Rohsenow W., Hartnett J., Young Cho. Handbook of Heat Transfer (776121), страница 35

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Thanks to E Teertstra,J. R. Culham, Y. S. Muzychka and M. Stevanovic for help with the figures and other assistance.NOMENCLATURESymbol, Definition, Sl UnitsA, Ac, Atarea, contact, and flux tube a r e a s : m 2Aa, Arapparent and real contact areas, Ar/A,, < 1: m 2As, ANAn, B,,AR~(~, ~)shape factor parametersFourier coefficients for temperature and heat lossaspect ratio of bodiesdimensionless ellipsoid area function = A / ( 2 / r , a 2)asemimajor axis of ellipse, hyperellipse, rectangle: mapoint contact radius, line contact half-width: ma,b,ca,bsemiaxes of ellipsoid: mal, a2, a3, a4, a5correlation coefficientsB(x, y)Bbeta functionBiBiot number = h~/k (typical)Biv~Biot number based on square root of areaBrinell hardness numberBHNCC*radii of bisphere: mlayer thicknesscapacitance: coulomb/voltcapacity: mc~c~,Cvdimensionless contact conductance = ohc/(mks)specific heat at constant pressure and volume: J/(kg.°C)Co, C1, C3, C5, C7correlation coefficientsCcT, CSTC1, C2, C3correlation coefficients for circular and square toroidsCcorrelation coefficientsspecific heat: J/(kg.°C)C1, c2Vickers microhardness correlation coefficientsCONDUCTION AND THERMAL CONTACTRESISTANCES(CONDUCTANCES)DDi, DoDGMdEE(~)E(¢, ~)E'eft, erfcerfc -1FF*F(¢, ~)FoFov~FocFI~fe~f~gi,HHRClibH~He~Hvh, ho hg, hjI(~3, ~[)InI,I,,~I~,,inerfcJ,K(~:)K~kkl, k2kak0ksL3.61diameter of cylinders, spheres, and toroids: minner and outer diameters of circular and square toroidsgeometric mean diameter: mdiameter of cylinders and toroids: mmodulus of elasticity: N/m 2complete elliptic integral of the second kindincomplete elliptic integral of second kindequivalent modulus = [(1 - v~)/E1 + (1 - v~ )/E2] -1error and complementary error functionsinverse complementary error functioncontact load: Ndimensionless line contact load = FA/(DL)incomplete elliptic integral of the first kindFourier number = ott/..T2(typical)Fourier number based on square root of areacritical value of Fourier numberradiative parameter for point contactelastoplastic contact parametergap conductance correlation equationmetric coefficients, jacobianheight of single and double conesRockwell C hardness numberBrinell hardnesssurface contact microhardness: MPaelastoplastic microhardness: MPaVickers microhardness: MPaheat transfer coefficient and contact, gap and joint conductances: W/(m2.K)shape factor integralnth-order modified Bessel function of first kindgap conductance integral = fg/(Y/~ + M/~)point contact gap integralline contact gap integralintegrals of complementary error functionnth-order Bessel function of first kindcomplete elliptic integral of first kind of modulus ~cnth-order modified Bessel function of first kindthermal conductivity: W/(m.K)layer and substrate conductivities: W/(m.K)conductivity at average temperature: W/(m.K)conductivity at reference temperature: W/(m-K)harmonic mean conductivity = 2klkE/(kl + k2): W/(m.K)length of circular and square cylinders and rectangular plate: m3.62CHAPTERTHREELdimensionless point contact parameter: D/(2a)L~ylcylinder length: mL1, L2, L3side dimensions of cuboid: mMgap gas parameter = (~I3A: mM~,M~gas and solid molecular weights: g/molemeffective surface asperity slope = V'ml2 + m 2ml, m2surface mean asperity slope of contacting rough surfacesNnumber sides of regular polygonsnoutward directed normal, shape parameterninterpolation parameterncontact spot density: m -20P, Pi, PoPareaarbitrary length scale: mperimeter, inner, and outer perimeters: mcontact pressure: MPaP~P,,ogas pressure: kPaPrPrandtl number = ~c/kP.-l,2(~)toroidal functionPcorrelation coefficientoheat flow rate: Wo~o,OeQ..-,,~(~)dimensionless heat flow rate = (Q~)/(kAOo) (typical)qqoRRoRc, R~reference gas pressure: kPainitial internal energy = pcVOi (typical): Jtotal electrical chargetoroidal functionheat flux: W/m 2surface heat flux: W/m 2thermal resistance, spreading resistance = (Ts.....- Tsink)/Oconstriction and spreading resistances: K/WR,,gap resistance: K/Wnjjoint or overall resistance = (1/R~ + 1/Rg + 1/R~)-I: K/WRrR*R,R*R,radiation resistance: K/Wdimensionless resistance - k ~ R (typical)dimensionless spreading resistance = 4k~R~ (typical)dimensionless contact, gap, or radiation resistance - k~DR (typical)dimensionless point contact resistance = Ln,dimensionless gap resistanceR*dimensionless radiative resistance = (3.82 x 101°)/T3n,,dimensionless joint resistance = (1/L + l/R*) -1Fradius, polar, and spherical coordinates: m7(typical): K/Wspreading resistanced based on centroid temperature: K/Wradius vector: mC O N D U C T I O N AND T H E R M A L CONTACT RESISTANCES (CONDUCTANCES)r0radius for hyperellipse: mSshape factor: ms~slmaterial yield or flow stress: MPaS0, S~shape factors for thin and thick conduction layers: ms~dimensionless shape factor = (S~)/ASactive surface area: m 2Sdimensionless parameter = 1/(1 + pz)dimensionless shape factorsdummy variabler,rrltemperature, area average temperature: Kfluid temperature: KT,initial solid temperature: Krmr~,omean temperature = ( T~ + 7"2)/2: Kreference gas temperature: Kttime: stvariable gap thickness: mtl, hcoating or layer thicknesses: mUdimensionless position variableudimensionless variable gap thickness = t/ouicurvilinear coordinatesVvolume: m 3v~electrical potential: VWwidth of annular area: mWwidth of rectangular plate: mx, y, zcartesian coordinates: mXdimensionless parameter for single and double conesxdimensionless mean plane separation = (1/X/2)(Y/o)Ymean plane separation = oN/2 erfc -~ (2P/Hc): mLnth-order Bessel function of second kindZdimensionless parameter = (Bi + 1)x/-F--oZdimensionless time p a r a m e t e r - 1/(2%/-~ x/-F--ov~)Greek Symbolsthermal diffusivity = k/(pc): m2/sO~accommodation coefficient = (20~1, 0~2accommodation coefficients for gap-0~1)/(~1 + (2-0~2)/0~bandwith surface roughness parameter 5 < ct < 100gas parameter: = (2"/)/[(), + 1)/Pr]dimensionless parameter in ellipsoidal integral = b/athermal conductivity temperature coefficient~,~integration limitsdimensionless parameter in ellipsoidal integral = b/a23.633.64CHAPTER THREEdimensionless p a r a m e t e r in ellipsoidal integral - b/ar(x)g a m m a function7Ydimensionless p a r a m e t e r in ellipsoidal integral = c/aspecific heat ratio = Cp/Cv7Adiffusion thickness = A/S: maspect ratio p a r a m e t e r = b/a < 1Aelastic p a r a m e t e r = [(1 - v2)/E1 + (1 - vZ)/E2]/2: (MPa) -18nnth eigenvalue81first eigenvalue51,0first eigenvalue for Bi -+ 051,0ofirst eigenvalue for Bi ---) oo8capproximate first eigenvalue = rt + 1/(X/-~)Epermittivity of space: faradErelative size = a/b (typical)Eccontact straindimensionless contact strain = 1.67(mE'/Sr)arguments of toroidal functions = D/ddimensionless position = x/L (typical)d u m m y variable11elliptic cylinder, bicylinder, oblate and prolate spheroidal coordinate]]1, ]'12integration limits]]min, ]]maxminimum and maximum integration limits0steady or transient t e m p e r a t u r e excess = [T(r, t) - Too] (typical): KOrm a x i m u m system t e m p e r a t u r e excess = Ty- Ti (typical): K0i0initial t e m p e r a t u r e excess = Ty- Ti (typical): KKirchhoff t e m p e r a t u r e = 1/ko f~o k dT (typical)" K0spherical, oblate, and prolate spheroidal coordinate0 -+multiple layer thickness and conductivity p a r a m e t e rKmodulus of complete and incomplete elliptic integrals of first and secondkindKthermal conductivity ratio = kl/k2 (typical)conductivity ratio = k2/kl1~32conductivity ratio = k3/k2Amolecular mean free path: mA~,omolecular m e a n free path at reference conditions: mZ,Z,1relative gap t h i c k n e s s - Y/oconductivity ratio p a r a m e t e r = (1 - k2/kl)/2conductivity ratio p a r a m e t e r = (1 + k2/k1)/2~theat flux distribution p a r a m e t e r q* = (1 - u2) ~~torder of Bessel function of the first kind J~ + (l/2)(n~E)ratio of molecular weights = Mg/MsVI, V2Poisson's ratiosCONDUCTION AND T H E R M A L CONTACT RESISTANCES (CONDUCTANCES)elliptic cylinder, bicylinder, oblate and prolate spheroidal coordinatedummy variable in ellipsoidal integralmass density: kg/m 3?t,EO'1, (Y2(Ydimensionless position in cylindrical and spherical coordinates = r/a(typical)heat flux parameter = 1, isothermal; = sin (8,,~)/[2J1(5,,~)], isofluxRMS surface rough of contacting surfaces: meffective RMS surface rough of interface: mrelative layer thickness - t/a (typical)l)*~,*r~/ave]/maxO3dimensionless temperature = [ T(r ) - Too]/(To - Too)nondimensional electrical potentialdimensionless temperatures for plate and cylinderapproximation function for compound cylindercompound cylinder functioncompound cylinder functiondimensionless spreading resistance = 4 k ~ R (typical)dimensionless spreading resistance = X/--~kaRcdimensionless spreading resistance = (1 -f_)3/Zk~c]2dimensionless spreading resistance = (1 - ~)kOc/X/-~dimensionless centroid based spreading resistance - k ~ / A R oangleSubscriptsv~aaveccylcontact planeCT, STeepffDGMgg,Og,lg,Pii,oscale length is square root of the areaapparentbased on average temperaturecontact, constriction, criticalcylinderapparentcircular and square toroidselectricalelastoplasticfluidflow stressDirichlet condition solutiongeometric meangap, gasat reference gas temperature and pressuregap for line contactgap for point contactinitial valueinner and outer3.653.66CHAPTER THREEiYL1mmaxrainNr/PPRrrectSSsourcesinkthickthinVr,~,zr,~,0n,~,zTI, ~, z~,0,~000,10, oooo1,01, oo1,21,2,3121Dinner body volume and areajointlongest straight line through inner bodyline contactmean valuebased on centroid temperaturebased on area average temperatureNeumann condition solutionnthpoint contactconstant pressure conditionRobin condition solutionradiativerectangular areaspreadingbased on harmonic mean valuebased on source temperaturebased on sink temperaturethick layerthin layerconstant volume conditioncylindrical coordinatesspherical coordinateselliptical cylinder coordinatesbicylinder coordinatesoblate and spheroidal coordinateszero thickness limitbased on centroid temperaturezeroeth order, first ordervalue on the surface and at infinityinfinite thickness limitfirst eigenvalue value at zero Biot number limitfirst eigenvalue value at infinite Biot number limitsolids 1 and 2; surfaces 1 and 2cuboid side dimensionsnet radiative transferone-dimensional conductionSuperscriptsdimensionless valuep+effective value of parameterparameter with positive or negative termsOverscoresmean valueCONDUCTION AND THERMAL CONTACT RESISTANCES (CONDUCTANCES)3.67REFERENCES1.

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions, Dover, New York, 1965.2. V.W. Antonetti, "On the Use of Metallic Coatings to Enhance Thermal Contact Conductance," PhDthesis, University of Waterloo, Waterloo, Ontario, Canada, 1983.3. V. W. Antonetti and M. M. Yovanovich, "Enhancement of Thermal Contact Conductance By Metallic Coatings: Theory and Experiments," Journal of Heat Transfer (107): 513-519, 1985.4. V.

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