Adrian Bejan(Editor), Allan D. Kraus (Editor). Heat transfer Handbok (776115), страница 34
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If heat enters the half-space, the flux0.60939ptPgVarlines spread apart as the heat is conducted away from the small source area (Fig. 4.2);———then the thermal resistance is called spreading resistance.Long PageIf the heat leaves the half-space through a small area, the flux lines are constricted* PgEnds: Ejectand the thermal resistance is called constriction resistance. The heat transfer may besteady or transient. The temperature field T in the half-space is, in general, threedimensional, and steady or transient.
The temperature in the source area may be two[271], (11)dimensional, and steady or transient.If heat transfer is into the half-space, the spreading resistance is defined as (Carslaw and Jaeger, 1959; Yovanovich, 1976c; Madhusudana, 1996; Yovanovich andAntonetti, 1988)Rs =T source − T sinkQ(K/W)(4.16)where T source is the source temperature and T sink is a convenient thermal sink temperature; and where Q is the steady or transient heat transfer rate:∂Tqn dA =−kQ=dA(W)(4.17)∂nAAwhere qn is the heat flux component normal to the area and ∂T /∂n is the temperaturegradient normal to the area.
If the heat flux distribution is uniform over the area,Q = qA. For singly and doubly connected source areas, three source temperatureshave been used in the definition: maximum temperature, centroid temperature, andarea-averaged temperature, which is defined according to Yovanovich (1976c) as1T source =T source dA(K)(4.18)AABOOKCOMP, Inc. — John Wiley & Sons / Page 271 / 2nd Proofs / Heat Transfer Handbook / Bejan272123456789101112131415161718192021222324252627282930313233343536373839404142434445THERMAL SPREADING AND CONTACT RESISTANCESwhere A is the source area. Because the sink area is much larger than the source area,it is, by convention, assumed to be isothermal (i.e., T sink = T∞ ).
The maximum andcentroid temperatures are identical for singly connected axisymmetric source areas;otherwise, they are different (Yovanovich, 1976c; Yovanovich and Burde, 1977);Yovanovich et al., 1977). For doubly connected source areas (e.g., circular annulus),the area-averaged source temperature is used (Yovanovich and Schneider, 1977). Ifthe source area is assumed to be isothermal, T source = T0 .The general definition of spreading (or constriction) resistance leads to the following relationship for the dimensionless spreading resistance:θ dALA(4.19)k LRs = A − (∂θ/∂z)z=0 dAAwhere θ = T (x,y) − T∞ , the rise of the source temperature above the sink temperature.
The arbitrary characteristic length scale of the source area is denoted asL . For convenience the dimensionless spreading resistance, denoted as ψ = k L Rs(Yovanovich, 1976c; Yovanovich and Antonetti, 1988), is called the spreading resistance parameter. This parameter depends on the heat flux distribution over the sourcearea and the shape and aspect ratio of the singly or doubly connected source area. Thespreading resistance definition holds for transient conduction into or out of the halfspace.
If the heat flux is uniform over the source area, the temperature is nonuniform;and if the temperature of the source area is uniform, the heat flux is nonuniform(Carslaw and Jaeger, 1959; Yovanovich, 1976c). The relation for the dimensionlessspreading resistance is mathematically identical to the dimensionless constriction resistance for identical boundary conditions on the source area. For a nonisothermalsingly connected area the spreading resistance can also be defined with respect to itsmaximum temperature or the temperature at its centroid (Carslaw and Jaeger, 1959;Yovanovich, 1976c; Yovanovich and Burde, 1977).
These temperatures, in general,are not identical and they are greater than the area-averaged temperature (Yovanovichand Burde, 1977).The definition of spreading resistance for the isotropic half-space is applicable forsingle and multiply isotropic layers which are placed in perfect thermal contact withthe half-space, and the heat that leaves the source area is conducted through the layeror layers before entering into the half-space. The conductance h cannot be definedfor the half-space problem because the corresponding area is not defined.4.2.2 Spreading and Constriction Resistances in Flux Tubesand ChannelsIf a circular heat source of area As is in contact with a very long circular flux tubeof cross-sectional area At (Fig.
4.3), the flux lines are constrained by the adiabaticsides to “bend” and then become parallel to the axis of the flux tube at some distancez = " from the contact plane at z = 0. The isotherms, shown as dashed lines,√areeverywhere orthogonal to the flux lines.
The temperature in planes z = " AtBOOKCOMP, Inc. — John Wiley & Sons / Page 272 / 2nd Proofs / Heat Transfer Handbook / Bejan[272], (12)Lines: 574 to 603———1.02235pt PgVar———Normal PagePgEnds: TEX[272], (12)DEFINITIONS OF SPREADING AND CONSTRICTION RESISTANCES123456789101112131415161718192021222324252627282930313233343536373839404142434445273Heat Source(Contact Area, As )z=0zz= "Flux Tube or Channel[273], (13)AtLines: 603 to 635Figure 4.3 Heat flow lines and isotherms for steady conduction from a finite heat source intoa flux tube or channel.
(From Yovanovich and Antonetti, 1988.)“far” from the contact plane z = 0 becomes isothermal, while the temperature inplanes near z = 0 are two- or three-dimensional. The thermal conductivity of theflux tube is assumed to be constant.The total thermal resistance R total for steady conduction from the heat source areain z = 0 to the arbitrary plane z = " is given by the relationshipQR total = T s − T z="(K)(4.20)where T s is the mean source temperature and T z=" is the mean temperature of thearbitrary plane. The one-dimensional resistance of the region bounded by z = 0 andz = " is given by the relationQR1D = T z=0 − T z="(K)(4.21)The total resistance is equal to the sum of the one-dimensional resistance and thespreading resistance:R total = R1D + RsorR total − R1D = Rs(K)(4.22)By substraction, the relationship for the spreading resistance, proposed by Mikic andRohsenow (1966), isRs =T s − T z=0QBOOKCOMP, Inc.
— John Wiley & Sons / Page 273 / 2nd Proofs / Heat Transfer Handbook / Bejan(K/W)(4.23)———0.57106pt PgVar———Normal Page* PgEnds: Eject[273], (13)274123456789101112131415161718192021222324252627282930313233343536373839404142434445THERMAL SPREADING AND CONTACT RESISTANCESwhere Q is the total heat transfer rate from the source area into the flux tube. It isgiven by∂T Q=−kdAs(W)(4.24)∂z z=0AsThe dimensionless spreading resistance parameter ψ = k LRs is introduced forconvenience. The arbitrary length scale L is related to some dimension of the sourcearea. In general, ψ depends on the shape and aspect ratio of the source area, theshape and aspect ratio of the flux tube cross section, the relative size of the sourcearea, the orientation of the source area relative to the cross section of the flux tube,the boundary condition on the source area, and the temperature basis for definition ofthe spreading resistance.The definitions given above are applicable to singly and doubly connected sourceareas; however, As /At < 1 in all cases.
The source area and flux tube cross-sectionalarea may be circular, square, elliptical, rectangular, or any other shape. The heat fluxand temperature on the source area may be uniform and constant. In general, bothheat flux and temperature on the source area are nonuniform. Numerous examplesare presented in subsequent sections.4.3 SPREADING AND CONSTRICTION RESISTANCESIN AN ISOTROPIC HALF-SPACE4.3.1IntroductionSteady or transient heat transfer occurs in a half-space z > 0 which may be isotropicor may consist of one or more thin isotropic layers bonded to the isotropic half-space.The heat source is some planar singly or doubly connected area such as a circularannulus located in the “free” surface z = 0 of the half-space. The dimensions of thehalf-space are much larger than the largest dimension of the source area.
The “free”surface z = 0 of the half-space outside the source area is adiabatic. If the sourcearea is isothermal, the heat flux over the source area is nonuniform. If the source issubjected to a uniform heat flux, the source area is nonisothermal.4.3.2Circular Area on a Half-SpaceThere are two classical steady-state solutions available for the circular source area ofradius a on the surface of a half-space of thermal conductivity k. The solutions arefor the isothermal and isoflux source areas.
In both problems the temperature field istwo-dimensional in circular-cylinder coordinates [i.e., θ(r, z)]. The important resultsare presented here.Isothermal Circular Source In this problem the mixed-boundary conditions(Sneddon, 1966) in the free surface areBOOKCOMP, Inc. — John Wiley & Sons / Page 274 / 2nd Proofs / Heat Transfer Handbook / Bejan[274], (14)Lines: 635 to 664———1.00616pt PgVar———Normal Page* PgEnds: Eject[274], (14)SPREADING AND CONSTRICTION RESISTANCES IN AN ISOTROPIC HALF-SPACE123456789101112131415161718192021222324252627282930313233343536373839404142434445z=00≤r<aθ = θ00r>a∂θ=0∂z275(4.25)√and the condition at remote points is: As r 2 + z2 → ∞, then θ → 0. Thetemperature distribution throughout the half-space z ≥ 0 is given by the infiniteintegral (Carslaw and Jaeger, 1959)θ=2θ0π∞e−λz J0 (λr) sin λa0dλλ(K)(4.26)where J0 (x) is the Bessel function of the first kind of order zero (Abramowitz andStegun, 1965) and λ is a dummy variable.
The solution can be written in the followingalternative form according to Carslaw and Jaeger (1959):θ=22aθ0 sin−1 22π(r − a) + z + (r + a)2 + z2(K)(4.27)[275], (15)Lines: 664 to 720———1.70782ptThe heat flow rate from the isothermal circular source into the half-space is found———fromNormal Page a∂θ* PgEnds: EjectQ=−k2πr dr∂z z=00 ∞dλ[275], (15)= 4kaθ0J1 (λa) sin λaλ0= 4kaθ0(W)(4.28)From the definition of spreading resistance one finds the relationship for thespreading resistance (Carslaw and Jaeger, 1959):Rs =1θ0=Q4ka(K/W)(4.29)The heat flux distribution over the isothermal heat source area is axisymmetric (Carslaw and Jaeger, 1959):q(r) =Q12πa 2 1 − (r/a)20≤r<a(W/m2)(4.30)This flux distribution is minimum at the centroid r = 0 and becomes unbounded atthe edge r = a.Isoflux Circular Sourcesurface areIn this problem the boundary conditions in the freeBOOKCOMP, Inc.
— John Wiley & Sons / Page 275 / 2nd Proofs / Heat Transfer Handbook / BejanPgVar276123456789101112131415161718192021222324252627282930313233343536373839404142434445THERMAL SPREADING AND CONTACT RESISTANCESz=00q0∂θ=−∂zk∂θ=0∂z0≤r<ar>a(4.31)where q0 = Q/πa 2 is the uniform heat flux. The condition at remote points isidentical.
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