PROOF COPY 009304JTQ
An Exact Explicit Analytical Solution of the Steady-State Temperature in a Half Space Subjected to a Moving Circular Heat Source Najib Laraqi e-mail:
[email protected] University of Paris 10, LEEE, EA 387, 1-Chemin Desvallie`res, 92410 Ville d’Avray, France
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So far in the literature, the distribution of stationary temperature over the surface of a half space subjected to a moving circular heat source has been reported in an integral or asymptotic form. In this paper, an exact explicit analytical solution is provided, which allows the determination of the temperatures over the contact area with a very short computational time, regardless of the value of the Peclet number. The solution is based on special functions (Bessel and hypergeometric functions) that are preprogrammed under a formal calculation software (e.g., Maple). The results of the proposed solution are in agreement with the asymptotic models available in the literature. 关DOI: 10.1115/1.1573233兴
Introduction
Proposed Model
We consider a semi-infinite body 共Fig. 1兲 with the thermal conductivity and the diffusivity denoted as k and ␣, respectively, and the zero reference temperature. The surface of this body is subjected to a uniform circular heat source q. The rest of this surface is adiabatic. The body is moving with regard to the heat source at a speed V in the x-direction. The temperature distribution in the body is three-dimensional, T(r, ,z). Re-defining the temperature function as T(r, ,z)⫽⌰(r, ,z)exp关Vr cos()/2␣ 兴 and introducing the dimensionless quantities r * ⫽r/a, and z * ⫽z/a, the governing equations can be written as follows:
⫺k
冉 冊 ⌰ z*
ⵜ 2 ⌰ 共 r * , ,z * 兲 ⫺ P 2e ⌰⫽0 ⫽ z * ⫽0
再
qae
⫺ P e r * cos共 兲
共 r⭐a 兲 共 r⬎a 兲
0
⌰ r * ⫽0 is finite, ⌰ r * →⬁ ⫽0,
冉 冊 ⌰
⫽0,
˜⫽1 ⌰
冕
M⫽ ⌰
⌰ cos共 m 兲 d ,
0
冕
⬁
(2) ⫽0 ⫽
˜ r * J 共  r * 兲 dr * ⌰ m
(3)
0
˜⫽ ⌰
冕
⬁
M J 共 r*兲d, ⌰ m
0
⬁
PY
⌰⫽
兺
m⫽0
˜, ⑀ m cos共 m 兲 ⌰
再
⑀ m ⫽1
共 m⫽0 兲
⑀ m ⫽2
共 m⫽0 兲
(4)
Then, we obtain the expression of temperature T(r * , ,z * ) as: ⬁
qa P r * cos共 兲 e e ⑀ m cos共 m 兲 k m⫽0
兺
00
T⫽
冋冕
⬁
0
册
5 e ⫺␥z G m  J m共  r * 兲 d  ␥ (5)
where
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and
␥ ⫽ 关  2 ⫹ P 2e 兴 1/2
5 ⫽ 共 ⫺1 兲 m P e I m⫹1 共 P e 兲 J m 共  兲 ⫹  I m 共 P e 兲 J m⫹1 共  兲 G m  2 ⫹ P 2e
(6)
(7)
Here J m ( or m⫹1 ) represents the Bessel function of first kind of order m 共or m⫹1) and I m ( or m⫹1 ) is the modified Bessel function of first kind of order m 共or m⫹1).
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2.1 Surface Temperature at the Center of Contact. In order to calculate the surface temperature at the center of contact we may consider two methods. The first one consists of using the expression provided par the Eq. 共5兲. The second one is based on the linear superposition method. The two methods are detailed below. 2.1.1 From the Analytical Solution. Putting r * ⫽0 in Eq. 共5兲, all terms J m (0)⫽0, except for the case m⫽0 for which J 0 (0)⫽1. The series is reduced to the only term m⫽0. Using the table of integrals given in Gradshteyn and Ryzhik 关关11兴, p. 710兴, we get the following simple expression: ⫺ Pe T* 关 I 1 共 P e 兲 ⫹I 0 共 P e 兲兴 c ⫽e
Contributed by the Tribology Division for publication in the ASME JOURNAL OF TRIBOLOGY. Manuscript received by the Tribology Division October 3, 2002 revised manuscript received January 30, 2003. Associate Editor: G. G. Adams.
PROOF COPY 009304JTQ
⌰ z * →⬁ ⫽0,
with the inverse transforms,
Temperature is an important parameter in various tribological applications. Several studies have been dedicated to this subject over the last decades. On a theoretical viewpoint, the predictive models use the moving heat source theory, which is derived from the pioneers, including Boussinesq 关1兴, Rosenthal 关2兴, Bolk 关3兴 and Jaeger 关4兴. This theory consists of calculating the temperature in a finite or semi-infinite body subjected to a heat flux on a region of its surface, with the body moving with regard to this source. The expression for the temperature is given under an integral form, where the controlling parameter is the Peclet number, P e ⫽Va/2␣ , which represents physically the ratio of the heat diffusion characteristic time 共i.e., a 2 / ␣ ) to the transit time of the heat source 共i.e. 2a/V). The particular case of P e ⫽0 共i.e., the stationary heat source兲 has an explicit solution 关5兴. When the Peclet number becomes large 共typically P e ⬎5), a solution may be obtained by neglecting the heat diffusion over the plan of heat source. Then, the heat equation becomes easy to solve and the evolution of temperatures is proportional to 1/冑P e 关3,4,6 – 8兴. For P e in the range of 0 to 5, the evolution of temperatures is determined by a numerical integration of the integral solution 关9,10兴. Recently, Tian and Kennedy 关10兴 proposed simple correlations to determine the steady-state temperature for any value of P e . The range of P e values between 0 and 5 is usually encountered in some Hertzian contacts or at the scale of asperities. Considering, for example, a steel body ( ␣ ⫽5 10⫺6 m2 •s⫺1 ) with an asperity of radius a⫽10⫺6 m, the value of P e ⫽5 requires a very large speed, such as V⫽50 m•s⫺1 . In this paper we provide an exact analytical solution under an explicit form, and present the results in the range 0–5 of Peclet values, knowing that the other values of the Peclet number have been widely investigated in the literature. Some comparisons with available results are also presented.
Journal of Tribology
,
冉 冊 ⌰
⫽0
(1)
To solve this problem we use the finite cosine Fourier transform with respect to -direction and the infinite Hankel transform with respect to r-direction, respectively, as
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1
2
(8)
For the particular case P e ⫽0, we have I 1 (0)⫽0 and I 0 (0) ⫽1. The dimensionless surface temperature at the center of the
Copyright © 2003 by ASME
OCTOBER 2003, Vol. 125 Õ 1
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Fig. 3 Principle of the linear superpostion technic to determine the temperature at the center of source
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point, results in an increase of temperature at the center of contact 共O兲 which is given by Carslaw and Jaeger 关关5兴, p. 267兴 as
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Fig. 1 Scheme of a circular heat source on a moving half space
T 共OP 兲 ⫽
T* c ⫽F
冉
1 ;2;⫺2 P e 2
冊
(10)
where F(1/2;2;⫺2 P e ) is an hypergeometric function which can be expressed as a function of Bessel functions 共Prudnikov et al. 关关13兴, p. 580兴兲 in the form of e ⫺ P e 关 I 0 ( P e )⫹I 1 ( P e ) 兴 . This expression is the same as Eq. 共8兲 obtained by the first method.
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2.2 Surface Temperature on the Contour of the Source. In order to calculate the surface temperature, T ct , over the contour of the source, we put r * ⫽1 in Eq. 共5兲. The dimensionless expression of this temperature may be written as follows:
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2.1.2 From Linear Superposition Method. Referring to Fig. 3, the punctual moving heat source of intensity Q, located at the P
(9)
In order to determine the surface temperature at the center of source, T c , we integrate Eq. 共9兲 over the whole contact area. Using the tables given in Gradshteyn and Ryzhik 关关11兴, p. 517 and p. 735兴, we get the following dimensionless expression:
PY
source then becomes equal to the known value, T * c ( P e →0)⫽1. If P e →⬁, the asymptotic expansions of I 1 ( P e ) and I 0 ( P e ) are the same, e P e / 冑2 P e 共Abramowitz and Stegun 关关12兴, p. 377兴兲, and we find: T c* ( P e →⬁)⫽ 冑2/ P e , which is consistent with the expression given by Tian and Kennedy 关10兴. Figure 2 provides the evolution of the dimensionless surface temperature at the center of the source as a function of P e . This temperature decreases from 1 to 0 as P e increases. The first half of this decrease occurs for P e in the range of 0 to 2.
Q e ⫺V/2␣ 共 ⫺ cos共 兲兲 2k
⬁
* ⫽e P e cos共 兲 T ct
兺
m⫽0
共 ⫺1 兲 m ⑀ m cos共 m 兲关 P e I m⫹1 共 P e 兲 C 1
⫹I m 共 P e 兲 C 2 兴
C 1⫽
C 2⫽
Fig. 2 Dimensionless surface temperature at the center of source versus Pe
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冉
Q
where
(11)
冊
3 3 共 ⫺1 兲 m F 1; ⫹m, ⫺m; P 2e ⫹ I 2m 共 2 P e 兲 2 2 Pe 共 4m ⫺1 兲 (12) 4
2
冉
冊
2 1 3 F 1; ⫺m, ⫹m; P 2e ⫹ 共 ⫺1 兲 m I 2m⫹1 共 2 P e 兲 共 2m⫹1 兲 2 2 (13)
The needed integrals have been determined using the Maple software. In Fig. 4 the temperatures at the contour of the source at ⫽ 共the entrance of the contact region兲 and at ⫽0 共the exit of the contact region兲 are plotted as a function of P e . The two temperatures decrease as P e increases. The decrease of the temperature at ⫽0 is almost linear and progressive 共about 37% of the decrease Transactions of the ASME
PROOF COPY 009304JTQ Table 1 Comparison of average surface temperatures between the exact solution and T&K †10‡ data T *a v 共T&K 关10兴兲num
Pe 0 0.01 0.1 0.3 0.5 0.7 1 1.5 2 2.5 3 3.5 4 4.5 5
共T&K 关10兴兲cor
Eq. 共14兲
0.849 0.842 0.791 0.703 0.640 0.591 0.535 0.469 0.422 0.387 0.360 0.338 0.319 0.303 0.289
0.849 0.844 0.802 0.725 0.663 0.614 0.555 0.485 0.435 0.398 0.369 0.345 0.325 0.308 0.292
0.844 0.802 0.724 0.614 0.556
0.368
0.294
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Fig. 4 Dimensionless temperatures at the enterance and exit of the heat source
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is reached at P e ⫽5). The decrease of the temperature at ⫽ is a lot faster since the smallest values of P e 共about 90% of the decrease is reached at P e ⫽5).
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2.3 Average Surface Temperature Over the Contact. Integrating Eq. 共5兲 we get the average dimensionless temperature over the contact area as ⬁
T* a v ⫽2
兺
m⫽0
2 共 ⫺1 兲 m ⑀ m 关 P 2e I m⫹1 共 P e 兲 D 1 ⫹2 P e I m⫹1 共 P e 兲 I m 共 P e 兲 D 2
2 ⫹I m 共 Pe兲D3兴
PY
(14)
This expression contains three integrals (D 1 ,D 2 ,D 3 ) having the following general form:

⬁
0
共  2 ⫹ P 2e 兲
J 共  兲 J 共  兲 d  5/2
(15)
32F 共 1;1/2⫺m,3/2⫹m; P 2e 兲 2 共 4m 2 ⫺1 兲共 9m 2 ⫺1 兲 ⫹
F 共 1/2⫹m;m⫺1/2,1⫹2m; P 2e 兲 共 ⫺1 兲 m 共 1⫺2m 兲 P 2m⫺3 e 3⌫ 共 2m⫹1 兲 (16)
3 共 4m ⫺1 兲共 2m⫹3 兲 2
⫹
F 共 3/2⫹m;m⫹1/2,2⫹2m; P 2e 兲 共 ⫺1 兲 m 共 2m⫹1 兲 P 2m⫺1 e 3⌫ 共 2m⫹2 兲 (17)
D 3⫽
4F 共 1,5/2;1/2⫺m,3/2,5/2⫹m; P 2e 兲
共 2m⫹1 兲共 2m⫹3 兲 ⫺
2 共 ⫺1 兲 m P 2m⫹1 F 共 3⫹m;2⫹m,3⫹2m; P 2e 兲 e 3⌫ 共 2m⫹3 兲
where ⌫ is the gamma function. Journal of Tribology PROOF COPY 009304JTQ
(18)
Nomenclature a k Pe Q q r, , z T T* V ␣ ⌰
⫽ ⫽ ⫽ ⫽ ⫽ ⫽ ⫽ ⫽ ⫽ ⫽ ⫽
Q
D 2⫽
16F 共 2;3/2⫺m,5/2⫹m; P 2e 兲
An exact analytical solution has been provided in this paper. It allowed the calculation of the steady-state temperature over the surface of a half space subject to a moving uniform circular heat source. The use of this solution is easy because the required special functions are preprogrammed in the Maple software and the Fourier series requires few terms to converge 共about 5 to 10兲. The results given by this solution are in agreement with the available asymptotic models. If one is interested in the temperature at the center of contact, the solution is simple and doesn’t require any calculation of the series solutions. The proposed solution also allows for the calculation of the thermal contact resistance, which requires the knowledge of the average temperature of the contact area that is given in this paper in an explicit form.
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D 1⫽
Conclusions
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with 共⫽1, ⫽m, ⫽m) for D 1 , 共⫽2, ⫽m, ⫽m⫹1) for D 2 , and 共⫽3, ⫽m⫹1, ⫽m⫹1) for D 3 . The expressions of those three integrals are given as follows 共Gradshteyn and Ryzhik 关关11兴, p. 704兴兲:
3
00
冕
Considering the particular case of P e ⫽0, we have I m⫹1 (0) ⫽0, I m⫽0 (0)⫽0 and I m⫽0 (0)⫽1, and Eq. 共15兲 gives, T a*v ( P e ⫽0)⫽8/3 ⬵0.8489, which is consistent with that given by Carslaw and Jeager 关关5兴, p. 216兴. The values of the dimensionless average surface temperature provided by the exact solution Eq. 共14兲 and from the correlation of Tian and Kennedy 关10兴 共subscript ‘cor’兲 versus P e are compared in Table 1. The results are in agreement. About 65% of the decrease of the average surface temperature is reached at P e ⫽5. Tian and Kennedy 关10兴 also numerically solved the integral solution. We report in Table 1 the values of average temperatures from the numerical integration 共subscript ‘‘num’’兲. The relative difference between the numerical integration and the exact solution 共14兲 is about 0.6% lower, those given by the correlation show a relative difference less than 4%.
radius of heat source thermal conductivity Peclet number⫽Va/2␣ heat flux⫽q a 2 heat flux by unit of area polar coordinates temperature dimensionless temperature⫽T/(qa/k) velocity thermal diffusivity temperature⫽T exp关⫺Vr cos()/2␣ 兴
Subscripts av ⫽ average c, ct ⫽ center, contour OCTOBER 2003, Vol. 125 Õ 3
PROOF COPY 009304JTQ
cor ⫽ correlation num ⫽ numerical integration
References 关1兴 Boussinesq, J., 1890, ‘‘Calcul des Tempe´ratures Successives d’un Milieu Homoge`ne et Atherme Inde´fini que Sillone une Source de Chaleur,’’ C.R., 110, pp. 1242–1244. 关2兴 Rosenthal, D., 1935, ‘‘Etude The´orique du Re´gime Thermique Pendant la Soudure a` l’Arc,’’ C.R., 2, pp. 1277–1292. 关3兴 Blok, H., 1937, ‘‘Les Tempe´ratures de Surface dans des Conditions de Graissage Sous Extreˆme Pression,’’ Proc. Sd. World Petrol. Cong., 3, pp. 471– 486. 关4兴 Jaeger, J. C., 1942, ‘‘Moving Sources of Heat and the Temperature of Sliding Contacts,’’ Proc. R. Soc. NSW 76, pp. 203–224. 关5兴 Carslaw, H. S., and Jaeger, J. C., 1959, Conduction of Heat in Solids, Oxford University Press, London.
关6兴 Archard, J. F., 1958 –1959, ‘‘The Temperature of Rubbing Surfaces,’’ Wear, 2, pp. 438 – 455. 关7兴 Holm, R., 1952, ‘‘Temperature Development in a Heated Contact With Application to Sliding Contacts,’’ J. Appl. Mech., 19, pp. 369–374. 关8兴 Francis, H. A., 1970, ‘‘Interfacial Temperature Distribution Within a Sliding Hertzian Contact,’’ ASLE Trans., 14, pp. 41–54. 关9兴 Gecim, B., and Winer, W. O., 1985, ‘‘Transient Temperature in the Vicinity of an Asperity Contact,’’ ASME J. Tribol., 107, pp. 333–342. 关10兴 Tian, X., and Kennedy, F. E., 1994, ‘‘Maximum and Average Flash Temperatures in Sliding Contacts,’’ ASME J. Tribol., 116, pp. 167–174. 关11兴 Gradshteyn, I. S., and Ryzhik, I. M., 1965, Table of Integrals Series and Products, Academic Press, New York and London. 关12兴 Abramowitz, M., and Stegun, I. A., 1972, Handbook of Mathematical Functions, Dover Publication, Inc., New York. 关13兴 Prudnikov, A. P., Brychkov, Yu. A., and Marichev, O. I, 1990, Integrals and Series, Gordon and Breach Science Publishers.
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