Adrian Bejan(Editor), Allan D. Kraus (Editor). Heat transfer Handbok (776115), страница 53
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He showed empirically that the higher the value of this parameter, the greater was the improvementin the joint conductance over a bare joint. Following this thought, Antonetti andYovanovich (1983) proposed (but did not prove experimentally) that the performanceof coated joints can be ranked by the parameter k /(H )0.93 . Table 4.21 shows thevariation in this parameter as the layer thickness is increased. Table 4.21 also suggests that even if the effective microhardness of the layer–substrate combinationsbeing considered is not known, the relative performance of coating materials can beestimated by assuming an infinitely thick coating (where the effective microhardnessis equal to the microhardness of the layer).In this section we have shown how a thermomechanical model for coated substrates can be used to predict enhancement in thermal joint conductance.
For theparticular case considered, an aluminum-to-aluminum joint, it was demonstrated thatup to an order of magnitude,TABLE 4.21Ranking the Effectiveness of Coatings k /(H )0.93Coating Thickness(µm)LeadTinSilver0124816∞3.053.727.0519.618.021.019.93.053.966.8110.510.812.912.23.053.533.984.686.248.168.38P = 2000 kN/m2σ = 4.0 µmBOOKCOMP, Inc. — John Wiley & Sons / Page 371 / 2nd Proofs / Heat Transfer Handbook / Bejanm = 0.20 radLines: 4188 to 4222———0.58879pt PgVar———Normal PagePgEnds: TEX[371], (111)372123456789101112131415161718192021222324252627282930313233343536373839404142434445THERMAL SPREADING AND CONTACT RESISTANCESimprovement in the contact conductance is possible, depending on the choice ofcoating material and the thickness employed.
It should also be noted that aluminumsubstrates are relatively soft and have a relatively high thermal conductivity, and ifthe joint in question had been, for example, steel against steel, improvement in thejoint conductance would have been even more impressive.4.17.3Elastomeric InsertsThin polymers and organic materials are being used to a greater extent in powergenerating systems. Frequently, these thin layers of relatively low thermal conductivity are inserted between two metallic rough surfaces assumed to be nominally flat.The joint that is formed consists of a single layer whose initial, unloaded thickness isdenoted as t0 and has thermal conductivity kp .
There are two mechanical interfacesthat consist of numerous microcontacts with associated gaps that are generally occupied with air. If radiation heat transfer across the two gaps is negligible, the overalljoint conductance is Yovanovich et al. (1997).kp111=++hjhc,1 + hg,1thc,2 + hg,2(m2 · K/W)(4.308)[372], (112)Lines: 4222 to 4255———6.48418pt PgVar———Short Pagewhere t is the polymer thickness under mechanical loading.
The contact and gapconductances at the mechanical interfaces are denoted as hc,i and hg,i , respectively, * PgEnds: Ejectand i = 1, 2.Since this joint is too complex to study, Fuller (2000) and Fuller and Marotta[372], (112)(2002) choose to investigate the simpler joint, which consisted of thermal greaseat interface 2, and the joint was placed in a vacuum.
Under these conditions, theyassumed that hg,2 → ∞ and hg,1 → 0. They assumed further that the compressionof the polymer layer under load may be approximated by the relationshipPt = t0 1 −(m)(4.309)Epwhere Ep is Young’s modulus of the polymer. Under these assumptions the jointconductance reduces to the simpler relationshiphj =1t0+hc,1kp1−PEp−1(W/m2 · K)(4.310)On further examination of the physical properties of polymers, Fuller (2000) concluded that the polymers will undergo elastic deformation of the contacting asperities.
Fuller examined use of the elastic contact model of Mikic (1974) and found thatthe disagreement between data and the predictions was large. To bring the model intoagreement with the data, it was found that the elastic hardness of the polymers shouldbe defined asBOOKCOMP, Inc. — John Wiley & Sons / Page 372 / 2nd Proofs / Heat Transfer Handbook / BejanJOINT CONDUCTANCE ENHANCEMENT METHODS123456789101112131415161718192021222324252627282930313233343536373839404142434445Hep =Ep m2.3(GPa)373(4.311)where m is the combined mean absolute asperity slope. The dimensionless contactconductance correlation equation was expressed ashc σ2.3P a2= a1(4.312)ks mmEpwhere a1 and a2 are correlation coefficients. Fuller and Marotta (2000) chose thecoefficient values a1 = 1.49 and a2 = 0.935, compared with the values that Mikic(1974) reported: a1 = 1.54 and a2 = 0.94.
In the Mikic (1974) elastic contact model,the elastic hardness was defined asmE He = √2[373], (113)(GPa)(4.313)where E is the effective Young’s modulus of the joint. For most polymer–metaljoints, E ≈ Ep because Ep E metal.Fuller (2000) conducted a series of vacuum tests for validation of the joint conductance model. The thickness, surface roughness and the thermophysical properties ofthe polymers and the aluminum alloy are given in Table 4.22.
The polymer thicknessin all cases is two to three orders of magnitude larger than the surface roughness (i.e.,t/σ > 100).The dimensionless joint conductance data for three polymers (delrin, polycarbonate, and PVC) are plotted in Fig. 4.35 against the dimensionless contact pressureover approximately three decades. Two sets of data are reported for delrin. The jointconductance model and the data show similar trends with respect to load. At thehigher loads, the data and the model approach asymptotes corresponding to the bulkresistance of the polymers.
The dimensionless joint conductance goes to differentasymptotes because the bulk resistance is defined by the thickness of the polymerTABLE 4.22 Thickness, Surface Roughness, and Thermophysical Propertiesof Test SpecimensMaterialDelrin BPolycarbonate APVC ATeflon AAluminumt0(103 m)σ(106 m)m(rad)k(W/m · K)E(GPa)ν1.881.991.831.89—1.020.7730.6500.6220.5110.4920.4700.4360.3050.2670.380.220.170.251833.592.384.140.13572.00.380.380.380.380.32Source: Fuller (2000).BOOKCOMP, Inc. — John Wiley & Sons / Page 373 / 2nd Proofs / Heat Transfer Handbook / BejanLines: 4255 to 4305———0.9843pt PgVar———Short PagePgEnds: TEX[373], (113)37410⫺2F6543210⫺3hjks m123456789101112131415161718192021222324252627282930313233343536373839404142434445THERMAL SPREADING AND CONTACT RESISTANCEStk1hc,1kphc,2k26543F2Delrin1; DataDelrin2; DataPolycarbonate; DataPVC; DataDelrin1; ModelDelrin2; ModelPolycarbonate; ModelPVC; Model10⫺46543210⫺510⫺5234 5 6 7 8 10⫺4234 5 6 7 8 10⫺32.3PEp m234 5 6 7 8 10⫺2Figure 4.35 Dimensionless joint conductance versus dimensionless contact pressure forpolymer layers.
(From Fuller and Marotta, 2000.)[374], (114)Lines: 4305 to 4321———0.097pt PgVar———Normal PagePgEnds: TEX[374], (114)layers. In general, there is acceptable agreement between the vacuum data and themodel predictions.4.17.4 Thermal Greases and PastesThere is much interest today in the use of thermal interface materials (TIMs) such asthermal greases and pastes to enhance joint conductance. Prasher (2001) and Savijaet al. (2002a,b) have reviewed the use of TIMs and the models that are availableto predict joint conductance. The thermal joint resistance or conductance of a jointformed by two nominally flat rough surfaces filled with grease (Fig.
4.22) dependon several geometric, physical, and thermal parameters. The resistance and conductance relations are obtained from a model that is based on the following simplifyingassumptions:••••Surfaces are nominally flat and rough with Gaussian height distributions.The load is supported by the contacting asperities only.The load is light; nominal contact pressure is small; P /Hc ≈ 10−3 to 10−5 .There is plastic deformation of the contacting asperities of the softer solid.BOOKCOMP, Inc. — John Wiley & Sons / Page 374 / 2nd Proofs / Heat Transfer Handbook / BejanJOINT CONDUCTANCE ENHANCEMENT METHODS123456789101112131415161718192021222324252627282930313233343536373839404142434445375• Grease is homogeneous, fills the interstitial gaps completely, and wets the bounding surfaces perfectly.In general, the joint conductance hj and joint resistance Rj depend on the contactand gap components.
The joint conductance is modeled ashj = hc + hg(W/m2 · K)(4.314)where hc represents the contact conductance and hg represents the gap conductance.The joint resistance is modeled as111=+RjRcRg(W/K)(4.315)where Rc is the contact resistance and Rg is the gap resistance.
For very light contactpressures it is assumed that hc hg and Rc Rg . The joint conductance andresistance depend on the gap only; therefore,hj = hgand11=RjRgwherehj =1Aa Rj[375], (115)Lines: 4321 to 4368———(4.316)0.89034pt PgVar———Normal PageBased on the assumptions given above, the gap conductance is modeled as an equivalent layer of thickness t = Y filled with grease having thermal conductivity kg . The * PgEnds: Ejectjoint conductance is given byhj =kgY(W/m2 · K)(4.317)The gap parameter Y is the distance between the mean planes passing through thetwo rough surfaces. This geometric parameter is related to the combined surfaceroughness σ = σ21 + σ22 , where σ1 and σ2 are the rms surface roughness of thetwo surfaces and the contact pressure P and effective microhardness of the softersolid, Hc .
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