数值仿真在金属和合金导热系数测定上的应用.pdf

http:/ The Application of Numerical Simulation in the Determination of Thermal conductivity of Metals and Alloys1 Ying Yang Jiemin Zhou School of Energy and Power Engineering School of Energy and Power Engineering Central South University Central South University Changsha,Hunan,PRC 410083 Changsha,Hunan,PRC 410083 yangying_ Prof_ yangying_ Abstract In this research,a transient experimental system was designed to determine the thermal conductivity of the metals and alloys whose theoretical foundation was an axial periodic heat flow technique.Meanwhile,mathematic calculation was applied to intuitively simulate the heat transfer and temperature periodic change process?to correct the error and then to increase the result precision to some extent.In verification experiment,compared with the recommended data from reference,the error of the corrected experimental results is within?5%at the temperature range of 300800K?which means the competitive method and the measuring system have the comparative reliability.The research on the method and the measuring system will pave the way to the subsequent research on the metals and new alloys such as lead-free eutectic solders.1 Introduction Although some traditional metal and alloys such as lead,Sn-Pb solders played a dominant role in jointing technique of all electronics packaging industry during the past several decades because of their better physical performances,it is an inevitable international trend to develop the replacers of lead solder out of the consideration of environmental protection and social long-term benefit 1,2.It is essential for the promising replacers to have enough and credible data to verify the equal or better reliability and the overall performances as compared with the Sn-Pb solders.With such a purpose,a transient experimental system was designed in this research to determine the thermal conductivity of the metals and alloys whose theoretical foundation was an axial periodic heat flow technique 3,4.Meanwhile,numerical method was applied to analysis the heat transfer and the deviation from the ideal condition and then to modify the initial experimental results.It is a significant attempt on acquiring more precise determining result by combining experimental determining and numerical analysis.2 Experimental principle and instrument 2.1 Methods for measuring thermal conductivity Based upon the assumptions that the specimen could be considered as a semi-infinite body and the heat conduction process in it is of one dimension the thermal diffusivity of the specimen could be calculated by 1 Support by the Special Foundation for Doctorate Discipline of China(Contract No.:20010533009)http:/ measuring the decreasing amplitude or increasing phase lag between the different positions as the heat wave propagated through the specimens in its axial direction.When the one end of the specimen was exposed to a periodic heat flow,the thermal diffusivity process could be formulated mathematically 5,6 as()()224tXPaL=(1)()()1222lnPXaa=(2)where aL is thermal diffusivity due to the phase lag,m2?s-1;aa is thermal diffusivity due to amplitude decrement,m2?s-1;P is the period of the wave of axial heat flow,s;Xis axial distance between the thermocouples,m;tis the phase time lag,s and?1,?are amplitude at thermocouple 1,2,K.The values of the diffusivity,calculated from the expression,may differ from each other due to radical heat transfer between the specimen and the surroundings,which can be taken into account by a correction factor,()()12ln2PtFL=(3)and LaFF1=(4)where FL,Fa are the correction factors of aL,aa.The corrected value is evaluated by LLFaa=oraaFaa=.The other correction can be described by following equation ()aLaaa+=21 (5)The relationship of thermal conductivity with thermal diffusivity is:Cka=(6)where a is thermal diffusivity,m2?s-1;k is thermal conductivity,W?m-1?K?is density,kg?m-3;and Cis specific heat capacity,J?kg-1?K?2.2 Instrument The specimens are cylindrical ingots with diameter 20mm and length 100mm.5mm(position 1)and 20mm(position 2)from bottom end of the cylinders respectably,radial measuring holes were grilled onto axis,in which thermocouples were inserted during the measurement.The thermocouples are type K with 0.5mm external diameters and the measuring region is between the two thermocouples.The specimen was mounted in a concentric position of a vertical tubular furnace kept at a given temperature initially.The bottom and top of the specimen connected with a small electrical heater and an independent temperature controller respectively.In additional,an auxiliary side heater was applied to reduce the radial heat dissipation to minimum.The scheme of the device is shown in Figure 1.http:/ 1-temperature control system;2-insulation;3-tubular furnace;4-specimen;5-thermcouple;6-sinus heater Figure1:Scheme of periodic method 2.3 Measuring process When0t,the bottom surface of the specimen was exposed to a periodical heat flow in sinus mode from the heater about 50s per period and the top of the specimen were kept at the initial temperature by temperature control system.The periodic heat flow was recognized by the thermocouples as a harmonic temperature variation over time with decreasing amplitude and increasing phase lag as the wave propagated through the specimen.The variation of temperature was illustrated in Figure 2.When the temperature changed regularly with the time,about a period curve was intercept to calculate the thermal diffusivity,then to determine the thermal conductivity.1-temperature at position 1;2-temperature at position 2 Figure 2:Measured temperature curve 3 Analysis and discussion 3.1 The asymmetry of the curve 2 3 4 5 6 1 http:/ In general,the heat transfer between the heater and the specimen is mainly composed of radiation and convection At the end of the heating sequence,the predominant way of the heat transfer is radiation because of the higher temperature,while at the end of the cooling sequence,convection is relatively more important.The transfer way of heat from the electric heater resulted in the temperature curves unsymmetrical about their maximum value.The deviation of the readings of the thermocouples from ideal curves depends on the period,emissivity of the surface of the specimen,the thermal time constant of the heater and the heat flux.3.2 The deviation from the ideal condition The other potential factors that resulted in the bias error are the temperature fluctuation on the top surface and the radial dissipation from the wall,caused by the limit of precision and sensitivity of the temperature control system and the heat-insulated property of the furnace.Figure 3 illustrates the qualitative result from mathematical analysis.It can be seen from Figure 3 that the curve 3 has obvious deviation from the curve 1 resulting from the ideal assumption while the curve 2 coincides with curve 1 well which means that the heat dissipation from the wall is crucial factors to influence the precision of determination.In order to compare the influence of bias error on the determination more visually,Figure 4 gives the contour maps,with and without radial heat dissipation respectively.1-ideal temperature curve;2-temperature curve with temperature fluctuation at the top surface;3-temperature curve with heat dissipation from the side wall Figure 3:The mathematical analysis 1-ideal temperature distribution;2-temperature distribution with heat dissipation from the wall Figure 4:The mathematical analysis http:/ 3.3 The correction of the experimental results The primary value of the thermal conductivity could be determined according the curve 1 and curve 2 in Figure 2,yet how to correct the asymmetry of the curve and how to increase the accuracy of the determination without more increasing the complexity of measuring device are the anticipant targets.In this investigation,the corrective coefficient of experimental data was successfully acquired by combining the experimental measurement and the mathematical analysis.The temperature fluctuation of the top surface and of the wall was measured synchronously with the temperature at position 1 and 2,then,data was treated to be time-dependent function such as periodic or polynomial expression which was applied to the mathematical analysis as the boundary condition.The experimental value of thermal conductivity,k0,was used for the first calculation,from which the temperature distribution was evaluated.According to the residual error between the simulating and the experimental result,the variable increment?k was determined and ki+1=ki+?k was applied to the next iteration.If the residual error was within the allowable range after the Nth iteration,the calculation was terminated and corresponding k was regarded as the corrective value.3.4 The results of determination In this research,the electrolytic pure iron was chosen as the measuring object because it has be recognized as standard reference materials with full and accurate recommended property value,part of which was listed in Table 1.Table 1:Thermal and physical properties of Fe 7 T/K k/(W?m-1?K-1)pc/(J?kg-1?K-1)/(kg?m-3)300 77.1 400 67.5 500 60.3 600 53.7 700 47.2 800 42.1 156.1+0.22T(273KT1033K)7870 It can be intuitively concluded from Figure 5 that more precise determining data could be acquired by combining experiment with numerical simulation.Compared with the recommended data,the creditability of the corrected thermal conductivity value is above 95%at the range of 300800K,which just coincides the normal measuring range of solder at solid state.1-data from literature;2-corrected data;3-purely experimental data Figure 5:Thermal conductivity of pure Fe http:/ 4 Conclusions The method presented in this paper is applicable to the determination of thermal diffusivity and thermal conductivity of metals and alloys.1)At temperatures between 300800K,the asymmetry of temperature curves due to the heating and heat transfer can be solved by curved fit;2)Compared with the recommended data,the creditability of the corrected thermal conductivity value is above 95%at 300800K;3)The mathematical analysis would visually illustrate the temperature distribution,the heat dissipation and decrease the blindness in improving the experimental device and procedure to some extent;4)The combination of experimental measurement and mathematical analysis would acquire the satisfying accuracy of the determination without more increasing the complexity of measuring device.So,the method for determining thermal conductivity of metals and alloys is feasible and successful.The research on the method and the measuring system will pave the way to the subsequent research on such new alloys as lead free solders.References 1 Fu H and Liu J,The Development of Lead-free Soldering,Proceedings of the Fourth International Symposium on Electronic packaging Technology,edited by Keyun Bi etc,Beijing,2001,449-454.2 Yoshitaka T,The latest Trends in Lead-Free Soldering,Proceedings of the Fourth International Symposium on Electronic packaging Technology,edited by Bi KY etc,Beijing,2001,434-438.3 Sabuga W and Hammerschmielt V,A New method for the evaluation of thermal conductivity and thermal diffusivity from transient hot strip measurements.The 12th Symposium on Thermaophysical Properties(Subsecond Thermophysics 2),NIST/ASME,1998.4 Righini D,Spisiak J,and bussoline GC,Measurement of thermophysical properties by a pulse-heating method,The 12th Symposium on Thermaophysical Properties(Subsecond Thermophysics 2),NIST/ASME.June 1998.5 Lamvik M,Determination of thermal diffusivity of solid by use of periodic heat flow,Intern J of thermophysics 1(2),1980,233.6 Zhou JM,Yang Y,Lamvik M and Wang G,Determination of thermal conductivity of magnesium-alloys.Journal of Central South University of Technology 8(4),2001,278-282.7 Chen H S,The Physical properties and Determining Methods of Metals,1,Beijing,Metallurgic Industry Press,1987,3,11,263,350.