Adrian Bejan(Editor), Allan D. Kraus (Editor). Heat transfer Handbok (776115), страница 33
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It is the convention to use the apparent contact area in the definition of the contact conductance.Since Ag = Aa − Ac and Ac /Aa 1, then Ag ≈ Aa . Finally, using the relationshipsgiven above, one can write the following relationships between the resistances andthe conductances:Lines: 354 to 4051111=++RjRcRgRr(W/K)hj = hc + hg + hr(W/m2 · K)(4.5)(4.6)111=+RjRcRg(W/K)(4.7)hj = hc + hg(W/m2 · K)(4.8)For joints (interfaces) placed in a vacuum where is no substance in the microgaps,Rg → ∞ and hg → 0 and the relationships becomehj = hc + hr(W/K)(W/m2 · K)(4.9)(4.10)In all cases there is heat transfer through the contacting asperities and hc and Rcare present in the relationships.
This heat transfer path is therefore very important.For most applications where the joint (interface) temperature level is below 600°C,radiation heat transfer becomes negligible, and therefore it is frequently ignored.4.1.3Nonconforming Smooth SolidsIf two smooth, nonconforming solids are in contact (Fig. 4.1a and d), heat transferacross the joint can be described by the relationships given in earlier sections. TheBOOKCOMP, Inc. — John Wiley & Sons / Page 267 / 2nd Proofs / Heat Transfer Handbook / Bejan———5.29628pt PgVarIf the gap substance is opaque, then Rr → ∞ and hr → 0, and the relationshipsreduce to111=+RjRcRr[267], (7)———Long PagePgEnds: TEX[267], (7)268123456789101112131415161718192021222324252627282930313233343536373839404142434445THERMAL SPREADING AND CONTACT RESISTANCESradiative path becomes more complex because the enclosure and its radiative properties must be considered.
If the apparent contact area is difficult to define, the use ofconductances should be avoided and resistances should be used. The joint resistance,neglecting radiation, is111=+RjRcRg4.1.4(W/K)(4.11)Nonconforming Rough SolidsIf two rough, nonconforming solids make contact (Fig. 4.1b and e), heat transferacross the joint is much more complex when a substance “fills” the microgaps associated with the microcontacts and the macrogap associated with the contour area.The joint resistance, neglecting radiative heat transfer, is defined by the relationship111=+RjRma,c + (1/Rmi,c + 1/Rmi,g )−1Rma,g(W/K)[268], (8)(4.12)Lines: 405 to 447———where the component resistances are Rmi,c and Rmi,g , the microcontact and microgap0.31018ptresistances, respectively, and Rma,c and Rma,g , the macrocontact and macrogap resis———tances, respectively.
If there is no interstitial substance in the microgaps and macroLong Pagegap, and the contact is in a vacuum, the joint resistance (neglecting radiation) consistsof the macro and micro resistances in series:* PgEnds: EjectRj = Rma,c + Rmi,c4.1.5(K/W)(4.13)Single Layer between Two Conforming Rough SolidsIf a single thin metallic or nonmetallic layer of uniform thickness is placed betweenthe contacting rough solids, the mechanical and thermal problems become morecomplex.
The layer thickness, thermal conductivity, and physical properties must alsobe included in the development of joint resistance (conductance) models. There arenow two interfaces formed, which are generally different.The presence of the layer can increase or decrease the joint resistance, dependingon several geometric, physical, and thermal parameters. A thin isotropic silver layerbonded to one of the solids can decrease the joint resistance because the layer isrelatively soft and has a high thermal conductivity. On the other hand, a relativelythick oxide coating, which is hard and has low thermal conductivity, can increasethe joint resistance.
The joint resistance, neglecting radiation, is given by the generalrelationshipRj =1Rmi,c1+1Rmi,g1−1+ R layer +1Rmi,c2+1Rmi,g2−1(K/W)(4.14)where Rmi,c1 , Rmi,g1 and Rmi,c2 , Rmi,g2 are the microcontact and microgap resistancesat the two interfaces formed by the two solids, which are separated by the layer. Thethermal resistance of the layer is modeled asBOOKCOMP, Inc. — John Wiley & Sons / Page 268 / 2nd Proofs / Heat Transfer Handbook / Bejan[268], (8)PgVar269INTRODUCTION123456789101112131415161718192021222324252627282930313233343536373839404142434445R layer =tk layerAa(K/W)(4.15)where t is the layer thickness under loading conditions. Except for very soft metals(e.g., indium, lead, tin) at or above room temperature, the layer thickness under loadconditions is close to the thickness before loading.
If the layers are nonmetallic, suchas elastomers, the thickness under load may be smaller than the preload thickness andelastic compression should be included in the mechanical model.To develop thermal models for the component resistances, it is necessary to consider single contacts on a half-space and on semi-infinite flux tubes and to find relations for the spreading–constriction resistances.4.1.6Parameters Influencing Contact Resistance or ConductanceReal surfaces are not perfectly smooth (specially prepared surfaces such as thosefound in ball and roller bearings can be considered to be almost ideal surfaces) butconsist of microscopic peaks and valleys.
Whenever two real surfaces are placedin contact, intimate solid-to-solid contact occurs only at discrete parts of the joint(interface) and the real contact area will represent a very small fraction (< 2%) of thenominal contact area. The real joint (interface) is characterized by several importantfactors:• Intimate contact occurs at numerous discrete parts of the nominal contact area.• The ratio of the real contact area to the nominal contact area is usually much lessthan 2%.• The pressure at the real contact area is much greater than the apparent contactpressure.
The real contact pressure is related to the flow pressure of the contactingasperities.• A very thin gap exists in the regions in which there is no solid–solid contact, andit is usually occupied by a third substance.• The third substance can be air, other gases, liquid, grease, grease filled with verysmall solid particles, and another metallic or nonmetallic substance.• The joint (interface) is idealized as a line; however, the actual “thickness” of thejoint (interface) ranges from 0.5 µm for very smooth surfaces to about 60 to 80µm for very rough surfaces.• Heat transfer across the interface can take place by conduction through the realcontact area, by conduction through the substance in the gap, or by radiationacross the gap if the substance in the gap is transparent to radiation or if the gapis under a vacuum.
All three modes of heat transfer may occur simultaneously;but usually, they occur in pairs, with solid–solid conduction always present.The process of heat transfer across a joint (interface) is complex because the jointresistance may depend on many geometrical, thermal, and mechanical parameters, ofwhich the following are very important:BOOKCOMP, Inc. — John Wiley & Sons / Page 269 / 2nd Proofs / Heat Transfer Handbook / Bejan[269], (9)Lines: 447 to 485———14.35603pt PgVar———Long PagePgEnds: TEX[269], (9)270123456789101112131415161718192021222324252627282930313233343536373839404142434445THERMAL SPREADING AND CONTACT RESISTANCES• Geometry of the contacting solids (surface roughness, asperity slope, and outof-flatness or waviness)• Thickness of the gap (noncontact region)• Type of interstitial fluid (gas, liquid, grease, or vacuum)• Interstitial gas pressure• Thermal conductivities of the contacting solids and the interstitial substance• Microhardness or flow pressure of the contacting asperities (plastic deformationof the highest peaks of the softer solid)• Modulus of elasticity and Poisson’s ratio of the contacting solids (elastic deformation of the wavy parts of the joint)• Average temperature of the joint influences radiation heat transfer as well as thethermophysical properties• Load or apparent contact pressure[270], (10)4.1.7 Assumptions for Resistance and Conductance ModelDevelopmentLines: 485 to 548Because thermal contact resistance is such a complex problem, it is necessary todevelop simple thermophysical models that can be analyzed and verified experimentally.
To achieve these goals the following assumptions have been made in the development of the several contact resistance models, which will be discussed later:-4.33pt PgVar• Contacting solids are isotropic: thermal conductivity and physical parameters areconstant.• Contacting solids are thick relative to the roughness or waviness.• Surfaces are clean: no oxide effect.• Contact is static: no vibration effects.• First loading cycle only: no hysteresis effect.• Relative apparent contact pressure (P /Hp for plastic deformation and P /He forelastic deformation) is neither too small (> 10−6 ) nor too large (< 10−1 ).• Radiation is small or negligible.• Heat flux at microcontacts is steady and not too large (< 107 W/m2).• Contact is in a vacuum or the interstitial fluid can be considered to be a continuumif it is not a gas.• Interstitial fluid perfectly wets both contacting solids.4.2 DEFINITIONS OF SPREADING AND CONSTRICTIONRESISTANCES4.2.1Spreading and Constriction Resistances in a Half-SpaceHeat may enter or leave an isotropic half-space (a region whose dimensions are muchlarger than the characteristic length of the heat source area) through planar singly orBOOKCOMP, Inc.
— John Wiley & Sons / Page 270 / 2nd Proofs / Heat Transfer Handbook / Bejan——————Long PagePgEnds: TEX[270], (10)DEFINITIONS OF SPREADING AND CONSTRICTION RESISTANCES123456789101112131415161718192021222324252627282930313233343536373839404142434445271Figure 4.2 Heat flow lines and isotherms for steady conduction from a finite heat source intoa half-space. (From Yovanovich and Antonetti, 1988.)[271], (11)Lines: 548 to 574doubly connected areas (e.g., circular or annular area). The free surface of the half———space is adiabatic except for the source area.
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