John H. Lienhard IV, John H. Lienhard V. A Heat Transfer Textbook (776116), страница 54
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The flow field degenerates to greater and greater degrees ofdisorder with each successive transition until, rather strangely, it regainsorder at the highest values of ReD .An important reflection of the complexity of the flow field is thevortex-shedding frequency, fv . Dimensional analysis shows that a dimensionless frequency called the Strouhal number, Str, depends on theReynolds number of the flow:Str ≡fv D= fn (ReD )u∞(7.64)Figure 7.11 defines this relationship experimentally on the basis of about550 of the best data available (see [7.24]). The Strouhal numbers stay alittle over 0.2 over most of the range of ReD . This means that behinda given object, the vortex-shedding frequency rises almost linearly withvelocity.Experiment 7.1When there is a gentle breeze blowing outdoors, go out and locate alarge tree with a straight trunk or the shaft of a water tower.
Wet your§7.6Heat transfer during cross flow over cylinders377Figure 7.12 Giedt’s local measurementsof heat transfer around a cylinder in anormal cross flow of air.finger and place it in the wake a couple of diameters downstream andabout one radius off center. Estimate the vortex-shedding frequency anduse Str 0.21 to estimate u∞ . Is your value of u∞ reasonable?Heat transferThe action of vortex shedding greatly complicates the heat removal process.
Giedt’s data [7.25] in Fig. 7.12 show how the heat removal changesas the constantly fluctuating motion of the fluid to the rear of the cylin-378Forced convection in a variety of configurations§7.6der changes with ReD . Notice, for example, that NuD is near its minimumat 110◦ when ReD = 71, 000, but it maximizes at the same place whenReD = 140, 000. Direct prediction by the sort of b.l. methods that wediscussed in Chapter 6 is out of the question. However, a great deal canbe done with the data using relations of the formNuD = fn (ReD , Pr)The broad study of Churchill and Bernstein [7.26] probably bringsthe correlation of heat transfer data from cylinders about as far as it ispossible.
For the entire range of the available data, they offer1/20.62 ReD Pr1/3NuD = 0.3 + !"1/41 + (0.4/Pr)2/3ReD1+282, 0005/8 4/5(7.65)This expression underpredicts most of the data by about 20% in the range20, 000 < ReD < 400, 000 but is quite good at other Reynolds numbersabove PeD ≡ ReD Pr = 0.2. This is evident in Fig. 7.13, where eqn. (7.65)is compared with data.Greater accuracy and, in most cases, greater convenience results frombreaking the correlation into component equations:• Below ReD = 4000, the bracketed term [1 + (ReD /282, 000)5/8 ]4/5is 1, so1/20.62 ReD Pr1/3"1/41 + (0.4/Pr)2/3NuD = 0.3 + !(7.66)• Below Pe = 0.2, the Nakai-Okazaki [7.27] relationNuD =1 0.8237 − ln Pe1/2(7.67)should be used.• In the range 20, 000 < ReD < 400, 000, somewhat better results aregiven by1/2 1/20.62 ReD Pr1/3ReDNuD = 0.3 + !(7.68)"1/4 1 +282, 0001 + (0.4/Pr)2/3than by eqn.
(7.65).Heat transfer during cross flow over cylinders§7.6Figure 7.13 Comparison of Churchill and Bernstein’s correlation with data by many workers from several countries for heattransfer during cross flow over a cylinder. (See [7.26] for datasources.) Fluids include air, water, and sodium, with both qwand Tw constant.All properties in eqns. (7.65)to (7.68) are to be evaluated at a film temperature Tf = (Tw + T∞ ) 2.Example 7.7An electric resistance wire heater 0.0001 m in diameter is placed perpendicular to an air flow.
It holds a temperature of 40◦ C in a 20◦ C airflow while it dissipates 17.8 W/m of heat to the flow. How fast is theair flowing?Solution. h = (17.8 W/m) [π (0.0001 m)(40 − 20) K] = 2833W/m2 K. Therefore, NuD = 2833(0.0001)/0.0264 = 10.75, where wehave evaluated k = 0.0264 at T = 30◦ C. We now want to find the ReDfor which NuD is 10.75. From Fig. 7.13 we see that ReD is around 300379380Forced convection in a variety of configurations§7.6when the ordinate is on the order of 10.
This means that we can solveeqn. (7.66) to get an accurate value of ReD :ReD =⎧⎨⎡(NuD − 0.3) ⎣1 +⎩0.4Pr2/3 1/4 :0.62 Pr1/3⎫2⎬⎭but Pr = 0.71, soReD =⎧⎨⎡(10.75 − 0.3) ⎣1 +⎩0.400.712/3 1/4 :1/30.62(0.71)⎫2⎬⎭= 463Thenu∞νReD ==D1.596 × 10−510−4463 = 73.9 m/sThe data scatter in ReD is quite small—less than 10%, it wouldappear—in Fig.
7.13. Therefore, this method can be used to measurelocal velocities with good accuracy. If the device is calibrated, itsaccuracy is improved further. Such an air speed indicator is called ahot-wire anemometer, as discussed further in Problem 7.45.Heat transfer during flow across tube bundlesA rod or tube bundle is an arrangement of parallel cylinders that heat, orare being heated by, a fluid that might flow normal to them, parallel withthem, or at some angle in between. The flow of coolant through the fuelelements of all nuclear reactors being used in this country is parallel tothe heating rods.
The flow on the shell side of most shell-and-tube heatexchangers is generally normal to the tube bundles.Figure 7.14 shows the two basic configurations of a tube bundle ina cross flow. In one, the tubes are in a line with the flow; in the other,the tubes are staggered in alternating rows. For either of these configurations, heat transfer data can be correlated reasonably well with power-lawrelations of the form1/3NuD = C RenD Pr(7.69)but in which the Reynolds number is based on the maximum velocity,umax = uav in the narrowest transverse area of the passage§7.6Heat transfer during cross flow over cylindersFigure 7.14 Aligned and staggered tube rows in tube bundles.Thus, the Nusselt number based on the average heat transfer coefficientover any particular isothermal tube isNuD =hDkandReD =umax DνŽukauskas at the Lithuanian Academy of Sciences Institute in Vilniushas written two comprehensive review articles on tube-bundle heat trans-381382Forced convection in a variety of configurations§7.6fer [7.28, 7.29].
In these he summarizes his work and that of other Sovietworkers, together with earlier work from the West. He was able to correlate data over very large ranges of Pr, ReD , ST /D, and SL /D (see Fig. 7.14)with an expression of the form⎧⎨0 for gases(7.70)NuD = Pr0.36 (Pr/Prw )n fn (ReD ) with n = 1⎩for liquids4where properties are to be evaluated at the local fluid bulk temperature,except for Prw , which is evaluated at the uniform tube wall temperature,Tw .The function fn(ReD ) takes the following form for the various circumstances of flow and tube configuration:100 ReD 103 :fn (ReD ) = 0.52 Re0.5D(7.71a)staggered rows: fn (ReD ) = 0.71 Re0.5D(7.71b)aligned rows:103 ReD 2 × 105 :aligned rows:fn (ReD ) = 0.27 Re0.63D , ST /SL 0.7(7.71c)For ST /SL < 0.7, heat exchange is much less effective.Therefore, aligned tube bundles are not designed in thisrange and no correlation is given.staggered rows: fn (ReD ) = 0.35 (ST /SL )0.2 Re0.6D ,ST /SL 2 (7.71d)fn (ReD ) = 0.40 Re0.6D , ST /SL > 2(7.71e)fn (ReD ) = 0.033 Re0.8D(7.71f)ReD > 2 × 105 :aligned rows:staggered rows: fn (ReD ) = 0.031 (ST /SL )0.2 Re0.8D ,Pr > 1(7.71g)NuD = 0.027 (ST /SL )0.2 Re0.8D ,Pr = 0.7(7.71h)All of the preceding relations apply to the inner rows of tube bundles.The heat transfer coefficient is smaller in the rows at the front of a bundle,§7.6Heat transfer during cross flow over cylinders383Figure 7.15 Correction for the heattransfer coefficients in the front rows of atube bundle [7.28].facing the oncoming flow.
The heat transfer coefficient can be correctedso that it will apply to any of the front rows using Fig. 7.15.Early in this chapter we alluded to the problem of predicting the heattransfer coefficient during the flow of a fluid at an angle other than 90◦to the axes of the tubes in a bundle. Žukauskas provides the empiricalcorrections in Fig. 7.16 to account for this problem.The work of Žukauskas does not extend to liquid metals. However,Kalish and Dwyer [7.30] present the results of an experimental study ofheat transfer to the liquid eutectic mixture of 77.2% potassium and 22.8%sodium (called NaK).
NaK is a fairly popular low-melting-point metalliccoolant which has received a good deal of attention for its potential use incertain kinds of nuclear reactors. For isothermal tubes in an equilateraltriangular array, as shown in Fig. 7.17, Kalish and Dwyer give>??sin φ + sin2 φ0.614 @ P − DC(7.72)NuD = 5.44 + 0.228 PeP1 + sin2 φFigure 7.16 Correction for the heattransfer coefficient in flows that are notperfectly perpendicular to heat exchangertubes [7.28].384Forced convection in a variety of configurations§7.7Figure 7.17 Geometric correction forthe Kalish-Dwyer equation (7.72).where• φ is the angle between the flow direction and the rod axis.• P is the “pitch” of the tube array, as shown in Fig.
7.17, and D isthe tube diameter.• C is the constant given in Fig. 7.17.• PeD is the Péclét number based on the mean flow velocity throughthe narrowest opening between the tubes.• For the same uniform heat flux around each tube, the constants ineqn. (7.72) change as follows: 5.44 becomes 4.60; 0.228 becomes0.193.7.7Other configurationsAt the outset, we noted that this chapter would move further and furtherbeyond the reach of analysis in the heat convection problems that it dealtwith. However, we must not forget that even the most completely empirical relations in Section 7.6 were devised by people who were keenlyaware of the theoretical framework into which these relationshad to fit.3Notice, for example, that eqn.
(7.66) reduces to NuD ∝ PeD as Pr becomes small. That sort of theoretical requirement did not just pop outof a data plot. Instead, it was a consideration that led the authors toselect an empirical equation that agreed with theory at low Pr.Thus, the theoretical considerations in Chapter 6 guide us in correlating limited data in situations that cannot be analyzed. Such correlationsOther configurations§7.7can be found for all kinds of situations, but all must be viewed critically.Many are based on limited data, and many incorporate systematic errorsof one kind or another.In the face of a heat transfer situation that has to be predicted, onecan often find a correlation of data from similar systems. This might involve flow in or across noncircular ducts; axial flow through tube or rodbundles; flow over such bluff bodies as spheres, cubes, or cones; or flowin circular and noncircular annuli.
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