John H. Lienhard IV, John H. Lienhard V. A Heat Transfer Textbook (776116), страница 51
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For a fixed heat flux, the change in Tb is given by eqn. (7.17),and a value of h is not needed.For an isothermal wall, the following curve fits are available for theNusselt number in thermally developing flow [7.4]:0.0018 Gz1/3NuD = 3.657 + 0.04 + Gz−2/3NuD = 3.657 +0.0668 Gz1/30.04 + Gz−2/32(7.28)(7.29)The error is less than 14% for Gz > 1000 and less than 7% for Gz < 1000.For fixed qw , a more complicated formula reproduces the exact resultfor local Nusselt number to within 1%:⎧1/3⎪−1for 2 × 104 ≤ Gz⎪⎨1.302 GzNuD = 1.302 Gz1/3 − 0.5for 667 ≤ Gz ≤ 2 × 104 (7.30)⎪⎪⎩4.364 + 0.263 Gz0.506 e−41/Gz for 0 ≤ Gz ≤ 667353354Forced convection in a variety of configurations§7.2Example 7.2A fully developed flow of air at 27◦ C moves at 2 m/s in a 1 cm I.D.
pipe.An electric resistance heater surrounds the last 20 cm of the pipe andsupplies a constant heat flux to bring the air out at Tb = 40◦ C. Whatpower input is needed to do this? What will be the wall temperatureat the exit?Solution. This is a case in which the wall heat flux is uniform alongthe pipe. We first must compute Gz20 cm , evaluating properties at(27 + 40) 2 34◦ C.Gz20cmReD Pr Dx(2 m/s)(0.01 m)(0.711)(0.01 m)16.4 × 10−6 m2 /s= 43.38=0.2 m=From eqn. 7.30, we compute NuD = 5.05, soTwexit − Tb =qw D5.05 kNotice that we still have two unknowns, qw and Tw . The bulktemperature is specified as 40◦ C, and qw is obtained from this numberby a simple energy balance:qw (2π Rx) = ρcp uav (Tb − Tentry )π R 2soqw = 1.159mkgJR· 2 · (40 − 27)◦ C ·· 1004= 378 W/m2m3kg·Ks2x1/80ThenTwexit = 40◦ C +(378 W/m2 )(0.01 m)= 68.1◦ C5.05(0.0266 W/m·K)Turbulent pipe flow§7.37.3Turbulent pipe flowTurbulent entry lengthThe entry lengths xe and xet are generally shorter in turbulent flow thanin laminar flow.
Table 7.1 gives the thermal entry length for various values of Pr and ReD , based on NuD lying within 5% of its fully developedvalue. These results are for a uniform wall heat flux imposed on a hydrodynamically fully developed flow. Similar results are obtained for auniform wall temperature.For Prandtl numbers typical of gases and nonmetallic liquids, the entry length is not strongly sensitive to the Reynolds number. For Pr > 1 inparticular, the entry length is just a few diameters. This is because theheat transfer rate is controlled by the thin thermal sublayer on the wall,which develops very quickly.Only liquid metals give fairly long thermal entrance lengths, and, forthese fluids, xet depends on both Re and Pr in a complicated way. Sinceliquid metals have very high thermal conductivities, the heat transferrate is also more strongly affected by the temperature distribution in thecenter of the pipe.
We discusss liquid metals in more detail at the end ofthis section.When heat transfer begins at the inlet to a pipe, the velocity and temperature profiles develop simultaneously. The entry length is then verystrongly affected by the shape of the inlet. For example, an inlet that induces vortices in the pipe, such as a sharp bend or contraction, can createTable 7.1 Thermal entry lengths, xet /D, for which NuD will beno more than 5% above its fully developed value in turbulentflowPr0.010.73.0ReD20,0007104100,00022123500,00032143355356Forced convection in a variety of configurations§7.3Table 7.2 Constants for the gas-flow simultaneous entrylength correlation, eqn. (7.31), for various inlet configurationsInlet configurationCnLong, straight pipeSquare-edged inlet180◦ circular bend90◦ circular bend90◦ sharp elbow0.97562.42540.97591.05172.01520.7600.6760.7000.6290.614a much longer entry length than occurs for a thermally developing flow.These vortices may require 20 to 40 diameters to die out.
For varioustypes of inlets, Bhatti and Shah [7.8] provide the following correlationfor NuD with L/D > 3 for air (or other fluids with Pr ≈ 0.7)CNuD=1+Nu∞(L/D)nfor Pr = 0.7(7.31)where Nu∞ is the fully developed value of the Nusselt number, and C andn depend on the inlet configuration as shown in Table 7.2.Whereas the entry effect on the local Nusselt number is confined toa few ten’s of diameters, the effect on the average Nusselt number maypersist for a hundred diameters.
This is because much additional lengthis needed to average out the higher heat transfer rates near the entry.The discussion that follows deals almost entirely with fully developedturbulent pipe flows.Illustrative experimentFigure 7.5 shows average heat transfer data given by Kreith [7.9, Chap. 8]for air flowing in a 1 in. I.D. isothermal pipe 60 in. in length. Let us seehow these data compare with what we know about pipe flows thus far.The data are plotted for a single Prandtl number on NuD vs. ReDcoordinates. This format is consistent with eqn. (7.25) in the fully developed range, but the actual pipe incorporates a significant entry region.Therefore, the data will reflect entry behavior.For laminar flow, NuD 3.66 at ReD = 750.
This is the correct valuefor an isothermal pipe. However, the pipe is too short for flow to be fullydeveloped over much, if any, of its length. Therefore NuD is not constant§7.3Turbulent pipe flow357Figure 7.5 Heat transfer to air flowing ina 1 in. I.D., 60 in. long pipe (afterKreith [7.9]).in the laminar range. The rate of rise of NuD with ReD becomes very greatin the transitional range, which lies between ReD = 2100 and about 5000in this case.
Above ReD 5000, the flow is turbulent and it turns outthat NuD Re0.8D .The Reynolds analogy and heat transferA form of the Reynolds analogy appropriate to fully developed turbulentpipe flow can be derived from eqn. (6.111)Cf (x) 2h4Stx ==ρcp u∞1 + 12.8 Pr0.68 − 1 Cf (x) 2(6.111)where h, in a pipe flow, is defined as qw /(Tw − Tb ). We merely replaceu∞ with uav and Cf (x) with the friction coefficient for fully developedpipe flow, Cf (which is constant), to getCf 2h4 =St =ρcp uav1 + 12.8 Pr0.68 − 1 Cf 2(7.32)This should not be used at very low Pr’s, but it can be used in eitheruniform qw or uniform Tw situations.
It applies only to smooth walls.358Forced convection in a variety of configurations§7.3The frictional resistance to flow in a pipe is normally expressed interms of the Darcy-Weisbach friction factor, f [recall eqn. (3.24)]:f ≡head lossu2avpipe lengthD2=∆pL ρu2avD 2(7.33)where ∆p is the pressure drop in a pipe of length L. However,!"∆p (π /4)D 2∆pDfrictional force on liquid==τw =surface area of pipeπ DL4Lsof =τw= 4Cfρu2av /8(7.34)Substituting eqn. (7.34) in eqn. (7.32) and rearranging the result, weobtain, for fully developed flow, f 8 ReD Pr4 (7.35)NuD =1 + 12.8 Pr0.68 − 1 f 8The friction factor is given graphically in Fig.
7.6 as a function of ReD andthe relative roughness, ε/D, where ε is the root-mean-square roughnessof the pipe wall. Equation (7.35) can be used directly along with Fig. 7.6to calculate the Nusselt number for smooth-walled pipes.Historical formulations. A number of the earliest equations for theNusselt number in turbulent pipe flow were based on Reynolds analogyin the form of eqn. (6.76), which for a pipe flow becomesSt =Cf2Pr−2/3 =fPr−2/38(7.36)or NuD = ReD Pr1/3 f /8(7.37)For smooth pipes, the curve ε/D = 0 in Fig.
7.6 is approximately givenby this equation:f0.046= Cf =4Re0.2D(7.38)359Figure 7.6 Pipe friction factors.360Forced convection in a variety of configurations§7.3in the range 20, 000 < ReD < 300, 000, so eqn. (7.37) becomesNuD = 0.023 Pr1/3 Re0.8Dfor smooth pipes. This result was given by Colburn [7.10] in 1933. Actually, it is quite similar to an earlier result developed by Dittus and Boelterin 1930 (see [7.11, pg. 552]) for smooth pipes.NuD = 0.0243 Pr0.4 Re0.8D(7.39)These equations are intended for reasonably low temperature differences under which properties can be evaluated at a mean temperature(Tb +Tw )/2. In 1936, a study by Sieder and Tate [7.12] showed that when|Tw −Tb | is large enough to cause serious changes of µ, the Colburn equation can be modified in the following way for liquids:0.14µb0.81/3(7.40)NuD = 0.023 ReD Prµwwhere all properties are evaluated at the local bulk temperature exceptµw , which is the viscosity evaluated at the wall temperature.These early relations proved to be reasonably accurate.
They gavemaximum errors of +25% and −40% in the range 0.67 Pr < 100 andusually were considerably more accurate than this. However, subsequentresearch has provided far more data, and better theoretical and physicalunderstanding of how to represent them accurately.Modern formulations. During the 1950s and 1960s, B. S.
Petukhov andhis co-workers at the Moscow Institute for High Temperature developeda vastly improved description of forced convection heat transfer in pipes.Much of this work is described in a 1970 survey article by Petukhov [7.13].Petukhov recommends the following equation, which is built fromeqn. (7.35), for the local Nusselt number in fully developed flow in smoothpipes where all properties are evaluated at Tb .NuD =(f /8) ReD Pr41.07 + 12.7 f /8 Pr2/3 − 1where104 < ReD < 5 × 1060.5 < Pr < 200200 Pr < 2000for 6% accuracyfor 10% accuracy(7.41)Turbulent pipe flow§7.3361and where the friction factor for smooth pipes is given by1f = 1.82 log10 ReD − 1.642(7.42)Gnielinski [7.14] later showed that the range of validity could be extendeddown to the transition Reynolds number by making a small adjustmentto eqn.
(7.41):NuD =(f /8) (ReD − 1000) Pr41 + 12.7 f /8 Pr2/3 − 1(7.43)for 2300 ≤ ReD ≤ 5 × 106 .Variations in physical properties. Sieder and Tate’s work on propertyvariations was also refined in later years [7.13]. The effect of variablephysical properties is dealt with differently for liquids and gases. In bothcases, the Nusselt number is first calculated with all properties evaluatedat Tb using eqn.
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