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BOOKCOMP, Inc. — John Wiley & Sons / Page 371 / 2nd Proofs / Heat Transfer Handbook / Bejan JOINT CONDUCTANCE ENHANCEMENT METHODS 371 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 [371], (111) Lines: 4188 to 4222 ——— 0.58879pt PgVar ——— Normal Page PgEnds: T E X [371], (111) the highest contact conductance, and silver with the highest microhardness has the lowest contact conductance. The thermal conductivity of the coating appears to play a secondary role. The unusual shape of the curves is attributable to the fact that the assumed effective hardness curve shown in Fig. 4.30 has three distinct zones. Moreover, because the microhardness of silver is much closer to aluminum than are the microhardness of lead and tin, respectively, the transition from one region to the next is not abrupt in the silver-on-aluminum effective microhardness curve, and this is reflected in the smoother contact conductance plot for the silver layer shown in Fig. 4.34. It should be noted that in the model that has been used, the load is assumed to be uniformly applied over the apparent contact area. 4.17.2 Ranking Metallic Coating Performance In his research on the effects of soft metallic foils on joint conductance, Yovanovich (1972) proposed that the performance of various foil materials may be ranked accord- ing to the parameter k/H, using the properties of the foil material. He showed empir- ically that the higher the value of this parameter, the greater was the improvement in the joint conductance over a bare joint. Following this thought, Antonetti and Yovanovich (1983) proposed (but did not prove experimentally) that the performance of coated joints can be ranked by the parameter k  /(H  ) 0.93 . Table 4.21 shows the variation in this parameter as the layer thickness is increased. Table 4.21 also sug- gests that even if the effective microhardness of the layer–substrate combinations being considered is not known, the relative performance of coating materials can be estimated by assuming an infinitely thick coating (where the effective microhardness is equal to the microhardness of the layer). In this section we have shown how a thermomechanical model for coated sub- strates can be used to predict enhancement in thermal joint conductance. For the particular case considered, an aluminum-to-aluminum joint, it was demonstrated that up to an order of magnitude, TABLE 4.21 Ranking the Effectiveness of Coatings  k  /(H  ) 0.93  Coating Thickness (µm) Lead Tin Silver 0 3.05 3.05 3.05 1 3.72 3.96 3.53 2 7.05 6.81 3.98 4 19.6 10.5 4.68 8 18.0 10.8 6.24 16 21.0 12.9 8.16 ∞ 19.9 12.2 8.38 P = 2000 kN/m 2 σ = 4.0 µm m = 0.20 rad BOOKCOMP, Inc. — John Wiley & Sons / Page 372 / 2nd Proofs / Heat Transfer Handbook / Bejan 372 THERMAL SPREADING AND CONTACT RESISTANCES 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 [372], (112) Lines: 4222 to 4255 ——— 6.48418pt PgVar ——— Short Page * PgEnds: Eject [372], (112) improvement in the contact conductance is possible, depending on the choice of coating material and the thickness employed. It should also be noted that aluminum substrates are relatively soft and have a relatively high thermal conductivity, and if the joint in question had been, for example, steel against steel, improvement in the joint conductance would have been even more impressive. 4.17.3 Elastomeric Inserts Thin polymers and organic materials are being used to a greater extent in power- generating systems. Frequently, these thin layers of relatively low thermal conduc- tivity 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 is denoted as t 0 and has thermal conductivity k p . There are two mechanical interfaces that consist of numerous microcontacts with associated gaps that are generally occu- pied with air. If radiation heat transfer across the two gaps is negligible, the overall joint conductance is Yovanovich et al. (1997). 1 h j = 1 h c,1 + h g,1 + k p t + 1 h c,2 + h g,2 (m 2 · K/W) (4.308) where t is the polymer thickness under mechanical loading. The contact and gap conductances at the mechanical interfaces are denoted as h c,i and h g,i , respectively, and i = 1, 2. Since this joint is too complex to study, Fuller (2000) and Fuller and Marotta (2002) choose to investigate the simpler joint, which consisted of thermal grease at interface 2, and the joint was placed in a vacuum. Under these conditions, they assumed that h g,2 →∞and h g,1 → 0. They assumed further that the compression of the polymer layer under load may be approximated by the relationship t = t 0  1 − P E p  (m) (4.309) where E p is Young’s modulus of the polymer. Under these assumptions the joint conductance reduces to the simpler relationship h j =  1 h c,1 + t 0 k p  1 − P E p  −1 (W/m 2 · K) (4.310) On further examination of the physical properties of polymers, Fuller (2000) con- cluded that the polymers will undergo elastic deformation of the contacting asperi- ties. Fuller examined use of the elastic contact model of Mikic (1974) and found that the disagreement between data and the predictions was large. To bring the model into agreement with the data, it was found that the elastic hardness of the polymers should be defined as BOOKCOMP, Inc. — John Wiley & Sons / Page 373 / 2nd Proofs / Heat Transfer Handbook / Bejan JOINT CONDUCTANCE ENHANCEMENT METHODS 373 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 [373], (113) Lines: 4255 to 4305 ——— 0.9843pt PgVar ——— Short Page PgEnds: T E X [373], (113) H ep = E p m 2.3 (GPa) (4.311) where m is the combined mean absolute asperity slope. The dimensionless contact conductance correlation equation was expressed as h c σ k s m = a 1  2.3P mE p  a 2 (4.312) where a 1 and a 2 are correlation coefficients. Fuller and Marotta (2000) chose the coefficient values a 1 = 1.49 and a 2 = 0.935, compared with the values that Mikic (1974) reported: a 1 = 1.54 and a 2 = 0.94. In the Mikic (1974) elastic contact model, the elastic hardness was defined as H e = mE  √ 2 (GPa) (4.313) where E  is the effective Young’s modulus of the joint. For most polymer–metal joints, E  ≈ E p because E p  E metal . Fuller (2000) conducted a series of vacuum tests for validation of the joint conduc- tance model. The thickness, surface roughness and the thermophysical properties of the polymers and the aluminum alloy are given in Table 4.22. The polymer thickness in 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, polycarbon- ate, and PVC) are plotted in Fig. 4.35 against the dimensionless contact pressure over approximately three decades. Two sets of data are reported for delrin. The joint conductance model and the data show similar trends with respect to load. At the higher loads, the data and the model approach asymptotes corresponding to the bulk resistance of the polymers. The dimensionless joint conductance goes to different asymptotes because the bulk resistance is defined by the thickness of the polymer TABLE 4.22 Thickness, Surface Roughness, and Thermophysical Properties of Test Specimens t 0 σ mkEν Material (10 3 m)(10 6 m) (rad) (W/m · K) (GPa) Delrin B 1.88 1.02 0.492 0.38 3.59 0.38 Polycarbonate A 1.99 0.773 0.470 0.22 2.38 0.38 PVC A 1.83 0.650 0.436 0.17 4.14 0.38 Teflon A 1.89 0.622 0.305 0.25 0.135 0.38 Aluminum — 0.511 0.267 183 72.0 0.32 Source: Fuller (2000). BOOKCOMP, Inc. — John Wiley & Sons / Page 374 / 2nd Proofs / Heat Transfer Handbook / Bejan 374 THERMAL SPREADING AND CONTACT RESISTANCES 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 [374], (114) Lines: 4305 to 4321 ——— 0.097pt PgVar ——— Normal Page PgEnds: T E X [374], (114) 10 Ϫ5 10 Ϫ4 10 Ϫ3 10 Ϫ2 10 Ϫ4 10 Ϫ3 10 Ϫ2 10 Ϫ5 2.3P Em p F F t h c,1 h c,2 k p Delrin1; Data Delrin1; Model Delrin2; Data Delrin2; Model Polycarbonate; Data Polycarbonate; Model PVC; Data PVC; Model h km j ␴ s 6 6 6 5 5 5 4 4 4 3 3 3 2 2 2 222333444555666777888 k 1 k 2 Figure 4.35 Dimensionless joint conductance versus dimensionless contact pressure for polymer layers. (From Fuller and Marotta, 2000.) layers. In general, there is acceptable agreement between the vacuum data and the model predictions. 4.17.4 Thermal Greases and Pastes There is much interest today in the use of thermal interface materials (TIMs) such as thermal greases and pastes to enhance joint conductance. Prasher (2001) and Savija et al. (2002a,b) have reviewed the use of TIMs and the models that are available to predict joint conductance. The thermal joint resistance or conductance of a joint formed by two nominally flat rough surfaces filled with grease (Fig. 4.22) depend on several geometric, physical, and thermal parameters. The resistance and conduc- tance relations are obtained from a model that is based on the following simplifying assumptions: • 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/H c ≈ 10 −3 to 10 −5 . • There is plastic deformation of the contacting asperities of the softer solid. BOOKCOMP, Inc. — John Wiley & Sons / Page 375 / 2nd Proofs / Heat Transfer Handbook / Bejan JOINT CONDUCTANCE ENHANCEMENT METHODS 375 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 [375], (115) Lines: 4321 to 4368 ——— 0.89034pt PgVar ——— Normal Page * PgEnds: Eject [375], (115) • Grease is homogeneous, fills the interstitial gaps completely, and wets the bound- ing surfaces perfectly. In general, the joint conductance h j and joint resistance R j depend on the contact and gap components. The joint conductance is modeled as h j = h c + h g (W/m 2 · K) (4.314) where h c represents the contact conductance and h g represents the gap conductance. The joint resistance is modeled as 1 R j = 1 R c + 1 R g (W/K) (4.315) where R c is the contact resistance and R g is the gap resistance. For very light contact pressures it is assumed that h c  h g and R c  R g . The joint conductance and resistance depend on the gap only; therefore, h j = h g and 1 R j = 1 R g where h j = 1 A a R j (4.316) Based on the assumptions given above, the gap conductance is modeled as an equiv- alent layer of thickness t = Y filled with grease having thermal conductivity k g . The joint conductance is given by h j = k g Y (W/m 2 · K) (4.317) The gap parameter Y is the distance between the mean planes passing through the two rough surfaces. This geometric parameter is related to the combined surface roughness σ =  σ 2 1 + σ 2 2 , where σ 1 and σ 2 are the rms surface roughness of the two surfaces and the contact pressure P and effective microhardness of the softer solid, H c . The mean plane separation Y , shown in Fig. 4.22, is given approximately by the simple power law relation (Antonetti, 1983) Y σ = 1.53  P H c  −0.097 (4.318) The power law relation shows that Y/σ is a relatively weak function of the relative contact pressure. Using this relation, the joint conductance may be expressed as h j = k g σ(Y/σ) = k g 1.53σ(P /H c ) −0.097 (W/m 2 · K) (4.319) which shows clearly how the geometric, physical, and thermal parameters influence the joint conductance. The relation for the specific joint resistance is BOOKCOMP, Inc. — John Wiley & Sons / Page 376 / 2nd Proofs / Heat Transfer Handbook / Bejan 376 THERMAL SPREADING AND CONTACT RESISTANCES 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 [376], (116) Lines: 4368 to 4419 ——— 0.86218pt PgVar ——— Normal Page PgEnds: T E X [376], (116) A a R j = 1 h j = 1.53  σ k g  P H c  −0.097 (m 2 · K/W) (4.320) In general, if the metals work-harden, the relative contact pressure P/H c is ob- tained from the relationship P H c =  P c 1 (1.62σ/m) c 2  1/(1+0.071c 2 ) (4.321) where the coefficients c 1 and c 2 are obtained from Vickers microhardness tests. The Vickers microhardness coefficients are related to the Brinell hardness H B for a wide range of metals. The units of σ in the relation above must be micrometers. The units of P and c 1 must be consistent. The approximation of Hegazy (1985) for microhardness is recommended: H c = (12.2 − 3.54H B )  σ m  −0.26 (GPa) (4.322) where H c , the effective contact microhardness, and H B , the Brinell hardness, are in GPa and the effective surface parameter (σ/m) is in micrometers. If the softer metal does not work-harden, H c ≈ H B . Since H B <H c , if we set H c = H B in the specific joint resistance relationship, this will give a lower bound for the joint resistance or an upper bound for the joint conductance. The simple grease model for joint conductance or specific joint resistance was compared against the specific joint resistance data reported by Prasher (2001). The surface roughness parameters of the bounding copper surfaces and the grease thermal conductivities are given in Table 4.23. All tests were conducted at an apparent contact pressure of 1 atm and in a vacuum. Prasher (2001) reported his data as specific joint resistance r j = A a R j versus the parameter σ/k g , where k g is the thermal conductivity of the grease. TABLE 4.23 Surface Roughness and Grease Thermal Conductivity Roughness, Conductivity, σ 1 = σ 2 k g Test (µm) (W/m · K) 1 0.12 3.13 2 1 3.13 3 3.5 3.13 4 1 0.4 5 3.5 0.4 6 3.5 0.25 7 3.5 0.22 Source: Prasher (2001). BOOKCOMP, Inc. — John Wiley & Sons / Page 377 / 2nd Proofs / Heat Transfer Handbook / Bejan JOINT CONDUCTANCE ENHANCEMENT METHODS 377 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 [377], (117) Lines: 4419 to 4431 ——— * 37.927pt PgVar ——— Normal Page PgEnds: T E X [377], (117) 4.17.5 Phase-Change Materials Phase-change materials (PCMs) are being used to reduce thermal joint resistance in microelectronic systems. PCMs may consist of a substrate such as an aluminum foil supporting a PCM such as paraffin. In some applications the paraffin may be filled with solid particles to increase the effective thermal conductivity of the paraffin. At some temperature T m above room temperature, the PCM melts, then flows through the microgaps, expels the air, and then fills the voids completely. After the temperature of the joint falls below T m , the PCM solidifies. Depending on the level of surface roughness, out-of-flatness, and thickness of the PCM, a complex joint is formed. Thermal tests reveal that the specific joint resistance is very small relative to the bare joint resistance with air occupying the microgaps (Fig. 4.36). Because of the complex nature of a joint with a PCM, no simple models are available for the several types of joints that can be formed when a PCM is used. Figure 4.36 Specific joint resistance versus σ/k for grease. (From Prasher, 2001.) BOOKCOMP, Inc. — John Wiley & Sons / Page 378 / 2nd Proofs / Heat Transfer Handbook / Bejan 378 THERMAL SPREADING AND CONTACT RESISTANCES 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 [378], (118) Lines: 4431 to 4609 ——— 0.20558pt PgVar ——— Short Page PgEnds: T E X [378], (118) 4.18 THERMAL RESISTANCE AT BOLTED JOINTS Bolted joints are frequently found in aerospace systems and less often in microelec- tronics systems. The bolted joints are complex because of their geometric configu- rations, the materials used, and the number of bolts and washers used. The pressure distributions near the location of the bolts are not uniform, and the region influenced by the bolts is difficult to predict. A number of papers are available to provide infor- mation on measured thermal resistances and to provide models to predict the thermal resistance under various conditions. Madhusudana (1996) and Johnson (1985) present material on the thermal and mechanical aspects of bolted joints. For bolted joints used in satellite thermal design, the publications of Mantelli and Yovanovich (1996, 1998a, b) are recommended. For bolted joints used in microelectronics cooling, the publications of Lee et al. (1993), Madhusudana et al. (1998) and Song et al. (1992b, 1993a) are recommended. Mikic (1970) describes variable contact pressure effects on joint conductance. NOMENCLATURE Roman Letter Symbols A area, m 2 Fourier coefficient, dimensionless geometric parameter related to radii of curvature, dimensionless A a apparent contact area, m 2 A g effective gap area, m 2 A n coefficient in summation, dimensionless A r real contact area, m 2 a radius of source area, m mean contact spot radius, m semimajor diameter of ellipse, m correlation coefficient, dimensionless radius of circle, m strip half-width, m side dimension of plate, m radius of flat contact, m a e elastic contact radius, m a ep composite elastic–plastic contact radius, m a L thick-layer limit of contact radius, m a p plastic contact radius, m a s thin-layer limit of contact radius, m a ∗ combination of terms, dimensionless BOOKCOMP, Inc. — John Wiley & Sons / Page 379 / 2nd Proofs / Heat Transfer Handbook / Bejan NOMENCLATURE 379 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 [379], (119) Lines: 4609 to 4609 ——— 0.00563pt PgVar ——— Short Page PgEnds: T E X [379], (119) B Fourier coefficient, dimensionless geometric parameter related to radii of curvature, dimensionless Bi Biot modulus, dimensionless Bi(x,y) beta function of arguments x and y, dimensionless B n coefficient in summation, dimensionless b semiminor diameter of ellipse, m side dimension of plate, m radius of compound disk, m correlation coefficient, dimensionless channel half-width, m b 1 radius of cylinder, m C Fourier coefficient, dimensionless correction factor, dimensionless C c contact conductance, dimensionless c correlation coefficient, dimensionless length dimension, m flux channel half-width, m side dimension of isoflux area, m D diameter of sphere, m diameter of circular cylinder, m d length dimension, m uniform gap thickness, m side dimension of isoflux area, m d o reference value for average diagonal, m d v Vickers diagonal, m E modulus of elasticity (Young’s modulus), N/m 2 modulus of elasticity for polymer, N/m 2 complete elliptic integral of second kind, dimensionless E  effective modulus of elasticity, N/m 2 erf(x) error function of argument x, dimensionless erfc(x) complementary error function of argument x, dimensionless F emissivity factor, dimensionless F i factor, dimensionless, i = 1, 2, 3 total normal load on a contact, N F(k  , ψ) incomplete elliptic integral of the first kind of modulus k  and amplitude ψ, dimensionless F ∗ combination of terms, dimensionless F  load per unit cylinder length, N Fo Fourier modulus, dimensionless f ep elastic–plastic parameter, dimensionless f g combination of terms, dimensionless f (u) axisymmetric heat flux distribution, dimensionless G parameter, dimensionless BOOKCOMP, Inc. — John Wiley & Sons / Page 380 / 2nd Proofs / Heat Transfer Handbook / Bejan 380 THERMAL SPREADING AND CONTACT RESISTANCES 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 [380], (120) Lines: 4609 to 4609 ——— 0.0059pt PgVar ——— Normal Page PgEnds: T E X [380], (120) H  effective microhardness, MPa H B Brinnell hardness, MPa bulk microhardness of substrate, MPa H ∗ B adjusted Brinnell hardness, MPa H e effective microhardness, MPa H ep elastic–plastic microhardness, MPa H p microhardness of softer contacting asperities, MPa substrate microhardness, MPa H S microhardness of softer material, MPa H V Vickers microhardness, MPa H 1 microhardness, layer 1, MPa H 2 microhardness, layer 2, MPa h conductance or heat transfer coefficient, W/m 2 · K h  coated joint conductance in vacuum, W/m 2 · K h c contact conductance, W/m 2 · K h g gap conductance, W/m 2 · K h j joint conductance, W/m 2 · K I g gap conductance integral, dimensionless I g, line contact elastogap integral, dimensionless I g,p point contact elastogap integral, dimensionless I o relative layer thickness, dimensionless I q layer thickness conductivity parameter, dimensionless I γ relative layer thickness conductivity parameter, dimensionless K thermal conductivity parameter, dimensionless complete elliptic integral of first kind, dimensionless Kn Knudsen number, dimensionless k modulus related to the ellipticity, dimensionless thermal conductivity, W/m · K k g effective gas thermal conductivity, W/m · K grease thermal conductivity, W/m · K k g,∞ gas thermal conductivity under continuum conditions, W/m ·K k s harmonic mean thermal conductivity of a joint, W/m · K k 1 layer 1 thermal conductivity, W/m · K k 2 layer 2 thermal conductivity, W/m · K L strip length, m L relative contact size, dimensionless  point in flux tube where flux lines are parallel, m L length scale, m M gas rarefaction parameter, m M g molecular weight of gas, g-mol M s molecular weight of solid, g-mol m counter, dimensionless Hertz elastic parameter, dimensionless absolute asperity slope, dimensionless . Proofs / Heat Transfer Handbook / Bejan JOINT CONDUCTANCE ENHANCEMENT METHODS 371 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 [371],. Proofs / Heat Transfer Handbook / Bejan 372 THERMAL SPREADING AND CONTACT RESISTANCES 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 [372],. Proofs / Heat Transfer Handbook / Bejan JOINT CONDUCTANCE ENHANCEMENT METHODS 373 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 [373],

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