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Three-dimensional transient temperature fi eld of brake shoe during hoists 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 shoes temperature fi eld during hoists 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 hoists 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 hoists emergency braking condition, the experiments for measuring brake shoes 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 hoists 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 13,6,10,11. The previous work focused on the brake pads temperature fi eld 14,10,12,13. 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 510. Therefore, it is necessary to investigate the brake shoes temperature fi eld with respect to investigating brake pads. Current theoretical models of brake shoes 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 pads 3-D transient temperature fi eld concentrated on fi nite element method 13,1417, approx- imate integration method 4,18, Greens 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 shoes temperature fi eldduringhoistsemergencybrakingandimprovethesafereliabil- 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 hoists braking friction pair. In or- der to analyze brake shoes 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.: +86 fax: +86 516 83590708. E-mail address: pengyuxing (Y.-x. Peng). Applied Thermal Engineering 29 (2009) 932937 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, its seen that a 6 r 6 b, 0 6 u 6 u0, 0 6 z 6 l. It is clear that the brake shoes 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 qsr;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 1:42DTl=Ll 1 4; 7a hu 0:59DTu=Lu 1 4; 7b Fig. 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) 932937933 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 shoes boundaries are of natural convection with the air. The boundary and initial condition can be represented by ? k oT or h1T h1T0 f1t;r a; t P 0; 0 6u6u0; 0 6 z 6 l;8a k oT or h2T h2T0 f2t;r b; t P 0; 0 6u6u0; 0 6 z 6 l;8b ? k oT oz h3T qs h3T0 f3t;z 0; t P 0; 0 6u6u0; a 6 r 6 b;8c k oT oz h4T h4T0 f4t;z l; t P 0; 0 6u6u0; a 6 r 6 b;8d ? k 1 r oT ou h5T h5T0 f5t;u 0; t P 0; 0 6 z 6 l; a 6 r 6 b;8e k 1 r oT ou h6T h6T0 f6t;uu0; t P 0; 0 6 z 6 l; a 6 r 6 b;8f Tr;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 Tr;u;z;t X 1 m1 Zbm;z Nbm Tr;u;bm;t;9 Tr;u;bm;t Z l 0 Zbm;z0 ? Tr;u;z0;tdz0;10 where Tr;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 Nbm 2 b2mH2 3 lb2mH2 3H3. 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 cosl ? bm ? b2 m? Tr;u;bm;t 1 a oTr;u;bm;t ot ;11a ?k oT or h1T ?f1t;r a; t P 0; 0 6u6u0;11b k oT or h2T ?f2t;r b; t P 0; 0 6u6u0;11c ?k 1 r oT ou h5T ?f5t;u 0; t P 0; a 6 r 6 b;11d k 1 r oT ou h6T ?f6t;uu0; t P 0; a 6 r 6 b;11e Tr;u;bm;t Z l 0 Zbm;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 Tr;u;bm;t X 1 n1 Uvn;u Nvn e Tr;vn;bm;t;12 e Tr;vn;bm;t Z u0 0 u0?Uvn;u0 ? Tr;u0;bm;tdu0;13 where e Tr;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, tanvnu0 vnH5H6 v2 n?H5H6 H5 h5 k ;H6 h6 k; N(vn) is the norm, 1 Nvn2 v 2 nH 2 5? u0 H6 v2 nH 2 6 ? H5 hi?1 . e Tr;vn;bm;t X 1 i1 Rvci;r Nci e T v ci;vn;bm;t;14 e T v ci;vn;bm;t Z b a Rvci;r0 ? e Tr0;vn;bm;tdr0;15 where e T v ci;vn;bm;t is the integral-transform of e Tr;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 Svci?Y0vci?bH2?Yvci?b;Uvci?J0vci?a?H1?Jvci?a; Vvci?J0vci?bH2?Jvci?b;Wvci?Y0vci?a?H1?Yvci?a; ci is the characteristic value which satisfi es the equation Uv? Sv? Wv? Vv= 0; N(ci) is the norm, 1 Nci p2 2 c2 iU 2 v B2?U2v?B1?V2v, where B1 H2 1 c2 i1 ? v=cia 2? and B 2 H 2 2c 2 i1 ? 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 ab2 mc 2 i e T v Aci;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) 932937 g1a? b ? Rvci;b k ? e? f2 a ? Rvci;a k ? e? f1 ? ; g2 Z b a v k ?f5? r2? Rvci;rdr Z b a v ? cosvnu0 H5? sinvnu0 k ?f6 ? r2? Rvci;rdr; g3 Z b a f3 k ? cosl ? bm ? sinvnbm H5 v 1 ? cosvnbm ? ? r ? Rvci;rdr: 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 shoes 3-D transient temperature fi eld is obtained Tr;u;z;t X 1 m1 X 1 n1 X 1 i1 Zbm;z Nbm Uvn;u Nvn Rvci;r Nci e?ab 2 mc2it ? e T v 0 Z t 0 e?ab 2 mt0Aci;vn;bm;tdt0 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 shoes 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. Whats shown in Figs. 59 are partial simulation results. What is shown in Fig. 5 is brake shoes 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 shoes 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) 932937935 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 surfaces 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 show