Adrian Bejan(Editor), Allan D. Kraus (Editor). Heat transfer Handbok (776115), страница 29
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The improved version ofeq. (3.374) isρL1 2 rft=r lnk(Tf − T0 ) 2 f r0 1 211 222r − r0 − St rf − r0 1 +(3.375)−4 f4ln(rf /r0 )If the surface of the cylinder is convectively cooled, the boundary condition isk∂T = h [T (r0 , t) − T∞ ]∂r r=r0(3.376)and the quasi-steady-state solutions for St = 0 in this case isTf − T∞rT =ln + Tf(k/ hr0 ) + ln(rf /r0 ) rfρL1 22k1 2 rf2t=r ln −r − r0 1 −k(Tf − T∞ ) 2 f r04 fhr0Lines: 4174 to 4225(3.377)Inward Cylindrical Freezing Consider a saturated liquid at the freezing temperature contained in a cylinder of inside radius ri .
If the surface temperature is suddenlyreduced to and kept at T0 such that T0 < Tf , the liquid freezes inward. The governingequation is1 ∂∂T1 ∂Tr=(3.379)r ∂r∂rα ∂twith initial and boundary conditionsT (rf , 0) = Tfdrf∂T k= ρL∂r r=rfdtBOOKCOMP, Inc. — John Wiley & Sons / Page 251 / 2nd Proofs / Heat Transfer Handbook / Bejan———5.4343pt PgVar———Short Page(3.378)* PgEnds: EjectNoting that the quasi-steady-state solutions such as eqs. (3.377) and (3.378) strictlyapply only when St = 0, Huang and Shih (1975) used them as zero-order solutions ina regular perturbation series in St and generated two additional terms. The three-termperturbation solution provides an improvement on eqs.
(3.377) and (3.378).T (ri , t) = T0[251], (91)(3.380a)(3.380b)(3.380c)[251], (91)252123456789101112131415161718192021222324252627282930313233343536373839404142434445CONDUCTION HEAT TRANSFEREquations (3.373) and (3.374) also give the quasi-steady-state solutions in this caseexcept that r0 now becomes ri .If the surface cooling is due to convection from a fluid at temperature T∞ , withheat transfer coefficient h, the quasi-steady-state solutions for T and t areTf − T∞rkT = T∞ +ln −(3.381)ln(rf /ri ) − (k/ hri )rihriρL2k1 2 rf1 22t=(3.382)r ln +r − rf 1 +k(Tf − T∞ ) 2 fri4 ihriOutward Spherical Freezing Consider a situation where saturated liquid atthe freezing temperature Tf is in contact with a sphere of radius r0 whose surfacetemperature T0 is less than Tf .
The differential equation for the solid phase is1 ∂T1 ∂ (T r)=2r ∂rα ∂t[252], (92)2(3.383)Lines: 4225 to 4295which is to be solved subject to the conditions of eqs. (3.380) (ri replaced by r0 ).In this case, the quasi-steady-state solution with St = 0 isTf − T011T = T0 +(3.384)−1/rf − 1/r0 rr0 ρLr021 rf 3 1 rf 2 1t=(3.385)−2+k(Tf − T0 ) 3 r0r06A regular perturbation analysis allows an improved version of eqs. (3.384) and(3.385) to be written as 2Rf − 3Rf + 2T − T011 − 1/RR 2 − 3R + 2(3.386)1−=+ St−Tf − T01 − 1/Rf6(Rf − 1)4R6Rf (Rf − 1)3andτ=161 + 2Rf3 − 3Rf2 + St 1 + Rf2 − 2Rf(3.387)whereR=rr0Rf =rfr0St =c(Tf − T0 )Lτ=k(Tf − T0 )tρLr02If the surface boundary condition is changed to eq. (3.376), the quasi-steady-statesolutions (St = 0) for T and t are(Tf − T0 )r011T = Tf +−(3.388)1 − r0 /rf + k/ hr0 rfrBOOKCOMP, Inc.
— John Wiley & Sons / Page 252 / 2nd Proofs / Heat Transfer Handbook / Bejan———-0.83556pt PgVar———Normal PagePgEnds: TEX[252], (92)CONDUCTION-CONTROLLED FREEZING AND MELTING123456789101112131415161718192021222324252627282930313233343536373839404142434445ρLr02t=k(Tf − T∞ ) rf 31 rf 2 1k1−−1 1++3r0hr02 r02253(3.389)A three-term solution to the perturbation solution which provides an improvementover eqs. (3.388) and (3.389) is provided by Huang and Shih (1975).Other Approximate Solutions Yan and Huang (1979) have developed perturbation solutions for planar freezing (melting) when the surface cooling or heating isby simultaneous convection and radiation. A similar analysis has been reported bySeniraj and Bose (1982). Lock (1971) developed a perturbation solution for planarfreezing with a sinusoidal temperature variation at the surface. Variable property planar freezing problems have been treated by Pedroso and Domato (1973) and Aziz(1978).
Parang et al. (1990) provide perturbation solutions for the inward cylindricaland spherical solidification when the surface cooling involves both convection andradiation.Alexiades and Solomon (1993) give several approximate equations for estimatingthe time needed to melt a simple solid body initially at its melting temperature Tm .For the situation when the surface temperature T0 is greater than Tm , the melt time tmcan be estimated bytm = l1 + 0.25 + 0.17ω0.70 St2αl (1 + ω)St2(0 ≤ St ≤ 4)(3.390)wherelAω=−1Vandcl (Tl − Tm )St =L(3.391)where Bi = hl/k.In this case, the surface temperature T (0, t) is given by the implicit relationship2T∞ − T mT (0, t) − Tm 1.83ρcl kl+−1(3.393)1.18Stt= 22h · StT∞ − T (0, t)T∞ − T (0, t)BOOKCOMP, Inc.
— John Wiley & Sons / Page 253 / 2nd Proofs / Heat Transfer Handbook / BejanLines: 4295 to 4343———0.35815pt PgVar———Normal PagePgEnds: TEX[253], (93)and l is the characteristic dimension of the body, A the surface area across which heatis transferred to the body, and V the volume of the body. For a plane solid heated atone end and insulated at the other, ω = 0 and l is equal to the thickness. For a solidcylinder and a solid sphere, l becomes the radius and ω = 1 for the cylinder andω = 2 for the sphere.If a hot fluid at temperature T∞ convects heat to the body with heat transfercoefficient h, the approximate melt time for 0 ≤ St ≤ 4 and Bi ≥ 0.10 isl220.701+(3.392)+ (0.25 + 0.17ω )Sttm =2αl (1 + ω)StBiEquations (3.390), (3.392), and (3.393) are accurate to within 10%.[253], (93)2541234567891011121314151617181920212223242526272829303132333435363738394041424344453.10.6CONDUCTION HEAT TRANSFERMultidimensional Freezing (Melting)In Sections 3.10.1 through 3.10.5 we have discussed one-dimensional freezing andmelting processes where natural convection effects were assumed to be absent andthe process was controlled entirely by conduction.
The conduction-controlled modelsdescribed have been found to mimic experimental data for freezing and meltingof water, n-octadecane, and some other phase-change materials used in latent heatenergy storage devices.Multidimensional freezing (melting) problems are far less amenable to exact solutions, and even approximate analytical solutions are sparse. Examples of approximate analytical solutions are those of Budhia and Kreith (1973) for freezing (melting)in a wedge, Riley and Duck (1997) for the freezing of a cuboid, and Shamshundar(1982) for freezing in square, elliptic, and polygonal containers. For the vast majorityof multidimensional phase-change problems, only a numerical approach is feasible.The available numerical methods include explicit finite-difference methods, implicitfinite-difference methods, moving boundary immobilization methods, the isothermmigration method, enthalpy-based methods, and finite elements.
Ozisik (1994) andAlexiades and Solomon (1993) are good sources for obtaining information on theimplementation of finite-difference schemes to solve phase-change problems. Papersby Comini et al. (1974) and Lynch and O’Neill (1981) discuss finite elements withreference to phase-change problems.3.11[254], (94)Lines: 4343 to 4352———6.0pt PgVar———Normal PagePgEnds: TEXCONTEMPORARY TOPICS[254], (94)A major topic of contemporary interest is microscale heat conduction, mentionedbriefly in Section 3.1, where we cited some important references on the topic. Anotherarea of active research is inverse conduction, which deals with estimation of thesurface heat flux history at the boundary of a heat-conducting solid from a knowledgeof transient temperature measurements inside the body.
A pioneering book on inverseheat conduction is that of Beck et al. (1985), and the book of Ozisik and Orlande(2000) is the most recent, covering not only inverse heat conduction but inverseconvection and inverse radiation as well.Biothermal engineering, in which heat conduction appears prominently in manyapplications, such as cryosurgery, continues to grow steadily. In view of the increasingly important role played by thermal contact resistance in the performance of electronic components, the topic is pursued actively by a number of research groups.The development of constructal theory and its application to heat and fluid flow discussed in Bejan (2000) offers a fresh avenue for research in heat conduction.
Although Green’s functions have been employed in heat conduction theory for manydecades, the codification by Beck et al. (1992) is likely to promote their use further.Similarly, hybrid analytic–numeric methodology incorporating the classical integraltransform approach has provided an alternative route to fully numerical methods.
Numerous heat conduction applications of this numerical approach are given by Cottaand Mikhailov (1997). Finally, symbolic algebra packages such as Maple V andBOOKCOMP, Inc. — John Wiley & Sons / Page 254 / 2nd Proofs / Heat Transfer Handbook / BejanNOMENCLATURE123456789101112131415161718192021222324252627282930313233343536373839404142434445255Mathematica are influencing both teaching and research in heat conduction, as shownby Aziz (2001), Cotta and Mikhailov (1997), and Beltzer (1995).NOMENCLATURERoman Letter SymbolsAcross-sectional area, m2area normal to heat flow path, m2Apfin profile area, m2Assurface area, m2aconstant, dimensions varyabsorption coefficient, m−1Bfrequency, dimensionlessbconstant, dimensions varyfin or spine height, m−1BiBiot number, dimensionlessCconstant, dimensions varycspecific heat, kJ/kg · Kdspine diameter, mrate of energy generation, WĖgFoFourier number, dimensionlessffrequency, s−1Hheight, mfin tip heat loss parameter, dimensionlesshheat transfer coefficient, W/m2 ·Kcontact conductance, W/m2 ·Khciunit vector along the x coordinate, dimensionlessjunit vector along the y coordinate, dimensionlesskthermal conductivity, W/m · Kkunit vector along the z coordinate, dimensionlessLthickness, length, or width, mlthickness, mcharacteristic dimension, mMfin parameter, m−1/2mfin parameter, m−1N1convection–conduction parameter, dimensionlessradiation–conduction parameter, dimensionlessN2nexponent, dimensionlessinteger, dimensionlessheat generation parameter, m−1normal direction, mparameter, s−1Pfin perimeter, mpinteger, dimensionlessBOOKCOMP, Inc.
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