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Three dimensional transient temperature fi eld of brake shoe during hoist s emergency braking Zhen cai Zhu Yu xing Peng Zhi yuan Shi Guo an Chen College of Mechanical and Electrical Engineering China University of Mining and Technology Xuzhou 221116 China a r t i c l ei n f o Article history Received 22 November 2007 Accepted 27 April 2008 Available online 6 May 2008 Keywords Brake shoe Three dimensional Transient temperature fi eld Integral transform method Emergency braking Hoist a b s t r a c t In order to exactly master the change rules of brake shoe s temperature fi eld during hoist s emergency braking the theoretical model of three dimensional 3 D transient temperature fi eld was established according to the theory of heat conduction the law of energy transformation and distribution and the operating condition of mining hoist s emergency braking An analytic solution of temperature fi eld was deduced by adopting integral transform method Furthermore simulation experiments of temperature fi eld were carried out and the variation regularities of temperature fi eld and internal temperature gradi ent were obtained At the same time by simulating hoist s emergency braking condition the experiments for measuring brake shoe s temperature were also conducted It is found by comparing simulation results with experimental data that the 3 D transient temperature fi eld 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 Introduction The hoist s emergency braking is a process of transforming mechanical energy into frictional heat energy of brake pair The emergency braking process of mining hoist has the characteristic of high speed and heavy load and this situation is worse than brak ing condition of vehicle train and so on 1 3 6 10 11 The previous work focused on the brake pad s temperature fi eld 1 4 10 12 13 Especially because the brake shoe is fi xed during the process of emergency braking so there is more intense temperature rise in brake shoe The brake shoe is kind of composite material and the temperature rise resulting from frictional heat energy is the most important factor affecting tribological behavior of brake shoe and the braking safety performance 5 10 Therefore it is necessary to investigate the brake shoe s temperature fi eld with respect to investigating brake pad s Current theoretical models of brake shoe s temperature fi eld are based 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 and brakes during operation Naji 12 established one dimensional mathematical model to describe the thermal behavior of a brake system Yevtushenko and Ivanyk 13 deduced the transient tem perature fi eld for an axi symmetrical heat conductivity problem with 2 D coordinates It is diffi cult for these models to refl ect the real temperature fi eld of brake shoe with 3 D geometry The methods solving brake pad s 3 D transient temperature fi eld concentrated on fi nite element method 1 3 14 17 approx imate integration method 4 18 Green s function method 12 and Laplace transformation method 9 13 etc The former three methods are numerical solution methods and are of low relative accuracy For example fi nite element method can solve the com plicate heat conduction problem but the accuracy of computa tional solution is relatively low which is affected by mesh density step length and so on Though the Laplace transformation method is an analytic solution method it is diffi cult to solve the equation of heat conduction with complicated boundaries There fore the analytic solution called integral transform method is adopted 19 because it is suitable for solving the problem of non homogeneous transient heat conduction In order to master the change rules of brake shoe s temperature fi eldduringhoist semergencybrakingandimprovethesafereliabil ity of braking a 3 D transient temperature fi eld of the brake shoe was studied based on integral transform method and the validity is proved by numerical simulation and experimental research 2 Theoretical analysis 2 1 Theoretical model Fig 1 shows the schematic of hoist s braking friction pair In or der to analyze brake shoe s 3 D temperature fi eld the cylindrical coordinates r u z is adopted to describe the geometric structure shown in Fig 2 where r is the distance between a point of brake shoe and the rotation axis of brake disc u is the central angle z 1359 4311 see front matter 2008 Elsevier Ltd All rights reserved doi 10 1016 j applthermaleng 2008 04 022 Corresponding author Tel 86fax 86 516 83590708 E mail address pengyuxing Y x Peng Applied Thermal Engineering 29 2009 932 937 Contents lists available at ScienceDirect Applied Thermal Engineering journal homepage is the distance between a point of brake shoe and the friction sur face As for the geometric structure and parameters shown in Fig 2 it s seen that a 6 r 6 b 0 6 u 6 u0 0 6 z 6 l It is clear that the brake shoe s temperature T is the function of the cylindrical coor dinates r u z and the time t According to the theory of heat conduction the differential equation of 3 D transient heat conduc tion is gained as follows o2T or2 1 r oT or 1 r2 o2T ou2 o2T oz2 1 a oT ot 1 whereais the thermal diffusivity a k q c k is the thermal con ductivity q is the density c is the specifi c heat capacity 2 2 Boundary condition 2 2 1 Heat fl ow and its distribution coeffi cient It is diffi cult for friction heat generated during emergency brak ing to emanate in a short time so it is almost totally absorbed by brake pair As the brake shoe is fi xed 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 real temperature fi eld of the brake shoe during emergency braking the heat fl ow and its distribution coeffi cient of friction surface must be determined with accuracy According to the operating condition of emergency braking suppose that the velocity of brake disc decreased linearly with time the heat fl ow is obtained with the form qs r t k l p v0 1 t t0 k l p w0 r 1 t t0 2 where q is the heat fl ow of friction surface p is the specifi c pressure betweenbrakepair v0andw0istheinitiallinearandangularvelocity of the brake disc l isthe frictioncoeffi cient betweenbrakepair t0is the whole braking time k is the distribution coeffi cient of heat fl ow Suppose the frictional heat is totally transferred to the brake shoe and brake disk and the distribution coeffi cient of heat fl ow is 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 with the form DT q k ffi ffi ffiffi p p ffi ffi ffi ffi ffi ffi ffiffi 4at p q ffiffi ffiffi ffiffi ffiffi ffiffi ffiffi pqck p ffiffi ffiffi ffi 4t p 3 where q is the heat fl ow absorbed by half plane And the heat fl ow is gained from Eq 3 q ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffiffi pqck p DT ffi ffi ffi ffiffi 4t p 4 Suppose the two half planes has the same temperature rise on the friction surface and then the ratio of heat fl ow entering the two half planes is given as qs qd ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffiffi pqscsks p DT ffi ffi ffi ffiffi 4t p ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffiffi pqdcdkd p DT ffi ffi ffi ffiffi 4t p ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi qscsks p ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffiffi qdcdkd p 5 where the subscript s and d mean the brake shoe and brake disc respectively According to Eq 5 the distribution coeffi cient of heat fl ow entering brake shoe is obtained with the form k qs qa qs qs qd 1 qd qs qd 1 1 qs qd 1 1 1 1 qscsks qdcdkd 1 2 6 2 2 2 Coeffi cient of convective heat transfer on the boundary With regard to the lateral surface and the top surface of the brake shoe their coeffi cients of convective heat transfer are ob tained respectively according to the natural heat convection boundary condition of upright plate and horizontal plate hl 1 42 DTl Ll 1 4 7a hu 0 59 DTu Lu 1 4 7b Fig 1 Schematic of hoist s 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 932 937933 where the subscript l and u represent the lateral surface and the top surface respectively h is the coeffi cient of convective heat transfer on the boundary DT is the temperature difference between the boundary and the ambient L is the shorter dimension of the boundary 2 2 3 Initial and boundary condition Contact surface between brake shoe and brake disc is subjected to continuous heat fl ow qsduring emergency braking process Brake shoe s boundaries are of natural convection with the air The boundary and initial condition can be represented by k oT or h1T h1T0 f1 t r a t P 0 0 6u6u0 0 6 z 6 l 8a k oT or h2T h2T0 f2 t r b t P 0 0 6u6u0 0 6 z 6 l 8b k oT oz h3T qs h3T0 f3 t z 0 t P 0 0 6u6u0 a 6 r 6 b 8c k oT oz h4T h4T0 f4 t z l t P 0 0 6u6u0 a 6 r 6 b 8d k 1 r oT ou h5T h5T0 f5 t u 0 t P 0 0 6 z 6 l a 6 r 6 b 8e k 1 r oT ou h6T h6T0 f6 t u u0 t P 0 0 6 z 6 l a 6 r 6 b 8f T r u z t T0 t 0 a 6 r 6 b 0 6u6u0 0 6 z 6 l 8g where T0is the initial temperature of the brake shoe at t 0 2 3 Integral transform solving method Integral transform method has two steps for solving the prob lem Firstly only by making suitable integral transform for space variable the original equation of heat conduction could be simpli fi ed as the ordinary differential equation with regard to the time variable t Then by taking inverse transform with regard to the solution of the ordinary differential equation the analytic solution of the temperature fi eld with regard to the space and time vari ables could be obtained Integral transform method is applied to solve Eq 1 with boundary condition Eq 8 By integral transform with regard to the space variables z u r in turn their partial differential could be eliminated Writing formulas to represent the operation of taking the inverse transform and the integral transform with re gard to z these are defi ned by T r u z t X 1 m 1 Z bm z N bm T r u bm t 9 T r u bm t Z l 0 Z bm z0 T r u z0 t dz0 10 where T r u bm t is the integral transform of T r u z t with regardtoz Z bm z isthecharacteristicfunction Z bm z cosbm l z bmis the characteristic value bmtanbml H3 and H3 h3 k N bm is the norm 1 N bm 2 b2m H2 3 l b2m H2 3 H3 Submit Eq 10 into Eqs 1 and 8 the following equations is obtained o2T or2 1 r oT or 1 r2 o2T ou2 f3 k cos l bm b2 m T r u bm t 1 a oT r u bm t ot 11a k oT or h1T f1 t r a t P 0 0 6u6u0 11b k oT or h2T f2 t r b t P 0 0 6u6u0 11c k 1 r oT ou h5T f5 t u 0 t P 0 a 6 r 6 b 11d k 1 r oT ou h6T f6 t u u0 t P 0 a 6 r 6 b 11e T r u bm t Z l 0 Z bm z0 T0dz0 t 0 a 6 r 6 b 0 6u6u0 11f In the same way the inverse transform and the integral transform with regard to u and r are defi ned by T r u bm t X 1 n 1 U vn u N vn e T r vn bm t 12 e T r vn bm t Z u0 0 u0 U vn u0 T r u0 bm t du0 13 where e T r vn bm t is the integral transform of T r 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 tanvnu0 vn H5 H6 v2 n H5H6 H5 h5 k H6 h6 k N vn is the norm 1 N vn 2 v 2 n H 2 5 u0 H6 v2 n H 2 6 H5 hi 1 e T r vn bm t X 1 i 1 Rv ci r N ci e T v ci vn bm t 14 e T v ci vn bm t Z b a Rv ci r0 e T r0 vn bm t dr0 15 where e T v ci vn bm t is the integral transform of e T r vn bm t with regard 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 functions of the fi rst and second kind with order v where Sv ci Y0v ci b H2 Yv ci b Uv ci J0v ci a H1 Jv ci a Vv ci J0v ci b H2 Jv ci b Wv ci Y0v ci a H1 Yv ci a ci is the characteristic value which satisfi es the equation Uv Sv Wv Vv 0 N ci is the norm 1 N ci p2 2 c2 iU 2 v B2 U2v B1 V2v where B1 H2 1 c2 i 1 v cia 2 and B 2 H 2 2 c 2 i 1 v cib 2 Finally according to the above integral transform Eqs 1 and 8 can be simplifi ed as follows d e T v dt a b2 m c 2 i e T v A ci vn bm t t 0 16a e T v ci vn bm t e T v 0 t 0 16b where A ci vn bm t g1 g2 g3 934Z c Zhu et al Applied Thermal Engineering 29 2009 932 937 g1 a b Rv ci b k e f2 a Rv ci a k e f1 g2 Z b a v k f5 r2 Rv ci r dr Z b a v cosvnu0 H5 sinvnu0 k f6 r2 Rv ci r dr g3 Z b a f3 k cos l bm sinvnbm H5 v 1 cosvnbm r Rv ci r dr The solution e T v ci vn bm t can be gained by solving the Eq 16 By taking the inverse transform with regard to e T v ci vn bm t according to Eqs 9 12 and 14 the analytic solution of brake shoe s 3 D transient temperature fi eld is obtained T r u z t X 1 m 1 X 1 n 1 X 1 i 1 Z bm z N bm U vn u N vn Rv ci r N ci e a b 2 m c2i t e T v 0 Z t 0 e ab 2 mt0A ci vn bm t dt0 2 4 3 5 17 3 Simulation and experiment Fig 4 shows the half section view of brake shoe sample Line c and 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 disc are asbestos free and 16Mn respectively Their parameters and the condition of emergency braking are shown in Table 1 Suppose that the friction coeffi cient and the specifi c pressure are constant during emergency braking process Based on the above analytic model simulation of brake shoe s 3 D temperature fi eld is carried out with t0 7 23 s The change rules of temperature fi eld and internal temperature gradient are analyzed What s shown in Figs 5 9 are partial simulation results What is shown in Fig 5 is brake shoe s 3 D temperature fi eld when 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 lowest temperature is 293 K And the heat energy is mainly concentrated Fig 4 Half section view of brake shoe s sample Table 1 Basic parameters of brake pair and the emergency braking condition q kg m 3 c J kg 1K 1 k W m 1K 1 T0 K v0 m s 1 p MPa l Brake shoe220625300 295293101 380 4 Brake disc786647353 212 51 58 Fig 5 3 D temperature fi eld 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 932 937935 on the layer of friction surface named thermal effect layer which indicates 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 surface s temperature with time t is simulated What is shown in Fig 6 re veals that the temperature of friction surface increases fi rstly then decreases This is because that the speed of brake disc is high in the beginning and this results in large heat fl ow while the coeffi cient of convective heat transfer is low on the boundary at the moment so the temperature increases at the late stage of brake the heat fl ow decreases with the speed while the coeffi cient of convective heat transfer is high due to large difference in temperature on the boundary which leads to decreasing in temperature Figs 6 and 7 refl ect 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 the friction layer is up to 3 739 105K m and decreases sharply along the direction z The lowest value is only 4 597 10 11K m In the beginning the temperature gradient of thermal effect layer is the highest while the temperature is close to the surrounding temper ature As the brake goes on the temperature gradient decreases gradually until the end Fig 9 shows the change of temperature at different depth on the line c with time t The temperature de creases sharply with the increasing z and the boundary condition has litter infl uence on the inner temperature The temperature in creases all the time when z P 0 0006 m Once the z is up to 0 002 m the difference in temperature during brake is less than 3 K It indicates that the heat energy focuses on the thermal effect layer and its thickness is about 0 002 m In order to prove the analytic model experiments were carried out on the friction tester in Fig 10 The experimental principle is as follows when the brake begins two brake shoes are pushed to brake the disc with certain pressure p and the temperature of point e on the friction surface is measured by thermocouple Because the specimen thickness is too thin and the structure of the friction tes ter is limited it is diffi cult to fi x the thermocouple in the brake shoe Therefore the thermocouple is fi xed directly on the brake disc which is closed to point e shown in Fig 10 Fig 11 shows the temperature s change rules at point e under two situations of emergency braking From Fig 11 it is observed that the temperature at point e in creases at fi rst then decreases the highest temperature by simula tion is lower than and also lags behind the experimental data In Fig 11a the simulation temperature reaches the maximum 427 14 K at 3 6 s while the experimental data comes up to the maximum 435 65 K at 3 8 s In Fig 11b the simulation result reaches the maximum 469 55 K at 4 5 s while the experimental data 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 at fi rst 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 t