calculation of the contact temperature of a friction...

Calculation of the contact temperature of a friction couple with a high-speed heat source 51

SCIENTIFIC PROBLEMS OF MACHINES OPERATION AND MAINTENANCE

4 (160) 2009

YU.G. KRAVCHENKO*, B.I. PELESHENKO**, A.I. BURYA**, O.YU. KUZNETSOVA2

Calculation of the contact temperature of a friction couple with a high-speed heat source

K e y w o r d s

Heat source density, high-speed sources of heat, temperature field.

S u m m a r y

The analytical calculation of the coefficient of heat flow distribution and the average contact temperature of the semi-limited body friction with the tetrahedral counterbody are presented, an example of calculating the friction conditions with high contact stresses and velocities is given.

1. Introduction

The durability of friction knots is very dependent on the level of their operation’s heat regime. The intensity of the working surfaces’ wear directly depends on the contact temperature of friction. It concerns the integral effect of the power and velocity of friction via the heat flows density and is an important evaluation index that can have a critical value for exact conditions of tribological processes.

In tribotechnics, high-speed sliding friction pairs where the temperature laws of high-speed sources (HSS) of heat (when the source’s transfer speed exceeds the speed of heat distribution in this body) is of high priority.

* National metallurgical academy of Ukraine, Gagarina avenue, 4, 49600, Dnepropetrovsk. ** Dnepropetrovsk State Agrarian University, Voroshilov st., 25, 49600, Dnepropetrovsk,

[email protected]

Yu.G. Kravchenko, B.I. Peleshenko, A.I. Burya, O.Yu. Kuznetsova

52

In this respect, the problem of improving the contact temperature calculation and the management of heat phenomena is always a priority in tribology.

The aim of this work is to obtain a formula for the temperature field of heat HSS concerning the behaviour of contact stress distribution, to determine the heat distribution between the body and the counterbody, to create the algorithm of calculating the average contact temperature of friction.

The work is performed on the basis of works by N. N. Rykalin [1], G. S. Karslow, D. K. Yeger [2], A. N. Reznikov [3] and their method of heat sources, which the system of point instant [1, 2] and fictive (reflected) [1, 3] sources.

2. Problem statement

The technique of calculating the contact temperature of friction consists of the following modules: 1. Deriving a formula of temperature fields and their average values for the

counterbody working surfaces and the heat source moving across the semi-limited body at secondary boundary conditions (at the known distribution behaviour and the numerical value of the heat sources density [1]);

2. Obtaining an expression for the coefficient of heat distribution between the friction bodies at quaternary boundary conditions (which lie in assigning the equation of the average temperature along the contact surface [3]); and,

3. Properly defining the heat flow density and the contact temperature of friction. The starting point for the calculation is the well-known formula [1, 2, 3] for

the temperature of the linear instant source QL I (J/m) on the surface of the semi-limited body (z=0, -∞ <y < ∞), which has been derived by means of the Kelvin heat conductivity equation for the point instant source within the non-limited body and by means of the fiction source (the semi-limited body artificially turns into the non-limited one)

21( )

expLIQ x xT

2 t 4atπλ −= −

(1)

Where, λ and a are the coefficients of heat- and temperature conductivity; t is the period of temperature observance; x1 is the abscissa of the linear instant source by x axis.

The situation of HSS essence (the released heat does not extend in front of

the source [1]) significantly simplifies the mathematical calculations of temperature fields. HSS classification incurs from the dimensionless Péclet criterion /eP v l a= ⋅ (v and l are the velocity and the length of the source in the

direction of movement, the coefficient of temperature conductivity a belongs to


the body material across which the source moves) and at eP 8 10> ÷ [3] the

source may refer to HSS. The formula for the average value of the counterbody’s temperature field is

obtained in [4]. It is accepted that there is no heat removal due to the convective heat

exchange with the environs (cooling). We do not account for heat accumulation by the counterbody across time and by the body due to the thermocyclicity of certain segments of the friction surface.

2. Main body

Reaching the stated aim includes two tasks. 1. Deriving a formula of the temperature field for the flat (band) HSS. First,

by means of integrating the initial expression (1) by x1 with the help of

substitution of uat

xx −=−4

1 ( 1 4dx atdu= , the integration limits u1= -∞ at

x1= -∞ and u2=∞ at x1=∞), we define the temperature of flat instant (FI) source with the density of QF I

FIFI

Q aТtλ π

=⋅

(2)

The further calculation is connected with two simplified statements at the passing of the linear constant (LC) source through the elementary band dx at the surface of the semi-limited body (Fig.1), indicating the following: – The period of the source’s affect on dx seems to be an instant dt. – All the heat flow during the period of dt propagates only as a perpendicular

to dx band.

Fig. 1. Calculation scheme of the linear HSS temperature


54

As a result, for dx band, we obtain the FI source QFI (J/m2); and, for the period dt, we obtain the LC source qLC (W/m). From the heat balance equation

QFI⋅dx = qLC⋅dt (J/m), we accept that QFI= qLC·dx

dt. Then, at QFI= LCq

v and

t = 1x x

v−

in (2) we obtain the temperature of the linear high-speed source

1( )LC

LH

q aT

x xvλ π=

⋅ ⋅ −⋅ (3)

The second integral transition leads to the dependency of the temperature field of the flat HSS on the heat source shape (the function of contact normal

stresses

l

xf 1 )

11

10T

xx

f dx1 l

Fl x x

⋅ =

−∫ (4)

The level of the average temperature with the introduction of the

dimensionless parameter 1x

lψ = is determined by the coefficient of the

source’s shape

( )1

0

S TAK F F dψ ψ= = ⋅∫ (5)

And this is calculated taking into account the general heat release density distribution

( )1

0

0

/Gq F S f d vvµ µ σ ψ ψ

= ⋅ = ⋅ ⋅ ⋅ ∫ (6)

(here, F Pµ µ= ⋅ is the friction force, µ is the coefficient of friction, P is the

pressure force; S=l·b is the contact area; σ0 are the peak values of normal

stresses at function ( )f ψ ) into the body via ε* coefficient

( )FH Gq 1 qε ∗= − ⋅ (7)


According to the formula for flat high-speed source (FH)

S FHFH

K q a lT

vλ π⋅ ⋅ ⋅=⋅ ⋅

(8)

For example, if the source’s density is equally distributed constl

xf =

1

(Fig. 2), then 1

0 1

x

T

dx1 2F x 2

l x x lψ= = =

−∫ , a 1

0

S

4K 2 d

3ψ ψ= ⋅ =∫ .

Fig. 2. Temperature field FT at equal density qFH

The value of FT function (4) and KS coefficient (5) for different shapes of heat sources are cited in Table 1.

Table 1. Dependency of FT function and KS coefficient of shape on the nature of the distribution of

contact normal stresses ( )0x fσ σ ψ= ⋅ at 1x

lψ =

Distribution of stresses Temperature field of HSS N

function )(ψf distribution formula FT (4) graph SK (5)

1 1 ( constx =σ )

xl

v

σ

σx

0

ψ2

0 0.5 1

1

2FT

ψ

1.333

2 ψ−1

0 0.5 1

0.5

1

)667.01(2 ψψ −

0 0.5 1

0.5

1

0.8


56

Distribution of stresses Temperature field of HSS N

function )(ψf distribution formula FT (4) graph SK (5)

3 ψ

0 0.5 1

0.5

1

ψψ333.1

0 0.5 1

1

2

0,533

4 )1( ψψ −

0 0.5 1

0.1

0.2

0.3

)267.0333.0(4 ψψψ −

0 0.5 1

0.2

0.4

0.229

5 21 ψ−

0 0.5 1

0.5

1

)2533.01(2 ψψ −

0 0.5 1

0.5

1

1.5

1.029

6 5.0)1( ψ− 0 0.5 1

0.5

1

ψψψψ

−+

−+1

1ln)1(

0 0.5 1

0.5

1

1.5

1.0

7 23ψ−e

0 0.5 1

0.5

1

)4829.126.11(2 ψψψ +−

0 0.5 1

1

2

3

1.084

8 2)1(3 ψ−−e

0 0.5 1

0.5

1

)483.1323.8

28.1285.2(2

ψψψψψ

+

−+−

0 0.5 1

1

2

1.255

9 )221(3 2ψψ +−−e 0 0.5 1

0.1

0.2

)431.7346.16

24.641(1.0

ψψψψψ

+

−++

0 0.5 1

0.2

0.176

10 ( )3 1eψψ

−−

0 0.5 1

0.5

1

)310.0

215.022.01(2

ψψψψ −−−

0 0.5 1

0.5

1

1.5

1.024


2. After obtaining the expression for the coefficient of heat distribution, we shall designate the share of friction heat absorption with the counterbody by ε∗

coefficient and with the body - by 1-ε∗. The solution issues from the equation of average contact temperatures of the HSS of the semi-limited body (8)

( ) S G B

B

1 K q a lT

v

ελ π

∗− ⋅ ⋅ ⋅ ⋅=

⋅ ⋅ (9)

And on the surface of the tetrahedron’s (counterbody’s) friction [4]

G

C

q l UT

επ λ

∗ ⋅ ⋅ ⋅=⋅

(10)

Whence,

B

C S

e

1

P1 U

K

ελλ π

∗ =+ ⋅ ⋅

⋅

(11)

Where, / BeP l av= ⋅ is Péclet criterion; 0

01

U 3,195 lnF6 F

= + +⋅

(20

Ca tF

l

⋅= is

Furie criterion, t is the source effect period); indices T and K at λ and a refer to the body and the counterbody.

The calculation analysis of formula (11) allows one to conclude that the

increase of heat dissipation via the counterbody ε∗ is connected with the ratio /B Cλ λ and /C Ba a , velocity v and length l of the source of heat release. Here,

the most significant effect is made by the ratio /B Cλ λ , and the necessary area of

the contact surface is expedient to be provided at the lowest possible length l at the account of increasing the width b.

3. Calculation results

The algorithm of calculation at the initial pressure force P, friction force v, length l and width b of the contact comes to the choice and the definition of the following: coefficients of heat conductivity λ and temperature conductivity a of the body and counterbody materials; criteria of Péclet eP (11) for the body and

of Furie 0F (11) for the counterbody; the coefficient of the heat source’s shape

KS (5, Table 1); functions of the counterbody’s temperature field U (11); the


58

coefficient of the heat flow distribution ε∗ (11); the coefficient of friction µ; density of heat release qG(6); average contact temperature T (9) or (10).

An example of calculating the coefficients of the heat flow distribution between the body and the counterbody and the contact temperature at equal normal pressure force (load) for the friction couple “steel (body) – anti-friction material (counterbody)” is cited in Table 2.

Additional calculations have shown the decrease of the coefficient of the distribution of heat release into the counterbody ε∗ with the increase of the period of the couple’s operation t at the stable temperature value T. Within formula (8) the period of time t is also absent as a factor of influence on temperature T.

The ε∗ (11) substitution into T (9) or (10) leads to a formula of the direct calculation

2

2

C B

C S B

C

PT

l ba t K al

3,195 ln ll 6 a t

v

v

µ

λ λ ππ

⋅ ⋅= + ⋅ ⋅ ⋅ ⋅ ⋅ + + ⋅ ⋅ ⋅

(12)

Which demonstrates the links between all the laws of the influence of incoming factors.

Table 2. The influence of the counterbody material on contact temperature (the body material being the hardened low-alloyed steel with λ=43 W/(m K), cp·ρ =3.6 106 J/(m3 K), a=11.9 10-6 m2/sec; length l=0.02m and width b=0.04m of the contact; the coefficient of the source’s shape

KS=0.8; pressure force P=20 103 N; velocity v =5 m/sec, Péclet criterion Pe = 8403.4; durability t=3.6 103 sec)

Thermophysical properties [5,

6, 7] at 20 °C Name of the

counter-body

material

λ,, W

m K⋅

(cp ρ) 106,

3

J

m K⋅

a 10-6, 2m

sec

Furie criterion,

F0

Distri-bution

function, U (11)

Distri-bution

coefficient,

ε∗ (11)

Coefficient of friction

of dry , boundary

µ

Density of heat release q0 106,

2

W

m

Contact temperat

ure T+20°C

Steel 57.4 3.7 15.5. 139.5 8.134 0.0025 0.35

0.25

43.75

31.25

119.9

90.5

Cast iron 51.5. 3.6 14.3 128.7 8.053 0.0023 0.30

0.20 37.5

25.0 105.6

77.2

Bronze 134 3.6 37.2 336.6 9.014 0.0053 0.25

0.15 31.25

18.75

90.9

62.5

Siliconized graphite

100 1.6 62.5 562.5 9.527 0.0038 0.15

0.10 18.75

12.5 63.2

48.8

Fluoroplas- tic

0.22 2.4 0.096 0.864 3.242 0.000024 0.10

0.05 12.5

6.25 48.1

34.1


The decrease of temperature T is reached at the decrease of the

ratio /P v l bµ ⋅ ⋅ ⋅ , the increase of heat conductivity λC, λB and the decrease of temperature conductivity aC, aB of the friction couple’s materials, as well as at the decrease of the coefficient of HSS shape’s value KS.

These investigations are of significant practical application during the construction of stress-speed friction knots.

Conclusion

1. The calculation of the temperature field of the band high-speed source of heat and the calculation of the source shape coefficient values for some functions of stress distribution within the friction contact have been completed.

2. The expression for defining the coefficient of the density distribution of the heat release between the body and the counterbody correspondingly with the high-speed and constant heat source has been obtained.

3. Using the example of calculating with equal pressure force, the influence of antifriction materials of the counterbody with different thermo-physical properties and coefficients of friction on the coefficient of density distribution of the heat release and the contact temperature has been demonstrated.

4. A generalised formula of significant practical application for the direct calculation of contact temperature with all the initial element thermo-physical, mechanical, geometrical factors of the tribological process has been derived.

5. The pressure force, coefficient of friction, contact zone width and coefficients of heat conductivity of the body and counterbody materials influence the friction temperature value to a greater extent than the friction velocity, contact length in the direction of friction velocity, the coefficients of temperature conductivity of the friction materials and the coefficient of high-speed source’s shape.

References

[1] Рыкалин Н.Н. Расчеты тепловых процессов при сварке. - М.:Машгиз, 1951. – 296 с. [2] Карслоу Г.С., Егер Д.К. Теплопроводность твердых тел. – М.:Наука, 1964. -487 с. [3] Резников А.Н. Теплофизика процессов механической обработки материалов. – М.:

Машиностроение, 1981. – 279 с. [4] Пелешенко Б.И., Кравченко Ю.Г., Буря А.И., Коваленко А.В. Расчет температурного

поля на полосовой поверхности трения контртела. – /Трение и износ, 2010. - № 6. – С.52-63.

[5] Чиркин В.С. Теплопроводность промышленных материалов. – М.: Машгиз, 1962. – 247с.


60

[6] Теплопроводность твердых тел: Справочник /Под ред. А.С. Охотина. – М.: Энергоатомиздат, 1984. – 320с.

[7] Физические величины: Справочник /А.П. Бабичев, Н.А. Бабушкина, А.М. Братковский и др.: Под ред. И.С. Григорьева, Е.З. Мейлихова. – М.: Энергоатомиздат, 1991. – 1232 с.

Manuscript received by Editorial Board, May 31st, 2010

calculation of the contact temperature of a friction...

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