ANALYTICAL STUDY OF THE TEMPERATURE DISTRIBUTION IN SOLIDS SUBJECTED TO NON-UNIFORM MOVING HEAT SOURCES

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1 Hamraoui, M., et al.: nalytical Study of the emperature Distribution in... HERML SCIENCE: Year 013, Vol. 17, No. 3, pp NLYICL SUDY OF HE EMPERURE DISRIUION IN SOLIDS SUJECED O NON-UNIFORM MOVING HE SOURCES by Mohamed HMROUI a,*, Mounir CHIKI b, Najib LRQI b, and Luis ROSEIRO c a Higher School of echnology, University Hassan II, Oasis, Casablanca, Morocco b Paris West University LIE, GE, Nanterre, France c Mechanical Engineering Department, Coimbra Institute of Engineering, Coimbra, Portugal Original scientific paper DOI: 10.98/SCI H We propose in this paper an analytical study of the temperature distribution in a solid subjected to moving heat sources. he power dissipated by the heat sources is considered non-uniform. he study was made in steady-state. he model is 3-D. It is valid regardless of the relative velocity of the source. We have considered three cases of semi-elliptic distribution of the power : (1) the maximum at the center of the source, () the maximum at the inlet of the source, (3) the maximum at the output of the source. hese configurations simulate the conformity imperfection of contact due to wear and/or the non-uniformity of contact pressure in frictional devices. We compare the temperature change for these different scenarios and for different relative velocities, considering the same total power dissipation. he reference case is that of a uniform source dissipating the same power. Key words: analytical solution, advection-diffusion problem, non-uniform moving heat sources, sliding contact Introduction Mechanical devices such as brake discs, ball or roller bearings, gears, and many others, are subjected to friction, which generates a heat flux that can reach a few MW/m. his phenomenon leads to high temperature rises which causes damage of materials. he calculation of temperature in solids subjected to frictional heating has been of great scientific interest over the past few decades. Since the pioneering works [1-5] concerning semi-infinite solids which are adiabatic outside the region heated by a moving heat source, many other research studies have been conducted. hey deal cooled semi-infinite bodies [6] and rotating cylinder subjected to a heat source and surface cooling [6-13]. In certain industrial applications, the solids are provided surface coating. Some studies have been carried out to analyse the effect of surface coating on the thermal behaviour of a solid subjected to friction process [14]. hese research studies showed the evolution of flash temperatures (notion introduced in [15]) according to the main characteristic parameters, i. e., Peclet and iot numbers. Peclet number is defined as: Pe = V/a, where V is the relative velocity, the characteristic length, and a the thermal diffusivity of the material. iot number is defined as: i = hl/l, where h is the convective heat transfer coefficient, L the characteristic length, and l the thermal conductivity of the material. * Corresponding author; hamraoui@est-uhc.ac.ma or hamraoui@hotmail.com

2 Hamraoui, M., et al.: nalytical Study of the emperature Distribution in HERML SCIENCE: Year 013, Vol. 17, No. 3, pp he theory of moving sources has long been used by several authors to simulate conditions of friction. he analytical solutions are often difficult to get because of: (1) the non-homogeneous boundary conditions (localized heating and surface cooling), () the relative motion (especially for the low values of Pe), and (3) the small size of the contact region in relation to other dimensions of solid. he dimensionless parameter (the Peclet number) is generally introduced to characterize the thermal behaviour of solids subjected to moving heat sources. When this parameter is lower than a certain threshold, the thermal behaviour is comparable to that of a fixed source, which is largely studied in the literature. When this number is higher than another given threshold, the terms of heat diffusion in the source plane become negligible what allows determining analytical solutions. he great difficulty of the analytical computation of temperatures relates to intermediate values of the Pe ranging between these two thresholds. his problem was solved in [16] for a single source and in [17, 18] for multiple sources. Some other configurations involving heat sources have been studied in the literature [19, 0]. From available studies, it is clearly shown that the increase of the Pe involves a decrease in temperature elevation over the contact region for a given heat flux. One of the difficulties encountered in the application of moving sources is the unknowledge of the spatial distribution of the power dissipated by these sources. Distributions of temperature and thermal gradients depend precisely of the spatial distribution of heat sources. Often, the models proposed in the literature consider that the power is dissipated uniformly over the entire area of contact. In practice, the power dissipation can be extremely nonuniform due to: (1) the non-conformity of the contacts, () the wear, (3) defect of the mechanical guiding, etc. his phenomenon generates localized spots of heat, which can damage the material. We propose in this paper an analytical study of the temperature distribution in a solid subjected to a non-uniform moving heat source. he study is made in steady state. he model is 3-D and it is valid regardless of the relative velocity of the source and its size. Mathematical model Governing euations Let us consider an homogenious finite medium thickness e and area as shown in fig. 1. he plans y = are assumed adiabatic. We consider a periodicity condition at the plans x = and x =. he surface z = 0 x a and y b is subjected to a heat source a flux density (x, y). he remainder of this area is assumed adiabatic. he temperature at the abscissa z = e is zero, (x, y, z = e) = 0. he solid is in relative motion respect to the heat source at a constant velocity V in the x-direction. We note l the thermal conductivity and a the thermal diffusivity of the solid. In the referencial related to the heat source, the steady-state temperature of the medium, (x, y, z), is governed by the following euations: x ( ) ( ), (, y, z) (, y, z) y V 0 y z a x x x (, y, z) (, y, z) (1) () 0, y 0 (3) ( x,, z) ( x,, z)

3 Hamraoui, M., et al.: nalytical Study of the emperature Distribution in... HERML SCIENCE: Year 013, Vol. 17, No. 3, pp Figure 1. Medium in relative motion respect to a heat source (case 0: uniform, cases 1 to 3: non-uniform) l xy (, ): for x a, y b, ( x, y, e) 0 (4) z ( x, y, 0) 0: elsewhere In e. (1), the term (V/a)/(/x) presents the advective component of heat transfer in the medium. When this term is zero, only the conduction phenomenon takes place. he mathematical difficulty of this problem occurs when this term is non-zero. We develop an analytical solution of this problem which is valid regardless of the value of the term (V/a)/(/x). General analytical solution aking into account the periodicity conditions respect to x-direction, which is given by e. (), it is appropriate to apply the finite complex integral Fourier transform to es. (1)-(4) as: * 1 jmpx exp d x (5) where j is the imaginary number (j = 1). Considering the adiabatic conditions at the edges y = respect to y-direction, it is appropriate to use the finite cosine integral Fourier transform as: 1 * npy cos d y (6) he application of the integral transforms (5) and (6) to euations (1-4) leads to a simple second order differential euation as: r s mn d dz s mn 0 m n jm V p p p a l d, ( z e) 0 (8) dz ( z 0) he term s mn can be written under the following form: s mn = r mn e j j mn : mn m n m V mpv p p p 4 and j a mn a mp n p tan 1 (7)

4 Hamraoui, M., et al.: nalytical Study of the emperature Distribution in HERML SCIENCE: Year 013, Vol. 17, No. 3, pp he solution of e. (7) reuires distinguishing between the two following cases: m = 0 and n = 0, for which s 00 = 0.hen: z 00 (9) Using the boundary conditions (8), we deduce: ( z e) l 1 b a 00 xy (, ) dd xy (9a) 4 b a Euation (9a) is the 1-D solution, where 00 is the average heat flux density over the solid area ( ). m 0 and/or n 0, for which s mn 0. hen: min mn smn z mn sinh( s mn z (10) he constants mn and mn are determined by using the boundary conditions (8). hat gives: mn sinh[ smn ( e z)] mn lsmn cosh( smne) 1 b a mn ( x,y) e jmpx/ npy cos dd 4 xy b a (10a) Expression of mn depends on the heat flux distribution of the source, (x, y). We give in that follows the exat expression of mn for the flux distributions considered in this study. Finally, we can decompose the solution of e. (7) into four terms such as: m n mn (11) When the expressions of the four terms of e. (11) are determined, we deduce the real temperature, (x, y, z), by using the inverse integral transforms such as: n y * p en cos n0 en 1( for n0) and en ( for n0) (1) jm x e em * p exp m0 (13) hat gives: em 1( for m 0) and em ( for m 0) e z e z e j 00 m m ( x, y, z) ( ) 0sinh[ s ( )] mpx/ 0 0nsinh[ s0n( e z)] e l m1 lsm0cosh( sm0e) m1 ls0n cosh( s 0 ne) npy n y cos 4 cos n1 p mn sinh[ s jm x/ m0 ( e z)] e p e n1 ls0ncosh( s0ne) (14) he notation e {...}denotes the real part. Complete solutions for different flux distributions (x, y) We consider three cases of non-uniform flux distributions (x, y) as shown in fig., knowing that the case of a uniform heat flux (x, y) = 0 is denoted Case 0.

5 Hamraoui, M., et al.: nalytical Study of the emperature Distribution in... HERML SCIENCE: Year 013, Vol. 17, No. 3, pp Case 0: (x, y) = 0 (15.0) x Case 1: xy (, ) 1 1 (15.1) a x a Case : xy (, ) 1 a (15.) x a Case 3: xy (, )3 1 (15.3) a Figure. Studied distributions of heat sources where 0, 1,, and 3 are constants. he application of integral transforms (5) and (6) to the boundary condition (4) leads to expressions of for the four cases studied under the form: Case 0 mpa npb 0 sin sin ab 0 00, mn (15.0a) ( mp)( np) mpa b a 0 sin 0 si m0 n n p b, 0n mp np Case 1 mpa npb J 1 1 sin 1pab 00, mn (15.1a) 4 m( np) Case Case 3 J mpa b, m 1 1 m0 0n 1 a npb sin 4n mpa mpa J1 jh1 pab, mn e jmp a/ npb sin 4 4m( np) mpa mpa a npb J1 jh1 sin b e jmpa/ 0n 4m 4n mpa mpa J1 jh1 3pab, mn 3 e jmp a/ npb sin 4 4m( np) 00 m0, 00 J b 3 mpa mpa jh e 4m 1 1 jmpa/ m0, 0n 3 a npb sin 4n (15.a) (15.3a) where J 1 is the essel function of the first kind of order 1 and H 1 is the Struve function of order 1.

6 Hamraoui, M., et al.: nalytical Study of the emperature Distribution in HERML SCIENCE: Year 013, Vol. 17, No. 3, pp Results and discussions We study the effect of spatial distribution of the heat dissipated by the source on the temperature for different values of Peclet number: Pe = V/a = 0, 0, and 00. he total power is the same for each distribution, that reuires: 1 = = 3 =4 0 /p. he temperature is given in a dimensionless form: + = /( 0 /l). We fixed arbitrarily: =,b=a, e = 0.5, a/ = 0.1 (i. e. ab/ = 1/100). he number of terms necessary to ensure the convergence of series is, in this case, eual to 150. It should be noted that this number decreases the increase of the ratio ab/. Figure 3 shows the change of the surface temperature of each studied case and for the three values of Pe. he heating zone is between 0.1 and 0.1. We can note two tendencies common to all the cases considered here: (1) the maximum surface temperature moves out of the source as the value of the Pe increases, and () the amplitude of the temperature peak (flash temperature) in the zone subjected to the heat source decreases the increase of the value of the Pe. Figure 3. Dimensionless surface temperature + (x, 0, 0) for the four studied cases and for different values of Pe = 0, 0, and 00

7 Hamraoui, M., et al.: nalytical Study of the emperature Distribution in... HERML SCIENCE: Year 013, Vol. 17, No. 3, pp Maximum surface temperatures are able 1. Comparison of maximum surface temperatures summarized in tab. 1. he highest surface temperature at Pe = 0 is obtained + max. Pe Case 0 Case 1 Case Case 3 for the semi-elliptic centered source (case 1). We can note that for Pe = 0 the cases of left and right eccentric sources give logically the same temperature When the value of Pe increases, the eccentric source at the right (case 3) gives the highest surface temperature. his result means that in the case of practical applications, such as brake pads, for example, temperatures will be much higher than the pads carry more at the output contact region. Here, the temperature difference is not significant because the semi-elliptical profile is relatively flat, but in the case of sharp profiles, this difference becomes more marked. Figure 4 shows the temperature change in the depth direction of the material at the abscissa x = 0 and y =0(i. e., at the center of solid) in the case 1. emperatures are plotted for three values of the Pe. We can note that for Pe close to zero, the 3-D effects are still present, but as soon as Pe becomes high, 3-D effects are localized in a thin region, which is located in the vicinity of the heated face (z = 0). eyond this region, the temperature becomes 1-D [ = (z) only]. his occurs at z/ 0. for Pe = 0 and at z/ 006. for Pe = 00. Figure 4. Dimensionless temperature + (0, 0, z) in the depth for the case 1 and for different values of Pe = 0, 0, 00 Conclusions nalytical solutions are presented in this paper to study the influence of non-uniformity of the distribution of heat sources on the thermal behavior of solids in relative motion (friction, welding, laser...). hese solutions are explicit and easy to use, since the functions which they use are available in different software such as Maple, Mathematica, etc. he proposed analytical method can be used for other configurations of the distribution of heat sources that those considered in this study. Nomenclature semi-width of the solid, [m] a semi-width of the source, [m] semi-length of the solid, [m] b semi-length of the source, [m] e thickness of the solid, [m] H Struve function J essel function Pe Peclet number, (= V/l) heat flux density, [Wm ] temperature, [K] + dimensionless temperature (= /( 0 /l) V velocity, [ms 1 ] x, y, z Cartesian co-ordinates, [m] Greek symbols a thermal diffusivity, [m s 1 ] l thermal conductivity, [Wm 1 K 1 ]

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