Packed-Tube Heat Trans MAX LEVA' Ofice of Synthetic Liquid Fuels, Bureau of Mines, Bruceton, Pa. New data pertaining to heat transfer through packed tubes are presented. The systems investigated were characterized by high ratios (>0.35) of particle to tube diameter. The heat transfer coefficients observed were unexpectedly high. A semiquantitative discussion is presented on channeling caused by wall effect in such systems. The use of high ratios of particle to tube diameter in contact catalysis is undesirable if degree of catalyst contacting and utilization are of prime importance.
Nu/(Re)" against DP/Dt and establishing the equation of the resulting straight line. An equally rigorous correlation of the present data is not quite so simple, because not only the displacement but also the slopes of the individual runs are dependent on Dp/Dt. Accurate correlation would therefore involve finding an expression showing slope in relation to the diameter ratio. Because the final correlation would include this function rn QII exponent to the Reynolds number, the resulting equation would probably be too involved.
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ARLIER investigators (1, 9,4,6,8) of heat transfer through packed tubes were concerned primarily with systems employing particles of such size that the resulting ratios of particle diameter to tube diameter (henceforth referred t o as diameter ratio) were within the limit 0.05 to 0.35. Undoubtedly this arbitrary limitation was influenced partly by commercial operation of fixed-bed catalytic reactors within this range. As investigations of fluid flow through packed vessels ( 5 ) disclosed that pressure drops decreased markedly with increasing diameter ratios, it seemed of interest to ascertain whether a sirhilar decrease in heattransfer film coefficients is also to be expected.
400 300 200
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DATA AND CORRELATION
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1,000
4,000
10,500
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The unit used for these studies has already been described (6) in detail. Steam-jacketed 0.5-inch, 0.75-inch, and 2-inch standard pipes were used with air as the fluid. Table I, the key to Figures 1 and 2, gives a brief orientation. On the assumption th'at the over-all temperature difference between air and steam is numerically equal to the mean logarithmic value obtained from the terminal fluid temperatures, an assumption which has since been verified by observed and integrated-point conditions, overall coefficienta were computed on the basis of the inside vessel diameter. With high steam-film coefficients on the outside, the over-all coefficients for practical engineering purpose: were equal to the inside air-film coefficients. Basic data are giveri in Table IT,
Figure 2.
Original Data Observed in 0.75-11ich Tube
T o arrive a t a simpler (and probably equally useful) form, an arithmetic average value of m = 0.75 was chosen and the ratio R' = N U / R ~ ~was . ? ~plotted against D,/Dt. The plot is shown in Figure 3; the coordinates are logarithmic and thr straight line has the equation:
After evaluation of the experimental constant anti rearrangement,
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TABLE I.
0RIEiXT.lTION O F EXPERIMEiXThL J v O R K A S I ) I ' . l ( ' K [ N l ;
CHARACTERISTICS Estiniatid
Run No. 1 2 3 5
Original Data Observed in 0.5-Inch Tubc
The first steps toniud correlation arc indicated in 1'iyurc.s 1 and 2, where Nusselt-numbers have heen plotted versus nio(lified Reynolds numbers. Because the individunl runs c:iii l)c represented as straight lines when using logarithmic coorcliii:itc~s, Nu = c ( R e ) m . This was also the case Tvith earlicr data ( O ) , and a satisfactory correlation resulted simply by plotting log I
Prebent address, 219 East G a r d e n R o a d . Pittsbiirgli 27, Pa.
2498
Inch 0.172
0.228
0.206
Dr'. Inch 0 622 0 ,824 0,622 0 ,824 2 ,067 0 ,622 0 ,824 0 ,821 0,824 0 ,622 0 . 824 0 ,622 0 622 0 ,632 0 .82 t 0 ,692
0.274 Copper cylinders Porcelain halls 0.733 ti Glass beads 0.228 0.328 7 C l a y halls 0.375 8 Brass rinys 0.373 9 Raschig r i n q Glass heads 0.298 10 11 Glans hcads 0.394 Clay balls 0.301 12 0.350 13 Clay halls 0.368 Glass h c a d s 14 0.499 I5 Porcelain halls 0.411 10 Glass beadv I /?-inch pipe 11.0 i n c h c s . a Pa, c k i n s irrights: 3 + i n c h p i p e s 1 1 . 0 inches. 2-incli p i j j i , 3 4 . 3 1ni.hcs. 4
Figure 1.
I'ackinK
Glass heads Glass heads Glass heads
DP,
Dp/Da 0.270 0.277 0.331 0,332 0.33: 0.366
0.394 0 , 4 5 .i 0.435 0.479 0,479
0.483
0,563 0.591 0.606 0.660
Values of 6
0.4.5 0.45
0.47 0.46 0.48 0.18
0.52 0.72 0.72 0.53 0 . 83 0.85
...
dlorx ?n Figures 1 and 2 0.93 0.88 0.95 0.85 0 . 90 1 00 0.713
0 ,i)!i O.Ffi 0.81) 0.78 0,80 0.80 0 . fi2 0.04 0 fi2
INDUSTRIAL AND ENGINEERING CHEMISTRY
December 1950
2499
A N D CALCULATED DATA TABLE 11. ORIQINAL Run
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Re At q h Nu Glass beads, Dp = 0.172 inch, D; = 0.622 inch, Dp/Dt = 0.276 1-a 12.4 5860 1632 86.4 632 49.5 137.2 b 11.28 5340 1484 83 570 46.3 128 C 8.31 3945 1092 74.6 407 36.7 101 d 6.27 2973 815 72.4 292 27.2 74.1 4.37 2072 566 66.9 199 20 54.4 e f 3.56 1688 454 63 151 16.1 43.5 Glass beads, Dp = 0.228 inch, Dt = 0.824 inch, 4/01= 0.277 1695 89.5 900 40.1 147 2-a 17.08 4605 1508 87.6 785 35.8 131 b 15.2 4100 3570 1313 83 675 13.23 32.5 119 1067 78 533 27.3 100 $ 10.75 2900 855 73.6 409 22.1 81.2 1' 8.57 2320 610 66.3 16.56 60.8 f 6.11 1655 275 Glass beads, D p = '0.206 inch, Dt = 0.622 inch, D p / D i = 0.331 5692 1905 81.4 614 50.9 140.5 3-a 12.0 b 10.48 4960 1665 77 526 46.1 127 4255 1414 76.4 C 8.96 435 38.3 105.8 1082 67.2 322 32.3 88 6.89 3272 d 780 66.6 225 22.1 62.1 5.05 2398 501 65.7 135 13.8 37.2 3.28 1558 Copper cylinders, D p = 0,274 inch, Dt = 0.824 inch, D p / D t = 0.332 23.2 2795 97.3 1247 51.1 19Q 4-a 2085 17.32 94.4 939 b 39.7 177.5 31.2 1506 12.52 87.3 681 116 1122 24.6 91.7 81.7 504 9.32 2625 21.9 189 91.0 1163 50.9 86.6 2045 41.3 896 17.02 153 f 1502 32 12.52 83.0 666 119 1120 25.6 9.32 494 95 77.0 0.733 inoh, D f = 2.067 inohes, D p / D t 0.355 Porcelain balls, D p 5300 6370 116 6260 34.8 329.2 5-a 123.6 b 124.3 5325 6410 110 6500 38.1 358.5 4485 5390 106 5300 32.2 302 104.5 3620 4330 108 4220 25.2 236 84.4 22.9 213.2 2890 3440 98 3480 67.4 2332 2763 92 2850 20 f 54.4 185 1579 1871 94 1845 12.66 117.2 1317 1561 93 1525 10.57 97.8 1069 1255 92 1220 8.55 i 24.9 78.8 j 19.46 835 971 90.5 925 6.59 60.4 k 12.51 537 625 92.5 550 3.84 35.2 Glass beada, Dp = 0.298 inch, Dt = 0.622 inch, 4 / 0 1 = 0.366 6550 2450 85 @-a 13.8 687 54 151 5900 2200 81 b 12.4 617 51 142 81.5 5340 1990 575 47.2 131 11.23 4750 1770 122 72.8 478 44 10.0 4130 1530 71 8.68 400 37.6 103.5 69 3630 1350 33.3 91.4 343 f 7.63 3140 1150 68 290 6.59 28.5 78 64.4 2700 980 245 25.4 69.1 5.69 i 4.50 2140 770 64 189 19.8 52.6 63.6 j 3.48 1655 595 139 14.6 39.4 Clay balls, Dp = 0.325 inch, D; = 0.824 inoh, Dp/Dt = 0.394 4280 120.5 1428 7-a 29.8 178 3685 115 1246 b 25.6 163 110.7 3250 1108 22.6' 149 106.4 985 2855 139 19.9 2420 100.5 16.85 849 126 2020 f 97 14.1 710 110 94.6 1650 92 579 11.48 f; 9.05 78.6 1302 87.1 456 56.1 6.11 i 884 81.8 305 Brass rings, D p = 0.375 inch, Di = 0.824 inch, Dp/Dt 0.465 146 2142 58.4 225 13900 8690 8-a 51.4 b 39.7 10720 6700 136 1757 51.3 198 23.9 6450 4025 121 1095 36.1 139 18.63 5035 3140 107.4 884 32.8 126.5 e 12.55 3380 2108 99.5 598 24.0 92.7 f 7.51 2030 1266 83.6 358 17.1 66
No.
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No. W G A NU Re At q Porcelain Raschig rings, Dp = 0.375 inch, Di = 0.824 inch, Dp/Dt = 0.455 9-a 49.4 13350 8350 149.6 2058 54.7 211 b 36.3 9790 6100 140.3 1630 46.2 178 28.4 7660 4780 130 1330 40.7 157 22.6 6100 3800 128 1080 33.5 129.5 15.0 4050 2520 117.2 747 25.4 98 9.66 2610 1624 104 497 19.04 73.6 0.298 inch, Dt = 0.622 inch, D p / D ; = 0.479 Glass beads, Dp 17.02 3960 107.3 860 53.8 152 10.a b 15.61 3620 104.5 782 50 141 C 3400 103.2 47.1 134 14.68 729 644 96.1 2972 d 12.85 44.8 126 2542 11.02 112 90.5 549 40.5 453 9.28 97.1 2138 86.3 35 334 79.5 7.03 77.5 1620 28 R 11 218 76.3 4.72 52.5 1080 19.1 Glass beads, D p = 0.394 inch, Dt = 0.824 inch, Dp/D; = 0.479 35.5 6190 11-a 1698 55.4 208 121.8 31.3 5450 116 b 1560 53.5 201 25.7 4480 110 1292 175 46.7 20.3 3535 105.2 1029. 38.9 146 15.8 2755 100.3 120 31.9 802 594 91.8 11.65 2030 97 25.8 7.94 86.4 1385 399 69.2 18.45 g Clay balls, Dp = 0.301 inch, DI = 0.622 inch, Dp/Dt = 0.483 12-a 16.1 7625 3763 97.7 841 57.9 164 6830 b 14.4 3365 97.0 742 51.4 144.5 12.8 6070 2985 95.4 656 46.2 129.5 11.2 5300 2597 88.9 574 42.9 119.5 9.77 4640 2260 86.4 492 38.2 106 f 8.09 3836 1872 82.6 32.7 90.5 402 2920 1420 R 6.15 77.1 298 26.0 71.5 h 4.36 2072 1000 75.6 18.24 50.2 205 Clay balls, D p = 0.350 inch, Dt = 0.622 inch, D p / D t = 0.563 13-a 20.4 5740 124.2 896 48.2 138 b 17.52 4735 115.2 130 790 45.8 15.28 4152 112.8 690 116 40.9 13.37 3616 105.3 611 109.3 38.8 12.0 3225 100.3 36.4 102 546 f 94 10.68 2862 97 34.8 489 9.16 2450 86.4 90.1 32.6 421 8.11 2162 83.2 81.6 369, 29.6 6.84 76 1820 74 ;I 27 308 5.56 1472 75.1 59.5 21.9 246 1 4.37 1154 68.2 50.7 18.6 190 822 3.12 63.1 36.7 13.6 128 Glass beads, D p 0.368 inch, Dt 0.622 inch, Dp/Di = 0.591 14-a 21.3 6150 131 138 928 45 b 130 18.25 5240 119 800 41.2 122 15.32 4420 37.6 108 686 33.4 113 564 12.74 3660 94.8 100.5 30.2 455 10.0 2850 84.8 f 7.55 333 2140 24.6 90.5 68.3 5.60 20.5 242 79 56.5 1570 B Poroelain balls, D p 0.499 inch, Dt = 0.824 inch, Dp/Dt = 0.606 15-8 54.6 14750 12250 151 2126 56.0 216 b 47.6 10720 12900 146.2 1905 51.9 200 41.8 11325 9440 143.2 1702 47.3 183 33.9 7620 9150 134.4 1442 42.6 165 27.6 7450 6200 129.8 1185 36.4 141 23.3 6290 5240 125.6 1015 3 2 . 1 124 g 18.7 5050 4205 118.2 821 27.7 107 h 13.12 3540 2945 107.7 566 21 81 Glass beads, D p 3 0.411 inch, Di 0.622 inch, Dp/Dt = 0.660 20.7 16-8 131 131 884 45.3 b 17.42 76 1 45.3 113 131 14.3 115 639 37.3 109 12.02 548 30.9 119 90 422 9.14 91.2 31.1 91 292 6.5 24 82 70 4.92 75.3 215 19.2 56 3.11 12.4 68.4 128 36
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For values of Dp/Dt varying from 0.35 to 0.60, (Di/Dp)o.ob= 1.045 ( *2.50/0), and Equation 2a reduces to
DlSCUSSION
Examination of Equation 2 reveals absence of the tube diameter term; therefore, plotting log hDp/k, against log Re should give a straight line of slope 0.75. This is verified in Figure 4. Different marks have been selected for the 0.5-inch and 0.75inch tube data, and the scatter seems iacking in orientation. For reliable extrapolation to systems of larger vessel diameter, tidditional data would he desirable.
For systems having diameter ratios less than 0.35,the following equation was found valid:
A comparison has been made between Equations 2 and 3, and the results are shown in Figure 5. The data pertain to an air-mass velocity of 3600 pounds per square foot hour, a 0.75-inch diameter tube, and a 212" F. bulk temperature. Using both equations, coefficienb have been calculated for the particlesize range 0.038 to 0.495 inch (diameter ratios 0.05 to 0.66). For zone a-b, Equation 3 is valid, and the new equation would give high values. For the next zone, b-c, Equation 3 is still the
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predicts high values for zone a-b, however. On the other hand, for zone c-d, Equation 3 is not at all in agreement with the data of Colburn and King, suggesting that a significant error would have resulted from an extrapolation of Equation 3 into t,he region df highdiameter ratios. From the preceding discussion, it would appear thiit, from the pressure drop-heat transfer point of view alone,, systems of large-diameter ratios offer considerable advantages over systems of small ratios. Despite this indication, large diameter-ratio systems have not found entry into the catalytic field, because the flow distrihution through a reactor becomes increasingly more unfavorable as the diameter ratio increases. To demonstrate this fact, let part a in Figure 7 be n cross section through a hypothetical packed tube assumed to be stacked with smooth spheres of uniform diameter in
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drop is caused by skin friction. However, the proportion of the pressure drop caused by skin friction on the wall of the vessel in packed tubes is unknown; no immediate theoretical development seems possible along have these lines. investigated Colburn the andpressure King ( q )drop-heat ,however, transfer relationships in packed and baffled tubes in ail empirical way. For baffled and empty tubes, coefficierits were proportional to the 0.44th power of the pressure drop. For packed tubes, on the other hand, this power was considerably smaller than 0.44; indications were that, at high flow rates, heat transfer is virtually independent of pressure
friction on the wall of the vessel decreases rapidly as the flow rate increases. The data from Figure 5 have been ad-
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