Adrian Bejan(Editor), Allan D. Kraus (Editor). Heat transfer Handbok (776115), страница 62
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This principle was first stated for bundles of verticalchannels with natural convection (Bejan, 1984; Bar-Cohen and Rohsenow, 1984). Itwas extended to bundles of channels with forced convection by Bejan and Sciubba(1992). It continues to generate an expanding class of geometric results, which was[414], (20)Lines: 792 to 800———0.927pt PgVar———Normal PagePgEnds: TEX[414], (20)Figure 5.10 Two-dimensional volume that generates heat and is cooled by forced convection.(From Bejan, 2000.)BOOKCOMP, Inc. — John Wiley & Sons / Page 414 / 2nd Proofs / Heat Transfer Handbook / BejanOPTIMAL CHANNEL SIZES FOR LAMINAR FLOW123456789101112131415161718192021222324252627282930313233343536373839404142434445415reviewed in Kim and Lee (1996) and Bejan (2000). In this section some of the forcedconvection results are reviewed.The opportunity for optimizing internal channel geometry becomes evident if Fig.5.10 is regarded as a single-stream heat exchanger intended for cooling electronics.The global thermal conductance of the electronics is the ratio between the total rateof heat generation in the package (q) and the maximum excess temperature registeredin the host spots (Tmax − T0 ).
The entrance temperature of the coolant is T0 . Designswith more components and circuitry installed in a given volume are desirable as isa larger q. This can be accommodated by increasing the ceiling temperature Tmax(usually limited by the design of electronics) and by increasing the conductance ratioq/(Tmax − Tmin ). The latter puts the design on a course of geometry optimization,because the conductance is dictated by the flow geometry.
The degree of freedom inthe design is the channel size D, on the number of channels, n = H /D.The existence of an optimal geometry is discovered by designing the stack of Fig.5.10 in its two extremes: a few large spacings, and many small spacings. When Dis large, the total heat transfer surface is small, the global thermal resistance is high,and consequently, each heat-generating surface is overheated. When D is small, thecoolant cannot flow through the package. The imposed heat generation rate can beremoved only by allowing the entire volume to overheat.
An optimal D exists inbetween and is located by intersecting the large D and small D asymptotes.For the two-dimensional parallel-plate stack of Fig. 5.10, where Pr ≥ 1 and thepressure differnce ∆P is fixed, the optimal spacing is (Bejan and Sciubba, 1992)Dopt 2.7Be−1/4L(5.47)whereBe = ∆PL2 /µαis the specified pressure drop number, which Bhattacharjee and Grosshandler (1988)and Petrescu (1994) have termed the Bejan number. Equation (5.47) underestimates(by only 12%) the value obtained by optimizing the stack geometry numerically. Furthermore, eq. (5.47) is robust, because it holds for both uniform-flux and isothermalplates. Equation (5.47) holds even when the plate thickness is not negligible relative tothe plate-to-plate spacing (Mereu et al., 1993).
The maximized thermal conductanceqmax /(Tmax − T0 ), or maximized heat transfer rate per unit volume that correspondsto eq. (5.47), isqmaxk 0.60 2 (Tmax − T0 )Be1/2HLWL(5.48)where W is the volume dimension in the direction perpendicular to the plane of Fig.5.10. Equation (5.48) overestimates (by 20%) the heat transfer density maximizednumerically (Bejan and Sciubba, 1992).The pressure drop number Be = ∆PL2 /µα is important, because in the fieldof internal forced convection it plays the same role that the Rayleigh number playsBOOKCOMP, Inc.
— John Wiley & Sons / Page 415 / 2nd Proofs / Heat Transfer Handbook / Bejan[415], (21)Lines: 800 to 831———0.3511pt PgVar———Normal PagePgEnds: TEX[415], (21)416123456789101112131415161718192021222324252627282930313233343536373839404142434445FORCED CONVECTION: INTERNAL FLOWSfor internal natural convection (Petrescu, 1994). Specifically, the equivalent of eq.(5.47) for natural convection through a H -tall stack of vertical D-wide channels witha Pr ≥ 1 fluid isDopt 2.3Ra−1/4H(5.49)where the Rayleigh number is specified (Bejan, 1984):Ra =gβH 3 (Tmax − T0 )ανEquations (5.47) and (5.49) make the Be–Ra analogy evident.
The Rayleigh numberis the dimensionless group that indicates the slenderness of vertical thermal boundarylayers in laminar natural convection,Ra ≈HδT4Lines: 831 to 874———where H is the wall height and δT is the thermal boundary layer thickness.Campo and Li (1996) considered the related problem where the parallel-platechannels are heated asymmetrically: for example, with adiabatic or nearly adiabaticregions. Campo (1999) used the intersection of asymptotes method in the optimization of the stack of Fig. 5.10, where the plates are with uniform heat flux.
His resultsconfirmed the correctness and robustness of the initial result of eq. (5.47). The geometric optimization of round tubes with steady and periodic flows and Pr 1 fluidswas performed by Rocha and Bejan (2001). Related studies are reviewed in Kim andLee (1996).The optimal internal spacings belong to the specified volume as a whole, withits purpose and constraints, not to the individual channel.
The robustness of thisconclusion becomes clear when we look at other elemental shapes for which optimalspacings have been determined. A volume heated by an array of staggered plates inforced convection (Fig. 5.11a) is characterized by an internal spacing D that scaleswith the swept length of the volume (Fowler et al., 1997):DoptL −1/2−1/4ReL 5.4PrLb(5.50)In this relation the Reynolds number is ReL = U∞ L/ν. The range in which this correlation was developed based on numerical simulations and laboratory experimentsis Pr = 0.72, 102 ≤ ReL ≤ 104 , and 0.5 ≤ N b/L ≤ 1.3, where N is the number ofplate surfaces that face one elemental channel; that is, N = 4 in Fig. 5.11a.Similarly, when the elements are cylinders in crossflow as in Fig.
5.11b, the optimal spacing S is influenced the most by the longitudinal dimension of the volume. Theoptimal spacing was determined based on the method of intersecting the asymptotesBOOKCOMP, Inc. — John Wiley & Sons / Page 416 / 2nd Proofs / Heat Transfer Handbook / Bejan[416], (22)4.87715pt PgVar———Normal PagePgEnds: TEX[416], (22)OPTIMAL CHANNEL SIZES FOR LAMINAR FLOW123456789101112131415161718192021222324252627282930313233343536373839404142434445417[417], (23)Lines: 874 to 876———1.097pt PgVar———Normal Page* PgEnds: Eject[417], (23)Figure 5.11 Forced-convection channels for cooling a heat-generating volume: (a) array ofstaggered plates; (b) array of horizontal cylinders; (c) square pins with impinging flow. (FromBejan, 2000.)(Stanescu et al., 1996; Bejan, 2000). The asymptotes were derived from the large volume of empirical data accumulated in the literature for single cylinders in crossflow(the large-S limit) and for arrays with many rows of cylinders (the small-S limit).
Inthe range 104 ≤ P̃ ≤ 108 , 25 ≤ H /D ≤ 200, and 0.72 ≤ Pr ≤ 50, the optimalspacing is correlated to within 5.6% byBOOKCOMP, Inc. — John Wiley & Sons / Page 417 / 2nd Proofs / Heat Transfer Handbook / Bejan418123456789101112131415161718192021222324252627282930313233343536373839404142434445FORCED CONVECTION: INTERNAL FLOWSSopt(H /D)0.52 1.59DP̃ 0.13 · Pr 0.24(5.51)where P̃ is an alternative dimensionless pressure drop number based on D: namely,P̃ = ∆PD 2 /µν. When the free stream velocity U is specified (instead of ∆P ), eq.(5.51) may be transformed by noting that with ∆P approximately equal to ∆P ≈2/2:ρU∞Sopt(H /D)0.52 1.70 0.26DReD · Pr 0.24(5.52)This correlation is valid in the range 140 ≤ ReD ≤ 14,000, where ReD = UD/ν.The minimized global thermal resistance that corresponds to this optimal spacing is4.5TD − T∞0.90qD/kLWReD · Pr 0.64[418], (24)(5.53)where TD is the cylinder temperature and q is the total rate of heat transfer from theH LW volume to the coolant (T ).
If the cylinders are arranged such that their centersform equilateral trianglesas in Fig. 5.11b,the total number of cylinders present inthe bundle is H W/ (S + D)2 cos 30° . This number and the contact area based on itmay be used to deduce from eq. (5.53) the volume-averaged heat transfer coefficientbetween the array and the stream.Fundamentally, these results show that there always is an optimal spacing for cylinders (or tubes) in crossflow heat exchangers when compactness is an objective. Thiswas demonstrated numerically by Matos et al.
(2001), who optimized numericallyassemblies of staggered round and elliptic tubes in crossflow. Matos et al. (2001) alsofound that the elliptic tubes perform better relative to round tubes: The heat transferdensity is larger by 13%, and the overall flow resistance is smaller by 25%.Optimal spacings emerge also when the flow is three-dimensional, as in an arrayof pin fins with impinging flow (Fig. 5.11c). The flow is initially aligned with thefins, and later makes a 90° turn to sweep along the base plate and across the fins.
Theoptimal spacings are correlated to within 16% by Ledezma et al. (1996):Sopt 0.81Pr −0.25 · Re−0.32LL(5.54)which is valid in the range 0.06 < D/L ≤ 0.14, 0.28 ≤ H /L ≤ 0.56, 0.72 ≤ Pr ≤7, 10 ≤ ReD ≤ 700, and 90 ≤ ReL ≤ 6000. Note that the spacing Sopt is controlledby the linear dimension of the volume L.
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