Rohsenow W., Hartnett J., Young Cho. Handbook of Heat Transfer (776121), страница 28
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The three definitions of dimensionless temperature are presented below.The three dimensionless solutions [11] are:Dirichlet Solutionlerfc/2 /p-1¢=pw h e r e , = ( T(r, t ) - Ti)/( To - Ti).3.30CHAPTERTHREENeumann Solution, = P erfc- P exp(p - 1 + Fo) erfc+ x/-FTo(3.94)w h e r e , = k(T(r, t) - Ti)/(aqo).Robin SolutionBi1~- Bi+lperfc( p - 1p-1Bi i exp[(Bi + 1)(p - 1) + (Bi + 1)2 Fo] erfc 2X/~o + (Bi + 1)%/-F--oo] (3.95)Bi+l pwhere ~ = ( T(r, t ) - Ti)/( T I - Ti).Instantaneous Surface Temperature and Heat Flux. The previous solutions give the following important results for the instantaneous surface temperature and surface heat flux.Dirichlet Condition.The instantaneous surface heat flux is given byaq(a, t)1k ( T o - Ti) = 1 + V~-----------~~o(3.96)The instantaneous surface temperature is given byNeumann Condition.k(T(a, t ) - Ti)= 1 - e F° erfc (~/-~o)aqo(3.97)Robin Condition.
The Robin solution given above yields expressions for the instantaneoussurface temperature and the instantaneous surface heat flux. They are as follows:(T(a, t ) - Ti)Bi11 - e (B'÷1)2F° erfc [(Bi + 1)v/-F--oo]lBi+l(r~- r/)andaq(a, t) _ Bi l1 + e (B~÷1)2F° erfc [(Bi + 1)X/~ollk ( T I - Ti) B i + l(3.98)(3.99)Instantaneous Thermal ResistanceResistance Definition.
The instantaneous thermal resistance for the three boundary conditions is defined as R = (T(a, t) - Ti)/Q where Q = q(a, t)4r~a2. The results given above yield thefollowing expressions.Dirichlet Condition Resistance14rckaRo = [1 + tvr~vro)l'l/"/--'~'"(3.100)Neumann Condition Resistance4~kaRN = 1 - eF° erfc (X~o)(3.101)CONDUCTION AND THERMAL CONTACTRESISTANCES(CONDUCTANCES)3.31Robin Condition Resistance1 - e z2 erfc4 r t k a R R = 1 + e z2 erfc(z)(z)(3.102)where z = (Bi + 1)V~o.The three previous expressions approach the steady-state result 4 r t k a R = 1 for largedimensionless time.The previous expression for the Robin condition can be calculated by the following rational approximation with a maximum error of less than 1.2 percent:1 - a~s - a2 $2 - a3 $34 r t k a R R = 1 + a l s + a2s 2 + a3 $3(3.103)where s = 1/(1 + p z ) and the coefficients are al = 0.3480242, a2 = -0.0958798, a3 = 0.7478556,p = 0.47047.The three solutions corresponding to the three boundary conditions can be used to obtainapproximate solutions for other convex bodies, such as a cube, for which there are no analytical solutions available.
The dimensionless parameters Bi and Fo are defined with respect to theequivalent sphere radius, which is obtained by setting the surface area of the sphere equal tothe surface area of the given body, i.e., a = VA/(4r0. This will be considered in the followingsection, which covers transient external conduction from isothermal convex bodies.Transient External Conduction From Isothermal Convex BodiesExternal transient conduction from an isothermal convex body into a surrounding space hasbeen solved numerically (Yovanovich et al. [149]) for several axisymmetric bodies: circulardisks, oblate and prolate spheroids, and cuboids such as square disks, cubes, and tall squarecuboids (Fig. 3.10).
The sphere has a complete analytical solution [11] that is applicable forall dimensionless times Fov~ - m / A . T h e dimensionless instantaneous heat transfer rate isQ~/-~a = Q V / - A / ( k A O o ) , where k is the thermal conductivity of the surrounding space, A is thetotal area of the convex body, and 00- To- Ti is the temperature excess of the body relativeto the initial temperature of the surrounding space. The analytical solution for the sphere isgiven byQ~/-~A= 2X/-~ +1(3.104)which consists of the linear superposition of the steady-state solution (dimensionless shapefactor) and the small-time solution (half-space solution). This observation was used to propose a simple approximate solution for all body shapes of the form1Q~AA= S~AA+(3.105)where S~A is the dimensionless shape factor for isothermal convex bodies in full space.
Thisparameter is a relatively weak function of shape and aspect ratio; its values lie in the range3.192 <_S~A < 4.195 for the wide range of bodies examined (Fig. 3.10). The simple model predicts values with RMS and maximum differences between the predicted and numerical valuesin the ranges 0.40-6.31 percent and 0.98-11.52 percent, respectively, as shown in Table 3.9. Themaximum differences lie in the intermediate range of dimensionless time: 10-3 < Fox/-x < 10-1.A more accurate model based on the method of Churchill and Usagi [13] was proposed forall bodies:[(1Q ~ = (s~)" + x/~ x/-F--~)nll/n(3.106)OBLATE SPHEROID(AR=0.5)SPHERE(An=l)PROLATE SPHEROID(AR=1.93)PROLATE SPHEROID(AR=IO)CIRCULAR DISKRECTANGULAR STRIP(AR=O)(AR=O)SQUARE DISK(AR=0.1)CUBE(AR=I)TALL CUBOID(AR=2)TALL CUBOID(AR=10)FIGURE 3.10 Various convex bodies.I~"r' ....."I.......'I......"I......."I........,~"I........,-7~0O:~ _ Q~/A"%.......0xsnherespin__,,(~"I......."I°~o~p~o~.voblam (AR=0.5)O .
. . . . .1 . I " ," ' , , ' ~,_-=-~f~; ~,v;7,~ " , , - ~ .,oI"' ' .....'phcr¢(AR=I)"......pro,=(AR=,O),~,~.~o!@*.. •v. ~ % •v 'o"~'=Q~ = 2V'~ +--li--,~#-~10°....... a ....... -, . . . . . . a . . . . . ,~ . . . . . . .a ........ , ....... a .......
., ........10 "610"s10 "410 "a10.210'10°10'10a10aFOcx =FIGURE 3.113.32(~)2Comparison of numerical results and proposed model.CONDUCTION AND THERMAL CONTACT RESISTANCES (CONDUCTANCES)TABLE 3.93.33Comparison of Superposition Model and Numerical ResultsI3odyS~Max. % diff.RMS % diff.Circular disk (AR = 0)Rectangular strip (AR = 0)Square disk (AR = 0.1)Cube (AR = 1)Tall cuboid (AR = 2)Oblate spheroid (AR = 0.5)Prolate spheroid (AR = 1.93)Tall cuboid (AR = 10)Prolate spheroid (AR = 10)3.1923.3033.3433.3883.4063.5293.5643.9454.1957.054.992.603.782.770.981.653.1511.523.362.941.912.521.870.400.731.636.31T h e values of the p a r a m e t e r n that reduce the R M S and m a x i m u m differences b e t w e e n them o d e l and the numerical data w e r e found by trial and e r r o r to lie in the range 0.87 < n < 1.10for R M S and m a x i m u m differences less than 1.34 percent and 2.06 percent, respectively.
Thelargest values of n were required for the thin bodies and the smallest values w e r e r e q u i r e d forthe tallest bodies. For bodies having aspect ratios close to unity, the values of n w e r e found tolie close to unity like the sphere. The values of n, and the c o r r e s p o n d i n g values of the maxim u m and R M S percent differences for the various bodies are given in Table 3.10.
The n u m e r ical data and the m o d e l predictions are shown in Fig. 3.11.Table 3.11 gives r e c o m m e n d e d values of n for axisymmetric bodies (spheroids) andcuboids for various aspect ratios.TABLE 3.10Comparison of Blended Model and Numerical ResultsBodynMax. % diff.RMS % diff.Circular disk (AR = 0)Rectangular strip (AR = 0)Square disk (AR = 0.1)Cube (AR = 1)Tall cuboid (AR - 2)Oblate spheroid (AR = 0.5)Prolate spheroid (AR - 1.93)Tall cuboid (AR - 10)Prolate spheroid (AR = 10)1.101.071.051.051.030.9940.990.960.871.831.281.441.951.650.821.592.082.060.800.620.671.040.810.360.531.131.34TABLE 3.11 Blending Parameter and RecommendationsBody shapeAspect ratioF/SpheroidsThin disksOblates and prolatesTall prolatesAR -- 00.5 < AR < 2AR >> 21.11.00.9CuboidsThin disksDisks and cubesTall cuboidsSquare cylindersAR =00.1 < AR < 1AR ---2AR >> 21.071.051.030.963.34CHAPTER THREESPREADING (CONSTRICTION) RESISTANCEIntroductionSpreading (constriction) resistance is an important thermal parameter that depends on several factors such as (1) geometry (singly or doubly connected areas, shape, aspect ratio),(2) domain (half-space, flux tube), (3) boundary condition (Dirchlet, Neumann, Robin), and(4) time (steady-state, transient).
The results are presented in the form of infinite series andintegrals that can be computed quickly and accurately by means of computer algebra systems.Accurate correlation equations are also provided.Definitions of Spreading ResistanceHalf-Space Spreading Resistance.When conduction occurs in a region whose dimensionsare two or more orders of magnitude larger than the largest dimension of the source area (Fig.3.12), the spreading resistance is defined as the difference between the heat source temperature and the heat sink temperature divided by the total heat transfer rate from the heatsource. If the flux over the heat source area is uniform, the source temperature may be basedon the area-average source temperature or the centroid temperature, which is the maximumtemperature or close to it.
If the heat sink temperature is spatially variable, then the areaaverage temperature is used in the definition. ThusRs =Tsource- Tsink(3.107)QFlux Tube or Channel Spreading Resistance. When conduction occurs in a confined regionsuch as a finite or an infinitely long flux tube or flux channel (Fig. 3.13), then the one-dimensionalconduction resistance and other resistances must be accounted for. The definition proposed byMikic and Rohsenow [65] can be used to define the spreading resistance. It is as follows:Rs__ Tso ....
- Tcontactplane(3.108)OHeat Sourcez=O/ / / / / / / / /~ / / l / / /////x.\2" 4 - - - l _ J _ _ k - - - ~ "%~".I I/Heat Source/////////////////I~,I/////////////////LI\....-"~\ /tl/J\.-~-... /!,::!F I G U R E 3.12Spreading in a half-space.Ii,'/'~.~..~iF I G U R E 3.13Spreading in a flux tube or channel.CONDUCTION AND THERMAL CONTACTRESISTANCES (CONDUCTANCES)3.35Dimensionless Spreading Resistance.Whichever definition is used, the dimensionlessspreading resistance is generally defined as R*= k~Rs, where k is the thermal conductivity ofthe region and ~ is some length scale related to the contact area. It will be shown for arbitrary,singly connected contact areas that ~ = X/-A, where A is the active area of the heat source andis the appropriate length scale.Spreading Resistance of Isoflux Arbitrary Areas on Half-SpaceCircular, Rectangular, and Square Areas.
The discussion of spreading resistance in isotropic half-spaces begins with the circular area, which has analytical solutions for the isothermal and isoflux boundary conditions, and the rectangular contact, which has an isofluxsolution. The solutions are reported in Carslaw and Jaeger [11]. From the circular contactsolutions, one finds that the spreading resistance (1) for the isothermal condition is Rs = 1/(4ka), where a is the contact radius, (2) for the isoflux condition is Rs = 8/(3x2ka) based on thearea-average temperature, and (3) for the isoflux condition is Rs = 1/(rtka) based on the centroid temperature.This geometry establishes the effect of boundary condition and the choice of source temperature used in the definition of the spreading resistance. The spreading resistances arerelated as follows:kaRs(centroid) > kaR,(area average) > kaR,(isothermal)(3.109)The temperature ratios are To/T = 1.1781 and T/T(isothermal)= 1.0807. These relationshipsare approximately valid for other geometries.The centroid and area-average temperatures of the rectangular contact area of length 2aand width 2b with a > b are given in Carslaw and Jaeger [11].
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