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The contact heat transfer during frictional heating in a three-element tribosystem
International Journal of Heat and Mass Transfer 53 (2010) 2740–2749
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The contact heat transfer during frictional heating in a three-element tribosystem A.A. Yevtushenko *, M. Kuciej, O. Yevtushenko Department of Applied Mechanics and Informatics, Faculty of Mechanical Engineering, Technical University of Bialystok, 45C Wiejska Street, Bialystok, 15-351, Poland
a r t i c l e
i n f o
Article history: Received 1 January 2010 Received in revised form 5 February 2010 Accepted 5 February 2010
Keywords: Frictional heating Temperature Heat transfer Braking
a b s t r a c t An analytical solution of a one-dimensional transient heat conduction problem for a tribosystem consisting of a semi-space sliding on a surface of a strip deposited on a semi-infinite foundation, is obtained. It is assumed, that the thermal contact of a strip and top semi-space is imperfect. Sliding with a constant speed or linearly decreasing with time (as at braking) is considered. Influence of Biot number, describing thermal conductivity of contact, on the temperature of a system component is studied. Ó 2010 Elsevier Ltd. All rights reserved.
1. Introduction The thermal analysis is an integral part of designing modern units of friction. Therefore, the thermal problem of friction – determination of temperature fields in sliding contacting bodies – represents one of the major problems of tribology and tribotechnics. The theoretical basis for the calculation of temperature fields, and for the effect of the relative sliding velocity and the ratio of thermal properties of bodies on the distribution of temperature was proposed in the pioneer works of Bowden and Tabor [1,2], Newcomb [3], Block [4], Jaeger [5], Shchedrov [6], Korovchinskij [7,8] and others authors. The main disadvantage of the statement of a thermal problem of friction was the fact that thermal processes were investigated in the interfaced bodies separately, and distribution of thermal energy between them was set a priori. In 1959, Ling formulated conditions of the perfect thermal contact, assuming equality of temperatures of bodies in the contact area and equality of the sum of intensities of heat flows directed to the body to a specific power of friction [9]. These conditions are associated with temperature fields of bodies and satisfy the energy conservation law. In fact, with the conditions of perfect thermal contact the formulation and solution of thermal problem of friction in the form of mathematical physics began. Reviews of research on the thermal problem of friction can be found in articles [10,11]. Despite of significant theoretical importance of conditions of the perfect thermal contact, their application for calculation of
* Corresponding author. Tel.: +48 085 746 9206; fax: +48 085 746 9210. E-mail address: [email protected] (A.A. Yevtushenko). 0017-9310/$ - see front matter Ó 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijheatmasstransfer.2010.02.028
temperatures in real tribosystems fraught with some difficulties. The most important problem is the measurement of temperature in the contact area. In a reality interaction of bodies is carried out not on all nominal surfaces of friction, but in discrete areas of contact. The sizes of these areas are very small, and the temperature in them essentially exceeds average temperature of nominal area of contact [12,13]. As the modern technology does not allow to reliably and accurately measure the temperature in the areas of real contact at friction, the theoretical equality of temperature cannot be received practically [14]. This led to the occurrence of various models of imperfect thermal contact at which contact temperature of bodies are assumed to be different. In 1963, Podstrigach offered conditions of imperfect thermal contact, which take into account the thermal resistance of a thin intermediate layer between bodies at friction [15]. These conditions have received wide theoretical and practical application. Reviews of such investigations are given in the articles [16,17]. The conditions of non-ideal thermal contact, based on the assumption on discreteness of the contact area and taking into account the temperature difference between the friction surface, in 1970 were received by Barber [18]. Although the methods used to obtain the conditions of imperfect thermal contact by Podstrigach and Barber are quite different, results are similar. Conditions of imperfect thermal contact of bodies with frictional heating are reduced to two equations. The first as well as in Ling, equates the sum of intensities of heat flows directed inside the body to specific power of friction. In the second equation affirms that the difference in the intensity of heat flux on the surface of contact is proportional to the difference of contact temperatures. The coefficient of proportionality is the contact heat transfer coefficient (the value, inversely proportional to the thermal resistance of contact). The
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Nomenclature Bi ¼ hd=K s Biot number d thickness of the strip erfðxÞ Gauss error function erfcðxÞ ¼ 1 erfðxÞ complementary error function ierfcðxÞ ¼ p1=2 expðx2 Þ x erfcðxÞ integral of the error function erfcðxÞ f coefficient of friction K coefficient of heat conduction k coefficient of thermal diffusivity pressure p0 q0 ¼ fV 0 p0 intensity of the frictional heat flux (the friction power) T temperature T 0 ¼ q0 d=K s temperature scaling factor T ¼ T=T 0 dimensionless temperature t time
finite element method to study transient heat conduction in two sliding bodies with frictional heat generated at the contact interface, has been used in article [19]. The numerical studies of the heat flux generated by friction, the thermal conductance and intrinsic heat partition coefficient in the dry friction between two solid have been made in papers [20,21], too. Previously, the authors obtained the exact solution of thermal problems of friction for two tribosystems. The first consists of two elements – a plane-parallel strip, sliding over the surface of the semi-space. The second tribosystem contains three elements – the semi-space, moving along the surface strip deposited on a semi-infinite foundation. Solutions to the first tribosystem were obtained under perfect [22,23] and imperfect [24] conditions of thermal contact. Frictional heat generation in the second tribosystem was investigated only under perfect conditions of thermal contact [25,26]. Therefore, the aim of this article is to obtain and analyse the analytical solution to one-dimensional transient thermal problem of friction for the second tribosystem under conditions of imperfect thermal contact.
time of braking sliding speed initial sliding speed spatial coordinate
ts V V0 z
Greek symbols
s ¼ ks t=d2 dimensionless time (Fourier’s number) ss ¼ ks ts =d2 dimensionless time of braking f ¼ z=d
dimensionless coordinate
Indexes f s t
bottom semi-space strip upper semi-space
Fig. 1. Scheme of a three-element brake system.
The transient temperature fields T t;s;f ðz; tÞ can be found from the solution to the following transient boundary-value heat conduction problem of friction:
o2 T t ðf; sÞ 2. Problem formulation The problem of contact interaction of two semi-spaces is considered, where one of them is homogeneous and the other is a semi-infinite foundation with a surface covered by a strip of thickness d. The perfect heat contact between the strip and the foundation takes place. It is supposed, that the constant compressive pressures p0 in direction of z-axis of the Cartesian system of coordinates Oxyz are applied to the infinities in semispaces (Fig. 1). The homogeneous upper semi-space slides with the constant velocity V in the direction of the y-axis on the strip surface. Due to friction the heat is generated on a contact plane z ¼ 0. It is assumed, that sum of the intensities of the frictional heat fluxes directed into each component of friction pair is equal with the specific friction power and that between surfaces of the top semi-space and the strip being in contact, the heat transfer takes place. The wear of the contact surface is negligible. Let us find the distribution of temperature fields in the frictional elements. Further, all values and the parameters concerning a top semi-space, strip and foundation will have bottom indexes ‘‘t”, ‘‘s” and ‘‘f”, respectively (Fig. 1).
of
2
o2 T s ðf; sÞ 2
of o2 T f ðf; sÞ
¼
1 oT t ðf; sÞ ; os kt
¼
oT s ðf; sÞ ; os
0 < f < 1;
1 < f < 0;
s > 0;
s > 0;
1 oT f ðf; sÞ ; 1 < f < 1; s > 0; os kf of2 oT s oT K t t ¼ 1; s > 0; of f¼0 of f¼0þ oT s oT þ K t t ¼ Bi½T t ð0; sÞ T s ð0; sÞ; s > 0; of f¼0 of f¼0þ
¼
T s ð1; sÞ ¼ T f ð1þ; sÞ; oT f oT s ¼ K ; f of f¼1þ of f¼1
s > 0; s > 0;
ð1Þ ð2Þ ð3Þ ð4Þ ð5Þ ð6Þ ð7Þ
T t ðf; sÞ ! 0;
f ! 1;
T f ðf; sÞ ! 0;
f ! 1;
T t ðf; 0Þ T s ðf; 0Þ T f ðf; 0Þ
¼ 0;
0 6 f < 1;
ð10Þ
¼ 0;
1 6 f 6 0;
ð11Þ
¼ 0;
1 < f 6 1;
ð12Þ
s > 0; s > 0;
ð8Þ ð9Þ
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ð13Þ
! fx Gt ðf; xÞ ¼ ½aðxÞDR ðxÞ þ bðxÞDI ðxÞ cos pffiffiffiffiffi þ ½bðxÞDR ðxÞ kt ! fx aðxÞDI ðxÞ sin pffiffiffiffiffi ; 0 6 f < 1; kt
ð14Þ
Gs ðf; xÞ ¼ ½BiDR ðxÞ þ et xDI ðxÞ cos½ð1 þ fÞx þ ef ½BiDI ðxÞ et xDR ðxÞ sin½ð1 þ fÞx; 1 6 f 6 0;
where
z f¼ ; d kt kt ¼ ; ks T t ¼
Tt ; T0
s¼
ks t 2
d
K f ¼
;
Kf ; Ks
K t ¼
Kt ; Ks
kf ¼
kf ; ks
hd ; Ks
Bi ¼
T s ¼
Ts ; T0
T f ¼
Tf ; T0
T0 ¼
q0 d : Ks
2
We perform the Laplace integral transform [27]
Z
1
0
T t;s;f ðf; sÞeps ds;
1 F t;s;f ðf; pÞ; p
ð16Þ
Dt;s;f ðf; pÞ F t;s;f ðf; pÞ ¼ pffiffiffi ; pDðpÞ
ð17Þ
pffiffiffi pffiffiffi DðpÞ ¼ Bið1 þ et ef Þ þ 2et p sh p pffiffiffi pffiffiffi þ Biðet þ ef Þ þ 2et ef p ch p;
ð18Þ qffiffiffi
0 6 f < 1;
p k t
ð20Þ
qffiffiffip k f
;
1 < f 6 1;
ð21Þ
where
K
K
f t ffi ffi: et ¼ pffiffiffiffi ef ¼ qffiffiffiffi ;
kt
ð22Þ
kf
The dimensionless quantities 0 < et;f < 1 (22), where et characterizes thermal activity of the material of the top semi-space relative to the material of the strip and ef – of the foundation to the strip, are known as ‘‘the coefficient of thermal activity” [28]. Inverting the Laplace transforms of the temperatures (16), we obtain [27]
T t;s;f ðf; sÞ ¼
Z s
F t;s;f ðf; sÞ ds;
s P 0;
1 < f 6 1;
bðxÞ ¼ ef ðBi sin x þ x cos xÞ:
ð23Þ
0
where, using the integration in the complex p-plane with cut along the negative direction of the axis Re p [22], the inverse of the transforms F t;s;f ðf; pÞ, (17)–(22), have the form
ð30Þ
ð31Þ
Substituting the functions, F t;s;f ðf; sÞ (24)–(31), into the solutions (23), after integration with respect to s, we finally obtain
T t;s;f ðf; sÞ ¼
2
p
Z
1
0
Gt;s;f ðf; sÞ Pðs; xÞ dx; DðxÞ
s P 0;
ð32Þ
where
1 2 1 ex s : 2 x
ð33Þ
The dimensionless intensity of the heat fluxes in the top semi-space, the strip and in the foundation we define as:
oT t ðf; sÞ ; 0 6 f < 1; s P 0; of oT ðf; sÞ qs ðf; sÞ ¼ s ; 1 6 f 6 0; s P 0; of oT f ðf; sÞ ; 1 < f 6 1; s P 0: qf ðf; sÞ ¼ K f of qt ðf; sÞ ¼ K t
; ð19Þ
pffiffiffi pffiffiffi pffiffiffi Ds ðf; pÞ ¼ ðBi þ et pÞ½et shð1 þ fÞ p þ chð1 þ fÞ p; 1 6 f 6 0; pffiffiffi ð1þfÞ Df ðf; pÞ ¼ ðBi þ et pÞe
aðxÞ ¼ Bi cos x x sin x;
Pðs; xÞ ¼
pffiffiffi pffiffiffi f pffiffiffi pffiffiffi Dt ðf; pÞ ¼ ðBiet þ pÞsh p þ Bi þ ef p ch p e
3
6ð1 þ fÞx7 et xDR ðxÞ sin 4 qffiffiffiffiffi 5; kf
ð15Þ
on the heat conduction equations (1)–(3) and the boundary conditions (4)–(9) with the homogeneous initial conditions (10)–(12) for the temperature. Thus, we obtain the Laplace transforms of the dimensionless temperatures in the following form:
T t;s;f ðf; pÞ ¼
ð29Þ
6ð1 þ fÞx7 Gf ðf; xÞ ¼ ½BiDR ðxÞ þ et xDI ðxÞ cos 4 qffiffiffiffiffi 5 þ ½BiDI ðxÞ kf 2 3
3. Solution to the problem
T t;s;f ðf; pÞ ¼
ð28Þ
ð34Þ ð35Þ ð36Þ
Taking the relations (34)–(36) into account, after differentiating the Laplace transforms of temperature (16)–(21) with respect to f we obtain the transforms of the intensities of the heat fluxes in the form:
t;s;f ðf; pÞ ¼ q
Q t;s;f ðf; pÞ ; pDðpÞ
ð37Þ
where qffiffiffi p pffiffiffi pffiffiffi pffiffiffi pffiffiffi f kt Q t ðf; pÞ ¼ et Bief þ p sh p þ Bi þ ef p ch p e ; 0 6 f < 1;
pffiffiffi
pffiffiffi pffiffiffi Q s ðf; pÞ ¼ Bi þ ef p ef ch½ð1 þ fÞ p þ sh½ð1 þ fÞ p ; 1 6 f 6 0;
pffiffiffi ð1þfÞ Q f ðf; pÞ ¼ ef ðBi þ et pÞe
ð38Þ ð39Þ
qffiffiffip k f
;
1 < f 6 1;
ð40Þ
ð24Þ
and the denominator DðpÞ is given by formula (18). The transforms (37)–(40) have a branch point at the origin p ¼ 0. Therefore, the integration contour in the complex p-plane was chosen the same as in the article [22]. As a result, we obtain
D2I ðxÞ;
ð25Þ
qt;s;f ðf; sÞ ¼ q0t;s;f
DR ðxÞ ¼ Biðet þ ef Þ cos x 2et x sin x;
ð26Þ
where
DI ðxÞ ¼ Bið1 þ et ef Þ sin x þ 2et ef cos x;
ð27Þ
q0t ¼
F t;s;f ðf; sÞ ¼
DðxÞ ¼
Z
2
p
D2R ðxÞ
þ
1
0
Gt;s;f ðf; xÞ x2 s dx; e DðxÞ
s P 0;
2
p
Z
1
0
Q t;s;f ðf; pÞ x2 s dx; e xDðxÞ
ef et ; q0s ¼ q0f ¼ ; et þ ef et þ ef
s P 0;
ð41Þ
ð42Þ
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(
!
fx Q t ðf; pÞ ¼ et ½aðxÞDI ðxÞ bðxÞDR ðxÞ cos pffiffiffiffiffi þ ½bðxÞDI ðxÞ kt !) fx þaðxÞDR ðxÞ sin pffiffiffiffiffi ; 0 6 f < 1; kt
the top semi-space and the strip ðBi ! 1Þ can be found from Eqs. (32) and (33) with integrands in the form 2
ð43Þ
Q s ðf; pÞ ¼ ef ½BiDI ðxÞ et ef xDR ðxÞ cos½ð1 þ fÞx ½BiDR ðxÞ þet xDI ðxÞ sin½ð1 þ fÞx;
1 6 f 6 0;
ð44Þ
8 2 3 > < 6ð1 þ fÞx7 Q f ðf; pÞ ¼ ef ½BiDI ðxÞ et xDR ðxÞ cos 4 qffiffiffiffiffi 5 ½BiDR ðxÞ > : kf 2 39 > = 6ð1 þ fÞx7 þet xDI ðxÞ sin 4 qffiffiffiffiffi 5 ; 1 < f 6 1: > ; kf
ð45Þ
ð46Þ
Z
qt ð0;
½Gt ð0þ; xÞ Gs ð0; xÞ 2 1 ex s dx; 2 x DðxÞ p 0 s P 0; ð47Þ
T t ð0þ; sÞ T s ð0; sÞ ¼
2
Z 0
1
2
ð48Þ
where, taking the formulae (28), (29) and (43), (44) into account, we have
2
p
1 0
1 < f 6 1:
ð56Þ
3
6ð1 þ fÞx7 þ ð1 þ et ef Þ sin x sin 4 qffiffiffiffiffi 5; kf
The analytical solution in the form of series on ierfcðxÞ functions for tribosystem considered, in the case of the perfect heat contact between the top semi-space and the strip has been obtained in article [25]. By comparing this solution and the solutions (32) and (33) obtained in the present article, we obtain the values of the following integrals:
1
p
Z
pffiffiffi 1 Gt;s;f ðf; sÞ s X Kn T t;s;f ;n ðf; sÞ; Pðs; xÞ dx ¼ ð1 þ et Þ n¼0 DðxÞ
1
0
where
" T t;n ðf; sÞ ¼ ierfc
ef et ½Q s ð0; xÞ Q t ð0þ; xÞ : dx ¼ xDðxÞ ef þ et
! # " ! # f 1 f 1 2n þ pffiffiffiffiffi pffiffiffi þ kf ierfc 2n þ 2 þ pffiffiffiffiffi pffiffiffi ; kt 2 s kt 2 s
0 6 f < 1;
ð58Þ
2n f 2n þ 2 þ f pffiffiffi þ kf ierfc pffiffiffi T s;n ðf; sÞ ¼ ierfc ; 2 s 2 s 1 6 f 6 0; 20
ð49Þ
ð50Þ
Calculations of the integral in the left side of Eq. (50) by procedure QAGI from a package QUADPACK [29] has shown that this equality is carried out identically. On the interface f ¼ 1 from Eqs. (29), (30) and (44), (45) follows:
s P 0; ð57Þ
T f ;n ðf;
1
ð59Þ 3
2 1 þ fC 1 7 6B sÞ ¼ ierfc4@2n þ 1 qffiffiffiffiffi A pffiffiffi5; ð1 þ ef Þ 2 s k f
Substituting Eqs. (47)–(49) into boundary condition (5), we obtain the relationship
Z
ð55Þ
6ð1 þ fÞx7 Gf ðf; xÞ ¼ ðet þ ef Þ cos x cos 4 qffiffiffiffiffi 5 kf 2 3
½Q s ð0; xÞ Q t ð0þ; xÞ x2 s e dx; xDðxÞ
x Gt ð0þ; xÞ Gs ð0; xÞ ¼ ½Q s ð0; xÞ Q t ð0þ; xÞ Bi ¼ x½ðef et ÞDI ðxÞ cos x ð1 et ef ÞDR ðxÞ sin x:
ð54Þ
1 6 f 6 0;
1
ef et 2 ef þ et p s P 0;
qs ð0; sÞ qt ð0þ; sÞ ¼
fx 2 Gt ðf; xÞ ¼ ½ðet þ ef Þ cos2 x þ ef ð1 þ et ef Þ sin x cos pffiffiffiffiffi kt ! fx ð1 e2f Þ sin x cos x sin pffiffiffiffiffi ; 0 6 f < 1; kt
þ ef ð1 þ et ef Þ sin x sin ½ð1 þ fÞx;
and from Eqs. (41) and (42) we find sÞ þ sÞ ¼ 1; s P 0, which means that boundary conditions (4) are satisfied. Difference in temperature and intensity of heat flux on the contact surface is found from formulae (32), (33) and (41) in the form qs ð0þ;
ð53Þ !
Gs ðf; xÞ ¼ ðet þ ef Þ cos x cos½ð1 þ fÞx
We note that the values of q0t;s;f (42) are the value of the intensities of the heat flows in steady state. In contrast to the heat flux, the temperature (32) does not reach the stationary state. On the friction surface f ¼ 0 from the formulae (43) and (44) leads
Q s ð0; xÞ þ Q t ð0þ;xÞ ¼ Biðet þ ef Þ cosx 2et x sin x DI ðxÞ Bið1 þ et ef Þsin x þ 2et ef x cos x DR ðxÞ ¼ DR ðxÞDI ðxÞ DI ðxÞDR ðxÞ ¼ 0;
DðxÞ ¼ ðet þ ef Þ2 cos2 x þ ð1 þ et ef Þ2 sin x;
1 < f 6 1;
Kn ¼
k¼
kn ;
ð60Þ
0 6 k < 1;
ð1Þn jkjn ; 1 < k 6 0;
ð1 et Þ ð1 ef Þ ; ð1 þ et Þ ð1 þ ef Þ
ð61Þ
ð62Þ
i.e. the boundary conditions (6) and (7) are satisfied.
and the integrands in the left-hand side of Eq. (57) have the form (53)–(56). Heat transfer between two homogeneous semi-spaces ðK s ¼ K f ; ks ¼ kf Þ. Solution to the thermal problem of friction for two homogeneous semi-spaces with heat transfer between them also give Eqs. (23) and (24) at K f ¼ 1; kf ¼ 1; ef ¼ 1:
4. Some particular solutions to the problem
T t;f ðf; sÞ ¼
Gs ð1; xÞ ¼ BiDR ðxÞ þ et xDI ðxÞ ¼ Gf ð1; xÞ;
ð51Þ
Q s ð1; xÞ ¼ ef ½BiDI ðxÞ et xDR ðxÞ ¼ Q f ð1; xÞ;
ð52Þ
Perfect heat contact between top semi-space and strip. Dimensionless temperatures in the case of the perfect heat contact between
¼
Z s 0
2
p
F t;f ðf; sÞds;
Z 0
1
F t;f ðf; sÞ
Gt;f ðf; xÞ x2 s dx; s P 0; e DðxÞ
ð63Þ
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where
T t ðf;
2
DðxÞ ¼ Bi ð1 þ et Þ2 þ 4e2t x2 ; h
2
Gt ðf; xÞ ¼ Bi ð1 þ et Þ þ 2et x
ð64Þ 2
i
! ! fx fx cos pffiffiffiffiffi þ Bið1 et Þx sin pffiffiffiffiffi ; kt kt
0 6 f < 1;
ð65Þ
! " ! pffiffiffi 2 s f ð1 et Þ f sÞ ¼ erfc pffiffiffiffiffi ierfc pffiffiffiffiffi ð1 þ et Þ 2 kt s 2 kt s ð1 þ et Þ2 Bi !# a2 sþpafffiffiffi p ffiffiffi f k t erfc þe ; 0 6 f < 1; s P 0; a s þ pffiffiffiffiffi 2 kt s
T f ðf; sÞ ¼
h i 2 Gf ðf; xÞ ¼ Bi ð1 þ et Þ þ 2e2t x2 cosðfxÞ þ Biet ð1 et Þx sinðfxÞ; 1 < f 6 0:
ð66Þ
Taking the formulae (64)–(66) into account, the integrands in the solutions (63) take the form:
"
2
#
!
Gt ðf; xÞ 1 ð1 e2 ÞBi fx ð1 et ÞBix 1 2 2 t 2 cos pffiffiffiffiffi þ 2 2 ¼ 2et DðxÞ 4et ða þ x Þ et ða þ x2 Þ 4 kt ! fx sin pffiffiffiffiffi ; 0 6 f < 1; kt
ð67Þ T f ð0; sÞ ¼
1
0
Z
1
0
Z
1
0
1 2 ex s cosðfxÞ dx ¼ 2
2 pffiffiffi xex s p a2 s ajfj jfj p ffiffiffi e erfc a s sin ð jfjx Þdx ¼ e ða2 þ x2 Þ 4 2 s
p ffiffiffi jfj eajfj erfc a s þ pffiffiffi ; 2 s
F t ðf; sÞ ¼
! 2 f f ð1 et Þ 2 p pffiffiffiffiffiffi e 4kt s ffiffiffiffiffiffiffi sÞ ¼ erfc 2ð1 þ et Þ a ps ð1 þ et Þ 2 kt s !# a2 sþpafffiffiffi pffiffiffi f k t e erfc a s þ pffiffiffiffiffiffiffi ; 2 kt s
ð71Þ
qt ð0; sÞ ¼ ð72Þ
ð74Þ
ð81Þ
f2 1 f ð1 et Þ 2 pffiffiffiffiffiffi e4s erfc pffiffiffi þ ð1 þ et Þ 2ð1 þ et Þ a ps 2 s
pffiffiffi f 2 ea saf erfc a s pffiffiffi ; 1 < f 6 0; 2 s
s P 0: ð82Þ
qf ð0; sÞ ¼
et ð1 þ et Þ s P 0;
pffiffiffi ð1 et Þ 2 2 pffiffiffiffiffiffi ea s erfc a s ; 2ð1 þ et Þ a ps ð83Þ
pffiffiffi 1 ð1 et Þ 2 2 pffiffiffiffiffiffi ea s erfc a s ; þ ð1 þ et Þ 2ð1 þ et Þ a ps s P 0:
ð84Þ
Taking the approximate relation for large values of time into account [32]
pffiffiffi 1 2 ea s erfc a s pffiffiffiffiffiffi ;
ð85Þ
a ps
the contact temperatures (79) and (80) and intensity of heat fluxes (83) and (84) are rewritten as
Taking into account the values of integrals [30,31]:
pffiffi jfj 2 ea sþajfj erfc a s þ pffiffi ds 2 s 0 pffiffiffi
pffiffiffi 2 s jfj 1 j fj jfj 2 ¼ ierfc pffiffiffi þ 2 erfc pffiffiffi þ ea sþajfj erfc a s þ pffiffiffi ; a a 2 s 2 s 2 s
s P 0;
On the contact surface f ¼ 0 from Eqs. (81) and (82) results
ð73Þ
Z s pffiffiffiffiffiffi 1 f2 jfj pffiffi e 4s ds ¼ 2 psierfc pffiffiffi ; s 2 s 0
et
0 6 f < 1; qf ðf; sÞ ¼
2 2 pffiffiffi f 1 ð1 et ÞBi a sþpffiffiffi f k t erfc pffiffiffiffiffiffi e 4kt s e a s þ pffiffiffiffiffiffiffi ; 2 4et 2et ps 2 kt s
pffiffiffi f2 1 ð1 et ÞBi a2 saf f e erfc a s pffiffiffi ; F f ðf; sÞ ¼ pffiffiffiffiffiffi e4s þ 4et 2 ps 2 s 1 < f 6 0:
ð80Þ
ð69Þ
!
0 6 f < 1;
pffiffiffii s et ð1 et Þ h 2 1 ea s erfc a s ; 2 p ð1 þ et Þ Bi
qt ðf;
functions F t;f ðf; sÞ (63) takes the form af
2 ð1 þ et Þ
ð79Þ rffiffiffiffi
After differentiating the dimensionless temperatures (77) and (78) with respect to f, from the relations (34) and (36) we obtain the dimensionless intensity of heat fluxes in the form:
ð70Þ
2 pffiffiffi ex s pea2 s af f e erfc a s pffiffiffi cosðfxÞ dx ¼ 2 2 ða þ x Þ 4a 2 s
pffiffiffi f af þe erfc a s þ pffiffiffi ; 2 s
pffiffiffii s ð1 et Þ h 2 1 ea s erfc a s ; þ 2 p ð1 þ et Þ Bi
ð68Þ
rffiffiffiffi
p f42s e ; s
rffiffiffiffi
s P 0:
By using the values of the integrals [30]
Z
2 sÞ ¼ ð1 þ et Þ
s P 0;
where
ð1 þ et Þ a¼ Bi: 2et
ð78Þ
The maximal temperatures are reached on a plane of friction f ¼ 0 and, as follows from the solutions (77) and (78), are equal:
T t ð0;
" # 2 Gf ðf; xÞ 1 ð1 e2 ÞBi ð1 et ÞBix 1 þ 2 2 t 2 cosðfxÞ þ ¼ sinðjfjxÞ; 2 4et ða2 þ x2 Þ DðxÞ 4et ða þ x Þ 1 < f 6 0;
pffiffiffi
2 s f et ð1 et Þ f p ffiffiffi erfc ierfc pffiffiffi þ ð1 þ et Þ 2 s 2 s ð1 þ et Þ2 Bi
p ffiffiffi f 2 þea saf erfc a s pffiffiffi ; 1 < f 6 0; s P 0: 2 s
ð77Þ
ð75Þ
s ð1 et Þ 1 1 pffiffiffiffiffiffi ; s 1; þ p ð1 þ et Þ2 Bi a ps
2 ð1 þ et Þ
T f ð0; sÞ
pffiffiffi
2 s e ð1 et Þ 1 pffiffiffiffi t p ffiffiffiffiffiffi ; 1 a ps ð1 þ et Þ p ð1 þ et Þ2 Bi
Z s
ð76Þ
from the first equation (63), we obtain the dimensionless temperatures in the semi-spaces in closed form:
rffiffiffiffi
T t ð0; sÞ
qt ð0; sÞ
et ð1 þ et Þ
ð1 et Þ 1 pffiffiffiffiffiffi ; 2ð1 þ et Þ a ps
s 1;
s 1;
ð86Þ
ð87Þ
ð88Þ
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qf ð0; sÞ
1 ð1 et Þ 1 pffiffiffiffiffiffi ; þ ð1 þ et Þ 2ð1 þ et Þ a ps
s 1:
ð89Þ
From Eqs. (83) and (84) or (88) and (89) follows, that qf ð0; sÞ þ qt ð0; sÞ ¼ 1; s P 0, i.e. boundary condition (4) is satisfied. In addition, from Eqs. (86)–(89) we obtain
qf ð0; sÞ qt ð0; sÞ ¼
h i ð1 et Þ 1 1 þ pffiffiffiffi ¼ Bi T t ð0; sÞ T f ð0; sÞ ; ð1 þ et Þ a ps
ð90Þ that testifies the satisfaction of a boundary condition (5). Perfect heat contact between two sliding uniformly homogeneous semi-spaces. Substituting in solutions (77), (78) and (81), (82) Bi ! 1 ða ! 1Þ, we find
T t ðf;
! pffiffiffi 2 s f sÞ ¼ ; qt ðf; sÞ ierfc pffiffiffiffiffiffiffi ð1 þ et Þ 2 kt s ! et f ¼ ; 0 6 f < 1; erfc pffiffiffiffiffiffiffi ð1 þ et Þ 2 kt s
s P 0;
2 s f T f ðf; sÞ ¼ ierfc pffiffiffi ; qf ðf; sÞ ð1 þ et Þ 2 s
1 f ¼ erfc pffiffiffi ; 1 < f 6 0; ð1 þ et Þ 2 s
ð91Þ
s P 0:
ð92Þ
1
Gf ðf; xÞ ¼ ð1 þ et Þ cosðfxÞ;
0 6 f < 1;
ð93Þ
1 < f < 0:
Substituting functions DðxÞ and Gt;f ðf; xÞ (93) into solutions (23) and (24), we obtain the temperature in the top semi-space in the form
T t ðf;
2 sÞ ¼ pð1 þ et Þ
Z sZ 0
0 6 f < 1;
1
e
x2 s
0
! x cos f pffiffiffiffiffi dx ds; kt
s P 0:
ð94Þ
Using successively the values of integrals (70) and (75) from Eq. (94), we obtain the first of the formulas (91). All the remained formulas (91) and (92) are received similarly.
5. Application to the solution to the thermal problem of friction during braking In the case of constant pressure p0 the sliding speed during braking decreases from initial value V 0 to zero at the stop time moment linearly with time (braking with constant deceleration) [34]. Then, the intensity of heat generation on the contact surface is equal to the power density of the friction force
qðtÞ ¼ q0 q ðsÞ;
q ðsÞ ¼ 1
o b T ðf; s sÞ ds; os t;s;f
0 6 s 6 ss ;
ð96Þ
b ðf; sÞ have the form (32) where the dimensionless temperatures T t;s;f and (33) with integrands (25)–(31). Substituting temperature b ðf; sÞ (32) to the right parts of Eq. (96) and changing the order T of the integration, we obtain
T t;s;f ðf; sÞ ¼
2
p
Z
1
0
Gt;s;f ðf; sÞ P0 ðs; xÞ dx; DðxÞ
0 6 s 6 ss ;
ð97Þ
where
P0 ðs; xÞ ¼ Pðs; xÞ
1 ½s Pðs; xÞ; x2 s s
0 6 s 6 ss ;
ð98Þ
s ; 0 6 s 6 ss ; ss
Pðs; 0Þ ¼ s;
P0 ðs; xÞ ¼ s 1
s ; 0 6 s 6 ss : 2ss
ð99Þ
We note that the solution to this problem in the limiting case of the perfect thermal contact of the top semi-space and the strip ðBi ! 1Þ was obtained earlier in article [26].
Eqs. (91) and (92) give the solution to the thermal problem of friction for two homogeneous semi-spaces [33]. The same solution can be obtained in a different way from solution in the case of perfect heat contact between the top semi-space and the strip. The formulae (53)–(56) under conditions of kf ¼ 1 and ef ¼ 1 will take the form DðxÞ ¼ ð1 þ et Þ2 and
B fx C Gt ðf; xÞ ¼ ð1 þ et Þ cos @qffiffiffiffiffiA; kf
q ð sÞ
0
and function Pðs; xÞ has the form (33). In limiting case x ! 0 from formulae (33) and (98) we find
pffiffiffi
0
T t;s;f ðf; sÞ ¼
Z s
ð95Þ
where ss is the dimensionless duration of braking. The solution to a boundary-value problem of heat conductivity (1)–(12) in case, when the intensity of the specific friction power on the right side of a boundary condition (4) has the form (95), we find using the Duhamel’s theorem [27]
6. Numerical analysis The influence of input parameters, like the thickness of the strip d, the physical constants K t;f ; kt;f (13) and the duration of braking ts on the temperature has been studied in the articles [25,26]. Therefore, the main purpose of numerical analysis in this article will be study the influence of the Biot number on temperature, which characterizes the heat transfer through the surface of contact between the strip and the top semi-space. We note that in a limiting case Bi ! 1 we should obtain results of the articles [25,26]. The numerical analysis of a problem shall be executed for the same materials, as in these articles. We have taken the upper semi-space as the cast iron, ChNMKh: Kt = 51 W m1 K1, kt = 14 106 m2 s1 and the foundation as steel, 30KhGSA: Kf = 37.2 W m1 K1, kf = 10.3 106 m2 s1 [34]. In case of uniform sliding of bodies, the strip materials are aluminum, Al: Ks = 209 W m1 K1, ks = 86 106 m2 s1 or zirconium dioxide, ZrO2: Ks = 2 W m1 K1, ks = 0.8 106 m2 s1 [25]. When sliding with a constant deceleration (braking) was considered, the material of a strip was the metal-ceramics, FMK-11: Ks = 34.3 W m1 K1, ks = 15.2 106 m2 s1 [26]. The thickness of a strip was equal d = 5 mm, the contact pressure was equal p0 ¼ 1 MPa and the initial temperature was equal to 20 °C. In the case of uniform sliding we set the initial sliding speed V0 = 5 m s1 and coefficient of friction f ¼ 0:3. At braking with constant deceleration the input parameters were equal V0 = 30 m s1, f ¼ 0:7 and, additionally, the time of braking ts ¼ 3:44 s [34]. The evolution of temperature on the contact surface z ¼ 0 at sliding with constant speed for a strip made from aluminum and zirconium dioxide is shown in Figs. 2 and 3, respectively. For the small values of Biot number ðBi ! 0Þ, the greatest jump of temperatures of the top semi-space and the strip are noticed. In the case of the aluminum strip increase in Biot number leads to a rapid equalization of temperature on the friction surface (Fig. 2a). Thus, for a value of Bi = 5 the contact temperatures of the upper semi-space and the strip are identical. For values of Bi P 5 we can use the solution to the considered thermal problem of friction, at perfect thermal contact [25]. Alignment of temperatures of a strip and top semi-space on a surface of contact at increase of Biot number in the case of ZrO2 strip occurs much more slowly, than for an aluminum strip (Fig. 3a). For the same values of Biot number Bi = 5, we see that the jump of the contact temperatures is still great. The
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Fig. 2. Evolution of the contact temperature Tð0; tÞ at uniform sliding for two tribosystems: (a) three element – cast iron/aluminum/steel; (b) cast iron/aluminum semi-spaces.
Fig. 3. Evolution of the contact temperature Tð0; tÞ at uniform sliding for two tribosystems: (a) three element – cast iron/zirconium dioxide/steel; (b) cast iron/ zirconium dioxide semi-spaces.
curves plotted for large values of Biot number ðBi ! 1Þ in Figs. 2a and 3a, coincide with the corresponding results obtained in article [25]. The evolutions of temperature on the contact surface between two semi-spaces are shown in Figs. 3b and 4b. Since aluminum has a high coefficient of thermal conductivity, the curves of the evolution of temperature, as shown in Fig. 2a and b, are different. This is proved, that there is, for example, much greater jump of temperatures at small values of the Biot number in the case of contact between two semi-spaces (Fig. 2b) than at the contact of the semi-space and strip (Fig. 2a). Since zirconium is a poor conductor of heat, the evolutions of temperatures in Fig. 3a and b are almost identical. Hence the practical conclusion, that in the case of a strip from zirconium dioxide, we can use the analytic solution for the two semi-spaces (77) and (78), instead of solutions for three-element system in the form of integrals (23)–(31).
Dependences of the contact temperatures of the top semi-space and strip on the Biot number are shown in Fig. 4. We see that the contact temperature is greater in that element of friction pair, which has a smaller coefficient of thermal conductivity. Such elements are: the top semi-space from the cast iron in Fig. 4a, and a dioxide zirconium strip in Fig. 4b. With the increase in the Biot number, the jump of temperatures decreases. In the case of an aluminum strip (Fig. 4a), we see that the value of Bi = 5 almost corresponds to the solution to the perfect thermal contact between the top semi-space and the strip, obtained in the article [25]. In the case of a strip from zirconium dioxide (Fig. 4b), alignment of temperatures on the contact surface with increasing of the Biot number is much slower. Even at Bi ¼ 100 the jump of temperatures are still significant. The evolution of the temperature (97) on a contact surface z ¼ 0 during braking for three values of Biot number is shown in Fig. 5.
A.A. Yevtushenko et al. / International Journal of Heat and Mass Transfer 53 (2010) 2740–2749
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Fig. 5. Evolution of the contact temperature Tð0; tÞ at braking for different values of the Biot number Bi.
Fig. 4. Dependence of the contact temperature Tð0; tÞ at uniform sliding on the Biot number Bi in time moment t ¼ 5 s for two strip materials: (a) aluminum; (b) dioxide zirconium.
The material of a strip in this case is the metal-ceramic FMK-11, used as a pad in brakes of planes [35]. We see that for fixed value of the parameter Bi temperature rapidly increases with the beginning of the process of braking. The maximal temperature T max is reached at the time moment tmax , which nearly corresponds to half of time braking ts ¼ 3:44 s: for values of the Biot number 0; 5 and Bi ! 1 we have tmax ¼ 1:5 s; 1:6 s and 1.6 s, respectively. Cooling of the contact surface, which started after reaching the maximum temperature occurs before the stop time moment ts ¼ 3:44 s. The greatest difference of temperatures of a top semi-space and a strip DT ¼ jT t ð0; t max Þ T s ð0; tmax Þj takes place for small values of the Biot number: DT = 232 °C at Bi ! 0. If Bi ! 1, then the temperatures of the top semi-space and the strip coincide. The maximal temperature in this case is equal 741 °C. The same value was obtained earlier by solving the considered problem at perfect thermal contact strip and the top semi-space [26].
Fig. 6. Dependence of the temperature on the distance jzj from surface of friction for three values of Biot number Bi in the time moment t ¼ 1:6 s for the strip thickness d = 5 mm.
The change of temperature in the elements of tribosystem in depth from a surface of friction is shown in Fig. 6. The maximal temperature is reached on a surface of friction and in the time moment t ¼ 1:6 s for values of the Biot number 0; 5 and Bi ! 1 is equal T t;max ¼ 868 C; 757 C; 741 C for the top semi-space and T s;max ¼ 636 C; 723 C; 741 C for the strip, respectively. The Biot number does not influence the character of temperature change over the thickness of a strip. Even at Bi ! 1 decrease in temperature with increasing distance from the contact surface is almost linearly. The cast iron top semi-space is heated to a depth of
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tion for two semi-spaces with imperfect or perfect thermal contact is obtained. We compared the results of calculation of temperature, obtained for the three-element system and at the sliding of two semi-spaces. It is established, that evolution of temperature on a surface of contact of the cast iron top semispace and a zirconium dioxide strip is almost the same, as in case of contact of two semi-spaces made from the same materials. This is caused by a small coefficient of thermal conductivity of zirconium dioxide. Influence of the steel foundation on temperature is noticeably in the case of the aluminum strip, which possesses significant heat conductivity. In the particular case when the values of Biot number Bi ! 1, we obtain the maximal temperature on the contact surface 741 °C, which coincides with the theoretical result of the article [26] and is only a little different from the experimental value 760 °C, published in the monograph [34]. If we take into account the heat transfer through the surface of contact (Bi = 5), then the calculated maximal temperature on a surface of the top semi-space will be equal 757 °C.
References Fig. 7. Dependence of the contact temperature Tð0; tÞ at braking on the Biot number Bi in time moment t ¼ 1:6 s for metal-ceramic FMK-11 strip material.
approximately equal thickness of the strip (d = 5 mm). Temperature in the steel foundation is insignificant. Dependences of the contact temperatures of the top semi-space and a strip on Biot number in the moment of time t ¼ 1:6 s, when the maximal temperature is reached, are shown in Fig. 7. We see that the thermal contact between these elements of the tribosystem can be considered as perfect already for Bi 80. Hence, for values of Biot number Bi > 80 it is possible to use the analytical solution obtained in the article [26] at calculation of the maximal temperature. 7. Conclusions The frictional heat generation in tribosystem consisting of three elements: a semi-space sliding on the surface of a strip deposited on a semi-infinite foundation is investigated. In contrast to the previously obtained solutions for the same tribosystem proposed in articles [25,26] here we take into account the heat transfer through the contact surface between the top semi-space and the strip. Influence of the dimensionless parameter – the Biot number Bi, describing a heat transfer through the contact surface, on temperature fields in the elements of tribosystem is studied. At relative sliding bodies with constant speed, the aluminum and zirconium dioxide is chosen as a material of a strip. These materials are one of most often used in tribotechnics as protective films and heatshielding coatings. The possibility of using the obtained solution to determinate temperature in friction components of brakes is shown. In this case, the sliding speed decreases linearly in time from the maximum value at the start of braking until stop. The metal-ceramic FMK-11 has been chosen as a material of a strip, since it is used at manufacturing of the pad in brakes of planes. From our point of view, the main result obtained in this article is the definition for the investigated class of materials and input parameters of the limit values of Biot number, in which it is possible to neglect the influence of heat transfer through the contact surface on the temperature and use the solutions of articles [25,26]. In case of identical materials of the strip and the foundation the analytical solutions to a corresponding heat problem of fric-
[1] F.P. Bowden, D. Tabor, The Friction and Lubrication of Solids: Part I, Clarendon Press, Oxford, 1950. [2] F.P. Bowden, D. Tabor, The Friction and Lubrication of Solids: Part II, Clarendon Press, Oxford, 1964. [3] T.P. Newcomb, Transient temperatures attained in disk brakes, Br. J. Appl. Phys. 10 (1959) 339–340. [4] H. Block, The flash temperature concept, Wear 6 (1963) 483–494. [5] J.C. Jaeger, Moving sources of heat and temperature of sliding contacts, Proc. Roy. Soc. New South Wales 76 (3) (1942) 203–224. [6] V.S. Shchedrov, Temperature at a sliding contact, in: Friction and Wear in Machines, issue 10, Moscow, 1955, pp. 155–296. [7] M.V. Korovchinskij, Questions of Friction and Lubrication Problems: Part I, Mir, Moscow, 1966 (in Russian). [8] M.V. Korovchinskij, Questions of Friction and Lubrication Problems: Part II, Mir, Moscow, 1968 (in Russian). [9] F.F. Ling, A quasi-iterative method for computing interface temperature distribution, Z. Angew. Math. Phys. (ZAMP) 10 (5) (1959) 461–475. [10] R.A. Burton, Thermal deformation in frictionally heated contact, Wear 59 (1980) 1–20. [11] R.A. Barber, M. Comninou, Thermoelastic contacts problems, in: R.R. Hetnarski (Ed.), Thermal Stresses III, Elsevier, Amsterdam, 1989, pp. 1–105. [12] B. Gesim, W.O. Winer, Transient temperatures in the vicinity of an asperity contact, Trans. ASME J. Tribol. 107 (3) (1985) 333–341. [13] L. Wallis, E. Leonardi, B. Milton, P. Joseph, Air flow and heat transfer in ventilated disc brake rotors with diamond and tear-drop pillars, Numer. Heat Transfer A 41 (6–7) (2002) 643–655. [14] Y. Harigaya, Measurement of the contact surface temperature, Jpn. J. Tribol. 35 (11) (1990) 1247–1254. [15] Ya.S. Podstrigach, The temperature field in a system of rigid bodies coupled by thin interface, Inzh.-Fiz. Zh. 6 (10) (1963) 129–136. [16] O.O. Evtushenko, E.H. Ivanyk, N.V. Horbachova, Analytic methods for thermal calculation of brakes (review), Mater. Sci. 36 (6) (2000) 857–862. [17] S.J. Matysiak, A.A. Yevtushenko, On heating problems of friction, J. Theoret. Appl. Mech. 39 (3) (2001) 577–588. [18] J.R. Barber, The conduction of heat from sliding solids, Int. J. Heat Mass Transfer 13 (1970) 857–869. [19] T.C. Kennedy, S. Traiviratana, Transient effects on heat conduction in sliding bodies, Numer. Heat Transfer A 47 (1) (2005) 57–77. [20] J.G. Bauzin, N. Laraqi, Simultaneous estimation of frictional heat flux and two thermal contact parameters for sliding contacts, Numer. Heat Transfer A 45 (4) (2004) 313–328. [21] S.C. Gupta, A numerical method for transient and thermal stresses in bonded anisotropic composite slabs with deforming moving boundaries, Numer. Heat Transfer A 24 (3) (1993) 287–303. [22] A.A. Yevtushenko, M. Kuciej, Influence of convective cooling on the temperature in a frictionally heated strip and foundation, Int. Commun. Heat Mass Transfer 36 (2) (2009) 129–136. [23] A.A. Yevtushenko, M. Kuciej, Temperature in a frictionally heated ceramicmetal patch and cast iron disc during braking, Numer. Heat Transfer A 56 (2) (2009) 97–108. [24] A.A. Yevtushenko, M. Kuciej, M. Rozniakowska, The contact heat transfer between the plane-parallel strip and the semi-infinite foundation, Int. Commun. Heat Mass Transfer 36 (2009) 787–793. [25] A.A. Yevtushenko, M. Kuciej, Influence of the protective strip properties on distribution of the temperature at transient frictional heating, Int. J. Heat Mass Transfer 52 (2009) 376–384.
A.A. Yevtushenko et al. / International Journal of Heat and Mass Transfer 53 (2010) 2740–2749 [26] A.A. Yevtushenko, M. Kuciej, Frictional heating during braking in a threeelement tribosystem, Int. J. Heat Mass Transfer 52 (2009) 2942–2948. [27] I.N. Sneddon, The Use of Integral Transforms, McGraw-Hill, New York, 1972. [28] A.V. Luikov, Analytical Heat Diffusion Theory, Academic Press, New York, 1968. [29] R. Piessens, E. de Doncker-Kapenga, C.W. Überhuber, D.K. Kahaner, QUADPACK: A Subroutine Package for Automatic Integration, Springer, Berlin, 1983. [30] A.P. Prudnikov, Yu.A. Brychkov, I.I. Marichev, Integrals and series, Elementary Functions, vol. 1, Gordon & Breach, New York, 1986. [31] A.P. Prudnikov, Yu.A. Brychkov, I.I. Marichev, Integrals and series, Special Functions, vol. 2, Gordon & Breach, New York, 1986.
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Contents lists available at ScienceDirect
International Journal of Heat and Mass Transfer journal homepage: www.elsevier.com/locate/ijhmt
The contact heat transfer during frictional heating in a three-element tribosystem A.A. Yevtushenko *, M. Kuciej, O. Yevtushenko Department of Applied Mechanics and Informatics, Faculty of Mechanical Engineering, Technical University of Bialystok, 45C Wiejska Street, Bialystok, 15-351, Poland
a r t i c l e
i n f o
Article history: Received 1 January 2010 Received in revised form 5 February 2010 Accepted 5 February 2010
Keywords: Frictional heating Temperature Heat transfer Braking
a b s t r a c t An analytical solution of a one-dimensional transient heat conduction problem for a tribosystem consisting of a semi-space sliding on a surface of a strip deposited on a semi-infinite foundation, is obtained. It is assumed, that the thermal contact of a strip and top semi-space is imperfect. Sliding with a constant speed or linearly decreasing with time (as at braking) is considered. Influence of Biot number, describing thermal conductivity of contact, on the temperature of a system component is studied. Ó 2010 Elsevier Ltd. All rights reserved.
1. Introduction The thermal analysis is an integral part of designing modern units of friction. Therefore, the thermal problem of friction – determination of temperature fields in sliding contacting bodies – represents one of the major problems of tribology and tribotechnics. The theoretical basis for the calculation of temperature fields, and for the effect of the relative sliding velocity and the ratio of thermal properties of bodies on the distribution of temperature was proposed in the pioneer works of Bowden and Tabor [1,2], Newcomb [3], Block [4], Jaeger [5], Shchedrov [6], Korovchinskij [7,8] and others authors. The main disadvantage of the statement of a thermal problem of friction was the fact that thermal processes were investigated in the interfaced bodies separately, and distribution of thermal energy between them was set a priori. In 1959, Ling formulated conditions of the perfect thermal contact, assuming equality of temperatures of bodies in the contact area and equality of the sum of intensities of heat flows directed to the body to a specific power of friction [9]. These conditions are associated with temperature fields of bodies and satisfy the energy conservation law. In fact, with the conditions of perfect thermal contact the formulation and solution of thermal problem of friction in the form of mathematical physics began. Reviews of research on the thermal problem of friction can be found in articles [10,11]. Despite of significant theoretical importance of conditions of the perfect thermal contact, their application for calculation of
* Corresponding author. Tel.: +48 085 746 9206; fax: +48 085 746 9210. E-mail address: [email protected] (A.A. Yevtushenko). 0017-9310/$ - see front matter Ó 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijheatmasstransfer.2010.02.028
temperatures in real tribosystems fraught with some difficulties. The most important problem is the measurement of temperature in the contact area. In a reality interaction of bodies is carried out not on all nominal surfaces of friction, but in discrete areas of contact. The sizes of these areas are very small, and the temperature in them essentially exceeds average temperature of nominal area of contact [12,13]. As the modern technology does not allow to reliably and accurately measure the temperature in the areas of real contact at friction, the theoretical equality of temperature cannot be received practically [14]. This led to the occurrence of various models of imperfect thermal contact at which contact temperature of bodies are assumed to be different. In 1963, Podstrigach offered conditions of imperfect thermal contact, which take into account the thermal resistance of a thin intermediate layer between bodies at friction [15]. These conditions have received wide theoretical and practical application. Reviews of such investigations are given in the articles [16,17]. The conditions of non-ideal thermal contact, based on the assumption on discreteness of the contact area and taking into account the temperature difference between the friction surface, in 1970 were received by Barber [18]. Although the methods used to obtain the conditions of imperfect thermal contact by Podstrigach and Barber are quite different, results are similar. Conditions of imperfect thermal contact of bodies with frictional heating are reduced to two equations. The first as well as in Ling, equates the sum of intensities of heat flows directed inside the body to specific power of friction. In the second equation affirms that the difference in the intensity of heat flux on the surface of contact is proportional to the difference of contact temperatures. The coefficient of proportionality is the contact heat transfer coefficient (the value, inversely proportional to the thermal resistance of contact). The
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Nomenclature Bi ¼ hd=K s Biot number d thickness of the strip erfðxÞ Gauss error function erfcðxÞ ¼ 1 erfðxÞ complementary error function ierfcðxÞ ¼ p1=2 expðx2 Þ x erfcðxÞ integral of the error function erfcðxÞ f coefficient of friction K coefficient of heat conduction k coefficient of thermal diffusivity pressure p0 q0 ¼ fV 0 p0 intensity of the frictional heat flux (the friction power) T temperature T 0 ¼ q0 d=K s temperature scaling factor T ¼ T=T 0 dimensionless temperature t time
finite element method to study transient heat conduction in two sliding bodies with frictional heat generated at the contact interface, has been used in article [19]. The numerical studies of the heat flux generated by friction, the thermal conductance and intrinsic heat partition coefficient in the dry friction between two solid have been made in papers [20,21], too. Previously, the authors obtained the exact solution of thermal problems of friction for two tribosystems. The first consists of two elements – a plane-parallel strip, sliding over the surface of the semi-space. The second tribosystem contains three elements – the semi-space, moving along the surface strip deposited on a semi-infinite foundation. Solutions to the first tribosystem were obtained under perfect [22,23] and imperfect [24] conditions of thermal contact. Frictional heat generation in the second tribosystem was investigated only under perfect conditions of thermal contact [25,26]. Therefore, the aim of this article is to obtain and analyse the analytical solution to one-dimensional transient thermal problem of friction for the second tribosystem under conditions of imperfect thermal contact.
time of braking sliding speed initial sliding speed spatial coordinate
ts V V0 z
Greek symbols
s ¼ ks t=d2 dimensionless time (Fourier’s number) ss ¼ ks ts =d2 dimensionless time of braking f ¼ z=d
dimensionless coordinate
Indexes f s t
bottom semi-space strip upper semi-space
Fig. 1. Scheme of a three-element brake system.
The transient temperature fields T t;s;f ðz; tÞ can be found from the solution to the following transient boundary-value heat conduction problem of friction:
o2 T t ðf; sÞ 2. Problem formulation The problem of contact interaction of two semi-spaces is considered, where one of them is homogeneous and the other is a semi-infinite foundation with a surface covered by a strip of thickness d. The perfect heat contact between the strip and the foundation takes place. It is supposed, that the constant compressive pressures p0 in direction of z-axis of the Cartesian system of coordinates Oxyz are applied to the infinities in semispaces (Fig. 1). The homogeneous upper semi-space slides with the constant velocity V in the direction of the y-axis on the strip surface. Due to friction the heat is generated on a contact plane z ¼ 0. It is assumed, that sum of the intensities of the frictional heat fluxes directed into each component of friction pair is equal with the specific friction power and that between surfaces of the top semi-space and the strip being in contact, the heat transfer takes place. The wear of the contact surface is negligible. Let us find the distribution of temperature fields in the frictional elements. Further, all values and the parameters concerning a top semi-space, strip and foundation will have bottom indexes ‘‘t”, ‘‘s” and ‘‘f”, respectively (Fig. 1).
of
2
o2 T s ðf; sÞ 2
of o2 T f ðf; sÞ
¼
1 oT t ðf; sÞ ; os kt
¼
oT s ðf; sÞ ; os
0 < f < 1;
1 < f < 0;
s > 0;
s > 0;
1 oT f ðf; sÞ ; 1 < f < 1; s > 0; os kf of2 oT s oT K t t ¼ 1; s > 0; of f¼0 of f¼0þ oT s oT þ K t t ¼ Bi½T t ð0; sÞ T s ð0; sÞ; s > 0; of f¼0 of f¼0þ
¼
T s ð1; sÞ ¼ T f ð1þ; sÞ; oT f oT s ¼ K ; f of f¼1þ of f¼1
s > 0; s > 0;
ð1Þ ð2Þ ð3Þ ð4Þ ð5Þ ð6Þ ð7Þ
T t ðf; sÞ ! 0;
f ! 1;
T f ðf; sÞ ! 0;
f ! 1;
T t ðf; 0Þ T s ðf; 0Þ T f ðf; 0Þ
¼ 0;
0 6 f < 1;
ð10Þ
¼ 0;
1 6 f 6 0;
ð11Þ
¼ 0;
1 < f 6 1;
ð12Þ
s > 0; s > 0;
ð8Þ ð9Þ
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ð13Þ
! fx Gt ðf; xÞ ¼ ½aðxÞDR ðxÞ þ bðxÞDI ðxÞ cos pffiffiffiffiffi þ ½bðxÞDR ðxÞ kt ! fx aðxÞDI ðxÞ sin pffiffiffiffiffi ; 0 6 f < 1; kt
ð14Þ
Gs ðf; xÞ ¼ ½BiDR ðxÞ þ et xDI ðxÞ cos½ð1 þ fÞx þ ef ½BiDI ðxÞ et xDR ðxÞ sin½ð1 þ fÞx; 1 6 f 6 0;
where
z f¼ ; d kt kt ¼ ; ks T t ¼
Tt ; T0
s¼
ks t 2
d
K f ¼
;
Kf ; Ks
K t ¼
Kt ; Ks
kf ¼
kf ; ks
hd ; Ks
Bi ¼
T s ¼
Ts ; T0
T f ¼
Tf ; T0
T0 ¼
q0 d : Ks
2
We perform the Laplace integral transform [27]
Z
1
0
T t;s;f ðf; sÞeps ds;
1 F t;s;f ðf; pÞ; p
ð16Þ
Dt;s;f ðf; pÞ F t;s;f ðf; pÞ ¼ pffiffiffi ; pDðpÞ
ð17Þ
pffiffiffi pffiffiffi DðpÞ ¼ Bið1 þ et ef Þ þ 2et p sh p pffiffiffi pffiffiffi þ Biðet þ ef Þ þ 2et ef p ch p;
ð18Þ qffiffiffi
0 6 f < 1;
p k t
ð20Þ
qffiffiffip k f
;
1 < f 6 1;
ð21Þ
where
K
K
f t ffi ffi: et ¼ pffiffiffiffi ef ¼ qffiffiffiffi ;
kt
ð22Þ
kf
The dimensionless quantities 0 < et;f < 1 (22), where et characterizes thermal activity of the material of the top semi-space relative to the material of the strip and ef – of the foundation to the strip, are known as ‘‘the coefficient of thermal activity” [28]. Inverting the Laplace transforms of the temperatures (16), we obtain [27]
T t;s;f ðf; sÞ ¼
Z s
F t;s;f ðf; sÞ ds;
s P 0;
1 < f 6 1;
bðxÞ ¼ ef ðBi sin x þ x cos xÞ:
ð23Þ
0
where, using the integration in the complex p-plane with cut along the negative direction of the axis Re p [22], the inverse of the transforms F t;s;f ðf; pÞ, (17)–(22), have the form
ð30Þ
ð31Þ
Substituting the functions, F t;s;f ðf; sÞ (24)–(31), into the solutions (23), after integration with respect to s, we finally obtain
T t;s;f ðf; sÞ ¼
2
p
Z
1
0
Gt;s;f ðf; sÞ Pðs; xÞ dx; DðxÞ
s P 0;
ð32Þ
where
1 2 1 ex s : 2 x
ð33Þ
The dimensionless intensity of the heat fluxes in the top semi-space, the strip and in the foundation we define as:
oT t ðf; sÞ ; 0 6 f < 1; s P 0; of oT ðf; sÞ qs ðf; sÞ ¼ s ; 1 6 f 6 0; s P 0; of oT f ðf; sÞ ; 1 < f 6 1; s P 0: qf ðf; sÞ ¼ K f of qt ðf; sÞ ¼ K t
; ð19Þ
pffiffiffi pffiffiffi pffiffiffi Ds ðf; pÞ ¼ ðBi þ et pÞ½et shð1 þ fÞ p þ chð1 þ fÞ p; 1 6 f 6 0; pffiffiffi ð1þfÞ Df ðf; pÞ ¼ ðBi þ et pÞe
aðxÞ ¼ Bi cos x x sin x;
Pðs; xÞ ¼
pffiffiffi pffiffiffi f pffiffiffi pffiffiffi Dt ðf; pÞ ¼ ðBiet þ pÞsh p þ Bi þ ef p ch p e
3
6ð1 þ fÞx7 et xDR ðxÞ sin 4 qffiffiffiffiffi 5; kf
ð15Þ
on the heat conduction equations (1)–(3) and the boundary conditions (4)–(9) with the homogeneous initial conditions (10)–(12) for the temperature. Thus, we obtain the Laplace transforms of the dimensionless temperatures in the following form:
T t;s;f ðf; pÞ ¼
ð29Þ
6ð1 þ fÞx7 Gf ðf; xÞ ¼ ½BiDR ðxÞ þ et xDI ðxÞ cos 4 qffiffiffiffiffi 5 þ ½BiDI ðxÞ kf 2 3
3. Solution to the problem
T t;s;f ðf; pÞ ¼
ð28Þ
ð34Þ ð35Þ ð36Þ
Taking the relations (34)–(36) into account, after differentiating the Laplace transforms of temperature (16)–(21) with respect to f we obtain the transforms of the intensities of the heat fluxes in the form:
t;s;f ðf; pÞ ¼ q
Q t;s;f ðf; pÞ ; pDðpÞ
ð37Þ
where qffiffiffi p pffiffiffi pffiffiffi pffiffiffi pffiffiffi f kt Q t ðf; pÞ ¼ et Bief þ p sh p þ Bi þ ef p ch p e ; 0 6 f < 1;
pffiffiffi
pffiffiffi pffiffiffi Q s ðf; pÞ ¼ Bi þ ef p ef ch½ð1 þ fÞ p þ sh½ð1 þ fÞ p ; 1 6 f 6 0;
pffiffiffi ð1þfÞ Q f ðf; pÞ ¼ ef ðBi þ et pÞe
ð38Þ ð39Þ
qffiffiffip k f
;
1 < f 6 1;
ð40Þ
ð24Þ
and the denominator DðpÞ is given by formula (18). The transforms (37)–(40) have a branch point at the origin p ¼ 0. Therefore, the integration contour in the complex p-plane was chosen the same as in the article [22]. As a result, we obtain
D2I ðxÞ;
ð25Þ
qt;s;f ðf; sÞ ¼ q0t;s;f
DR ðxÞ ¼ Biðet þ ef Þ cos x 2et x sin x;
ð26Þ
where
DI ðxÞ ¼ Bið1 þ et ef Þ sin x þ 2et ef cos x;
ð27Þ
q0t ¼
F t;s;f ðf; sÞ ¼
DðxÞ ¼
Z
2
p
D2R ðxÞ
þ
1
0
Gt;s;f ðf; xÞ x2 s dx; e DðxÞ
s P 0;
2
p
Z
1
0
Q t;s;f ðf; pÞ x2 s dx; e xDðxÞ
ef et ; q0s ¼ q0f ¼ ; et þ ef et þ ef
s P 0;
ð41Þ
ð42Þ
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(
!
fx Q t ðf; pÞ ¼ et ½aðxÞDI ðxÞ bðxÞDR ðxÞ cos pffiffiffiffiffi þ ½bðxÞDI ðxÞ kt !) fx þaðxÞDR ðxÞ sin pffiffiffiffiffi ; 0 6 f < 1; kt
the top semi-space and the strip ðBi ! 1Þ can be found from Eqs. (32) and (33) with integrands in the form 2
ð43Þ
Q s ðf; pÞ ¼ ef ½BiDI ðxÞ et ef xDR ðxÞ cos½ð1 þ fÞx ½BiDR ðxÞ þet xDI ðxÞ sin½ð1 þ fÞx;
1 6 f 6 0;
ð44Þ
8 2 3 > < 6ð1 þ fÞx7 Q f ðf; pÞ ¼ ef ½BiDI ðxÞ et xDR ðxÞ cos 4 qffiffiffiffiffi 5 ½BiDR ðxÞ > : kf 2 39 > = 6ð1 þ fÞx7 þet xDI ðxÞ sin 4 qffiffiffiffiffi 5 ; 1 < f 6 1: > ; kf
ð45Þ
ð46Þ
Z
qt ð0;
½Gt ð0þ; xÞ Gs ð0; xÞ 2 1 ex s dx; 2 x DðxÞ p 0 s P 0; ð47Þ
T t ð0þ; sÞ T s ð0; sÞ ¼
2
Z 0
1
2
ð48Þ
where, taking the formulae (28), (29) and (43), (44) into account, we have
2
p
1 0
1 < f 6 1:
ð56Þ
3
6ð1 þ fÞx7 þ ð1 þ et ef Þ sin x sin 4 qffiffiffiffiffi 5; kf
The analytical solution in the form of series on ierfcðxÞ functions for tribosystem considered, in the case of the perfect heat contact between the top semi-space and the strip has been obtained in article [25]. By comparing this solution and the solutions (32) and (33) obtained in the present article, we obtain the values of the following integrals:
1
p
Z
pffiffiffi 1 Gt;s;f ðf; sÞ s X Kn T t;s;f ;n ðf; sÞ; Pðs; xÞ dx ¼ ð1 þ et Þ n¼0 DðxÞ
1
0
where
" T t;n ðf; sÞ ¼ ierfc
ef et ½Q s ð0; xÞ Q t ð0þ; xÞ : dx ¼ xDðxÞ ef þ et
! # " ! # f 1 f 1 2n þ pffiffiffiffiffi pffiffiffi þ kf ierfc 2n þ 2 þ pffiffiffiffiffi pffiffiffi ; kt 2 s kt 2 s
0 6 f < 1;
ð58Þ
2n f 2n þ 2 þ f pffiffiffi þ kf ierfc pffiffiffi T s;n ðf; sÞ ¼ ierfc ; 2 s 2 s 1 6 f 6 0; 20
ð49Þ
ð50Þ
Calculations of the integral in the left side of Eq. (50) by procedure QAGI from a package QUADPACK [29] has shown that this equality is carried out identically. On the interface f ¼ 1 from Eqs. (29), (30) and (44), (45) follows:
s P 0; ð57Þ
T f ;n ðf;
1
ð59Þ 3
2 1 þ fC 1 7 6B sÞ ¼ ierfc4@2n þ 1 qffiffiffiffiffi A pffiffiffi5; ð1 þ ef Þ 2 s k f
Substituting Eqs. (47)–(49) into boundary condition (5), we obtain the relationship
Z
ð55Þ
6ð1 þ fÞx7 Gf ðf; xÞ ¼ ðet þ ef Þ cos x cos 4 qffiffiffiffiffi 5 kf 2 3
½Q s ð0; xÞ Q t ð0þ; xÞ x2 s e dx; xDðxÞ
x Gt ð0þ; xÞ Gs ð0; xÞ ¼ ½Q s ð0; xÞ Q t ð0þ; xÞ Bi ¼ x½ðef et ÞDI ðxÞ cos x ð1 et ef ÞDR ðxÞ sin x:
ð54Þ
1 6 f 6 0;
1
ef et 2 ef þ et p s P 0;
qs ð0; sÞ qt ð0þ; sÞ ¼
fx 2 Gt ðf; xÞ ¼ ½ðet þ ef Þ cos2 x þ ef ð1 þ et ef Þ sin x cos pffiffiffiffiffi kt ! fx ð1 e2f Þ sin x cos x sin pffiffiffiffiffi ; 0 6 f < 1; kt
þ ef ð1 þ et ef Þ sin x sin ½ð1 þ fÞx;
and from Eqs. (41) and (42) we find sÞ þ sÞ ¼ 1; s P 0, which means that boundary conditions (4) are satisfied. Difference in temperature and intensity of heat flux on the contact surface is found from formulae (32), (33) and (41) in the form qs ð0þ;
ð53Þ !
Gs ðf; xÞ ¼ ðet þ ef Þ cos x cos½ð1 þ fÞx
We note that the values of q0t;s;f (42) are the value of the intensities of the heat flows in steady state. In contrast to the heat flux, the temperature (32) does not reach the stationary state. On the friction surface f ¼ 0 from the formulae (43) and (44) leads
Q s ð0; xÞ þ Q t ð0þ;xÞ ¼ Biðet þ ef Þ cosx 2et x sin x DI ðxÞ Bið1 þ et ef Þsin x þ 2et ef x cos x DR ðxÞ ¼ DR ðxÞDI ðxÞ DI ðxÞDR ðxÞ ¼ 0;
DðxÞ ¼ ðet þ ef Þ2 cos2 x þ ð1 þ et ef Þ2 sin x;
1 < f 6 1;
Kn ¼
k¼
kn ;
ð60Þ
0 6 k < 1;
ð1Þn jkjn ; 1 < k 6 0;
ð1 et Þ ð1 ef Þ ; ð1 þ et Þ ð1 þ ef Þ
ð61Þ
ð62Þ
i.e. the boundary conditions (6) and (7) are satisfied.
and the integrands in the left-hand side of Eq. (57) have the form (53)–(56). Heat transfer between two homogeneous semi-spaces ðK s ¼ K f ; ks ¼ kf Þ. Solution to the thermal problem of friction for two homogeneous semi-spaces with heat transfer between them also give Eqs. (23) and (24) at K f ¼ 1; kf ¼ 1; ef ¼ 1:
4. Some particular solutions to the problem
T t;f ðf; sÞ ¼
Gs ð1; xÞ ¼ BiDR ðxÞ þ et xDI ðxÞ ¼ Gf ð1; xÞ;
ð51Þ
Q s ð1; xÞ ¼ ef ½BiDI ðxÞ et xDR ðxÞ ¼ Q f ð1; xÞ;
ð52Þ
Perfect heat contact between top semi-space and strip. Dimensionless temperatures in the case of the perfect heat contact between
¼
Z s 0
2
p
F t;f ðf; sÞds;
Z 0
1
F t;f ðf; sÞ
Gt;f ðf; xÞ x2 s dx; s P 0; e DðxÞ
ð63Þ
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where
T t ðf;
2
DðxÞ ¼ Bi ð1 þ et Þ2 þ 4e2t x2 ; h
2
Gt ðf; xÞ ¼ Bi ð1 þ et Þ þ 2et x
ð64Þ 2
i
! ! fx fx cos pffiffiffiffiffi þ Bið1 et Þx sin pffiffiffiffiffi ; kt kt
0 6 f < 1;
ð65Þ
! " ! pffiffiffi 2 s f ð1 et Þ f sÞ ¼ erfc pffiffiffiffiffi ierfc pffiffiffiffiffi ð1 þ et Þ 2 kt s 2 kt s ð1 þ et Þ2 Bi !# a2 sþpafffiffiffi p ffiffiffi f k t erfc þe ; 0 6 f < 1; s P 0; a s þ pffiffiffiffiffi 2 kt s
T f ðf; sÞ ¼
h i 2 Gf ðf; xÞ ¼ Bi ð1 þ et Þ þ 2e2t x2 cosðfxÞ þ Biet ð1 et Þx sinðfxÞ; 1 < f 6 0:
ð66Þ
Taking the formulae (64)–(66) into account, the integrands in the solutions (63) take the form:
"
2
#
!
Gt ðf; xÞ 1 ð1 e2 ÞBi fx ð1 et ÞBix 1 2 2 t 2 cos pffiffiffiffiffi þ 2 2 ¼ 2et DðxÞ 4et ða þ x Þ et ða þ x2 Þ 4 kt ! fx sin pffiffiffiffiffi ; 0 6 f < 1; kt
ð67Þ T f ð0; sÞ ¼
1
0
Z
1
0
Z
1
0
1 2 ex s cosðfxÞ dx ¼ 2
2 pffiffiffi xex s p a2 s ajfj jfj p ffiffiffi e erfc a s sin ð jfjx Þdx ¼ e ða2 þ x2 Þ 4 2 s
p ffiffiffi jfj eajfj erfc a s þ pffiffiffi ; 2 s
F t ðf; sÞ ¼
! 2 f f ð1 et Þ 2 p pffiffiffiffiffiffi e 4kt s ffiffiffiffiffiffiffi sÞ ¼ erfc 2ð1 þ et Þ a ps ð1 þ et Þ 2 kt s !# a2 sþpafffiffiffi pffiffiffi f k t e erfc a s þ pffiffiffiffiffiffiffi ; 2 kt s
ð71Þ
qt ð0; sÞ ¼ ð72Þ
ð74Þ
ð81Þ
f2 1 f ð1 et Þ 2 pffiffiffiffiffiffi e4s erfc pffiffiffi þ ð1 þ et Þ 2ð1 þ et Þ a ps 2 s
pffiffiffi f 2 ea saf erfc a s pffiffiffi ; 1 < f 6 0; 2 s
s P 0: ð82Þ
qf ð0; sÞ ¼
et ð1 þ et Þ s P 0;
pffiffiffi ð1 et Þ 2 2 pffiffiffiffiffiffi ea s erfc a s ; 2ð1 þ et Þ a ps ð83Þ
pffiffiffi 1 ð1 et Þ 2 2 pffiffiffiffiffiffi ea s erfc a s ; þ ð1 þ et Þ 2ð1 þ et Þ a ps s P 0:
ð84Þ
Taking the approximate relation for large values of time into account [32]
pffiffiffi 1 2 ea s erfc a s pffiffiffiffiffiffi ;
ð85Þ
a ps
the contact temperatures (79) and (80) and intensity of heat fluxes (83) and (84) are rewritten as
Taking into account the values of integrals [30,31]:
pffiffi jfj 2 ea sþajfj erfc a s þ pffiffi ds 2 s 0 pffiffiffi
pffiffiffi 2 s jfj 1 j fj jfj 2 ¼ ierfc pffiffiffi þ 2 erfc pffiffiffi þ ea sþajfj erfc a s þ pffiffiffi ; a a 2 s 2 s 2 s
s P 0;
On the contact surface f ¼ 0 from Eqs. (81) and (82) results
ð73Þ
Z s pffiffiffiffiffiffi 1 f2 jfj pffiffi e 4s ds ¼ 2 psierfc pffiffiffi ; s 2 s 0
et
0 6 f < 1; qf ðf; sÞ ¼
2 2 pffiffiffi f 1 ð1 et ÞBi a sþpffiffiffi f k t erfc pffiffiffiffiffiffi e 4kt s e a s þ pffiffiffiffiffiffiffi ; 2 4et 2et ps 2 kt s
pffiffiffi f2 1 ð1 et ÞBi a2 saf f e erfc a s pffiffiffi ; F f ðf; sÞ ¼ pffiffiffiffiffiffi e4s þ 4et 2 ps 2 s 1 < f 6 0:
ð80Þ
ð69Þ
!
0 6 f < 1;
pffiffiffii s et ð1 et Þ h 2 1 ea s erfc a s ; 2 p ð1 þ et Þ Bi
qt ðf;
functions F t;f ðf; sÞ (63) takes the form af
2 ð1 þ et Þ
ð79Þ rffiffiffiffi
After differentiating the dimensionless temperatures (77) and (78) with respect to f, from the relations (34) and (36) we obtain the dimensionless intensity of heat fluxes in the form:
ð70Þ
2 pffiffiffi ex s pea2 s af f e erfc a s pffiffiffi cosðfxÞ dx ¼ 2 2 ða þ x Þ 4a 2 s
pffiffiffi f af þe erfc a s þ pffiffiffi ; 2 s
pffiffiffii s ð1 et Þ h 2 1 ea s erfc a s ; þ 2 p ð1 þ et Þ Bi
ð68Þ
rffiffiffiffi
p f42s e ; s
rffiffiffiffi
s P 0:
By using the values of the integrals [30]
Z
2 sÞ ¼ ð1 þ et Þ
s P 0;
where
ð1 þ et Þ a¼ Bi: 2et
ð78Þ
The maximal temperatures are reached on a plane of friction f ¼ 0 and, as follows from the solutions (77) and (78), are equal:
T t ð0;
" # 2 Gf ðf; xÞ 1 ð1 e2 ÞBi ð1 et ÞBix 1 þ 2 2 t 2 cosðfxÞ þ ¼ sinðjfjxÞ; 2 4et ða2 þ x2 Þ DðxÞ 4et ða þ x Þ 1 < f 6 0;
pffiffiffi
2 s f et ð1 et Þ f p ffiffiffi erfc ierfc pffiffiffi þ ð1 þ et Þ 2 s 2 s ð1 þ et Þ2 Bi
p ffiffiffi f 2 þea saf erfc a s pffiffiffi ; 1 < f 6 0; s P 0: 2 s
ð77Þ
ð75Þ
s ð1 et Þ 1 1 pffiffiffiffiffiffi ; s 1; þ p ð1 þ et Þ2 Bi a ps
2 ð1 þ et Þ
T f ð0; sÞ
pffiffiffi
2 s e ð1 et Þ 1 pffiffiffiffi t p ffiffiffiffiffiffi ; 1 a ps ð1 þ et Þ p ð1 þ et Þ2 Bi
Z s
ð76Þ
from the first equation (63), we obtain the dimensionless temperatures in the semi-spaces in closed form:
rffiffiffiffi
T t ð0; sÞ
qt ð0; sÞ
et ð1 þ et Þ
ð1 et Þ 1 pffiffiffiffiffiffi ; 2ð1 þ et Þ a ps
s 1;
s 1;
ð86Þ
ð87Þ
ð88Þ
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qf ð0; sÞ
1 ð1 et Þ 1 pffiffiffiffiffiffi ; þ ð1 þ et Þ 2ð1 þ et Þ a ps
s 1:
ð89Þ
From Eqs. (83) and (84) or (88) and (89) follows, that qf ð0; sÞ þ qt ð0; sÞ ¼ 1; s P 0, i.e. boundary condition (4) is satisfied. In addition, from Eqs. (86)–(89) we obtain
qf ð0; sÞ qt ð0; sÞ ¼
h i ð1 et Þ 1 1 þ pffiffiffiffi ¼ Bi T t ð0; sÞ T f ð0; sÞ ; ð1 þ et Þ a ps
ð90Þ that testifies the satisfaction of a boundary condition (5). Perfect heat contact between two sliding uniformly homogeneous semi-spaces. Substituting in solutions (77), (78) and (81), (82) Bi ! 1 ða ! 1Þ, we find
T t ðf;
! pffiffiffi 2 s f sÞ ¼ ; qt ðf; sÞ ierfc pffiffiffiffiffiffiffi ð1 þ et Þ 2 kt s ! et f ¼ ; 0 6 f < 1; erfc pffiffiffiffiffiffiffi ð1 þ et Þ 2 kt s
s P 0;
2 s f T f ðf; sÞ ¼ ierfc pffiffiffi ; qf ðf; sÞ ð1 þ et Þ 2 s
1 f ¼ erfc pffiffiffi ; 1 < f 6 0; ð1 þ et Þ 2 s
ð91Þ
s P 0:
ð92Þ
1
Gf ðf; xÞ ¼ ð1 þ et Þ cosðfxÞ;
0 6 f < 1;
ð93Þ
1 < f < 0:
Substituting functions DðxÞ and Gt;f ðf; xÞ (93) into solutions (23) and (24), we obtain the temperature in the top semi-space in the form
T t ðf;
2 sÞ ¼ pð1 þ et Þ
Z sZ 0
0 6 f < 1;
1
e
x2 s
0
! x cos f pffiffiffiffiffi dx ds; kt
s P 0:
ð94Þ
Using successively the values of integrals (70) and (75) from Eq. (94), we obtain the first of the formulas (91). All the remained formulas (91) and (92) are received similarly.
5. Application to the solution to the thermal problem of friction during braking In the case of constant pressure p0 the sliding speed during braking decreases from initial value V 0 to zero at the stop time moment linearly with time (braking with constant deceleration) [34]. Then, the intensity of heat generation on the contact surface is equal to the power density of the friction force
qðtÞ ¼ q0 q ðsÞ;
q ðsÞ ¼ 1
o b T ðf; s sÞ ds; os t;s;f
0 6 s 6 ss ;
ð96Þ
b ðf; sÞ have the form (32) where the dimensionless temperatures T t;s;f and (33) with integrands (25)–(31). Substituting temperature b ðf; sÞ (32) to the right parts of Eq. (96) and changing the order T of the integration, we obtain
T t;s;f ðf; sÞ ¼
2
p
Z
1
0
Gt;s;f ðf; sÞ P0 ðs; xÞ dx; DðxÞ
0 6 s 6 ss ;
ð97Þ
where
P0 ðs; xÞ ¼ Pðs; xÞ
1 ½s Pðs; xÞ; x2 s s
0 6 s 6 ss ;
ð98Þ
s ; 0 6 s 6 ss ; ss
Pðs; 0Þ ¼ s;
P0 ðs; xÞ ¼ s 1
s ; 0 6 s 6 ss : 2ss
ð99Þ
We note that the solution to this problem in the limiting case of the perfect thermal contact of the top semi-space and the strip ðBi ! 1Þ was obtained earlier in article [26].
Eqs. (91) and (92) give the solution to the thermal problem of friction for two homogeneous semi-spaces [33]. The same solution can be obtained in a different way from solution in the case of perfect heat contact between the top semi-space and the strip. The formulae (53)–(56) under conditions of kf ¼ 1 and ef ¼ 1 will take the form DðxÞ ¼ ð1 þ et Þ2 and
B fx C Gt ðf; xÞ ¼ ð1 þ et Þ cos @qffiffiffiffiffiA; kf
q ð sÞ
0
and function Pðs; xÞ has the form (33). In limiting case x ! 0 from formulae (33) and (98) we find
pffiffiffi
0
T t;s;f ðf; sÞ ¼
Z s
ð95Þ
where ss is the dimensionless duration of braking. The solution to a boundary-value problem of heat conductivity (1)–(12) in case, when the intensity of the specific friction power on the right side of a boundary condition (4) has the form (95), we find using the Duhamel’s theorem [27]
6. Numerical analysis The influence of input parameters, like the thickness of the strip d, the physical constants K t;f ; kt;f (13) and the duration of braking ts on the temperature has been studied in the articles [25,26]. Therefore, the main purpose of numerical analysis in this article will be study the influence of the Biot number on temperature, which characterizes the heat transfer through the surface of contact between the strip and the top semi-space. We note that in a limiting case Bi ! 1 we should obtain results of the articles [25,26]. The numerical analysis of a problem shall be executed for the same materials, as in these articles. We have taken the upper semi-space as the cast iron, ChNMKh: Kt = 51 W m1 K1, kt = 14 106 m2 s1 and the foundation as steel, 30KhGSA: Kf = 37.2 W m1 K1, kf = 10.3 106 m2 s1 [34]. In case of uniform sliding of bodies, the strip materials are aluminum, Al: Ks = 209 W m1 K1, ks = 86 106 m2 s1 or zirconium dioxide, ZrO2: Ks = 2 W m1 K1, ks = 0.8 106 m2 s1 [25]. When sliding with a constant deceleration (braking) was considered, the material of a strip was the metal-ceramics, FMK-11: Ks = 34.3 W m1 K1, ks = 15.2 106 m2 s1 [26]. The thickness of a strip was equal d = 5 mm, the contact pressure was equal p0 ¼ 1 MPa and the initial temperature was equal to 20 °C. In the case of uniform sliding we set the initial sliding speed V0 = 5 m s1 and coefficient of friction f ¼ 0:3. At braking with constant deceleration the input parameters were equal V0 = 30 m s1, f ¼ 0:7 and, additionally, the time of braking ts ¼ 3:44 s [34]. The evolution of temperature on the contact surface z ¼ 0 at sliding with constant speed for a strip made from aluminum and zirconium dioxide is shown in Figs. 2 and 3, respectively. For the small values of Biot number ðBi ! 0Þ, the greatest jump of temperatures of the top semi-space and the strip are noticed. In the case of the aluminum strip increase in Biot number leads to a rapid equalization of temperature on the friction surface (Fig. 2a). Thus, for a value of Bi = 5 the contact temperatures of the upper semi-space and the strip are identical. For values of Bi P 5 we can use the solution to the considered thermal problem of friction, at perfect thermal contact [25]. Alignment of temperatures of a strip and top semi-space on a surface of contact at increase of Biot number in the case of ZrO2 strip occurs much more slowly, than for an aluminum strip (Fig. 3a). For the same values of Biot number Bi = 5, we see that the jump of the contact temperatures is still great. The
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Fig. 2. Evolution of the contact temperature Tð0; tÞ at uniform sliding for two tribosystems: (a) three element – cast iron/aluminum/steel; (b) cast iron/aluminum semi-spaces.
Fig. 3. Evolution of the contact temperature Tð0; tÞ at uniform sliding for two tribosystems: (a) three element – cast iron/zirconium dioxide/steel; (b) cast iron/ zirconium dioxide semi-spaces.
curves plotted for large values of Biot number ðBi ! 1Þ in Figs. 2a and 3a, coincide with the corresponding results obtained in article [25]. The evolutions of temperature on the contact surface between two semi-spaces are shown in Figs. 3b and 4b. Since aluminum has a high coefficient of thermal conductivity, the curves of the evolution of temperature, as shown in Fig. 2a and b, are different. This is proved, that there is, for example, much greater jump of temperatures at small values of the Biot number in the case of contact between two semi-spaces (Fig. 2b) than at the contact of the semi-space and strip (Fig. 2a). Since zirconium is a poor conductor of heat, the evolutions of temperatures in Fig. 3a and b are almost identical. Hence the practical conclusion, that in the case of a strip from zirconium dioxide, we can use the analytic solution for the two semi-spaces (77) and (78), instead of solutions for three-element system in the form of integrals (23)–(31).
Dependences of the contact temperatures of the top semi-space and strip on the Biot number are shown in Fig. 4. We see that the contact temperature is greater in that element of friction pair, which has a smaller coefficient of thermal conductivity. Such elements are: the top semi-space from the cast iron in Fig. 4a, and a dioxide zirconium strip in Fig. 4b. With the increase in the Biot number, the jump of temperatures decreases. In the case of an aluminum strip (Fig. 4a), we see that the value of Bi = 5 almost corresponds to the solution to the perfect thermal contact between the top semi-space and the strip, obtained in the article [25]. In the case of a strip from zirconium dioxide (Fig. 4b), alignment of temperatures on the contact surface with increasing of the Biot number is much slower. Even at Bi ¼ 100 the jump of temperatures are still significant. The evolution of the temperature (97) on a contact surface z ¼ 0 during braking for three values of Biot number is shown in Fig. 5.
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Fig. 5. Evolution of the contact temperature Tð0; tÞ at braking for different values of the Biot number Bi.
Fig. 4. Dependence of the contact temperature Tð0; tÞ at uniform sliding on the Biot number Bi in time moment t ¼ 5 s for two strip materials: (a) aluminum; (b) dioxide zirconium.
The material of a strip in this case is the metal-ceramic FMK-11, used as a pad in brakes of planes [35]. We see that for fixed value of the parameter Bi temperature rapidly increases with the beginning of the process of braking. The maximal temperature T max is reached at the time moment tmax , which nearly corresponds to half of time braking ts ¼ 3:44 s: for values of the Biot number 0; 5 and Bi ! 1 we have tmax ¼ 1:5 s; 1:6 s and 1.6 s, respectively. Cooling of the contact surface, which started after reaching the maximum temperature occurs before the stop time moment ts ¼ 3:44 s. The greatest difference of temperatures of a top semi-space and a strip DT ¼ jT t ð0; t max Þ T s ð0; tmax Þj takes place for small values of the Biot number: DT = 232 °C at Bi ! 0. If Bi ! 1, then the temperatures of the top semi-space and the strip coincide. The maximal temperature in this case is equal 741 °C. The same value was obtained earlier by solving the considered problem at perfect thermal contact strip and the top semi-space [26].
Fig. 6. Dependence of the temperature on the distance jzj from surface of friction for three values of Biot number Bi in the time moment t ¼ 1:6 s for the strip thickness d = 5 mm.
The change of temperature in the elements of tribosystem in depth from a surface of friction is shown in Fig. 6. The maximal temperature is reached on a surface of friction and in the time moment t ¼ 1:6 s for values of the Biot number 0; 5 and Bi ! 1 is equal T t;max ¼ 868 C; 757 C; 741 C for the top semi-space and T s;max ¼ 636 C; 723 C; 741 C for the strip, respectively. The Biot number does not influence the character of temperature change over the thickness of a strip. Even at Bi ! 1 decrease in temperature with increasing distance from the contact surface is almost linearly. The cast iron top semi-space is heated to a depth of
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tion for two semi-spaces with imperfect or perfect thermal contact is obtained. We compared the results of calculation of temperature, obtained for the three-element system and at the sliding of two semi-spaces. It is established, that evolution of temperature on a surface of contact of the cast iron top semispace and a zirconium dioxide strip is almost the same, as in case of contact of two semi-spaces made from the same materials. This is caused by a small coefficient of thermal conductivity of zirconium dioxide. Influence of the steel foundation on temperature is noticeably in the case of the aluminum strip, which possesses significant heat conductivity. In the particular case when the values of Biot number Bi ! 1, we obtain the maximal temperature on the contact surface 741 °C, which coincides with the theoretical result of the article [26] and is only a little different from the experimental value 760 °C, published in the monograph [34]. If we take into account the heat transfer through the surface of contact (Bi = 5), then the calculated maximal temperature on a surface of the top semi-space will be equal 757 °C.
References Fig. 7. Dependence of the contact temperature Tð0; tÞ at braking on the Biot number Bi in time moment t ¼ 1:6 s for metal-ceramic FMK-11 strip material.
approximately equal thickness of the strip (d = 5 mm). Temperature in the steel foundation is insignificant. Dependences of the contact temperatures of the top semi-space and a strip on Biot number in the moment of time t ¼ 1:6 s, when the maximal temperature is reached, are shown in Fig. 7. We see that the thermal contact between these elements of the tribosystem can be considered as perfect already for Bi 80. Hence, for values of Biot number Bi > 80 it is possible to use the analytical solution obtained in the article [26] at calculation of the maximal temperature. 7. Conclusions The frictional heat generation in tribosystem consisting of three elements: a semi-space sliding on the surface of a strip deposited on a semi-infinite foundation is investigated. In contrast to the previously obtained solutions for the same tribosystem proposed in articles [25,26] here we take into account the heat transfer through the contact surface between the top semi-space and the strip. Influence of the dimensionless parameter – the Biot number Bi, describing a heat transfer through the contact surface, on temperature fields in the elements of tribosystem is studied. At relative sliding bodies with constant speed, the aluminum and zirconium dioxide is chosen as a material of a strip. These materials are one of most often used in tribotechnics as protective films and heatshielding coatings. The possibility of using the obtained solution to determinate temperature in friction components of brakes is shown. In this case, the sliding speed decreases linearly in time from the maximum value at the start of braking until stop. The metal-ceramic FMK-11 has been chosen as a material of a strip, since it is used at manufacturing of the pad in brakes of planes. From our point of view, the main result obtained in this article is the definition for the investigated class of materials and input parameters of the limit values of Biot number, in which it is possible to neglect the influence of heat transfer through the contact surface on the temperature and use the solutions of articles [25,26]. In case of identical materials of the strip and the foundation the analytical solutions to a corresponding heat problem of fric-
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