Adrian Bejan(Editor), Allan D. Kraus (Editor). Heat transfer Handbok (776115), страница 32
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— John Wiley & Sons / Page 263 / 2nd Proofs / Heat Transfer Handbook / Bejan[263], (3)Lines: 195 to 294———2.0pt PgVar———Long PagePgEnds: TEX[263], (3)2641234567891011121314151617181920212223242526272829303132333435363738394041424344454.1THERMAL SPREADING AND CONTACT RESISTANCESINTRODUCTIONWhen two solids are joined, imperfect joints (interfaces) are formed.
The imperfectjoints occur because “real” surfaces are not perfectly smooth and flat. A mechanicaljoint consists of numerous discrete microcontacts that may be distributed in a randompattern over the apparent contact area if the contacting solids are nominally flat (conforming) and rough, or they may be distributed over a certain portion of the apparentcontact area, called the contour area, if the contacting solids are nonconforming andrough.
The contact spot size and density depend on surface roughness parameters,physical properties of the contacting asperities, and the apparent contact pressure. Thedistribution of the contact spots over the apparent contact area depends on the localout-of-flatness of the two solids, their elastic or plastic or elastic–plastic properties,and the mechanical load. Microgaps and macrogaps appear whenever there is absenceof solid-to-solid contact.
The microgaps and macrogaps are frequently occupied by athird substance, such as gas (e.g., air), liquid (e.g., oil, water), or grease, whose thermal conductivities are frequently much smaller than those of the contacting solids.The joint formed by explosive bonding may appear to be perfect because thereis metal-to-metal contact at all points in the interface that are not perfectly flat andperpendicular to the local heat flux vector.
When two metals are brazed, soldered,or welded, a joint is formed that has a small but finite thickness and it consists of acomplex alloy whose thermal conductivity is lower than that of the joined metals. Acomplex joint is formed when the solids are bonded or epoxied.As a result of the “imperfect” joint, whenever heat is transferred across the joint,there is a measurable temperature drop across the joint that is related directly to thejoint resistance and the heat transfer rate.There are several review articles by Fletcher (1972, 1988, 1990), Kraus and BarCohen (1983), Madhusudana and Fletcher (1986), Yovanovich (1986, 1991), Madhusudana (1996), Lambert and Fletcher (1996), and Yovanovich and Antonetti (1988),that should be consulted for details of thermal joint resistance and conductance ofdifferent types of joints.4.1.1 Types of Joints or InterfacesSeveral definitions are required to define heat transfer across joints (interfaces)formed by two solids that are brought together under a static mechanical load.
Theheat transfer across the joint is frequently related to contact resistances or contactconductances and the effective temperature drop across the joint (interface). The definitions are based on the type of joint (interface), which depends on the macro- andmicrogeometry of the contacting solids, the physical properties of the substrate andthe contacting asperities, and the applied load or apparent contact pressure.Figure 4.1 illustrates six types of joints that are characterized by whether thecontacting surfaces are smooth and nonconforming (Fig. 4.1a), rough and conforming(nominally flat) (Fig. 4.1c), or rough and nonconforming (Fig.
4.1b). One or morelayers may also be present in the joint, as shown in Fig. 4.1d–f.If the contacting solids are nonconforming (e.g., convex solids) and their surfacesare smooth (Fig. 4.1a and d), the joint will consist of a single macrocontact and aBOOKCOMP, Inc. — John Wiley & Sons / Page 264 / 2nd Proofs / Heat Transfer Handbook / Bejan[264], (4)Lines: 294 to 321———0.0pt PgVar———Long PagePgEnds: TEX[264], (4)INTRODUCTION123456789101112131415161718192021222324252627282930313233343536373839404142434445265[265], (5)Lines: 321 to 323———-0.903pt PgVar———Long PagePgEnds: TEX[265], (5)Figure 4.1 Six types of joints.macrogap. The macrocontact may be formed by elastic, plastic, or elastic–plasticdeformation of the substrate (bulk).
The presence of a single “layer” will alter thenature of the joint according to its physical and thermal properties relative to thoseof the contacting solids. Thermomechanical models are available for finding the jointresistance of these types of joints.The surfaces of the solids may be conforming (nominally flat) and rough (Fig.4.1c and f ). Under a static load, elastic, plastic, or elastic–plastic deformation of thecontacting surface asperities occurs. The joint (interface) is characterized by manyBOOKCOMP, Inc. — John Wiley & Sons / Page 265 / 2nd Proofs / Heat Transfer Handbook / Bejan266123456789101112131415161718192021222324252627282930313233343536373839404142434445THERMAL SPREADING AND CONTACT RESISTANCESdiscrete microcontacts with associated microgaps that are more or less uniformlydistributed in the apparent (nominal) contact area.
The sum of the microcontactareas, called the real area of contact, is a small fraction of the apparent contactarea. Thermomechanical models are available for obtaining the contact, gap, and jointconductances (or resistances) of these types of joints.A third type of joint is formed when nonconforming solids with surface roughnesson one or both solids (Fig. 4.1b and e) are brought together under load. In thismore complex case the microcontacts with associated microgaps are formed in aregion called the contour area, which is some fraction of the apparent contact area.The substrate may undergo elastic, plastic, or elastic–plastic deformation, while themicrocontacts may experience elastic, plastic, or elastic–plastic deformation.
A fewthermomechanical models have been developed for this type of joint.The substance in the microgaps and macrogaps may be a gas (air, helium, etc.), a[266], (6)liquid (water, oil, etc.), grease, or some compound that consists of grease filled withmany micrometer-sized solid particles (zinc oxide, etc.) that increase its effective thermal conductivity and alter its rheology. The interstitial substance is assumed to wetLines: 323 to 354the surfaces of the bounding solids completely, and its effective thermal conductivityis assumed to be isotropic.———If one (or more) layers are present in the joint, the contact problem is much more-1.7679pt PgVarcomplex and the associated mechanical and thermal problems are more difficult to———model because the layer thickness, and its physical and thermal properties and surfaceLong Pagecharacteristics, must be taken into account.* PgEnds: EjectThe total (joint) heat transfer rate across the interface may take place by conductionthrough the microcontacts, conduction through the interstitial substance, and radiation across the microgaps and macrogaps if the interstitial substance is transparent to[266], (6)radiation.
Definitions of thermal contact, gap, and joint resistances and contact, gap,and joint conductances for several types of joints are given below.4.1.2Conforming Rough SolidsIf the solids are conforming and their surfaces are rough (Fig. 4.1c and f ), heat transferacross the joint (interface) occurs by conduction through the contacting microcontactsand through the microgap substance and by radiation across the microgap if thesubstance is transparent (e.g., dry air).
The total or joint heat transfer rate Qj , ingeneral, is the sum of three separate heat transfer rates:Qj = Qc + Qg + Qr(W)(4.1)where Qj , Qc , Qg , and Qr represent the joint, contact, gap, and radiative heat transferrates, respectively. The heat transfer rates are generally coupled in some complexmanner; however, in many important problems, the coupling is relatively weak. Thejoint heat transfer rate is related to the effective temperature drop across the joint∆Tj , nominal contact area Aa , joint resistance Rj , and joint conductance hj by thedefinitionsQj = hj Aa ∆TjandBOOKCOMP, Inc. — John Wiley & Sons / Page 266 / 2nd Proofs / Heat Transfer Handbook / BejanQj =∆TjRj(W)(4.2)INTRODUCTION123456789101112131415161718192021222324252627282930313233343536373839404142434445267These definitions result in the following relationships between joint conductance andjoint resistance:hj =1Aa Rj(W/m2 · K)andRj =1Aa hj(K/W)(4.3)The component heat transfer rates are defined by the relationshipsQc = hc Aa ∆TjQg = hg Ag ∆Tj ,Qr = hr Ag ∆Tj(4.4)which are all based on the effective joint temperature drop ∆Tj and their respectiveheat transfer areas: Aa and Ag , the apparent and gap areas, respectively.
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