Heat Transfer Handbook part 37 potx

Sridhar and Yovanovich 1996a showed that the elastic deformation model was in better agreement with the vacuum data obtained for joints formed by conforming rough surfaces of tool steel,

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λ = −0.5444 − 0.6636

ln P

H e



− 0.03204

ln P

H e

2

− 0.000771

ln P

H e

3 (4.262) Sridhar and Yovanovich (1994) reviewed the plastic and elastic deformation con-tact conductance correlation equations and compared them against vacuum data (Mi-kic and Rohsenow, 1966; Antonetti, 1983; Hegazy, 1985; Nho, 1989; McWaid and Marschall, 1992a, b) for several metal types, having a range of surface roughnesses, over a wide range ofapparent contact pressure Sridhar and Yovanovich (1996a) showed that the elastic deformation model was in better agreement with the vacuum data obtained for joints formed by conforming rough surfaces of tool steel, which is very hard

The elastic asperity contact and thermal conductance models ofGreenwood and Williamson (1966), Greenwood (1967), Greenwood and Tripp (1967, 1970), Bush et

al (1975), and Bush and Gibson (1979) are different from the Mikic (1974) elastic contact model presented in this chapter However, they predict similar trends of contact conductance as a function apparent contact pressure

4.16.4 Conforming Rough Surface Model: Elastic–Plastic Deformation

Sridhar and Yovanovich (1996c) developed an elastic–plastic contact conductance model which is based on the plastic contact model ofCooper et al (1969) and the elastic contact model ofMikic (1974) The results are summarized below in terms ofthe geometric parametersA r /A a, the real-to-apparent area ratio;n, the contact

spot density;a, the mean contact spot radius; and λ, the dimensionless mean plane

separation:

A r

A a =

f ep

2 erfc

λ

√ 2



(4.263)

16

m

σ

2 exp(−λ2)

a =

 8 π

f epσ

mexp

λ2 2

 erfc

λ

√ 2



(4.265)

na = 1

8

 2 π

f ep m

σ exp

−λ2 2



(4.266)

h cσ

k s m =

1

2√

2π

f epexp(−λ2/2)

1−(f ep /2)erfc(λ/√2)

1.5 (4.267)

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λ =√2erfc−1

1

f ep

2P

H ep



(4.268)

The important elastic–plastic parameter f ep is a function of the dimensionless contact strain ∗, which depends on the amount ofwork hardening This physical parameter lies in the range 0.5 ≤ f ep ≤ 1.0 The smallest and largest values

correspond to zero and infinitely large contact strain, respectively The elastic–plastic parameter is related to the contact strain:

f ep=



1 ∗)21/2



The dimensionless contact strain is defined as

∗

c = 1.67

mE

S f



(4.270)

whereS f is the material yield or flow stress (Johnson, 1985), which is a complex physical parameter that must be determined by experiment for each metal

The elastic–plastic microhardnessH epcan be determined by means ofan iterative procedure which requires the following relationship:

H ep=  2.76S f

The elastoplastic contact conductance model moves smoothly between the elastic contact model ofMikic (1974) and the plastic contact conductance model ofCooper

et al (1969), which was modified by Yovanovich (1982), Yovanovich et al (1982a), and Song and Yovanovich (1988) to include the effect of work-hardened layers on the deformation of the contacting asperities The dimensionless contact pressure for elastic–plastic deformation of the contacting asperities is obtained from the following approximate explicit relationship:

P

H ep =

0.9272P

c1(1.43 σ/m) c2

1/(1+0.071c2)

(4.272)

where the coefficientsc1andc2are obtained from Vickers microhardness tests The Vickers microhardness coefficients are related to Brinell and Rockwell hardness for

a wide range ofmetals

Correlation Equations for Dimensionless Contact Conductance: Elastic–

Plastic Model The complex elastic–plastic contact model proposed by Sridhar and Yovanovich (1996a, b, c, d) may be approximated by the following correlation equations for the dimensionless contact conductance:

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C c=















1.54

P

H ep

0.94

1.245b1

P

H ep

b2

1.25

P

H ep

0.95

where the elastic–plastic correlation coefficientsb1andb2depend on the dimension-less contact strain:

b1= 1+46,690.2∗)2.49

1/30

(4.276)

1 ∗)1.842

1/600

(4.277)

4.16.5 Gap Conductance for Large Parallel Isothermal Plates

Two infinite isothermal surfaces form a gap of uniform thicknessd which is much

greater than the roughness ofboth surfaces:d σ1andσ2 The gap is filled with

a stationary monatomic or diatomic gas The boundary temperatures areT1andT2, whereT1 > T2 The Knudsen number for the gap is defined as Kn= Λ/d, where

Λ is the molecular mean free path of the gas, which depends on the gas temperature and its pressure The gap can be separated into three zones: two boundary zones, which are associated with the two solid boundaries, and a central zone The boundary zones have thicknesses that are related to the molecular mean free pathsΛ1andΛ2, where

Λ1= Λ0T1

T0

P g,0

P g and Λ2= Λ0T2

T0

P g,0

andΛ0, T0, andP g,0represent the molecular mean free path and the reference temper-ature and gas pressure In the boundary zones the heat transfer is due to gas molecules that move back and forth between the solid surface and other gas molecules located at distancesΛ1andΛ2from both solid boundaries The energy exchange between the

gas and solid molecules is imperfect At the hot solid surface at temperature T1, the gas molecules that leave the surface after contact are at some temperatureT g,1 < T1, and at the cold solid surface at temperatureT2, the gas molecules that leave the surface after contact are at a temperatureT g,2 > T2 The two boundary zones are called slip regions.

In the central zone whose thickness is modeled as d − Λ1 − Λ2, and whose temperature range isT g,1 ≥ T ≥ Tg,2, heat transfer occurs primarily by molecular

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diffusion Fourier’s law of conduction can be used to determine heat transfer across the central zone

There are two heat flux asymptotes, corresponding to very small and very large Knudsen numbers They are: for a continuum,

Kn→ 0 q → q0= k g T1− T2

and for free molecules,

where

M = αβΛ =

2− α1

α1

+2− α2

α2



2γ

andk gis the thermal conductivity,α1andα2the accommodation coefficients,γ the ratio ofspecific heats, and Pr the Prandtl number

The gap conductance, defined ash g = q/(T1− T2), has two asymptotes:

for Kn→ 0, hg → k g

d for Kn→ ∞, hg→

k g M

For the entire range ofthe Knudsen number, the gap conductance is given by the relationship

h g = k g

This relatively simple relationship covers the continuum, 0 < Kn < 0.1, slip,

0.1 < Kn < 10, and free molecule, 10 < Kn < ∞, regimes Song (1988) introduced

the dimensionless parameters

G = k g

h g d and M∗=

M

and recast the relationship above as

The accuracy ofthe simple parallel-plate gap model was compared against the data (argon and nitrogen) ofTeagan and Springer (1968), and the data (argon and helium) ofBraun and Frohn (1976) The excellent agreement between the simple gap model and all data is shown in Fig 4.26 The simple gap model forms the basis of the gap model for the joint formed by two conforming rough surfaces

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10⫺3

10⫺1

100

101

102

100

M *

He (Braun & Frohn, 1976) (Braun & Frohn, 1976) (Teagan & Springer, 1968) (Teagan & Springer, 1968)

Ar Ar

N2 Interpolated ModelG =1⫹M *

Figure 4.26 Gap conductance model and data for two large parallel isothermal plates (From Song et al., 1992a.)

4.16.6 Gap Conductance for Joints between Conforming Rough Surfaces

Ifthe gap between two conforming rough surfaces as shown in Fig 4.22 is occupied

by a gas, conduction heat transfer will occur across the gap This heat transfer is characterized by the gap conductance, defined as

h g = ∆T Q j

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with∆T j as the effective temperature drop across the gas gap andQ gthe heat transfer rate across the gap Because the local gap thickness and local temperature drop vary

in very complicated ways throughout the gap, it is difficult to develop a simple gap conductance model

Several gap conductance models and correlation equations have been presented by

a number ofresearchers (Cetinkale and Fishenden, 1951; Rapier et al., 1963; Shlykov, 1965; Veziroglu, 1967; Lloyd et al., 1973; Garnier and Begej, 1979; Loyalka, 1982;

Yovanovich et al., 1982b); they are given in Table 4.19 The parameters that appear in Table 4.19 areb t = 2(CLA1+ CLA2), where CLA iis the centerline-average surface

TABLE 4.19 Models and Correlation Equations for Gap Conductance for Conforming RoughSurfaces

0.305b t + M

Fishenden (1951) Rapier et al (1963) h g = k g 1.2

2b t + M +

0.8

2b t ln

1+2M b t



t

 10

3 +10X +X42− 4 X13 +X32+X2 ln(1 + X)









k g

0.264 b t + M for b t > 15 µm

kg

1.78 b t + M for b t < 15 µm

Lloyd et al (1973) h g= δ + βΛ/(α k g

1+ α2) δ not given

Garnier and Begej (1979) h g = k g exp(−1/Kn)

1− exp(−1/Kn)

δ + M



δ not given

Loyalka (1982) h g= δ + M + 0.162(4 − α k g

1− α2)βΛ δ not given

Yovanovich et al (1982b) h g= √k g /σ

2π

 ∞

0 exp

−(Y/σ − t/σ)2/2

t σ



Y

σ =

√

2 erfc−1

2P

H p



P

H p =

P

c1(1.62σ/m) c2

Source: Song (1988).

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roughness ofthe two contacting surfaces,M = αβΛ, X = b t /M, σ = σ2

1+ σ2

2, where the units ofσ are µm The Knudsen number Kn that appears in the Garnier and Begej (1979) correlation equation is not defined

Song and Yovanovich (1987), Song (1988), and Song et al (1993b) reviewed the models and correlation equations given in Table 4.19 They found that for some ofthe correlation equations the required gap thicknessδ was not defined, and for other correlation equations an empirically based average gap thickness was specified that is constant, independent ofvariations ofthe apparent contact pressure The gap conductance model developed by Yovanovich et al (1982b) is the only one that accounts for the effect of mechanical load and physical properties of the contacting asperities on the gap conductance This model is presented below

The gap conductance model for conforming rough surfaces was developed, modi-fied, and verified by Yovanovich and co-workers (Yovanovich et al., 1982b; Hegazy, 1985; Song and Yovanovich, 1987; Negus and Yovanovich, 1988; Song et al., 1992a, 1993b)

The gap contact model is based on surfaces having Gaussian height distributions and also accounts for mechanical deformation of the contacting surface asperities

Development ofthe gap conductance model is presented in Yovanovich (1982, 1986), Yovanovich et al (1982b), and Yovanovich and Antonetti (1988)

The gap conductance model is expressed in terms ofan integral:

h g =k g σ

1

√

2π

 ∞ 0 exp

− (Y/σ − u)2/2

k g

σI g (W/m2· K) (4.286) wherek gis the thermal conductivity ofthe gas trapped in the gap andσ is the effective surface roughness of the joint, andu = t/σ is the dimensionless local gap thickness.

The integralI gdepends on two independent dimensionless parameters:Y/σ, the mean

plane separation; andM/σ, the relative gas rarefaction parameter.

The relative mean planes separation for plastic and elastic contact are given by the relationships

Y σ

 plastic=√2 erfc−1

2P

H p



Y σ

 elastic=√2 erfc−1

4P

H e

The relative contact pressuresP /H p for plastic deformation andP /H e for elastic deformation can be determined by means of appropriate relationships

The gas rarefaction parameter isM = αβΛ, where the gas parameters are defined

as:

α = 2− α1

α1

+2− α2

α2

(4.288)

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Λ = Λ0

T g

T g,0

P g

whereα is the accommodation coefficient, which accounts for the efficiency of gas–

surface energy exchange There is a large body of research dealing with experimental and theoretical aspects ofα for various gases in contact with metallic surfaces under various surface conditions and temperatures (Wiedmann and Trumpler, 1946; Hart-nett, 1961; Wachman, 1962; Thomas, 1967; Semyonov et al., 1984; Loyalka, 1982)

Song and Yovanovich (1987) and Song et al (1992a, 1993b) examined the several gap conductance models available in the literature and the experimental data and models for the accommodation coefficients

Song and Yovanovich (1987) developed a correlation for the accommodation for engineering surfaces (i.e., surfaces with absorbed layers of gases and oxides) They proposed a correlation that is based on experimental results ofnumerous investiga-tors for monatomic gases The relationship was extended by the introduction of a monatomic equivalent molecular weight to diatomic and polyatomic gases The final correlation is

α = exp(C0T )

g

C1+ M g + [1 − exp(C0T )] (1 + µ)2.4µ 2

 (4.291)

withC0= −0.57, T = (T s − T0)/T0, M g = M gfor monatomic gases (= 1.4M g for diatomic and polyatomic gases),C1 = 6.8 in units of Mg(g/mol), andµ = Mg /M s

whereT s andT0 = 273 K are the absolute temperatures ofthe surface and the gas, andM g andM s are the molecular weights ofthe gas and the solid, respectively

The agreement between the predictions according to the correlation above and the published data for diatomic and polyatomic gases was within±25%

The gas parameter β depends on the specific heat ratio γ = Cp /C v and the Prandtl number Pr The molecular mean free path of the gas moleculesΛ depends

on the type ofgas, the gas temperatureT g and gas pressure P g, and the reference values ofthe mean free pathΛ0, the gas temperatureT g,0, and the gas pressureP g,0, respectively

Wesley and Yovanovich (1986) compared the predictions ofthe gap conductance model and experimental measurements ofgaseous gap conductance between the fuel and clad ofa nuclear fuel rod The agreement was very good and the model was recommended for fuel pin analysis codes

The gap integral can be computed accurately and easily by means ofcomputer algebra systems Negus and Yovanovich (1988) developed the following correlation equations for the gap integral:

In the range 2≤ Y/σ ≤ 4:

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f g =







1.063 + 0.0471

4−Y σ

1.68

ln σ

M

0.84

for 0.01 ≤ M

σ ≤ 1

1+ 0.06 σ

M

0.8

for 1≤ M

σ < ∞ The correlation equations have a maximum error ofapproximately 2%

4.16.7 Joint Conductance for Conforming Rough Surfaces

The joint conductance for a joint between two conforming rough surfaces is

h j = hc + hg (W/m2· K) (4.293)

when radiation heat transfer across the gap is neglected The relationship is applicable

to joints that are formed by elastic, plastic, or elastic–plastic deformation of the contacting asperities The mode ofdeformation will influenceh candh gthrough the relative mean plane separation parameterY/σ.

The gap and joint conductances are compared against data (Song, 1988) obtained for three types of gases, argon, helium, and nitrogen, over a gas pressure range between 1 and 700 torr The gases occupied gaps formed by conforming rough Ni

200 and stainless steel type 304 metals In all tests the metals forming the joint were identical, and one surface was flat and lapped while the other surface was flat and glass bead blasted

The gap and joint conductance models were compared against data obtained for relatively light contact pressures where the gap and contact conductances were com-parable Figure 4.27 shows plots ofthe joint conductance data and the model predic-tions for very rough stainless steel type 304 surfaces atY/σ = 1.6×10−4 Agreement among the data for argon, helium, and nitrogen is very good for gas pressures be-tween approximately 1 and 700 torr At the low gas pressure of1 torr, the measured and predicted joint conductance values for the three gases differ by a few percent becauseh g  hcandh j ≈ hc As the gas pressure increases there is a large increase

in the joint conductances because the gap conductances are increasing rapidly The joint conductances for argon and nitrogen approach asymptotes for gas pressures ap-proaching 1 atm The joint conductances for helium are greater than for argon and nitrogen, and the values do not approach an asymptote in the same pressure range

The asymptote for helium is approached at gas pressures greater than 1 atm

Figure 4.28 shows the experimental and theoretical gap conductances as points and curves for nitrogen and helium for gas pressures between approximately 10 and 700 torr The relative contact pressure is 1.7 × 10−4is based on the plastic deformation model The joint was formed by Ni 200 surfaces (one flat and lapped and the second flat and glass bead blasted) The data were obtained by subtracting the theoretical value ofh cfrom the measured values ofh j to get the values ofh gthat appear on the

plots The agreement between the data and the predicted curves is very good

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100

102

103

104

105

102

P g(torr)

h j

K)

Experiment

Theory

He

He

Ar

Ar

N2

N2

Vacuum

Vacuum

Stainless Steel 304

␴= 4.83 m␮

R = p 14.7 m␮

Y/ = 3.02␴

P H/ = 1.6 10e ⫻ ⫺4

Figure 4.27 Joint conductance model and data for conforming rough stainless steel 304 surfaces (From Song, 1988.)

Figure 4.29 shows the experimental data for argon, nitrogen, and helium and the dimensionless theoretical curve for the gap model recast as (Song et al., 1993b)

where

G = k g

h g Y and M∗=

M

αβΛ

There is excellent agreement between the model and the data over the entire range ofthe gas–gap parameterM∗ The joint was formed by very rough conforming Ni

by a gas, conduction heat transfer will occur across the gap This heat transfer is characterized by the gap conductance, defined as

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diffusion Fourier’s law of conduction can be used to determine heat transfer across the central zone

There are two heat flux asymptotes, corresponding to very small and very large Knudsen