外文翻译--三维制动器瞬态温度场的紧急制动 英文版

fieldChenHoisttheheatthetemperatureC211 2008 Elsevier Ltd. All rights reserved.is a processenergyhoistsituationon 13,6,10,11is fixedtion of temperature perturbations in multi-disk clutches andbrakes during operation. Naji 12 established one-dimensionalmathematical model to describe the thermal behavior of a brakesystem. Yevtushenko and Ivanyk 13 deduced the transient tem-perature field for an axi-symmetrical heat conductivity problemwith 2-D coordinates. It is difficult for these models to reflect thereal temperature field of brake shoe with 3-D geometry.2. Theoretical analysis2.1. Theoretical modelFig. 1 shows the schematic of hoists braking friction pair. In or-der to analyze brake shoes 3-D temperature field, the cylindricalcoordinates (r,u,z) is adopted to describe the geometric structureshown in Fig. 2, where r is the distance between a point of brakeshoe and the rotation axis of brake disc； u is the central angle； z* Corresponding author. Tel.: +86 fax: +86 516 83590708.Applied Thermal Engineering 29 (2009) 932937Contents lists availableE-mail address: (Y.-x. Peng).emergency braking, so there is more intense temperature rise inbrake shoe. The brake shoe is kind of composite material, and thetemperature rise resulting from frictional heat energy is the mostimportant factor affecting tribological behavior of brake shoe andthe braking safety performance 510. Therefore, it is necessaryto investigate the brake shoes temperature field with respect toinvestigating brake pads.Current theoretical models of brake shoes temperature field arebased on one dimension or two. Afferrante 11 built a two-dimen-sional (2-D) multilayered model to estimate the transient evolu-method is an analytic solution method, it is difficult to solve theequation of heat conduction with complicated boundaries. There-fore, the analytic solution called integral-transform method isadopted 19, because it is suitable for solving the problem ofnon-homogeneous transient heat conduction.In order to master the change rules of brake shoes temperaturefieldduringhoistsemergencybrakingandimprovethesafereliabil-ity of braking, a 3-D transient temperature field of the brake shoewas studied based on integral-transform method, and the validityis proved by numerical simulation and experimental research.1. IntroductionThe hoists emergency brakingmechanical energy into frictional heatemergency braking process of miningof high speed and heavy load, and thising condition of vehicle, train and sowork focused on the brake pads temperatureEspecially, because the brake shoe1359-4311/$ - see front matter C211 2008 Elsevier Ltd. Alldoi:10.1016/j.applthermaleng.2008.04.022of transformingof brake pair. Thehas the characteristicis worse than brak-. The previousfield 14,10,12,13.during the process ofThe methods solving brake pads 3-D transient temperaturefield concentrated on finite element method 13,1417, approx-imate integration method 4,18, Greens function method 12 andLaplace transformation method 9,13, etc. The former threemethods are numerical solution methods and are of low relativeaccuracy. For example, finite element method can solve the com-plicate heat conduction problem, but the accuracy of computa-tional solution is relatively low, which is affected by meshdensity, step length and so on. Though the Laplace transformationIntegral-transform methodEmergency brakingwith experimental data, that the 3-D transient temperature field model of brake shoe is valid and prac-tical, and analytic solution solved by integral-transform method is correct.Three-dimensional transient temperatureemergency brakingZhen-cai Zhu, Yu-xing Peng*, Zhi-yuan Shi, Guo-anCollege of Mechanical and Electrical Engineering, China University of Mining and Technology,article infoArticle history:Received 22 November 2007Accepted 27 April 2008Available online 6 May 2008Keywords:Brake shoeThree-dimensionalTransient temperature fieldabstractIn order to exactly masterbraking, the theoretical modelaccording to the theory ofoperating condition of miningdeduced by adopting integral-transformfield were carried out andent were obtained. At the samefor measuring brake shoesApplied Thermaljournal homepage: www.elsevirights reserved.of brake shoe during hoistsXuzhou 221116, Chinachange rules of brake shoes temperature field during hoists emergencyof three-dimensional (3-D) transient temperature field was establishedconduction, the law of energy transformation and distribution, and thehoists emergency braking. An analytic solution of temperature field wasmethod. Furthermore, simulation experiments of temperaturevariation regularities of temperature field and internal temperature gradi-time, by simulating hoists emergency braking condition, the experimentswere also conducted. It is found, by comparing simulation resultsat ScienceDirectE/locate/apthermengis the distance between a point of brake shoe and the friction sur-face. As for the geometric structure and parameters shown in Fig. 2,its seen that a6r6 b,06u6u0,06z6l. It is clear that thebrake shoes temperature T is the function of the cylindrical coor-dinates (r,u,z) and the time (t). According to the theory of heatconduction, the differential equation of 3-D transient heat conduc-tion is gained as follows:o2Tor21roTor1r2o2Tou2o2Toz21aoTot； 1wherea is the thermal diffusivity,a = k /(qC1 c)； k is the thermal con-ductivity； q is the density； c is the specific heat capacity.2.2. Boundary condition2.2.1. Heat-flow and its distribution coefficientIt is difficult for friction heat generated during emergency brak-ing to emanate in a short time, so it is almost totally absorbed bybrake pair. As the brake shoe is fixed, the temperature of the fric-tion surface rises much sharply, and this eventually affects its tri-bological behavior more seriously. In order to master the realtemperature field of the brake shoe during emergency braking,the heat-flow and its distribution coefficient of friction surfacemust be determined with accuracy. According to the operatingcondition of emergency braking, suppose that the velocity of brakedisc decreased linearly with time, the heat-flow is obtained withthe formqsr；tk C1lC1 pC1 v0C11C0 t=t0k C1lC1 p C1 w0C1 r:1C0 t=t0； 2where q is the heat-flow of friction surface； p is the specific pressurebetweenbrakepair；v0andw0istheinitiallinearandangularvelocityofthebrakedisc；listhefrictioncoefficientbetweenbrakepair；t0isthe whole braking time, k is the distribution coefficient of heat-flow.Suppose the frictional heat is totally transferred to the brakeshoe and brake disk, and the distribution coefficient of heat-flowis obtained according to the analysis of one-dimensional heat con-duction. Fig. 3 shows the contact schematic of two half-planes.Under the condition of one-dimensional transient heat conduc-tion, the temperature rise of friction surface (z = 0) is obtained withthe formDT qkpp4atpqpqckp4tp； 3where q is the heat-flow absorbed by half-plane. And the heat-flowis gained from Eq. (3)p prespectively. According to Eq. (5), the distribution coefficient ofZ.-c. Zhu et al./Applied Thermal Engineering 29 (2009) 932937 933Fig. 1. Schematic of hoists braking friction pair.Fig. 2. 3-D geometrical model of brake shoe.heat-flow entering brake shoe is obtained with the formk qsqaqsqs qd 1C0qdqs qd 1 C01qsqd 1 1 C011qscsksqdcdkdC16C1712: 62.2.2. Coefficient of convective heat transfer on the boundaryWith regard to the lateral surface and the top surface of thebrake shoe, their coefficients of convective heat transfer are ob-tained, respectively, according to the natural heat convectionboundary condition of upright plate and horizontal platehl 1:42DTl=Ll14； 7ahu 0:59DTu=Lu14； 7bq pqckDT= 4t: 4Suppose the two half-planes has the same temperature rise onthe friction surface, and then the ratio of heat-flow entering thetwo half-planes is given asqsqdpqscskspDT=4tppqdcdkdpDT=4tp qscskspqdcdkdp ； 5where the subscript s and d mean the brake shoe and brake disc,Fig. 3. Contact schematic of two half-planes.Engineeringwhere the subscript l and u represent the lateral surface and the topsurface, respectively； h is the coefficient of convective heat transferon the boundary, DT is the temperature difference between theboundary and the ambient, L is the shorter dimension of theboundary.2.2.3. Initial and boundary conditionContact surface between brake shoe and brake disc is subjectedto continuous heat-flow qsduring emergency braking process.Brake shoes boundaries are of natural convection with the air.The boundary and initial condition can be represented byC0koTor h1T h1T0 f1t； r a； t P0； 0 6u6u0；0 6 z 6 l； 8akoTor h2T h2T0 f2t； r b； t P0； 0 6u6u0；0 6 z 6 l； 8bC0koToz h3T qs h3T0 f3t； z 0； t P0；0 6u6u0； a 6 r 6 b； 8ckoToz h4T h4T0 f4t； z l； t P0； 0 6u6u0；a 6 r 6 b； 8dC0k1roTou h5T h5T0 f5t； u 0； t P0； 0 6 z 6 l；a 6 r 6 b； 8ek1roTou h6T h6T0 f6t； u u0； t P0； 0 6 z 6 l；a 6 r 6 b； 8fTr；u；z；tT0； t 0； a 6 r 6 b； 0 6u6u0；0 6 z 6 l； 8gwhere T0is the initial temperature of the brake shoe at t =0.2.3. Integral-transform solving methodIntegral-transform method has two steps for solving the prob-lem. Firstly, only by making suitable integral-transform for spacevariable, the original equation of heat conduction could be simpli-fied as the ordinary differential equation with regard to the timevariable t. Then, by taking inverse transform with regard to thesolution of the ordinary differential equation, the analytic solutionof the temperature field with regard to the space and time vari-ables could be obtained.Integral-transform method is applied to solve Eq. (1) withboundary condition Eq. (8). By integral-transform with regard tothe space variables (z,u,r) in turn, their partial differential couldbe eliminated”. Writing formulas to represent the operation oftaking the inverse transform and the integral-transform with re-gard to z, these are defined byTr；u；z；tX1m1Zbm；zNbmTr；u；bm；t； 9Tr；u；bm；tZl0Zbm；z0C1Tr；u；z0；tdz0； 10934 Z.-c. Zhu et al./Applied Thermalwhere Tr；u；bm；t is the integral-transform of T(r,u,z,t) withregard to z； Z(bm,z) is the characteristic function, Z(bm,z)=cosbm(l C0 z)； bmis the characteristic value, bmtanbml = H3, andH3h3k； N(bm) is the norm,1Nbm 2b2mH23lb2mH23H3.Submit Eq. (10) into Eqs. (1) and (8), the following equations isobtained:o2Tor21roTor1r2o2Tou2f3kcosl C1 bmC0b2mC1 Tr；u；bm；t1aoTr；u；bm；tot； 11aC0koTor h1T C22f1t； r a； t P0； 0 6u6u0； 11bkoTor h2T C22f2t； r b； t P0； 0 6u6u0； 11cC0k1roTou h5T C22f5t； u 0； t P0； a 6 r 6 b； 11dk1roTou h6T C22f6t； u u0； t P0； a 6 r 6 b； 11eTr；u；bm；tZl0Zbm；z0C1T0dz0； t 0；a 6 r 6 b； 0 6u6u0: 11fIn the same way, the inverse transform and the integral-transformwith regard to u and r are defined byTr；u；bm；tX1n1Uvn；uNvneTr；vn；bm；t； 12eTr；vn；bm；tZu00u0C1Uvn；u0C1Tr；u0；bm；tdu0； 13whereeTr；vn；bm；t is the integral-transform of Tr；u；bm；t with re-gard to u； U(vn,u) is the characteristic function, U(vn,u)=vnC1 cosvnu +H5C1 sinvnu； vnis the characteristic value, tanvnu0vnH5H6v2nC0H5H6H5h5k；H6h6k； N(vn) is the norm,1Nvn2 v2nH25C1 u0H6v2nH26C16C17H5hiC01.eTr；vn；bm；tX1i1Rvci；rNcieTvci；vn；bm；t； 14eTvci；vn；bm；tZbaRvci；r0C1eTr0；vn；bm；tdr0； 15whereeTvci；vn；bm；t is the integral-transform ofeTr；vn；bm；t withregard to r； Rv(ci,r) is the characteristic function, Rv(ci,r)=SvC1Jv(ciC1 r) C0 VvC1 Yv(ciC1 r), Jv(ciC1 r) and Yv(ciC1 r) are the Bessel functionsof the first and second kind with order v, whereSvciC1Y0vciC1bH2C1YvciC1b； UvciC1J0vciC1aC0H1C1JvciC1a；VvciC1J0vciC1bH2C1JvciC1b； WvciC1Y0vciC1aC0H1C1YvciC1a；ciis the characteristic value which satisfies the equation UvC1 SvC0WvC1 Vv=0； N(ci) is the norm,1Ncip22c2iU2vB2C1U2vC0B1C1V2v, where B1 H21c2i1 C0v=cia2C138 and B2 H22c2i1 C0v=cib2C138.Finally, according to the above integral-transform, Eqs. (1) and(8) can be simplified as follows:deTvdtab2mc2ieTv Aci；vn；bm；t； t 0； 16av v29 (2009) 932937eTci；vn；bm；teT0； t 0； 16bwhere A(ci,vn,bm,t)=g1+ g2+ g3,g1 aC1b C1 Rvci；bkC1eC22f2a C1 Rvci；akC1eC22f1C18C19；g2ZbavkC1C22f5C1 r2C1 Rvci；rdr Zbav C1cosvnu0 H5C1sinvnu0kC1C22f6C1 r2C1 Rvci；rdr；g3Zbaf3kC1 cosl C1bmC1 sinvnbmH5v1C0 cosvnbmC20C21C1 rC1 Rvci；rdr:The solutioneTvci；vn；bm；t can be gained by solving the Eq. (16).Bytaking the inverse transform with regard toeTvci；vn；bm；t accordingto Eqs. (9), (12) and (14), the analytic solution of brake shoes 3-Dtransient temperature field is obtainedTr；u；z；tX1m1X1n1X1i1Zbm；zNbmUvn；uNvnRvci；rNcieC0ab2mc2itC1eTv0Zt0eC0ab2mt0Aci；vn；bm；tdt02435: 17field is carried out with t0= 7.23 s. The change rules of temperaturefield and internal temperature gradient are analyzed. Whatsshown in Figs. 59 are partial simulation results.What is shown in Fig. 5 is brake shoes 3-D temperature fieldwhen time is 7.23 s. It is seen from Fig. 5 that the highest temper-ature of the brake shoe is 396.534 K after braking, and its lowesttemperature is 293 K. And the heat energy is mainly concentratedFig. 5. 3-D temperature field of brake shoe (t = 7.23 s).Fig. 6. The change of temperature on friction surface with time t.Z.-c. Zhu et al./Applied Thermal Engineering 29 (2009) 932937 935Fig. 4. Half section view of brake shoes sample.Table 1Basic parameters of brake pair and the emergency braking conditionq(kg mC03)c(J kgC01KC01)k(W mC01KC01)T0(K)v0(m sC01)p(MPa)l3. Simulation and experimentFig. 4 shows the half section view of brake shoe sample. Line cand d are the center line and bottom line of the cross section,respectively. The sample dimension is: a = 137.5 mm, b = 162.5 mm,u0= 1/6 rad, l = 6 mm. The material of brake shoe and brake discare asbestos-free and 16Mn, respectively. Their parameters andthe condition of emergency braking are shown in Table 1.Suppose that the friction coefficient and the specific pressureare constant during emergency braking process. Based on theabove analytic model, simulation of brake shoes 3-D temperatureBrake shoe 2206 2530 0.295 293 10 1.38 0.4Brake disc 7866 473 53.2 12.5 1.58Fig. 7. The change of temperature on line d with time t.creases all the time when zP0.0006 m. Once the z is up to0.002 m, the difference in temperature during brake is less than3 K. It indicates that the heat energy focuses on the thermal effectlayer, and its thickness is about 0.002 m.In order to prove the analytic model, experiments were carriedout on the friction tester in Fig. 10. The experimental principle is asfollows: when the brake begins, two brake shoes are pushed tobrake the disc with certain pressure p and the temperature of pointe on the friction surface is measured by thermocouple. Because thespecimen thickness is too thin and the structure of the friction tes-ter is limited, it is difficult to fix the thermocouple in the brakeshoe. Therefore, the thermocouple is fixed directly on the brakedisc which is closed to point e shown in Fig. 10. Fig. 11 showsthe temperatures change rules at point e under two situations ofemergency braking.From Fig. 11, it is observed that the temperature at point e in-at first, then decreases； the highest temperature by simula-is lower than and also lags behind the experimental data. In11a, the simulation temperature reaches the maximumK at 3.6 s while the experimental data comes up to the435.65 K at 3.8 s. In Fig. 11b, the simulation resultthe maximum 469.55 K at 4.5 s while the experimentalcomes up to 479.68 K at 5 s. It is seen from Fig. 11, the temper-measured by experiment is lower than simulation results atEngineering 29 (2009) 932937Fig. 8. The change of temperature gradient on line c with time t.936 Z.-c. Zhu et al./Applied Thermalon the layer of friction surface (named thermal effect layer), whichindicates the thermal diffusibility of the brake shoe is poor. In or-der to mater the temperature change rules of friction surface dur-ing emergency braking process, the variation of friction surfacestemperature with time t is simulated. What is shown in Fig. 6 re-veals that the temperature of friction surface increases firstly, thendecreases. This is because that the speed of brake disc is high in thebeginning and this results in large heat-flow while the coefficientof convective heat transfer is low on the boundary at the moment,so the temperature increases； at the late stage of brake the heat-flow decreases with the speed while the coefficient of convectiveheat transfer is high due to large difference in temperature onthe boundary, which leads to decreasing in temperature. Figs. 6and 7 reflect the temperature change rules in the radial dimension:the temperature at the outside of brake shoe is higher than that in-side, and the outside temperature changes more greatly.Fig. 8 demonstrates the change rules of the temperature gradi-ent along the direction z. The highest temperature gradient of thefriction layer is up to 3.739 C2 105K/m and decreases sharply alongthe direction z. The lowest value is only 4.597 C2 10C011K/m. In thebeginning the temperature gradient of thermal effect layer is thehighest while the temperature is close to the surrounding temper-ature. As the brake goes on, the temperature gradient decreasesgradually until the end. Fig. 9 shows the change of temperatureat different depth on the line c with time t. The temperature de-creases sharply with the increasing z, and the boundary conditionhas litter influence on the inner temperature. The temperature in-then it inverses. This is because the thermocouple itself ab-heat energy from the brake shoe in the beginning, then re-to the brake shoe when the temperature decreases.on between the experimental data and the simulation re-indicates that the simulation shows good agreement with thent, and the errors of their highest temperature are 1.99%Fig. 9. The change of temperature at different depth on the line c with time t.Fig. 10. Schematic of friction tester.creasestionFig.427.14maximumreachesdataaturefirst,sorbsleasesComparissultsexperimeFig. 11a. Temperatures change rules at point e with time t (p = 1.38 MPa, v0=1-0 m/s).beginning the temperature gradient of thermal effect layerchange rules of brake shoes 3-D transient temperature fieldduring emergency braking.AcknowledgementsThis project is supported by the Key Project of Chinese Ministryof Education (Grant No. 107054) and Program for New CenturyExcellent Talents in University (Grant No. NCET-04-0488).Z.-c. Zhu et al./Applied Thermal Engineering 29 (2009) 932937 937was the highest, the temperature increased swiftly； as thebraking process going on, the temperature gradientdecreased while the temperature increased； the boundaryand 2.16%, respectively. It indicates that the analytic solution of 3-D transient temperature field is correct.4. Conclusion(1) The theoretical model of 3-D transient temperature field wasestablished according to the theory of heat conduction andthe emergency braking condition of mining hoist. The inte-gral-transform method was applied to solve the theoreticalmodel, and the analytic solution of temperature field wasdeduced. It indicates that integral-transform method iseffective to solve the problem of 3-D transient temperaturefield with regard to cylindrical coordinates.(2) Based on the analytic solution of the theoretical model, thenumerical analysis was adopted to simulate the change rulesof temperature distribution under the emergency brakingcondition. Simulation results showed: the temperature offriction surface increased firstly and then decreased； in theFig. 11b. Temperatures change rules at point e with time t (p = 1.58 MPa, v0=1-2.5 m/s).condition had litter influence on the internal temperaturerise； the heat energy was concentrated on the thermal effectlayer and its thickness is about 2 mm.(3) The experimental data has good agreement with the simula-tion results, and the errors of their highest temperature areabout 2%, which prove the correctness of the integral-trans-form method solving the theoretical model of 3-D transienttemperature field. The analytical model can reflect theReferences1 Y. Yang, J.M. Zhou, Numerical simulation study of 3-D thermal stress field withcomplex boundary, Journal of Engineering Thermophysics 27 (3) (2007)487489.2 L. Li, J. Song, Z.Y. Guo, Study on fast finite element simulation model of thermalanalysis of vehicle brake, Journal of System Simulation 17 (12) (2005) 28692872. 2877.3 C.H. Gao, X.Z. 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