4.15.3 Contact Resistance of Isothermal Elliptical Contact Areas The general spreading–constriction resistance model, as proposed by Yovanovich 1971, 1986, is based on the assumption tha
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B
A =
(1/k2)E(k) − K(k) K(k) − E(k) (4.151)
whereK(k) and E(k) are complete elliptic integrals ofthe first and second kind of
modulusk.
The Hertz solution requires the calculation ofk, the ellipticity, K(k), and E(k).
This requires a numerical solution ofthe transcendental equation that relatesk, K(k),
andE(k) to the local geometry ofthe contacting solids through the geometric
pa-rametersA and B This is usually accomplished by an iterative numerical procedure.
To this end, additional geometric parameters have been defined (Timoshenko and Goodier, 1970):
cosτ = B − A B + A and ω = A B ≤ 1 (4.152) Computed values of m and n, or m/n and n, are presented with τ or ω as the
independent parameter Table 4.15 shows howk, m, and n depend on the parameter
ω over a range ofvalues that should cover most practical contact problems The
parameterkmay be computed accurately and efficiently by means of the Newton–
Raphson iteration method applied to the following relationships (Yovanovich, 1986):
k
new= k+N(k)
TABLE 4.15 Hertz Contact Parameters and Elastoconstriction Parameter
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where
N(k) = k2E(k)
K(k)
k2+A B
− k4
1+A B
(4.154)
D(k) = E(k)
K(k)
kk2− 2kA
B
+A
B k2k (4.155)
Ifthe initial guess forkis based on the following correlation of the results given in
Table 4.15, the convergence will occur within two to three iterations:
k=
1−
0.9446
A
B
0.61352
1/2
(4.156)
Polynomial approximations ofthe complete elliptic integrals (Abramowitz and Ste-gun, 1965) may be used to evaluate them with an absolute error less than 10−7over the full range ofk.
The local gap thickness is required for the elastogap resistance model developed by Yovanovich (1986) The general relationship for the gap thickness can be determined
by means ofthe following surface displacements (Johnson, 1985; Timoshenko and Goodier, 1970):
δ(x,y) = δ0+ w(x,y) − w0 (m) (4.157) whereδ0(x,y) is the local gap thickness under zero load conditions, w(x,y) is the total
local displacement ofthe surfaces ofthe bodies outside the loaded area, andw0is the approach ofthe contact bodies due to loading
The total local displacement ofthe two bodies is given by
3F∆
2π
∞
µ
1− x2
a2+ t −
y2
b2+ t
[(a2+ t)(b2+ t)t]1/2 (4.158)
whereµ is the positive root ofthe equation
x2
a2+ µ +
y2
b2+ µ = 1 (4.159)
Whenµ > 0, the point ofinterest lies outside the elliptical contact area:
x2
a2 +y2
Whenµ = 0, the point ofinterest lies inside the contact area, and when x = y =
0, w(0, 0) = w0, the total approach ofthe contacting bodies is
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w0= 3F∆
2π
∞ 0
dt
[(a2+ t)(b2+ t)t]1/2
= 3πa F∆ K(k) (m) (4.161) The relationships for the semiaxes and the local gap thickness are used in the following subsections to develop the general relationships for the contact and gap resistances
4.15.3 Contact Resistance of Isothermal Elliptical Contact Areas
The general spreading–constriction resistance model, as proposed by Yovanovich (1971, 1986), is based on the assumption that both bodies forming an elliptical contact area can be taken to be a conducting half-space This approximation of actual bodies
is reasonable because the dimensions ofthe contact area are very small relative to the characteristic dimensions ofthe contacting bodies
Ifthe free (noncontacting) surfaces ofthe contacting bodies are adiabatic, the total ellipsoidal spreading–constriction resistance ofan isothermal elliptical contact area witha ≥ b is (Yovanovich, 1971, 1986)
R c= ψ
2k s a (K/W) (4.162)
wherea is the semimajor axis, k s is the harmonic mean thermal conductivity ofthe joint,
k s = k22k1k2 + k
2
(W/m· K) (4.163)
andψ is the spreading/constriction parameter ofthe isothermal elliptical contact area
developed in the section for spreading resistance of an isothermal elliptical area on
an isotropic half-space:
ψ = 2
in whichK(k) is the complete elliptic integral ofthe first kind ofmodulus kand is
related to the semiaxes
k=
1−
b a
21/2
The complete elliptic integral can be computed accurately by means ofaccurate polynomial approximations and by computer algebra systems This important special function can also be approximated by means of the following simple relationships:
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K(k) =
ln4a
2π
(1 +√b/a)2 0.1736 < k ≤ 1
(4.165)
These approximations have a maximum error less than 0.8%, which occurs atk =
0.1736 The ellipsoidal spreading–constriction parameter approaches the value of1
whena = b, the circular contact area.
When the results ofthe Hertz elastic deformation analysis are substituted into the results ofthe ellipsoidal constriction analysis, one obtains the elastoconstriction resistance relationship developed by Yovanovich (1971, 1986):
k s (24F ∆ρ∗)1/3 R c= π2 K(k m) ≡ ψ∗ (4.166)
where the effective radius of the ellipsoidal contact is defined asρ∗= [2(A + B)]−1.
The left-hand side is a dimensionless group consisting of the known total mechanical load F , the effective thermal conductivity k s ofthe joint, the physical parameter
∆, and the isothermal elliptical spreading/constriction resistance R c The right-hand side is defined to beψ∗, which is called the thermal elastoconstriction parameter
(Yovanovich, 1971, 1986) Typical values ofψ∗for a range of values ofω are given
in Table 4.15 The elastoconstriction parameterψ∗→ 1 when k = b/a = 1, the case
ofthe circular contact area
The thermal resistance ofthe gas-filled gap depends on three local quantities: the local gap thickness, thermal conductivity ofthe gas, and temperature difference between the bounding solid surfaces The gap model is based on the subdivision of the gap into elemental heat flow channels (flux tubes) having isothermal upper and lower boundaries and adiabatic sides (Yovanovich and Kitscha, 1974) The heat flow lines
in each channel (tube) are assumed to be straight and perpendicular to the plane of contact
Ifthe local effective gas conductivityk g (x,y) in each elemental channel is assumed
to be uniform across the local gap thicknessδ(x,y), the differential gap heat flow
rate is
dQ g =k g (x,y) ∆T δ(x,y) g (x,y) dxdy (W) (4.167) The total gap heat flow rate is given by the double integralQ g =Ag dQ g, where
the integration is performed over the entire effective gap areaA g.
The thermal resistance ofthe gap, R g, is defined in terms ofthe overall joint
temperature drop∆Tj (Yovanovich and Kitscha, 1974):
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1
R g =
Q g
∆T j =
Ag
k g (x,y) ∆T g (x,y) δ(x,y) ∆T j dA g (W/K) (4.168)
The local gap thickness in the general case oftwo bodies in elastic contact forming
an elliptical contact area is given above
The local effective gas conductivity is based on a model for the effective thermal conductivity ofa gaseous layer bounded by two infinite isothermal parallel plates
Therefore, for each heat flow channel (tube) the effective thermal conductivity is approximated by the relation (Yovanovich and Kitscha, 1974)
k g (x,y) = k g,∞
wherek g,∞is the gas conductivity under continuum conditions at STP The accom-modation parameterα is defined as
α = 2− α1
α1
+2− α2
α2
(4.170)
whereα1andα2are the accommodation coefficients at the solid–gas interfaces (Wied-mann and Trumpler, 1946; Hartnett, 1961; Wachman, 1962; Thomas, 1967; Kitscha and Yovanovich, 1975; Madhusudana, 1975, 1996; Semyonov et al., 1984; Wesley and Yovanovich, 1986; Song and Yovanovich, 1987; Song, 1988; Song et al., 1992a, 1993b) The fluid property parameterβ is defined by
whereγ is the ratio ofthe specific heats and Pr is the Prandtl number The mean free
pathΛ ofthe gas molecules is given in terms ofΛg,∞, the mean free path at STP, as follows:
Λ = Λg,∞T T g
g,∞
P g,∞
Two models for determining the local temperature difference,∆T g (x,y), are proposed
(Yovanovich and Kitscha, 1974) In the first model it is assumed that the bounding solid surfaces are isothermal at their respective contact temperatures; hence
This is called the thermally decoupled model (Yovanovich and Kitscha, 1974), since
it assumes that the surface temperature at the solid–gas interface is independent of the temperature field within each solid
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In the second model (Yovanovich and Kitscha, 1974), it is assumed that the tem-perature distribution ofthe solid–gas interface is induced by conduction through the solid–solid contact, under vacuum conditions This temperature distribution is ap-proximated by the temperature distribution immediately below the surface of an in-sulated half-space that receives heat from an isothermal elliptical contact Solving for this temperature distribution, using ellipsoidal coordinates it was found that
∆Tg (x,y)
F (k, ψ) K(k) (4.174)
whereF (k, ψ) is the incomplete elliptic integral ofthe first kind ofmodulus kand
amplitude angleψ (Abramowitz and Stegun, 1965; Byrd and Friedman, 1971) The
moduluskis given above and the amplitude angle is
ψ = sin−1 a2
a2+ µ
1/2
(4.175)
where the parameterµ is defined above It ranges between µ = 0, the edge ofthe
elliptical contact area, toµ = ∞, the distant points within the half-space Since the
solid–gas interface temperature is coupled to the interior temperature distribution, it
is called the coupled half-space model temperature drop
The general elastogap model has not been solved Two special cases ofthe general model have been examined They are the sphere-flat contact, studied by Yovanovich and Kitscha (1974) and Kitscha and Yovanovich (1975), and the cylinder-flat contact, studied by McGee et al (1985) The two special cases are discussed below
4.15.5 Joint Radiative Resistance
Ifthe joint is in a vacuum, or the gap is filled with a transparent substance such as dry air, there is heat transfer across the gap by radiation It is difficult to develop a general relationship that would be applicable for all point contact problems because radiation heat transfer occurs in a complex enclosure that consists of at least three nonisother-mal convex surfaces The two contacting surfaces are usually metallic, and the third surface forming the enclosure is frequently a reradiating surface such as insulation
Yovanovich and Kitscha (1974) and Kitscha and Yovanovich (1975) examined an enclosure that was formed by the contact of a metallic hemisphere and a metallic circular disk ofdiameterD The third boundary ofthe enclosure was a nonmetallic
circular cylinder ofdiameterD and height D/2 The metallic surfaces were assumed
to be isothermal at temperaturesT1andT2withT1 > T2 These temperatures corre-spond to the extrapolated temperatures from temperatures measured on both sides of the joint The joint temperature was defined asT j = (T1+ T2)/2 The dimensionless
radiation resistance was found to have the relationship
Dk s R r = k s
πσDT3
j
1 2
2 1 + 1.103
(4.176)
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whereσ = 5.67×10−8W/m2·K4is the Stefan–Boltzmann constant, 1and 2are the emissivities ofthe hemisphere and disk, respectively, andk sis the effective thermal conductivity ofthe joint
4.15.6 Joint Resistance of Sphere–Flat Contact
The contact, gap, radiative, and joint resistances ofthe sphere–flat contact shown in Fig 4.17 are presented here The contact radiusa is much smaller than the sphere
diameter D Assuming an isothermal contact area, the general elastoconstriction
resistance model (Yovanovich, 1971, 1986; Yovanovich and Kitscha, 1974), becomes
R c= 1
2k s a (K/W) (4.177)
wherek s = 2k1k2/(k1 + k2) is the harmonic mean thermal conductivity ofthe
contact, and the contact radius is obtained from the Hertz elastic model (Timoshenko and Goodier, 1970):
2a
D =
3F∆
D2
1/3
(4.178)
whereF is the mechanical load at the contact and ∆ is the joint physical parameter
defined above
The general-coupled elastogap resistance model for point contacts reduces, for the sphere–flat contact, to (Yovanovich and Kitscha, 1974; Kitscha and Yovanovich 1975; Yovanovich, 1986):
1
R g =
D
L k g,0 I g,p (W/K) (4.179)
whereL = D/2a is the relative contact size The gap integral for point contacts
proposed by Yovanovich and Kitscha (1974) and Yovanovich (1975) is defined as
I g,p=
L 1
2x tan−1√
x2− 1
The local gap thicknessδ is obtained from the relationship
2δ
D = 1 − 1−
x L
21/2
+πL12 (2 − x2) sin−11
x +
x2− 1
−L12 (4.181)
wherex = r/a and 1 ≤ x ≤ L The gap gas rarefaction parameter is defined as
where the gas parametersα, β, and Γ are as defined above.
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is (Yovanovich and Kitscha, 1974; Kitscha and Yovanovich, 1975)
1
R j =
1
R c+
1
The models proposed were verified by experiments conducted by Kitscha (1982) The test conditions were: sphere diameterD = 25.4 mm; vacuum pressure P g = 10−6
torr; mean interface temperature range 316≤ Tm ≤ 321 K; harmonic mean thermal
conductivity ofsphere-flat contactk s = 51.5 W/(m · K); emissivities ofvery smooth
sphere and lapped flat (rms roughness is 1 2 = 0.8,
respec-tively; elastic properties ofsphere and flatE1 = E2= 206 GPa and ν1= ν2 = 0.3.
The dimensionless joint resistance is given by the relationship
1
R∗
j =R1∗ +R1∗
r
(4.184)
where
R∗
j = Dks R j R∗
c = Dks R c = L R∗
r = 1415
300
T m
3
(4.185)
The model and vacuum data are compared for a load range in Table 4.16 The agree-ment between the joint resistance model and the data is excellent over the full range oftests
Yovanovich and Kitscha (1974), the dimensionless joint resistance with a gas in the gap is given by
1
R∗
j = R1∗ +R1∗
r +R1∗ (4.186)
TABLE 4.16 Dimensionless Load, Constriction, Radiative, and Joint Resistances
j
Source: Kitscha (1982).
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where
R∗
g = Dk s R g= k s L2
k g,∞ I g,p (4.187)
The joint model for the sphere–flat contact is compared against data obtained for the following test conditions: sphere diameter D = 25.4 mm; load is 16 N;
dimensionless loadL = 115.1; mean interface temperature range 309 ≤ T m ≤ 321
K; harmonic mean thermal conductivity ofsphere–flat contactk s = 51.5 W/m · K);
emissivities ofsmooth sphere and lapped flat are 1 2= 0.8, respectively.
The load was fixed such thatL = 115.1 for all tests, while the air pressure was
varied from 400 mmHg down to a vacuum The dimensionless resistances are given
in Table 4.17 It can be seen that the dimensionless radiative resistance was relatively large with respect to the dimensionless gap and contact resistances The dimension-less gap resistance values varied greatly with the gas pressure The agreement be-tween the joint resistance model and the data is very good for all test points
4.15.7 Joint Resistance of a Sphere and a Layered Substrate
Figure 4.18 shows three joints: contact between a hemisphere and a substrate, contact between a hemisphere and a layer offinite thickness bounded to a substrate, and contact between a hemisphere and a very thick layer wheret/a 1.
In the general case, contact is between an elastic hemisphere ofradiusρ and elastic
properties:E3, ν3 and an elastic layer ofthicknesst and elastic properties: E1, ν1, which is bonded to an elastic substrate ofelastic properties:E2, ν2 The axial load is
F It is assumed that E1 < E2for layers that are less rigid than the substrate
The contact radiusa is much smaller than the dimensions ofthe hemisphere and
the substrate The solution for arbitrary layer thickness is complex because the contact radius depends on several parameters [i.e.,a = f (F, ρ, t, E i , ν i ), i = 1, 2, 3] The
contact radius lies in the rangea S ≤ a ≤ a L, wherea Scorresponds to the very thin
TABLE 4.17 Effect of Gas Pressure on Gap and Joint Resistances for Air
j
Source: Kitscha (1982).
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Figure 4.18 Contact between a hemisphere and a layer on a substrate: (a) hemisphere and substrate; (b) hemisphere and layer offinite thickness; (c) hemisphere and very thick layer.
(From Stevanovi´c et al., 2001.)
layer limit,t/a → 0 (Fig 4.18a) and a L corresponds to the very thick layer limit,
t/a → ∞ (Fig 4.18c).
For the general case, a contact in a vacuum, and ifthere is negligible radiation heat transfer across the gap, the joint resistance is equal to the contact resistance, which
is equal to the sum ofthe spreading–constriction resistances in the hemisphere and layer–substrate, respectively
The joint resistance is given by Fisher (1985), Fisher and Yovanovich (1989), and Stevanovi´c et al (2001, 2002)
R j = R c= 1
4k3a +
ψ12
4k2a (K/W) (4.188)
wherea is the contact radius The first term on the right-hand side represents the
con-striction resistance in the hemisphere, andψ12 is the spreading resistance parameter
in the layer–substrate The thermal conductivities ofthe hemisphere and the substrate appear in the first and second terms, respectively The layer–substrate spreading resis-tance parameter depends on two dimensionless parameters:τ = t/a and κ = k1/k2 This parameter was presented above under spreading resistance in a layer on a half-space To calculate the joint resistance the contact radius must be found
A special case arises when the rigidity ofthe layer is much smaller than the rigidity ofthe hemisphere and the layer This corresponds to “soft” metallic layers such as indium, lead, and tin; or nonmetallic layers such as rubber or elastomers In this case, sinceE1 E2 andE1 E3, the hemisphere and substrate may be modeled as perfectly rigid while the layer deforms elastically
The dimensionless numerical values fora/a L obtained from the elastic contact model ofChen and Engel (1972) according to Stevanovi´c et al (2001) are plotted
in Fig 4.19 for a wide range of relative layer thicknessτ = t/a and for a range of
values ofthe layer Young’s modulusE1 The contact model, which is represented by the correlation equation ofthe numerical values, is (Stevanovi´c et al., 2002)
... 4.18c).For the general case, a contact in a vacuum, and ifthere is negligible radiation heat transfer across the gap, the joint resistance is equal to the contact resistance, which
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