracy as well as the capability of the CVMLPG method.

An analytic model for transient heat conduction in bi-layered structures with flexible serpentine heaters

Uniform heating of complex surfaces,especially non-developable surfaces,is a crucial problem that traditional rigid heaters cannot solve.Inspired by flexible electronic devices,a novel design for the stretchable heating film is proposed with the flexible serpentine wire embedded in the soft polymer film,which can be attached to non-developable surfaces conformally.It provides a new way for the stretchable heaters to realize uniform heating of complex surfaces.However,the thermal field of flexible serpentine heaters(FSHs)depends on the configurations of the embedded serpentine heating wire,which requires accurate theoretical prediction of real-time temperature distribution.Therefore,the analytical model for the transient heat conduction in FSHs is solved by the separation of variables method and validated by the finite element analysis(FEA)in this paper.Based on this model,the effects of the geometric parameters,such as the radius and the length of the serpentine heaters,on the thermal uniformity are systematically investigated.This study can help to design and fabricate flexible heaters with uniform heating in the future.