Heat Transfer Handbook part 35 potx

and Flat Surfaces in a Vacuum A model is available for calculating the joint resistance of an elastic–plastic contact ofa portion ofa hemisphere whose radius ofcurvature isρ attached to

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1.1 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0

Chen and Engel Model E MPa1( )

0.05 0.5 50 100 500

␶ MODEL

Figure 4.19 Comparison ofdata and model for contact between a rigid hemisphere and an elastic layer on a rigid substrate (From Stevanovi´c et al., 2001.)

a

a L = 1 − c3 exp(c1τc2 ) (4.189) with correlation coefficients:c1 = −1.73, c2= 0.734, and c3= 1.04 The reference

contact radius isa L, which corresponds to the very thick layer limit given by

a L=

3 4

F ρ

E13

1/3

for t

a → ∞ (4.190)

The maximum difference between the correlation equation and the numerical values obtained from the model of Chen and Engel (1972) is approximately 1.9%

forτ = 0.02 The following relationship, based on the Newton–Raphson method, is

recommended for calculation of the contact radius (Stevanovi´c et al., 2001):

a n+1= a n − a L {1 − 1.04 exp[−1.73(t/a n )0.734]}

1+ 1.321(a L /a S )(t/a n )0.734 exp[−1.73(t/a n )0.734] (4.191)

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Ifthe first guess isa0= a L, fewer than six iterations are required to give eight-digit accuracy

In the general case where the hemisphere, layer, and substrate are elastic, the contact radius lies in the rangea S ≤ a ≤ a LforE2 < E1 The two limiting values

ofa are, according to Stevanovi´c et al (2002),

a =











a S=

3 4

F ρ

E23

1/3

for t

a → 0

a L=

3 4

F ρ

E13

1/3

for t

(4.192)

where the effective Young’s modulus for the two limits are defined as

E13=

1− ν2 1

E1

+1− ν23

E3

−1

E23=

1− ν2 2

E2

+1− ν23

E3

−1

(4.193)

The dimensionless contact radius and dimensionless layer thickness were defined

as (Stevanovi´c et al., 2002)

a∗= a − a S

a L − a S where 0< a∗< 1 (4.194)

τ∗=

t a

√ α

1/3

whereα = a L

a S =

E23

E13

1/3

(4.195)

The dimensionless numerical values obtained from the full model of Chen and Engel (1972) for values ofα in the range 1.136 ≤ α ≤ 2.037 are shown in Fig 4.20.

The correlation equation is (Stevanovi´c et al., 2002)

a − a S

a L − a S = 1 − exp



−π1/4

t√α

a

π/4

(4.196)

Since the unknown contact radiusa appears on both sides, the numerical solution

ofthe correlation equation requires an iterative method (Newton–Raphson method)

to find its root For all metal combinations, the following solution is recommended (Stevanovi´c et al., 2002):

a = a S + (a L − a S )



1− exp



−π1/4

t√α

a0

π/4

(4.197)

where

a0= a S + (a L − a S )



1− exp



−π1/4

2t√α

a S + a L

π/4

(4.198)

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Figure 4.20 Comparison ofthe data and model for elastic contact between a hemisphere and

a layer on a substrate (From Stevanovi´c et al., 2002.)

and Flat Surfaces in a Vacuum

A model is available for calculating the joint resistance of an elastic–plastic contact ofa portion ofa hemisphere whose radius ofcurvature isρ attached to a cylinder

whose radius isb1and a cylindrical flat whose radius isb2 The elastic properties of the hemisphere areE1, ν1, and the elastic properties ofthe flat areE2, ν2 The thermal conductivities arek1andk2, respectively

Ifthe contact strain is very small, the contact is elastic and the Hertz model can

be used to predict the elastic contact radius denoted asa e On the other hand, if the contact strain is very large, plastic deformation may occur in the flat, which is

assumed to be fully work hardened, and the plastic contact radius is denoted a p Between the fully elastic and fully plastic contact regions there is a transition called

the elastic–plastic contact region, which is very difficult to model In the region the

contact radius is denoted asa ep, the elastic–plastic contact radius The relationship

betweena e , a p, anda ep isa e ≤ a ep ≤ a p

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The elastic–plastic radius is related to the elastic and plastic contact radii by means ofthe composite model based on the method ofChurchill and Usagi (1972) for combining asymptotes (Sridhar and Yovanovich, 1994):

a ep=a n

e + a n p

1/n

wheren is the combination parameter, which is found empirically to have the value

n = 5 The elastic and plastic contact radii may be obtained from the relationships

(Sridhar and Yovanovich, 1994)

a e=

3 4

Fρ

E

1/3

and a p =

F

πH B

1/2

(m) (4.200) with the effective modulus

1

E = 1− νE 2

1 +1− νE 2

2 (m2/N) (4.201)

The plastic parameter is the Brinell hardness H B ofthe flat The elastic–plastic deformation model assumes that the hemispherical solid is harder than the flat The static axial load isF

The joint resistance for a smooth hemispherical solid in elastic–plastic contact with smooth flat is given by (Sridhar and Yovanovich, 1994)

R j = ψ1

4k1a ep + ψ2

4k2a ep (K/W) (4.202)

The spreading–constriction resistance parameters for the hemisphere and flat are

ψ1 =

1−a b ep

1

1.5

and ψ2=

1−a b ep

2

1.5

(4.203)

Alternative Constriction Parameter for a Hemisphere The following spreading–constriction parameter can be derived from the hemisphere solution:

where 1 Ifthe contact is in a vacuum and the radiation heat transfer across the gap is negligible,R j = R c Also, ifb1= b2= b,

ψ1= ψ2=1−a

b

1.5

(4.205) The joint and dimensionless joint resistances for this case become

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R j = ψ

2k s a and R j∗= 2bk s R j =(1 − a/b) a/b 1.5 (4.206) wherek s = 2k1k2/(k1+ k2).

Sridhar and Yovanovich (1994) compared the dimensionless joint resistance against data obtained for contacts between a carbon steel ball and several flats of

Ni 200, carbon steel, and tool steel The nondimensional data and the dimensionless joint resistance model are compared in Fig 4.21 for a range ofvalues ofthe recipro-cal contact strainb/a The agreement between the model and the data over the entire

range 20< b/a < 120 is very good The points near b/a ≈ 100 are in the elastic

contact region, and the points nearb/a ≈ 20 are close to the plastic contact region.

In between the points are in the transition region, called the elastic–plastic contact

region.

Ifthe material ofthe flat work-hardens as the deformation takes place, the model for predicting the contact radius is much more complex, as described by Sridhar and Yovanovich (1994) and Johnson (1985) This case is not given here

Figure 4.21 Comparison ofthe data and model for an elastic–plastic contact between a hemisphere and a flat (From Sridhar and Yovanovich, 1994.)

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Models have been presented (Yovanovich, 1967, 1971, 1978) for calculating the overall thermal resistance ofslowly rotating instrument bearings, which consist of many very smooth balls contained by very smooth inner and outer races The thermal resistance models for bearings are based on elastic contact of the balls with the inner and outer races and spreading and constriction resistances in the balls and in the inner and outer races For each ball there are two elliptical contact areas, one at the inner race and one at the outer race The local thickness ofthe adjoining gap is very complex

to model There are four spreading–constriction zones associated with each ball The full elastoconstriction resistance model must be used to obtain the overall thermal resistance ofthe bearing Since these are complex systems, the contact resistance models are also complex; therefore, they are not presented here The references above should be consulted for the development of the contact resistance models and other pertinent references

Ifa long smooth circular cylinder with radius ofcurvatureρ1= D1/2, length L1, and elastic properties:E1, ν1makes contact with another long smooth circular cylinder with radius ofcurvatureρ2= D2/2, length L1, and elastic properties:E2, ν2, then in general, ifthe axes ofthe cylinders are not aligned (i.e., they are crossed), an elliptical contact area is formed with semiaxesa and b, where it is assumed that a < b If the

cylinder axes are aligned, the contact area becomes a strip ofwidth 2a, and the larger

axes are equal to the length ofthe cylinder The general Hertz model presented may

be used to find the semiaxes and the local gap thickness ifthe axes are not aligned For aligned axes, the general equations reduce to simple relationships, which are given below

Contact Strip and Local Gap Thickness Ifthe two cylinder axes are aligned,

the contact area is a strip ofwidth 2a and length L1, where (Timoshenko and Goodier, 1970; Walowit and Anno, 1975)

a = 2

2F ρ∆

πL1

1/2

where the effective curvature is

1

ρ =

1

ρ1

+ 1

ρ2

and the contact parameter is

∆ = 1

2

1− ν2 1

E1

+1− ν22

E2



(m2/N) (4.209)

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The contact pressure is maximum along the axis ofthe contact strip, and it is given

by the relationship

P0= π2aL F

1 =

2πL1ρ∆

1/2

(N/m2) (4.210)

and the pressure distribution has the form (Timoshenko and Goodier, 1970; Walowit and Anno, 1975)

P (x ) = P0



1−x a

2 for 0≤ x ≤ a (N/m2) (4.211)

The mean contact area pressure is

P m= F

2aL1

=4P0

π (N/m2) (4.212)

The normal approach ofthe two aligned cylinders is (Timoshenko and Goodier, 1970;

Walowit and Anno, 1975)

α = 2F π

1− ν2 1

E1

ln4ρ1

1 2

 +1− ν22

E2

ln4ρ2

1 2



(m) (4.213)

whereF= F/L1is the load per unit cylinder length The general local gap thickness relationship is (Timoshenko and Goodier, 1970; Walowit and Anno, 1975)

2δ

ρ =

1−L12

1/2

−



1−Lξ22

1/2

+ξ(ξ2− 1)1/2− cosh−1ξ − ξ2+ 12L (4.214)

where

2a ξ =

x

Ifa single circular cylinder ofdiameterD or (ρ1 = D1/2 = D/2) is in elastic

contact with a flat (ρ2= ∞), put ρ = D/2 in the relationships above.

Contact Resistance at a Line Contact The thermal contact resistance for the very narrow contact strip ofwidth 2a formed by the elastic contact of a long smooth

circular cylinder ofdiameterD and a smooth flat whose width is 2b and whose length

L1is identical to the cylinder length is given by the approximate relationship (McGee

et al., 1985)

πL1k1

ln 4 1

−π

2



πL1k2

ln 2 2 (K/W) (4.216)

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where the thermal conductivities ofthe cylinder and flat arek1andk2, respectively

The contact parameters are 1 2 = a/b for the flat For

elastic contacts, 2a/D  1 and 2a/b  1 for most engineering applications The

approximate relationship forR cbecomes more accurate for very narrow strips

The width ofthe flat relative to the cylinder diameter may be 2b > D, 2b = D,

or 2b < D McGee et al (1985) proposed the use ofthe dimensionless form ofthe

contact resistance:

R∗

c = L1k s R c= 1

2π

k s

k1

ln π

F∗ − k s

2k1

+ 1

2π

k s

k2

ln 1

4πF∗ (4.217)

where

F∗= L F∆

1D and k s=

2k1k2

k1+ k2

(4.218)

Gap Resistance at a Line Contact The general elastogap resistance model for line contacts proposed by Yovanovich (1986) reduces for the circular cylinder–flat contact to

1

R g =

4aL1

D k g,∞ I g,l (W/K) (4.219)

wherek g,∞is the gas thermal conductivity and the line contact elastogap integral is defined as (Yovanovich, 1986)

I g,l =π2

 L

1

cosh−1(ξ)dξ

where

2a ξ =

x

This is the coupled elastogap model Numerical integration is required to calculate values ofI g,l

The gas rarefaction parameter that appears in the gap integral is

where the accommodation parameter and other gas parameters are defined as

α = 2− α1

α1

+2− α2

α2

(γ + 1) Pr γ =

C p

C v (4.223)

and the molecular mean free path is

Λ = Λg,∞ T g

T g,∞

P g,∞

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whereΛg,∞is the value of the molecular mean free path at the reference temperature

T g,∞and gas pressureP g,∞

Joint Resistance at a Line Contact The joint resistance at a line contact, neglecting radiation heat transfer across the gap, is

1

R j = 1

R c + 1

R g (W/K) (4.225)

McGee et al (1985) examined the accuracy ofthe contact, gap, and joint resistance relationships for helium and argon for gas pressures between 10−6torr and 1 atm

The effect of contact load was investigated for mechanical loads between 80 and

8000 N on specimens fabricated from Keewatin tool steel, type 304 stainless steel, and Zircaloy 4 The experimental data were compared with the model predictions, and good agreement was obtained over a limited range ofexperimental parameters

Discrepancies were observed at the very light mechanical loads due to slight amounts ofform error (crowning) along the contacting surfaces

Joint Resistance of Nonconforming Rough Surfaces There is ample

em-pirical evidence that surfaces may not be conforming and rough, as shown in Fig 4.1c and f The surfaces may be both nonconforming and rough, as shown in Fig 4.1b and

e, where a smooth hemispherical surface is in contact with a flat, rough surface.

If surfaces are nonconforming and rough, the joint that is formed is more complex from the standpoint of defining the micro- and macrogeometry before load is applied, and the definition of the micro- and macrocontacts that are formed after load is ap-plied The deformation of the contacting asperities may be elastic, plastic, or elastic–

plastic The deformation of the bulk may also be elastic, plastic, or elastic–plastic

The mode ofdeformation ofthe micro- and macrogeometry are closely connected under conditions that are not understood today

The thermal joint resistance ofsuch a contact is complex because heat can cross the joint by conduction through the microcontacts and the associated microgaps and

by conduction across the macrogap Ifthe temperature level ofthe joint is suffi-ciently high, there may be significant radiation across the microgaps and macrogap

Clearly, this type ofjoint represents complex thermal and mechanical problems that are coupled

Many vacuum data have been reported (Clausing and Chao, 1965; Burde, 1977;

Kitscha, 1982) that show that the presence ofroughness can alter the joint resistance

of a nonconforming surface under light mechanical loads and have negligible effects

at higher loads Also, the presence of out-of-flatness can have significant effects on the joint resistance ofa rough surface under vacuum conditions

It is generally accepted that the joint resistance under vacuum conditions may be modeled as the superposition ofmicroscopic and macroscopic resistance (Clausing and Chao, 1965; Greenwood and Tripp, 1967; Holm, 1967; Yovanovich, 1969; Burde and Yovanovich, 1978; Lambert, 1995; Lambert and Fletcher, 1997) The joint resis-tance can be modeled as

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R j = Rmic+ Rmac (K/W) (4.226) The microscopic resistance is given by the relationship

Rmic= ψmic

2k s Na S (K/W) (4.227)

whereψmicis the average spreading–constriction resistance parameter,N is the

num-ber ofmicrocontacts that are distribution in some complex manner over the contour area ofradiusa Landa Srepresents some average microcontact spot radius, and the harmonic mean thermal conductivity ofthe joint isk s = 2k1k2/(k1+ k2).

The macroscopic resistance is given by the relationship

Rmac= ψmac

2k s a L (K/W) (4.228)

whereψmacis the spreading–constriction resistance parameter for the contour area of radiusa L

The mechanical model should be capable ofpredicting the contact parameters:

a S , a L, andN These parameters are also required for the determination of the thermal

spreading–constriction parametersψmicandψmac

At this time there is no simple mechanical model available for prediction of the ge-ometric parameters required in microscopic and macroscopic resistance relationships

There are publications (e.g., Greenwood and Tripp, 1967; Holm, 1967; Burde and Yovanovich, 1978; Johnson, 1985; Lambert and Fletcher, 1997; Marotta and Fletcher, 2001) that deal with various aspects ofthis very complex problem

There are models for predicting contact, gap, and joint conductances between con-forming (nominally flat) rough surfaces developed by Greenwood and Williamson (1966), Greenwood (1967), Greenwood and Tripp (1970), Cooper et al (1969), Mi-kic (1974), Sayles and Thomas (1976), Yovanovich (1982), and DeVaal (1988)

The three mechanical models—elastic, plastic, or elastic–plastic deformation of the contacting asperities—are based on the assumptions that the surface asperities have Gaussian height distributions about some mean plane passing through each surface and that the surface asperities are distributed randomly over the apparent contact areaA a Figure 4.22 shows a very small portion ofa typical joint formed between two nominally flat rough surfaces under a mechanical load

Each surface has a mean plane, and the distance between them, denoted asY ,

is related to the effective surface roughness, the apparent contact pressure, and the plastic or elastic physical properties ofthe contacting asperities

A very important surface roughness parameter is the surface roughness: either the rms (root-mean-square) roughness or the CLA (centerline-average) roughness, which are defined as (Whitehouse and Archard, 1970; Onions and Archard, 1973; Thomas, 1982)

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[ 335] ,(75)... Resistance at a Line Contact The joint resistance at a line contact, neglecting radiation heat transfer across the gap, is

1

R j = 1

R... class="page_container" data-page="7">

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