Computational Fluid Mechanics and Heat Transfer Third Edition_11 docx

§7.6 Heat transfer during cross flow over cylinders 379 Figure 7.13 Comparison of Churchill and Bernstein’s correla- tion with data by many workers from several countries for heat transfer during cross flow over a cylinder. (See [7.24] for data sources.) Fluids include air, water, and sodium, with both q w and T w constant. All properties in eqns. (7.65)to(7.68) are to be evaluated at a film tem- perature T f = (T w +T ∞ )  2. Example 7.7 An electric resistance wire heater 0.0001 m in diameter is placed per- pendicular to an air flow. It holds a temperature of 40 ◦ Cina20 ◦ C air flow while it dissipates 17.8W/m of heat to the flow. How fast is the air flowing? Solution. h = (17.8W/m)  [π(0.0001 m)(40 − 20)K]= 2833 W/m 2 K. Therefore, Nu D = 2833(0.0001)/0.0264 = 10.75, where we have evaluated k = 0.0264 at T = 30 ◦ C. We now want to find the Re D for which Nu D is 10.75. From Fig. 7.13 we see that Re D is around 300 380 Forced convection in a variety of configurations §7.6 when the ordinate is on the order of 10. This means that we can solve eqn. (7.66) to get an accurate value of Re D : Re D =    ( Nu D −0.3)   1 +  0.4 Pr  2/3  1/4  0.62 Pr 1/3    2 but Pr = 0.71, so Re D =    (10.75 −0.3)   1 +  0.40 0.71  2/3  1/4  0.62(0.71) 1/3    2 = 463 Then u ∞ = ν D Re D =  1.596 ×10 −5 10 −4  463 = 73.9m/s The data scatter in Re D is quite small—less than 10%, it would appear—in Fig. 7.13. Therefore, this method can be used to measure local velocities with good accuracy. If the device is calibrated, its accuracy is improved further. Such an air speed indicator is called a hot-wire anemometer, as discussed further in Problem 7.45. Heat transfer during flow across tube bundles A rod or tube bundle is an arrangement of parallel cylinders that heat, or are being heated by, a fluid that might flow normal to them, parallel with them, or at some angle in between. The flow of coolant through the fuel elements of all nuclear reactors being used in this country is parallel to the heating rods. The flow on the shell side of most shell-and-tube heat exchangers is generally normal to the tube bundles. Figure 7.14 shows the two basic configurations of a tube bundle in a 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 configura- tions, heat transfer data can be correlated reasonably well with power-law relations of the form Nu D = C Re n D Pr 1/3 (7.69) but in which the Reynolds number is based on the maximum velocity, u max = u av in the narrowest transverse area of the passage §7.6 Heat transfer during cross flow over cylinders 381 Figure 7.14 Aligned and staggered tube rows in tube bundles. Thus, the Nusselt number based on the average heat transfer coefficient over any particular isothermal tube is Nu D = hD k and Re D = u max D ν Žukauskas at the Lithuanian Academy of Sciences Institute in Vilnius has written two comprehensive review articles on tube-bundle heat trans- 382 Forced convection in a variety of configurations §7.6 fer [7.26, 7.27]. In these he summarizes his work and that of other Soviet workers, together with earlier work from the West. He was able to corre- late data over very large ranges of Pr, Re D , S T /D, and S L /D (see Fig. 7.14) with an expression of the form Nu D = Pr 0.36 ( Pr/Pr w ) n fn ( Re D ) with n =    0 for gases 1 4 for liquids (7.70) where properties are to be evaluated at the local fluid bulk temperature, except for Pr w , which is evaluated at the uniform tube wall temperature, T w . The function fn(Re D ) takes the following form for the various circum- stances of flow and tube configuration: 100  Re D  10 3 : aligned rows: fn ( Re D ) = 0.52 Re 0.5 D (7.71a) staggered rows: fn ( Re D ) = 0.71 Re 0.5 D (7.71b) 10 3  Re D  2 ×10 5 : aligned rows: fn ( Re D ) = 0.27 Re 0.63 D ,S T /S L  0.7 (7.71c) For S T /S L < 0.7, heat exchange is much less effective. Therefore, aligned tube bundles are not designed in this range and no correlation is given. staggered rows: fn ( Re D ) = 0.35 ( S T /S L ) 0.2 Re 0.6 D , S T /S L  2 (7.71d) fn ( Re D ) = 0.40 Re 0.6 D ,S T /S L > 2 (7.71e) Re D > 2 ×10 5 : aligned rows: fn ( Re D ) = 0.033 Re 0.8 D (7.71f) staggered rows: fn ( Re D ) = 0.031 ( S T /S L ) 0.2 Re 0.8 D , Pr > 1 (7.71g) Nu D = 0.027 ( S T /S L ) 0.2 Re 0.8 D , 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, §7.6 Heat transfer during cross flow over cylinders 383 Figure 7.15 Correction for the heat transfer coefficients in the front rows of a tube bundle [7.26]. facing the oncoming flow. The heat transfer coefficient can be corrected so 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 heat transfer 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 empirical corrections in Fig. 7.16 to account for this problem. The work of Žukauskas does not extend to liquid metals. However, Kalish and Dwyer [7.28] present the results of an experimental study of heat transfer to the liquid eutectic mixture of 77.2% potassium and 22.8% sodium (called NaK). NaK is a fairly popular low-melting-point metallic coolant which has received a good deal of attention for its potential use in certain kinds of nuclear reactors. For isothermal tubes in an equilateral triangular array, as shown in Fig. 7.17, Kalish and Dwyer give Nu D =  5.44 +0.228 Pe 0.614      C P −D P  sin φ + sin 2 φ 1 +sin 2 φ  (7.72) Figure 7.16 Correction for the heat transfer coefficient in flows that are not perfectly perpendicular to heat exchanger tubes [7.26]. 384 Forced convection in a variety of configurations §7.7 Figure 7.17 Geometric correction for the 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 is the tube diameter. • C is the constant given in Fig. 7.17. • Pe D is the Péclét number based on the mean flow velocity through the narrowest opening between the tubes. • For the same uniform heat flux around each tube, the constants in eqn. (7.72) change as follows: 5.44 becomes 4.60; 0.228 becomes 0.193. 7.7 Other configurations At the outset, we noted that this chapter would move further and further beyond the reach of analysis in the heat convection problems that it dealt with. However, we must not forget that even the most completely em- pirical relations in Section 7.6 were devised by people who were keenly aware of the theoretical framework into which these relations had to fit. Notice, for example, that eqn. (7.66) reduces to Nu D ∝  Pe D as Pr be- comes small. That sort of theoretical requirement did not just pop out of a data plot. Instead, it was a consideration that led the authors to select an empirical equation that agreed with theory at low Pr. Thus, the theoretical considerations in Chapter 6 guide us in correlat- ing limited data in situations that cannot be analyzed. Such correlations §7.7 Other configurations 385 can be found for all kinds of situations, but all must be viewed critically. Many are based on limited data, and many incorporate systematic errors of one kind or another. In the face of a heat transfer situation that has to be predicted, one can often find a correlation of data from similar systems. This might in- volve flow in or across noncircular ducts; axial flow through tube or rod bundles; flow over such bluff bodies as spheres, cubes, or cones; or flow in circular and noncircular annuli. The Handbook of Heat Transfer [7.29], the shelf of heat transfer texts in your library, or the journals referred to by the Engineering Index are among the first places to look for a cor- relation curve or equation. When you find a correlation, there are many questions that you should ask yourself: • Is my case included within the range of dimensionless parameters upon which the correlation is based, or must I extrapolate to reach my case? • What geometric differences exist between the situation represented in the correlation and the one I am dealing with? (Such elements as these might differ: (a) inlet flow conditions; (b) small but important differences in hardware, mounting brack- ets, and so on; (c) minor aspect ratio or other geometric nonsimilarities • Does the form of the correlating equation that represents the data, if there is one, have any basis in theory? (If it is only a curve fit to the existing data, one might be unjustified in using it for more than interpolation of those data.) • What nuisance variables might make our systems different? For example: (a) surface roughness; (b) fluid purity; (c) problems of surface wetting • To what extend do the data scatter around the correlation line? Are error limits reported? Can I actually see the data points? (In this regard, you must notice whether you are looking at a correlation 386 Chapter 7: Forced convection in a variety of configurations on linear or logarithmic coordinates. Errors usually appear smaller than they really are on logarithmic coordinates. Compare, for ex- ample, the data of Figs. 8.3 and 8.10.) • Are the ranges of physical variables large enough to guarantee that I can rely on the correlation for the full range of dimensionless groups that it purports to embrace? • Am I looking at a primary or secondary source (i.e., is this the au- thor’s original presentation or someone’s report of the original)? If it is a secondary source, have I been given enough information to question it? • Has the correlation been signed by the persons who formulated it? (If not, why haven’t the authors taken responsibility for the work?) Has it been subjected to critical review by independent experts in the field? Problems 7.1 Prove that in fully developed laminar pipe flow, (−dp/dx)R 2  4µ is twice the average velocity in the pipe. To do this, set the mass flow rate through the pipe equal to (ρu av )(area). 7.2 A flow of air at 27 ◦ C and 1 atm is hydrodynamically fully de- veloped ina1cmI.D. pipe with u av = 2m/s. Plot (to scale) T w , q w , and T b as a function of the distance x after T w is changed or q w is imposed: a. In the case for which T w = 68.4 ◦ C = constant. b. In the case for which q w = 378 W/m 2 = constant. Indicate x e t on your graphs. 7.3 Prove that C f is 16/Re D in fully developed laminar pipe flow. 7.4 Air at 200 ◦ C flows at 4 m/sovera3cmO.D. pipe that is kept at 240 ◦ C. (a) Find h. (b) If the flow were pressurized water at 200 ◦ C, what velocities would give the same h, the same Nu D , and the same Re D ? (c) If someone asked if you could model the water flow with an air experiment, how would you answer? [u ∞ = 0.0156 m/s for same Nu D .] Problems 387 7.5 Compare the h value calculated in Example 7.3 with those calculated from the Dittus-Boelter, Colburn, and Sieder-Tate equations. Comment on the comparison. 7.6 Water at T b local = 10 ◦ C flows ina3cmI.D. pipe at 1 m/s. The pipe walls are kept at 70 ◦ C and the flow is fully developed. Evaluate h and the local value of dT b /dx at the point of inter- est. The relative roughness is 0.001. 7.7 Water at 10 ◦ C flows overa3cmO.D. cylinder at 70 ◦ C. The velocity is 1 m/s. Evaluate h. 7.8 Consider the hot wire anemometer in Example 7.7. Suppose that 17.8W/m is the constant heat input, and plot u ∞ vs. T wire over a reasonable range of variables. Must you deal with any changes in the flow regime over the range of interest? 7.9 Water at 20 ◦ C flows at 2 m/s over a 2 m length of pipe, 10 cm in diameter, at 60 ◦ C. Compare h for flow normal to the pipe with that for flow parallel to the pipe. What does the comparison suggest about baffling in a heat exchanger? 7.10 A thermally fully developed flow of NaK in a 5 cm I.D. pipe moves at u av = 8m/s. If T b = 395 ◦ C and T w is constant at 403 ◦ C, what is the local heat transfer coefficient? Is the flow laminar or turbulent? 7.11 Water entersa7cmI.D. pipe at 5 ◦ C and moves through it at an average speed of 0.86 m/s. The pipe wall is kept at 73 ◦ C. Plot T b against the position in the pipe until (T w − T b )/68 = 0.01. Neglect the entry problem and consider property variations. 7.12 Air at 20 ◦ C flows over a very large bank of 2 cm O.D. tubes that are kept at 100 ◦ C. The air approaches at an angle 15 ◦ off normal to the tubes. The tube array is staggered, with S L = 3.5cmandS T = 2.8 cm. Find h on the first tubes and on the tubes deep in the array if the air velocity is 4.3m/s before it enters the array. [ h deep = 118 W/m 2 K.] 7.13 Rework Problem 7.11 using a single value of h evaluated at 3(73 − 5)/4 = 51 ◦ C and treating the pipe as a heat exchan- ger. At what length would you judge that the pipe is no longer efficient as an exchanger? Explain. 388 Chapter 7: Forced convection in a variety of configurations 7.14 Go to the periodical engineering literature in your library. Find a correlation of heat transfer data. Evaluate the applicability of the correlation according to the criteria outlined in Section 7.7. 7.15 Water at 24 ◦ C flows at 0.8m/s in a smooth, 1.5 cm I.D. tube that is kept at 27 ◦ C. The system is extremely clean and quiet, and the flow stays laminar until a noisy air compressor is turned on in the laboratory. Then it suddenly goes turbulent. Calcu- late the ratio of the turbulent h to the laminar h.[h turb = 4429 W/m 2 K.] 7.16 Laboratory observations of heat transfer during the forced flow of air at 27 ◦ C over a bluff body, 12 cm wide, kept at 77 ◦ C yield q = 646 W/m 2 when the air moves 2 m/s and q = 3590 W/m 2 when it moves 18 m/s. In another test, everything else is the same, but now 17 ◦ C water flowing 0.4m/s yields 131,000 W/m 2 . The correlations in Chapter 7 suggest that, with such limited data, we can probably create a fairly good correlation in the form: Nu L = CRe a Pr b . Estimate the constants C, a, and b by cross-plotting the data on log-log paper. 7.17 Air at 200 psia flows at 12 m/s in an 11 cm I.D. duct. Its bulk temperature is 40 ◦ C and the pipe wall is at 268 ◦ C. Evaluate h if ε/D = 0.00006. 7.18 How does h during cross flow over a cylindrical heat vary with the diameter when Re D is very large? 7.19 Air enters a 0.8 cm I.D. tube at 20 ◦ C with an average velocity of 0.8m/s. The tube wall is kept at 40 ◦ C. Plot T b (x) until it reaches 39 ◦ C. Use properties evaluated at [(20 +40)/2] ◦ C for the whole problem, but report the local error in h at the end to get a sense of the error incurred by the simplification. 7.20 Write Re D in terms of ˙ m in pipe flow and explain why this rep- resentation could be particularly useful in dealing with com- pressible pipe flows. 7.21 NaK at 394 ◦ C flows at 0.57 m/s across a 1.82 m length of 0.036 m O.D. tube. The tube is kept at 404 ◦ C. Find h and the heat removal rate from the tube. 7.22 Verify the value of h specified in Problem 3.22. [...]... M Jacobi, and R K Shah Fluid ow and heat transfer at micro- and meso-scales with application to heat exchanger design Appl Mech Revs., 53(7):175193, 2000 [7.3] W M Kays and M E Crawford Convective Heat and Mass Transfer McGraw-Hill Book Company, New York, 3rd edition, 1993 394 Chapter 7: Forced convection in a variety of congurations [7.4] R K Shah and M S Bhatti Laminar convective heat transfer in... correlating forced convection heat transfer data and a comparison with uid friction Trans AIChE, 29:174, 1933 [7.11] L M K Boelter, V H Cherry, H A Johnson, and R C Martinelli Heat Transfer Notes McGraw-Hill Book Company, New York, 1965 [7.12] E N Sieder and G E Tate Heat transfer and pressure drop of liquids in tubes Ind Eng Chem., 28:1429, 1936 [7.13] B S Petukhov Heat transfer and friction in turbulent... Jr and J P Hartnett, editors, Advances in Heat Transfer, volume 8, pages 93160 Academic Press, Inc., New York, 1972 [7.27] A ukauskas Heat transfer from tubes in crossow In T F Irvine, Jr and J P Hartnett, editors, Advances in Heat Transfer, volume 18, pages 87159 Academic Press, Inc., New York, 1987 [7.28] S Kalish and O E Dwyer Heat transfer to NaK owing through unbaed rod bundles Int J Heat Mass Transfer, ... 105:8990, 1983 [7.16] T S Ravigururajan and A E Bergles Development and verication of general correlations for pressure drop and heat transfer in single-phase turbulent ow in enhanced tubes Exptl Thermal Fluid Sci., 13:5570, 1996 [7.17] R L Webb Enhancement of single-phase heat transfer In S Kakaỗ, R K Shah, and W Aung, editors, Handbook of Single-Phase Convective Heat Transfer, chapter 17 Wiley-Interscience,... York, 1987 [7.18] B Lubarsky and S J Kaufman Review of experimental investigations of liquid-metal heat transfer NACA Tech Note 3336, 1955 [7.19] C B Reed Convective heat transfer in liquid metals In S Kakaỗ, R K Shah, and W Aung, editors, Handbook of Single-Phase Convective Heat Transfer, chapter 8 Wiley-Interscience, New York, 1987 [7.20] R A Seban and T T Shimazaki Heat transfer to a uid owing turbulently... and J S Klein Heat transfer to laminar ow in a round tube or a at platethe Graetz problem extended Trans ASME, 78:441448, 1956 [7.8] M S Bhatti and R K Shah Turbulent and transition ow convective heat transfer in ducts In S Kakaỗ, R K Shah, and W Aung, editors, Handbook of Single-Phase Convective Heat Transfer, chapter 4 Wiley-Interscience, New York, 1987 [7.9] F Kreith Principles of Heat Transfer Intext... convection from gases and liquids to a circular cylinder in crossow J Heat Transfer, Trans ASME, Ser C, 99:300306, 1977 [7.25] S Nakai and T Okazaki Heat transfer from a horizontal circular wire at small Reynolds and Grashof numbers1 pure convection Int J Heat Mass Transfer, 18:387396, 1975 395 396 Chapter 7: Forced convection in a variety of congurations [7.26] A ukauskas Heat transfer from tubes in... properties In T.F Irvine, Jr and J P Hartnett, editors, Advances in Heat Transfer, volume 6, pages 504564 Academic Press, Inc., New York, 1970 [7.14] V Gnielinski New equations for heat and mass transfer in turbulent pipe and channel ow Int Chemical Engineering, 16:359368, 1976 References [7.15] S E Haaland Simple and explicit formulas for the friction factor in turbulent pipe ow J Fluids Engr., 105:8990,... Shah, and W Aung, editors, Handbook of Single-Phase Convective Heat Transfer, chapter 3 WileyInterscience, New York, 1987 [7.5] R K Shah and A L London Laminar Flow Forced Convection in Ducts Academic Press, Inc., New York, 1978 Supplement 1 to the series Advances in Heat Transfer [7.6] L Graetz ĩber die wọrmeleitfọhigkeit von ỹssigkeiten Ann Phys., 25:337, 1885 [7.7] S R Sellars, M Tribus, and J S... sections Then each mechanism is developed independently in Sections 8.3 and 8.4 and in Section 8.5, respectively Chapter 9 deals with other natural convection heat transfer processes that involve phase changefor example: 397 398 Natural convection in single-phase uids and during lm condensation Đ8.2 Nucleate boiling This heat transfer process is highly disordered as opposed to the processes described . Viscous Fluid Flow. McGraw-Hill Book Company, New York, 1974. [7.2] S. S. Mehendale, A. M. Jacobi, and R. K. Shah. Fluid flow and heat transfer at micro- and meso-scales with application to heat. S. Bhatti and R. K. Shah. Turbulent and transition flow convec- tive heat transfer in ducts. In S. Kakaç, R. K. Shah, and W. Aung, editors, Handbook of Single-Phase Convective Heat Transfer, chap- ter. 1933. [7 .11] L. M. K. Boelter, V. H. Cherry, H. A. Johnson, and R. C. Martinelli. Heat Transfer Notes. McGraw-Hill Book Company, New York, 1965. [7.12] E. N. Sieder and G. E. Tate. Heat transfer and