中雙鏈刮板輸送機(jī)設(shè)計【全套含8張CAD圖紙】,全套含8張CAD圖紙,中雙鏈刮板,輸送,設(shè)計,全套,CAD,圖紙 Three-dimensional transient temperature field of brake shoe during hoistsemergency brakingZhen-cai Zhu, Yu-xing Peng*, Zhi-yuan Shi, Guo-an ChenCollege of Mechanical and Electrical Engineering, China University of Mining and Technology, Xuzhou 221116, Chinaa r t i c l ei n f oArticle history:Received 22 November 2007Accepted 27 April 2008Available online 6 May 2008Keywords:Brake shoeThree-dimensionalTransient temperature fieldIntegral-transform methodEmergency brakingHoista b s t r a c tIn order to exactly master the change rules of brake shoes temperature field during hoists emergencybraking, the theoretical model of three-dimensional (3-D) transient temperature field was establishedaccording to the theory of heat conduction, the law of energy transformation and distribution, and theoperating condition of mining hoists emergency braking. An analytic solution of temperature field wasdeduced by adopting integral-transform method. Furthermore, simulation experiments of temperaturefield were carried out and the variation regularities of temperature field and internal temperature gradi-ent were obtained. At the same time, by simulating hoists emergency braking condition, the experimentsfor measuring brake shoes temperature were also conducted. It is found, by comparing simulation resultswith 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.? 2008 Elsevier Ltd. All rights reserved.1. IntroductionThe hoists emergency braking is a process of transformingmechanical energy into frictional heat energy of brake pair. Theemergency braking process of mining hoist has the characteristicof high speed and heavy load, and this situation is worse than brak-ing condition of vehicle, train and so on 13,6,10,11. The previouswork focused on the brake pads temperature field 14,10,12,13.Especially, because the brake shoe is fixed during the process ofemergency 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-tion 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.The 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 transformationmethod 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.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; z1359-4311/$ - see front matter ? 2008 Elsevier Ltd. All rights reserved.doi:10.1016/j.applthermaleng.2008.04.022* Corresponding author. Tel.: +86 13805209649; fax: +86 516 83590708.E-mail address: (Y.-x. Peng).Applied Thermal Engineering 29 (2009) 932937Contents lists available at ScienceDirectApplied Thermal Engineeringjournal homepage: 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 a 6 r 6 b, 0 6 u 6 u0, 0 6 z 6 l. 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;1whereais the thermal diffusivity,a= k /(q? c); k is the thermal con-ductivity;qis 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;t k ?l? p ? v0? 1 ? t=t0 k ?l? p ? w0? r:1 ? t=t0;2where q is the heat-flow of friction surface; p is the specific pressurebetweenbrakepair;v0andw0istheinitiallinearandangularvelocityof the brake disc;listhe frictioncoefficient betweenbrakepair; 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 qkffiffiffiffippffiffiffiffiffiffiffiffi4atpqffiffiffiffiffiffiffiffiffiffiffiffipqckpffiffiffiffiffi4tp;3where q is the heat-flow absorbed by half-plane. And the heat-flowis gained from Eq. (3)q ffiffiffiffiffiffiffiffiffiffiffiffipqckpDT=ffiffiffiffiffi4tp: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 asqsqdffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffipqscskspDT=ffiffiffiffiffi4tpffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffipqdcdkdpDT=ffiffiffiffiffi4tpffiffiffiffiffiffiffiffiffiffiffiffiffiqscskspffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiqdcdkdp;5where the subscript s and d mean the brake shoe and brake disc,respectively. According to Eq. (5), the distribution coefficient ofheat-flow entering brake shoe is obtained with the formk qsqaqsqs qd 1 ?qdqs qd 1 ?1qsqd 1 1 ?11 qscsksqdcdkd?12:62.2.2. 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;7bFig. 1. Schematic of hoists braking friction pair.Fig. 2. 3-D geometrical model of brake shoe.Fig. 3. Contact schematic of two half-planes.Z.-c. Zhu et al./Applied Thermal Engineering 29 (2009) 932937933where 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 by? koTor h1T h1T0 f1t;r a; t P 0; 0 6u6u0;0 6 z 6 l;8akoTor h2T h2T0 f2t;r b; t P 0; 0 6u6u0;0 6 z 6 l;8b? koToz h3T qs h3T0 f3t;z 0; t P 0;0 6u6u0; a 6 r 6 b;8ckoToz h4T h4T0 f4t;z l; t P 0; 0 6u6u0;a 6 r 6 b;8d? k1roTou h5T h5T0 f5t;u 0; t P 0; 0 6 z 6 l;a 6 r 6 b;8ek1roTou h6T h6T0 f6t;uu0; t P 0; 0 6 z 6 l;a 6 r 6 b;8fTr;u;z;t T0;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;t X1m1Zbm;zNbmTr;u;bm;t;9Tr;u;bm;t Zl0Zbm;z0 ? Tr;u;z0;tdz0;10where Tr;u;bm;t is the integral-transform of T(r,u,z,t) withregardtoz;Z(bm,z)isthecharacteristicfunction,Z(bm,z) =cosbm(l ? 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 ? bm ? b2m? Tr;u;bm;t1aoTr;u;bm;tot;11a?koTor h1T ?f1t;r a; t P 0; 0 6u6u0;11bkoTor h2T ?f2t;r b; t P 0; 0 6u6u0;11c?k1roTou h5T ?f5t;u 0; t P 0; a 6 r 6 b;11dk1roTou h6T ?f6t;uu0; t P 0; a 6 r 6 b;11eTr;u;bm;t Zl0Zbm;z0 ? T0dz0;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;t X1n1Uvn;uNvneTr;vn;bm;t;12eTr;vn;bm;t Zu00u0?Uvn;u0 ? Tr;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) = vn? cosvnu +H5? sinvnu; vnis the characteristic value, tanvnu0vnH5H6v2n?H5H6H5h5k;H6h6k; N(vn) is the norm,1Nvn2 v2nH25?u0H6v2nH26?H5hi?1.eTr;vn;bm;t X1i1Rvci;rNcieTvci;vn;bm;t;14eTvci;vn;bm;t ZbaRvci;r0 ?eTr0;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) = Sv?Jv(ci? r) ? Vv? Yv(ci? r), Jv(ci? r) and Yv(ci? r) are the Bessel functionsof the first and second kind with order v, whereSvci?Y0vci?bH2?Yvci?b;Uvci?J0vci?a?H1?Jvci?a;Vvci?J0vci?bH2?Jvci?b;Wvci?Y0vci?a?H1?Yvci?a;ciis the characteristic value which satisfies the equation Uv? Sv?Wv? Vv= 0; N(ci) is the norm,1Ncip22c2iU2vB2?U2v?B1?V2v, where B1 H21c2i1 ? v=cia2? and B2 H22c2i1 ? v=cib2?.Finally, according to the above integral-transform, Eqs. (1) and(8) can be simplified as follows:deTvdtab2mc2ieTv Aci;vn;bm;t;t 0;16aeTvci;vn;bm;t eTv0;t 0;16bwhere A(ci,vn,bm,t) = g1+ g2+ g3,934Z.-c. Zhu et al./Applied Thermal Engineering 29 (2009) 932937g1a?b ? Rvci;bk?e?f2a ? Rvci;ak?e?f1?;g2Zbavk?f5? r2? Rvci;rdr Zbav ? cosvnu0 H5? sinvnu0k?f6? r2? Rvci;rdr;g3Zbaf3k? cosl ? bm ? sinvnbmH5v1 ? cosvnbm? r? 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;t X1m1X1n1X1i1Zbm;zNbmUvn;uNvnRvci;rNcie?ab2mc2it?eTv0Zt0e?ab2mt0Aci;vn;bm;tdt02435:173. 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 temperaturefield 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. 4. Half section view of brake shoes sample.Table 1Basic parameters of brake pair and the emergency braking conditionq(kg m?3)c(J kg?1K?1)k(W m?1K?1)T0(K)v0(m s?1)p(MPa)lBrake shoe220625300.295293101.380.4Brake disc786647353.212.51.58Fig. 5. 3-D temperature field of brake shoe (t = 7.23 s).Fig. 6. The change of temperature on friction surface with time t.Fig. 7. The change of temperature on line d with time t.Z.-c. Zhu et al./Applied Thermal Engineering 29 (2009) 932937935on 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 ? 105K/m and decreases sharply alongthe direction z. The lowest value is only 4.597 ? 10?11K/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-creases all the time when z P 0.0006 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-creases at first, then decreases; the highest temperature by simula-tion is lower than and also lags behind the experimental data. InFig. 11a, the simulation temperature reaches the maximum427.14 K at 3.6 s while the experimental data comes up to themaximum 435.65 K at 3.8 s. In Fig. 11b, the simulation resultreaches the maximum 469.55 K at 4.5 s while the experimentaldata comes up to 479.68 K at 5 s. It is seen from Fig. 11, the temper-ature measured by experiment is lower than simulation results atfirst, then it inverses. This is because the thermocouple itself ab-sorbs heat energy from the brake shoe in the beginning, then re-leases to the brake shoe when the temperature decreases.Comparison between the experimental data and the simulation re-sults indicates that the simulation shows good agreement with theexperiment, and the errors of their highest temperature are 1.99%Fig. 8. The change of temperature gradient on line c with time t.Fig. 9. The change of temperature at different depth on the line c with time t.Fig. 10. Schematic of friction tester.Fig. 11a. Temperatures change rules at point e with time t (p = 1.38 MPa, v0= 1-0 m/s).936Z.-c. Zhu et al./Applied Thermal Engineering 29 (2009) 932937and 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 thebeginning the temperature gradient of thermal effect layerwas the highest, the temperature increased swiftly; as thebrakingprocessgoingon,thetemperaturegradientdecreased while the temperature increased; the boundarycondition 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 thechange 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).References1 Y. Yang, J.M. Zhou, Numer