Adrian Bejan(Editor), Allan D. Kraus (Editor). Heat transfer Handbok (776115), страница 73
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ForReD ≤ 2 × 105 , the boundary layer remains laminar until the separation point, whichBOOKCOMP, Inc. — John Wiley & Sons / Page 482 / 2nd Proofs / Heat Transfer Handbook / BejanLines: 2114 to 2165———Normal PagePgEnds: TEX[482], (44)HEAT TRANSFER FROM ARRAYS OF OBJECTS123456789101112131415161718192021222324252627282930313233343536373839404142434445483occurs at an arc of about 81°. For higher values of ReD , the boundary layer undergoestransition to turbulence, which results in higher fluid momentum and the pushing backof the separation point to about 140°.The foregoing changes result in large variations in the transport behavior withthe angle θ. If the interest is on an average heat transfer coefficient, many empiricalequations are available. One such correlation for an isothermal cylinder that is basedon the data of many investigators for various ReD and Pr, such that ReD · Pr > 0.2was proposed by Churchill and Bernstein (1977):5/8 4/51/2ReDh̄D0.62ReD · Pr1/31+NuD ≡(6.155)= 0.30 +k[1 + (0.40/Pr)2/3 ]1/4282,000The fluid properties are evaluated at the film temperature, which is the average of thesurface and ambient fluid values.Flow Over an Isothermal SphereWhitaker (1972) recommends the correlation1/22/3NuD = 2 + 0.4ReD + 0.06ReDPr0.4µµs1/4(6.156)The uncertainty band around this correlation is ±30% in the range: 0.71 < Pr 380,3.5 < ReD < 7.6 × 104 , and 1.0 < µ/µs < 3.2.
All properties except µs areevaluated at the ambient temperature.[483], (45)Lines: 2165 to 2208———1.36206pt PgVar———Normal PagePgEnds: TEX[483], (45)6.56.5.1HEAT TRANSFER FROM ARRAYS OF OBJECTSCrossflow across Tube BanksThis configuration is encountered in many practical heat transfer applications, including shell-and-tube heat exchangers. Zhukauskas (1972, 1987) has provided comprehensive heat transfer and pressure drop correlations for tube banks in aligned andstaggered configurations (Fig.
6.15). The heat transfer from a tube depends on its location within the bank. In the range of lower Reynolds numbers, typically the tubes inthe first row show similar heat transfer to those in the inner rows. For higher Reynoldsnumbers, flow turbulence leads to higher heat transfer from inner tubes than from thefirst row. The heat transfer becomes invariant with tube location following the third orfourth row in the mixed-flow regime, occurring above ReD,max . The average Nusseltnumber from a tube is then of the formNuD = C ·RemD,max· Pr0.36PrPrs1/4(6.157)for 0.7 ≤ Pr < 5000, 1 < ReD,max < 2 × 104 , and NL ≥ 20 and where m is as givenin Table 6.2. The Reynolds number ReD,max in eq. (6.157) is based on the maximumBOOKCOMP, Inc. — John Wiley & Sons / Page 483 / 2nd Proofs / Heat Transfer Handbook / Bejan484123456789101112131415161718192021222324252627282930313233343536373839404142434445FORCED CONVECTION: EXTERNAL FLOWSSLSLA1A1V, T⬁V, T⬁STSTDA2A2DSD(a) Aligned tube bank(b) Staggered tube bank[484], (46)Figure 6.15 Tube bundle configurations studied by Zhukaukas (1987).Lines: 2208 to 2271fluid velocity within the tube bank which occurs at the mimimum-flow cross section.For the aligned arrangement, it occurs at A1 in Fig.
6.15 and is given byVmaxST=VST − D(6.158)where V is the upstream fluid velocity.For the staggered configuration, the maximum fluid velocity occurs at A2 if 2(SD −D) < (ST − D), in which caseVmax =STV2(SD − D)(6.159)Otherwise, the aligned tube expression of eq. (6.158) can be used.TABLE 6.2 Parameters ReD,max , C, and m for Various Aligned and Staggered TubeArrangementsConfigurationAlignedStaggeredAlignedStaggeredAligned (ST /SL > 0.7)Staggered (ST /SL < 2)Staggered (ST /SL > 2)AlignedStaggered (Pr > 0.70)Staggered (Pr = 0.70)ReD,maxCm16–1001.6–40100–100040–10001000–2 × 1051000–2 × 1051000–2 × 1052 × 105 –2 × 1062 × 105 –2 × 1062 × 105 –2 × 1060.901.040.520.710.270.35 (ST /SL )0.200.400.0330.031 (ST /SL )0.200.027 (ST /SL )0.200.400.400.500.500.630.600.600.800.800.80Source: Zhukauskas (1987).BOOKCOMP, Inc.
— John Wiley & Sons / Page 484 / 2nd Proofs / Heat Transfer Handbook / Bejan———-3.1679pt PgVar———Long PagePgEnds: TEX[484], (46)HEAT TRANSFER FROM ARRAYS OF OBJECTS123456789101112131415161718192021222324252627282930313233343536373839404142434445TABLE 6.3485Parameter C2 for Various Tube Rows and ConfigurationsNL123457101316Aligned(ReD,max > 1000)Staggered(ReD,max 100–1000)Staggered(ReD,max > 1000)0.700.800.860.900.920.950.970.980.990.830.870.910.940.950.970.980.980.990.640.760.840.890.920.950.970.980.99Source: Zhukauskas (1987).It is noted that all properties except Prs are evaluated at the arithmetic mean ofthe fluid inlet and outlet temperatures. The constants C and m are listed in Table 6.2,where it can be observed that for ST /SL < 0.7, heat transfer is poor and the alignedtube configuration should not be employed.If NL < 20, a corrected expression can be employed:NuL = C2 · NuL NL <20NL ≥20[485], (47)Lines: 2271 to 2293———-1.06493pt PgVarwhere C2 = 1.0 for aligned tubes in the range ReD,max and is provided for otherconditions in Table 6.3.———Long PagePgEnds: TEX6.5.2[485], (47)Flat PlatesStack of Parallel Plates Consider a stack of parallel plates placed in a freestream (Fig.
6.16). The envelope for the plate stack has a fixed volume, W (width)×L (length) ×H (height). Morega et al. (1995) conducted numerical analyses on atwo-dimensional model where W L, t (the plate thickness), equal to L/20 and q (the heat flux) uniform over the plate surfaces, excluding the edges. The questionaddressed by Morega et al. is the number of plates needed to maximize the heattransfer performance from the entire plate stack.
Fewer plates than the optimumFigure 6.16 Plate stack. (From Morega et al., 1995.)BOOKCOMP, Inc. — John Wiley & Sons / Page 485 / 2nd Proofs / Heat Transfer Handbook / Bejan486123456789101112131415161718192021222324252627282930313233343536373839404142434445FORCED CONVECTION: EXTERNAL FLOWSreduce the amount of heat transfer area in a given volume. A greater populationof plates than the optimum increases the resistance to flow through the stack andhence diverts the flow to the free stream zone outside the stack.
The heat transferperformance is represented by the nondimensional hot-spot temperature,θhot =k(Tmax − T0 )q (2nL)(6.160)where Tmax is the maximum surface temperature on the plate surface, T0 is the freestream temperature, and k is the fluid thermal conductivity.Figure 6.17 shows θhot versus the number of plates in the stack, n. The parameteris the Reynolds number ReL = U0 L/ν, where U0 is the free stream velocity andν is the kinematic viscosity of the fluid. Figure 6.17 also shows the curves derivedfrom Nakayama et al. (1988), where the experimental data were obtained with thefinned heat sinks having nearly isothermal surfaces. Despite the difference in thermalboundary conditions, the two groups of curves imply a smooth transition of optimumn from low to high Reynolds number regimes.
Morega et al. (1995) proposed arelationship for optimum n:nopt 1/20.26(H /L)Pr1/4 · ReL1/21 + 0.26(t/L)Pr1/4 · ReLLines: 2293 to 2317———4.51717pt PgVar(6.161)where Pr is the Prandtl number. Equation (6.161) is applicable to cases where Pr ≥0.7 and n 1.Figure 6.17 Nondimensional hot spot temperature (θhot ) versus the number of plates (n).(From Morega et al., 1995.)BOOKCOMP, Inc. — John Wiley & Sons / Page 486 / 2nd Proofs / Heat Transfer Handbook / Bejan[486], (48)———Normal PagePgEnds: TEX[486], (48)HEAT TRANSFER FROM ARRAYS OF OBJECTS123456789101112131415161718192021222324252627282930313233343536373839404142434445487Offset Strips Offset arrangements of plates or strips (Fig.
6.1d) reproduced withthe addition of the symbols in Fig. 6.18 offer the advantage of heat transfer enhancement. The boundary layer development is interrupted at the end of the strip and thenresumes from the leading edge of the strip in the downstream row. This arrangementallows the boundary layer to be held thin everywhere in the strip array. A similareffect can be obtained with the in-line arrangement of strips, but offsetting the stripsfrom row to row introduces additional favorable effects on heat transfer.
The fluid inthe offset strips has a longer distance to travel after leaving the trailing edge of a stripto the leading edge of the next strip than in the in-line counterpart. This elongateddistance results in a longer elapsed time for the diffusion (or dispersion) of momentum and heat. When the fluid velocity is high enough, vortex shedding or turbulenceattains a high level in the intervening space between the strips. Thus, the strips afterthe third row come to be exposed to highly dynamic flow; thermal wakes shed fromthe upstream strips tend to be diluted by increased level of turbulence so that the individual strip is washed by a cooler stream than in the in-line strip arrangement.
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