Adrian Bejan(Editor), Allan D. Kraus (Editor). Heat transfer Handbok (776115), страница 61

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If thex-independent velocity profile of Hagen–Poiseuille flow is assumed, it is possibleto solve the energy conservation equation and determine, as an infinite series, thetemperature field (Graetz, 1883). The Pr = ∞ curve in Fig. 5.7 shows the mainfeatures of the Graetz solution for heat transfer in the entrance region of a roundtube with an isothermal wall (T0 ). The Reynolds number ReD = UD/ν is based onthe tube diameter D and the mean velocity U . The bulk dimensionless temperatureof the stream (θ∗m ), the local Nusselt number (Nux ), and the averaged Nusselt number(Nu0−x ) are defined by[408], (14)Lines: 535 to 553———1.927pt PgVar———Normal Page* PgEnds: Eject1100[408], (14)␪*m (Pr = ⬁)␪*mNu0⫺x (Pr = ⬁)Nux123456789101112131415161718192021222324252627282930313233343536373839404142434445FORCED CONVECTION: INTERNAL FLOWS10Pr = ⬁50.120.730.013.66(0.1x/DReD Pr(1/21Figure 5.7 Heat transfer in the entrance region of a round tube with isothermal wall. (FromBejan, 1995; drawn based on data from Shah and London, 1978, and Hornbeck, 1965.)BOOKCOMP, Inc.

— John Wiley & Sons / Page 408 / 2nd Proofs / Heat Transfer Handbook / BejanHEAT TRANSFER IN DEVELOPING FLOW123456789101112131415161718192021222324252627282930313233343536373839404142434445θ∗m =Nux =Nu0−x =409Tm − T0Tin − T0(5.27)qx Dk(T 0 − Tm )(5.28)Dq0−xk ∆Tm(5.29)In these expressions, Tm (x) is the local mean temperature, Tin the stream inlet temthe heat flux averaged from x = 0 toperature, qx the local wall heat flux, and q0−xx, and ∆Tlm the logarithmic mean temperature difference,∆Tlm =[T0 − Tin (x)] − (T0 − Tm )ln [T0 − Tm (x)/(T0 − Tin )](5.30)The dimensionless longitudinal position plotted on the abscissa is also known as x∗ :x/Dx∗ =ReD · Pr(5.31)[409], (15)Lines: 553 to 618———1.28441pt PgVar———1/2This group, and the fact that the knees of the Nu curves occur at x∗ ≈ 10−1 , supportNormal Pagethe XT estimate anticipated by eq.

(5.26).* PgEnds: EjectThe following analytical expressions are recommended by a simplified alternativeto Graetz’s series solution (Lévêque, 1928; Drew, 1931). The relationships for thePr = ∞ curves shown in Fig. 5.7 are (Shah and London, 1978)[409], (15)−1/3− 0.70x∗ ≤ 0.011.077x∗(5.32)Nux =3−0.488 −57.2x∗3.657 + 6.874(10 x∗ )ex∗ > 0.01−1/3− 0.70x∗ ≤ 0.005 1.615x∗−1/3Nu0−x = 1.615x∗(5.33)− 0.200.005 < x∗ < 0.033.657 + 0.0499/x∗x∗ > 0.03The thermally developing Hagen–Poiseuille flow in a round tube with uniformheat flux q can be analyzed by applying Graetz’s method (the Pr = ∞ curves in Fig.5.8).

The results for the local and overall Nusselt numbers are represented within 3%by the equations (see also Shah and London, 1978)−1/3− 1.00x∗ ≤ 0.00005 3.302x∗−1/3(5.34)Nux∗ = 1.302x∗− 0.500.00005 < x∗ ≤ 0.00153−0.506 −41x∗ex∗ > 0.0014.364 + 8.68(10 x∗ )−1/3x∗ ≤ 0.031.953x∗Nu0−x =(5.35)4.364 + 0.0722/x∗x∗ > 0.03BOOKCOMP, Inc. — John Wiley & Sons / Page 409 / 2nd Proofs / Heat Transfer Handbook / Bejan410100Nux123456789101112131415161718192021222324252627282930313233343536373839404142434445FORCED CONVECTION: INTERNAL FLOWS10Pr = ⬁Nu0⫺x (Pr = ⬁)5[410], (16)20.74.3630.01(0.1x/DReD Pr(1/21Figure 5.8 Heat transfer in the entrance region of a round tube with uniform heat flux.

(FromBejan, 1995; drawn based on data from Shah and London, 1978, and Hornbeck, 1965.)Lines: 618 to 654———2.2831pt PgVar———Normal PagePgEnds: TEX[410], (16)whereNux∗ = q Nu0−x = q with∆Tavg =Dk [T0 (x) − Tm (x)]Dk ∆Tavg x−1dx1x 0 T0 (x) − Tm (x)(5.36)Analogous results are available for the heat transfer to thermally developingHagen–Poiseuille flow in ducts with other cross-sectional shapes. The Nusselt numbers for a parallel-plate channel are shown in Fig. 5.9. The curves for a channel withisothermal surfaces are approximated by (Shah and London, 1978)−1/31.233x∗+ 0.40x∗ ≤ 0.001(5.37)Nu0−x =3−0.488 −245x∗7.541 + 6.874(10 x∗ )ex∗ > 0.001BOOKCOMP, Inc.

— John Wiley & Sons / Page 410 / 2nd Proofs / Heat Transfer Handbook / Bejan411HEAT TRANSFER IN DEVELOPING FLOWNu0−x−1/3 1.849x∗= 1.849x∗−1/3 + 0.607.541 + 0.0235/x∗x∗ ≤ 0.00050.0005 < x∗ ≤ 0.006x∗ > 0.006(5.38)If the plate-to-plate spacing is D, the Nusselt numbers are defined asq (x)Dhk [T0 − Tm (x)]Nux =q0−xDhk ∆TlmNu0−x =where Dh = 2D and ∆Tlm is given by eq.

(5.30).The thermal entrance region of the parallel-plate channel with uniform heat flux[411], (17)and Hagen–Poiseuille flow is characterized by (Shah and London, 1978)−1/3x∗ ≤ 0.0002 1.490x∗Lines: 654 to 688−1/3Nux = 1.490x∗− 0.400.0002 < x∗ ≤ 0.001 (5.39)———8.235 + 8.68(103 x∗ )0.506 e−164x∗x∗ > 0.001* 16.42421pt PgVar———Normal Page100PgEnds: TEXNuxNu 0⫺x[411], (17)Uniformwall heatfluxNux , Nu 0⫺ x123456789101112131415161718192021222324252627282930313233343536373839404142434445108.237.54NuxUniformwalltemperature Nu 0⫺x30.01(0.1x/Dh 1/2ReDh Pr(1Figure 5.9 Heat transfer in the thermal entrance region of a parallel-plate channel withHagen–Poiseuille flow. (From Bejan, 1995; drawn based on data from Shah and London, 1978.)BOOKCOMP, Inc.

— John Wiley & Sons / Page 411 / 2nd Proofs / Heat Transfer Handbook / Bejan412123456789101112131415161718192021222324252627282930313233343536373839404142434445FORCED CONVECTION: INTERNAL FLOWSNu0−x−1/3 2.236x∗−1/3= 2.236x∗+ 0.908.235 + 0.0364/x∗x∗ ≤ 0.0010.001 < x∗ ≤ 0.01(5.40)x∗ ≥ 0.01The Nusselt number definitions areNux =Nu0−x =q Dhk [T0 (x) − Tm (x)]q Dhk ∆Tavgwhere ∆Tavg is furnished by eq. (5.36). It is worth repeating that x∗ represents thedimensionless longitudinal coordinate in the thermal entrance region, eq. (5.31),which in the case of the parallel-plate channel becomesx∗ =x/DhReDh · PrLines: 688 to 732———All of the results compiled in this section hold in the limit Pr → ∞.3.9784pt PgVar———Normal Page* PgEnds: Eject5.4.3 Thermally and Hydraulically Developing FlowWhen the Prandtl number is not much greater than 1, especially when XT and Xare comparable, the temperature and velocity profiles develop together, at the samelongitudinal location x from the duct entrance.

This is the most general case, andheat transfer results for many duct geometries have been developed numerically by anumber of investigators. Their work is reviewed by Shah and London (1978). Here,a few leading examples of analytical correlations of the numerical data are shown.Figure 5.7 shows a sample of the finite-Pr data available for a round tube withan isothermal wall. The recommended analytical expressions for the local (Shah andBhatti, 1987) and overall (Stephan, 1959) Nusselt numbers in the range 0.1 ≤ Pr ≤1000 in parallel-plate channels are0.024x∗−1.14 0.0179Pr 0.17 x∗−0.64 − 0.14Nux = 7.55 +21 + 0.0358Pr 0.17 x∗−0.64Nu0−x = 7.55 +0.024x∗−1.141 + 0.0358Pr 0.17 x∗−0.64(5.41)(5.42)The pressure drop over the hydrodynamically developing length x, or ∆P =P (0) − P (x), can be calculated using (Shah and London, 1978)∆P1.25 + 64x+ − 13.74(x+ )1/21/2=13.74(x)++11 + 0.00021(x+ )−2ρU 22BOOKCOMP, Inc.

— John Wiley & Sons / Page 412 / 2nd Proofs / Heat Transfer Handbook / Bejan[412], (18)(5.43)[412], (18)OPTIMAL CHANNEL SIZES FOR LAMINAR FLOW123456789101112131415161718192021222324252627282930313233343536373839404142434445413On the right-hand side, x+ is the dimensionless coordinate for the hydrodynamicentrance region,x+ =x/DReD(5.44)which also appears on the abscissa of Fig. 5.3. Note the difference beween X+ andx∗ . Equation (5.43) can be used instead of the (Cf )0−x · ReD curve shown in Fig. 5.3by noting the force balance∆PorπD 2= τ0−x πDx4∆P= 4x+ Cf · ReD12ρU0−x2(5.45)Figures 5.8 and 5.9 show, respectively, several finite-Pr solutions for the localNusselt number in the entrance region of a tube with uniform heat flux.

A closed-formexpression that covers both the entrance and fully developed regions was developedby Churchill and Ozoe (1973):Nux1/64.364 1 + (Gz/29.6)23/2 1/3Gz/19.04= 1+ 1/21/31 + (Pr/0.0207)2/31 + (Gz/29.6)2q Dk [T0 (x) − Tm (x)]The Graetz number is defined asGz =ππ ReD · Pr=4x∗4 x/DEquation (5.46) agrees within 5% with numerical data for Pr = 0.7 and Pr = 10 andhas the correct asymptotic behavior for large and small Gz and Pr.5.5OPTIMAL CHANNEL SIZES FOR LAMINAR FLOWThe geometry of the boundary layers in the entrance region of a duct (Fig. 5.6) bringswith it a fundamental property of great importance in numerous and diverse heatBOOKCOMP, Inc. — John Wiley & Sons / Page 413 / 2nd Proofs / Heat Transfer Handbook / BejanLines: 732 to 792———1.0923pt PgVar———Normal PagePgEnds: TEX[413], (19)(5.46)whereNux =[413], (19)414123456789101112131415161718192021222324252627282930313233343536373839404142434445FORCED CONVECTION: INTERNAL FLOWStransfer applications.

When the objective is to use many channels in parallel, for thepurpose of cooling or heating a specified volume that is penetrated by these channels,there is an optimal channel geometry such that the heat transfer rate integrated overthe volume is maximal. Alternatively, because the volume is fixed, use of a bundleof channels of optimal sizes means that the heat transfer density (the average heattransfer per unit volume) is maximal.In brief, the optimal channel size (D, in Fig. 5.10 must be such that the thermalboundary layers merge just as the streams leave the channels.

The streamwise lengthof the given volume (L) matches the scale of XT . In this case most of the space occupied by fluid is busy transferring heat between the solid walls and the fluid spaces.This geometric optimization principle is widely applicable, as this brief review shows.The reason is that every duct begins with an entrance region, and from the constructalpoint of view of deducing flow architecture by maximizing global performance subject to global constaints (such as the volume) (Bejan, 2000), the most useful portionof the duct is its entrance length.

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