Rohsenow W., Hartnett J., Young Cho. Handbook of Heat Transfer (776121), страница 33
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The gap integral for pointcontacts proposed by Kitscha and Yovanovich [46] is defined asIg,p =IL 2x tan -1 V'x 2 - 1(28/D) + (2M/D) dx(3.174)The local gap thickness ~i is obtained from28D -1-((L)2)1-1/2+~1 [( 2 - x 2) sin -~(1)+V'x 2 - 111- L2(3.175)where L - D/(2a), x - r/a, and 1 _<x ___L. The gap gas rarefaction parameter is defined asM = alSA(3.176)with ct = (2 - Ctl)/txl + (2 - Ctz)/Ct2, where O~l and ix2 are the accommodation coefficients at thegas-solid interfaces, respectively. The gas parameter 13= (27)/[(7 + 1)/Pr], where 7= Cp/Cv andPr is the Prandtl number. The mean free path A of the gas molecules is given in terms of Ag,0,the mean free path at some reference gas temperature To and reference gas pressure P0, asfollows:Tg Pg,oA=Ag.0 Tg,o PgC O N D U C T I O N A N D T H E R M A L CONTACT R E S I S T A N C E S ( C O N D U C T A N C E S )3.53Gap Resistance Model of Cylinder-Flat Contacts.
The general elastogap resistance modelfor line contacts [143] reduces for the circular cylinder-fiat contact to1 _ 4aLcyl kgolgtRgO' '(3.177)where D and L are the diameter and length of the cylinder, and a is the half-width of the contact strip. The gap integral for line contacts [63] isIgl = % i I"c°sh-1 (;)d;' rt(28/0) + (2M/D)(3.178)The local gap thickness for line contacts [121] is28 (1___/1/2 (~2~1/2 2 ~-~ - l - L 2 ]- 1---~] +[~(~2-1)1/2-cosh-1(~)-~2+1](3.179)where L = D/(2a), ~ = xlL and 1 < x < L.
The dimensionless contact strip width is obtainedfrom the Hertz theory [117]2a/FAD - 4~/ rtDLcyl(3.180)where F is the total load at the contact and A is the physical parameter defined earlier.Radiative Resistance Model of Sphere-Flat Contacts. The radiative resistance of a gapformed by two bodies in elastic contact, such as a sphere-flat or cylinder-flat contact, respectively, is complex because it depends on the geometry of the gap--the surface emissivities ofthe boundaries, which includes the side walls that form the enclosure.Kitscha and Yovanovich [46] and Kitscha [47] proposed the following radiative resistancemodel for a sphere-disk contact with bounding side walls. All surfaces were assumed to begray with constant emissivity values 61, 62, and 6 3 for the sphere, disk, and side walls, respectively. The sphere and disk were assumed to be isothermal at temperatures 7'1 and T2, respectively.
The following expression was proposed:1Rr = A2ff ,24(yT3(3.181)where A2 is the surface area of the disk, ~ is the Stefan-Boltzmann constant,_and Tm = (T1 + T2)/2is the mean absolute temperature of the contact. The radiative parameter F12 is defined as1MF121 --611 --6 2- +~26162+ 1.104(3.182)Joint Resistance Model of Sphere-Flat Contacts.
The joint thermal resistance of a contactformed by elastic bodies, such as a sphere-flat contact is obtained from the model proposedby Kitscha and Yovanovich [46] and Kitscha [47]"1111Rj -- R c ÷ l~g~ ÷ rR--(3.183)which is a function of the constriction resistance Rc, the gap resistance R s, and the radiativeresistance Rr, which are in parallel. The accuracy of the proposed model was verified bynumerous experiments.3.54CHAPTER THREEExperimental Verification of Elastoconstriction and Elastogap Models.Experimentaldata have been obtained for the elastoconstriction resistance of point contacts [47] and linecontacts [63] for a range of sphere and cylinder diameters, material properties, and mechanical loads.
Data were obtained for the verification of the elastogap model for the point contact[46] and line contact [63]. The elastogap models have been tested with air, argon, helium, andnitrogen as the gap fluid at gas pressures from 10-6 torr to atmospheric pressure.Some representative test data for the elastoconstriction and elastogap resistances compared with the theoretical values are given in the following sections.Sphere-Flat Test Results.Kitscha [47] performed experiments on steady heat conductionthrough 25.4- and 50.8-mm sphere-flat contacts in an air and argon environment at pressuresbetween 10-5 torr and atmospheric pressure. He obtained vacuum data for the 25.4-mmdiameter smooth sphere in contact with a polished flat having a surface roughness of approximately 0.13 ktm RMS.
The mechanical load ranged from 16 to 46 N. The mean contacttemperature ranged between 321 and 316 K. The harmonic mean thermal conductivity of thesphere-flat contact was found to be 51.5 W/mK. The emissivities of the sphere and flat wereestimated to be ~ = 0.2 and ~2 = 0.8, respectively.The contact, gap, radiative, and joint resistances were nondimensionalized as R* = DksR.The dimensionless radiative resistance for the sphere-fiat contact given above becomes3.82 x 101°Rr*=T~mThe dimensionless constriction resistance is R* = L and the dimensionless joint resistance in avacuum is1-1+~1(3.184)The model predictions and the vacuum experimental results are compared in Table 3.16.TABLE 3.16Dimensionless Load, Constriction, Radiative,and Joint Resistances [47]N, newtonsL, D/2aTin, KR*rR~R j test16.022.255.687.2195.7266.9467.0115.1103.276.065.450.045.137.43213213213203193183161155115511551164117711881211104.794.771.361.948.043.436.4107.099.470.961.948.842.635.4The radiative resistance was approximately 10 times the constriction resistance at the lightest load and 30 times at the highest load.
The largest difference between the theory and experiments is approximately -4.7 percent, within the probable experimental error. These and othervacuum tests [47] verified the accuracy of the elastoconstriction and the radiation model~The elastogap model for a point contact was verified by Kitscha [47] and Ogniewicz [159].For air, the gas parameter M depends on Tm, Pg as follows:M = 1.373 x 10-4 DPgTm(3.185twhere D is in cm, Tm in K and Pg in mmHg. The numerical value M is based on air propertiesat Tg,0= 288K and Pg,0= 760 mmHg.CONDUCTION AND THERMAL CONTACTRESISTANCES (CONDUCTANCES)3.55TABLE 3.17 Elastogap Resistance Theory and Air Data [47]:D = 25.4 mm, D/(2a) - 115.1Tin, KPg, mm HgR •gR*rRj*theoryRj*test309310311316318321322325321400.0100.040.04.41.80.60.50.2Vacuum76.987.497.1137.2167.2227.9246.6345.4oo12931280126812091186115311431111115544.547.850.659.564.571.773.480.1104.746.849.652.359.065.773.174.380.3107.0The elastogap model and the experimental results are compared over a range of gas pressures in Table 3.17.
Although tests were conducted at smaller values of the dimensionlessparameter L over a range of gas pressures, sphere diameters, and gases, the results given inTable 3.17 are representative of the other data and they also correspond to the case that challenges the validity of the proposed elastoconstriction and elastogap models. First note that theradiative resistance is approximately 10 times the constriction resistance.
Second, observethat the gap resistance is approximately % of the constriction resistance at the highest gaspressure, approximately equal to the constriction resistance at a gas pressure between 4 and40 mmHg, and finally 3 times the constriction resistance at Pg = 0.2 mmHg. The agreementbetween the theory and the tests is very good to excellent.The largest difference occurs at the highest gas pressures, where the theory predicts lowerjoint resistances by approximately 5 percent. The agreement between theory and experimentimproves with decreasing gas pressure.It can also be seen in Table 3.17 that the air within the sphere-fiat gap significantlydecreases the joint resistance when compared with the vacuum result.Thermal Contact, Gap, and Joint Conductance ModelsThermal contact, gap, and joint conductance models developed by many researchers over thepast five decades are reviewed and summarized in several articles [20, 23, 50, 58,143,147,148]and in the recent text of Madhusudana [59].
The models are, in general, based on the assumption that the contacting surfaces are conforming (or fiat) and that the surface asperities haveparticular height and asperity slope distributions [26, 116, 125]. The models assume eitherplastic or elastic deformation of the contacting asperities, and require the thermal spreading(constriction) resistance results presented above.Plastic Contact Conductance Model of Cooper, Mikic, and Yovanovich.The thermal contact conductance models are based on three fundamental models: (1) the metrology model(surface roughness and asperity slope), (2) the contact mechanics model (deformation of thesofter contacting surface asperities), and (3) the thermal constriction (spreading) resistancemodel for the microcontact areas. Cooper et al.
[14] presented the contact conductancemodel, which is based on the Gaussian distribution of the asperity heights and slopes, theplastic deformation of the contacting asperities, and the constriction resistance, which is basedon the isothermal circular contact area on a circular flux tube result. The development of thedimensionless contact conductance model for conforming rough surfaces has been presentedin several publications [14, 65, 139, 143, 147, 148]. The theoretical dimensionless contact conductance has the formhc1exp(-x 2)Cc (3.186)m ks 2V~n (1-e)153.56CHAPTER THREEwith x = erfc -1 (2P/Hc) and e = ~ c ,where P is the contactpressure and H~ is the flow pressure at the plasticallydeformed surface asperities.
The surface parameters are 6 =V'621 + 6 2 and m = kTm 2 + m 2, where 61 and 62 represent theRMS surface roughness of the two contacting surfaces andrnl and m2 represent the absolute mean slopes of the surfaceasperities of the contacting surfaces (Fig. 3.24). The interface effective thermal conductivity is defined as k s - 2klk2/(kl + k2). The metrology model also gives the following geometric relationships [139-141].YI....,!:iii!!~i~Relative Real Contact Area.......:~,~i#: :i.f#i~.'.........Ni:.J~...N .....i;iiN:;:~i~i~i!!!~F I G U R E 3.24parameters.~.2= Ar _ 1 erfc (x)Aa 2~.,~!iii:.i:m =,' m ] + m 2where At is the total real contact area and Aa is the corresponding total apparent area.Contact Spot Density.C o n f o r m i n g r o u g h surface g e o m e t r i c1 ( ~ ) 2 exp(-2x2)n=~erfc (x)Mean Contact Spot Radius.a: g ( )exp x' er,c x,where x = (1/V2)(Y/o) and Y is the mean plane separation (Fig.
3.24). The relative meanplane separation is obtained from13which is approximated by [139-141][ ( :/1Y - 1.184 -ln 3.132oHe]_!which is valid in the ranges 4.75 > Y/o > 2.0 and 10 -6 < P/Hc -< 2 × 10-2, and it has a maximumerror of approximately 1 percent. The relative mean plane separation appears in the gap conductance model.Yovanovich [139] proposed the correlation equation for the contact conductance model:[ p '~0.95Co= 1 . 2 5 ( ~ [ )(3.187)which is valid for the wide range 10-6 _<P/Hc -< 2 × 10-z and has a maximum error of approximately 1.5 percent.Yovanovich et al. [140] and Hegazy [35] recognized the importance of the effect of workhardened layers on the deformation of the contacting asperities.
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