Rohsenow W., Hartnett J., Young Cho. Handbook of Heat Transfer (776121), страница 34
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They proposed an iterativeprocedure to obtain the appropriate value of the contact microhardness Hc from Vickersmicrohardness Hv measurements [41,114]. Song and Yovanovich [102] reported the explicitexpression for calculating the relative contact pressure:P [ O.9272P ]1/(1+°°71c2)H~-c1(1.626/m) c2(3.188)CONDUCTION AND THERMAL CONTACTRESISTANCES(CONDUCTANCES)3.57where the correlation coefficients c1 and C2 are obtained from Vickers microhardness measurements. Sridhar and Yovanovich [108] developed correlation equations for the Vickerscoefficients:c~ _ [4.0 - 5.77H~ + 4.0(H~ 2 - 0.61(H~ 313178andc2 =-0.370 + 0.442(-~1B )where HB is the Brinell hardness [41,114] and H~ = HB/3178. The correlation equations arevalid for the Brinell hardness range of 1300-7600 MPa.
The above correlation equations weredeveloped for a range of metals: Ni200, SS304, Zr alloys, Ti alloys, and tool steel. Sridhar andYovanovich [108] also reported a correlation equation that relates the Brinell hardness number to the Rockwell C hardness number:B H N = 43.7 + l O . 9 2 H R C -HRC 2HRC 3+ ~5.18340.26for the range 20 < H R C < 65. It has been demonstrated that the above plastic contact conductance model predicts accurate values of h~ for a range of surface roughness o/m, a range ofmetals (SS304, Ni200, Zr alloys, etc.), and a range of relative contact pressure P/H~ [2, 35,105,106, 109].Plastic Contact Conductance Model o f Greenwood and Williamson.
Sridhar and Yovanovich [109] developed correlation equations for the contact conductance of conforming roughsurfaces based on the Greenwood and Williamson [26] surface model using the plastic deformation model described above. The dimensionless contact conductance correlation isCc=Ohc{0.9272P}(0"971°0/25193)/(1+0"038c2)m ks - 0"91°~°31 cl [(2.47/0~°269)(0/m ) ]C2(3.189)which is valid in the relative contact pressure range 10-5 < P/Hc < 10-2. The surface parametero~is the bandwidth parameter, which depends on the variance of surface heights, the varianceof surface slopes, and the variance of the second derivative of surface heights.
The correlationis valid for the range 5 _<a < 100. c and m are the surface roughness and the surface slope asdefined previously. The parameters c1 and c2 are the Vickers microhardness correlation coefficients, which can be determined by means of the relationships just presented.Sridhar and Yovanovich [109] found that the Greenwood and Williamson (GW) model[26] and the Cooper, Mikic, and Yovanovich (CMY) model [14] are in very good agreementwhen a = 5. For all values of ot > 5, the GW model predicts values of Cc that are greater thanthose predicted by the CMY model.Elastic Contact Conductance Models of Mikic and Greenwood and Williamson.
Sridharand Yovanovich [106] reviewed the elastic contact models proposed by Greenwood andWilliamson [26] and Mikic [66] and compared the correlation equation with data obtained forfive different metals. The models were developed for conforming rough surfaces; they differin the description of the surface metrology and the contact mechanics. The thermal modeldeveloped by Cooper et al.
[14] was used. The details of the development of the models andthe correlation equations are reviewed by Sridhar and Yovanovich [106]. The correlationequation derived from the Mikic [66] surface and asperity contact models ism ~-1.54( )094mE(3.190)3.58CHAPTER THREEwhere 0 and m are the contact surface roughness and slope. The equivalent modulus isdefined as E ' = [(1 - v2)/E~ + (1 - vZ2)/E2]-1 where Vl and v2 are the Poisson's ratio, and E1 andE2 are the elastic modulus of the contacting asperities.The correlation equation derived from the Greenwood and Williamson [26] surface andasperity contact models is0"922°t'/2°ss4mECc = ° m hCk,-(1.18+0.161 In 00(V/-2P)(3.191)where cx is the bandwidth parameter discussed above.The two elastic contact correlation equations [106] were developed for the ranges 10-s <X/2P/mE' < 10 -2 and 5 < ot < 100.
The two correlation equations predict values of Cc that arein close agreement when ot = 5 and for the largest value of o~ the difference between the predicted values is approximately 35 percent.The correlation equation developed from the Mikic [66] models is simpler; therefore it isrecommended for predicting contact conductance for elastic contacts.Elastoplastic Contact Model and Relationships.Sridhar and Yovanovich [106] developedan elastoplastic contact conductance model that is summarized below in terms of the geometric parameters (1) Ar/Aa, the real to apparent area ratio; (2) n, the contact spot density; (3) a,the mean contact spot radius; and (4) ~,, which is the dimensionless mean plane separation:A r _ fep erfc (~,/V~)Ao21 ( ~ ) 2 exp(-)v2)n=~erfc (L/V~)a = ~ / ~ . ~ e p " ~m exp(k2/2) erfc (~,/X/2)hcksmV~ep " exp(-K2/2)2 X / ~ r~ [1 - V'(fep/2) erfc (~X/2)] ~s, erfc,( 1The elastoplastic parameter fep is a function of the dimensionless contact strain e*, whichdepends on the amount of work hardening.
This physical parameter lies in the range 0.5 <fep < 1.0. The smallest and largest values correspond to zero and infinitely large contact strain,respectively. The dimensionless contact strain is defined as/mE'\where Sy is the material yield or flow stress [41], which is a complex physical parameter thatmust be determined by experiment for each metal.
The elastoplastic microhardness Hep canbe determined by means of an iterative procedure that requires the following relationship:2"76SIHep = [1 + (6.5/e*)2] 1/2(3.192)The proposed elastoplastic contact conductance model moves smoothly between the elasticcontact model of Mikic [66] and the plastic contact conductance model of Cooper, Mikic, andCONDUCTION AND THERMAL CONTACT RESISTANCES (CONDUCTANCES)3.59Yovanovich [14], which was modified by Yovanovich and co-workers [102, 139, 140] toinclude the effect of work-hardened layers on the deformation of the contacting asperities.Gap Conductance Model and IntegralThe gap conductance model for conforming rough surfaces was developed, modified, and verified by Yovanovich and co-workers [35, 73, 100-104].
The gap contact model is based on surfaces having Gaussian height distributions. It also accounts for the mechanical deformation ofthe contacting surface asperities. The development of the gap conductance model appears inseveral papers [139, 143,147].The gap conductance model is expressed in terms of an integral:hg- ke,o k/~nl I~ exp[-(Y/O-u+ M/oU)2/2] du(3.193)where kg is the thermal conductivity of the gas trapped in the gap, o is the effective surfaceroughness of the interface, and u = t/o is the dimensionless local gap thickness. The integraldepends on two parameters: Y/G, which is the mean plane separation, and M/G, which is therelative gas rarefaction parameter defined and discussed previously. The gas rarefactionparameter M = ~[3A has also been defined and discussed previously.Accommodation Coefficients.The accommodation coefficient 0t accounts for the efficiencyof gas-surface energy exchange.
There is a large body of research dealing with experimentaland theoretical aspects of a for different gases in contact with metallic surfaces under varioussurface conditions and temperatures [32, 92, 120, 126]. Song and Yovanovich [100] and Songet al. [101,104] examined the gap conductance models available in the literature, the experimental data, and the models for the accommodation coefficients. Song and Yovanovich [100]developed a correlation for the accommodation coefficient for "engineering" surfaces (i.e.,surfaces with adsorbed layers of gases and oxides). They proposed a correlation based onexperimental results of numerous investigators for monatomic gases. The relation wasextended by the introduction of a "monatomic equivalent molecular weight" to diatomic andpolyatomic gases.
The correlation equation has the form: exp(CoT)[Mg/(C1 + Me,) + 11 - exp(CoT)}{2.4g/(1 + g)2}]where C0 = -0.57T : ( T~ - To)/ToMe, -- Mg for monatomic gases= 1.4 Me, for diatomic and polyatomic gasesC1 = 6.8, units of Mg [g/mole]g : Me,/M,where Ts and To are the absolute temperatures of the surface and the gas, and Mg and Ms arethe molecular weights of the gas and the solid, respectively. The agreement between the predictions according to this correlation and the published data for diatomic and polyatomicgases was within +25 percent.Wesley and Yovanovich [127] compared the predictions of the proposed gap conductancemodel and experimental measurements of gaseous gap conductance between the fuel andclad of a nuclear fuel rod.
The agreement was very good and the model was recommended forfuel pin analysis codes.Gap Conductance Correlation Equations. Although the gap integral can be computedaccurately and easily by means of computer algebra systems, Negus and Yovanovich [73] proposed the following correlation equations for the gap integral:3.60CHAPTERTHREE~g ~( Y/o) + (M/o)withfg = 1.063 + 0.0471(4- Y/c)168[ln (a/M)] °84for 2 < Y/c < 4andfg = 1 + 0.06(o/M) °81 < M/o < oofor 2 < Y/o < 4andand0.01 < M/c <_1The correlations have a maximum error of approximately 2 percent.ACKNOWLEDGMENTSThe author is grateful to the Natural Science and Engineering Research Council for its continued financial support of the research for the preparation of this chapter.
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