The completely empirical velocity distribution expression presented here has been used to calculate velocity distributions for 57 cases of water flow in smooth and rough pipes over a Reynolds number range of 9200 to 3,240,000. The error between experimental and calculated ratios of point velocity to maximum velocity averaged well below +l%; in only a few cases the error was as much as +2%. The expression meets the physical flow boundary conditions and because of its relative simplicity should be of use to both the industrial and research worker. Nomenclature
C = aconstant D = pipe diameter e
=
pipe roughness
E, = turbulent momentum diffusivity f = Fanning- friction factor g. = Newton’s gravitational conversion factor M = slope of velocity distribution a t wall
S
= = =
u
=
7
R
ue =
U
=
Y
= = ~=
D
T ,
radial position measured from center line pipe radius square of position parameter, (r/R)* time-averaged point velocity time-averaged center line velocity averag-e bulk velocity molecular momentum diffusivity fluid densitv total shear stress in radial direction caused by flow in axial direction
r.. = total shear stress a t wall = flow constant, C, P = flow parameter, C,
CL
Literature Cited
Capps, D. O., master’s thesis, University of Denver, 1966. Carcaran, W. H., Opfell, J. B., Sage, B. H., “Momentum Transfer In Fluids,” pp. 199-204, Academic Press, New York, 1956. Aeronaut.Sci. 1. 1 ,~ , 1 il93lal. (1931a). Karman,T.von, JJ.. Aeronout..Tci. Karman.T.von. ~~..~-,. Karman, T. vo”, Karman; “on, Natl. Advis&y Advisory Comm. Aeronaut., NACA T M 611 (1931b). Laufer, J., Natl. Advisory Comm. Aeronaut., NACA Rept.
1174 (1954). Longwil, P. A.,“Mechanics of Fluid Flow,” Chap. 8, McGrawHill, New York, 1966. Nikuradse, J., Forrchun~shcjt1932, p. 356. Nikuradse, J., Fomhschun&ejt 1933, p. 361. Page, F., Schlinger, W. G., Breaux, D. K., Sage, B. H., I d . E q . Cham. 44, 424 (1952). Prandtl, L., Z. Ver. Dcut. In#. 77, 105 (1933); NACA TM 720 (19111 ~.~
-~,.
Venezian, E., doctoral dissertation, California Institute of Technology, 1962.
DUANE 0. CAPPS Universtty of Denuer
Dcnver, Colo. THOMAS R . R E H M
University of Arizona Tucson, Ariz. RECEIVED for review October 25, 1966 ACCEPTED December 18, 1967
EFFICIENCY STUDY OF JET T R A Y S I N A 6-INCH DIAMETER LABORATORY COLUMN Over-all tray efficiency data were obtained on a 6-inch diameter jet tray laboratory column as a function of vapor rate, hole diameter, and tab angle. Perforated tray efficiencies using the same system were determined in the same column.
little quantitative information bas been puhlished concerning the characteristics of jet trays (although Forgrieve, 1960, reported the results of plant tests) a series of experiments was run to determine pressure drop and entrainment characteristics by air-water testing (Todd and Van Winkle, 1967). I n addition, over-all jet tray efficiencies were determined for four different trays in a 6-inch diameter column using the methyl ethyl ketone-toluene system. Comparative data were determined on a perforated tray with the same system and column. The hole diameters for the jet trays were 0.25 and 0.375 inch and the tab angles were 45“ and 60’ from the plate surface. Hole-free areas were 5.47% for the 0.25-inch and 5.27% for the 0.375-inch diameters. The column had a tray spacing of 18 inches, downcomer area of 5.66%, and weir height of 1 inch, and was operated a t atmospheric pressure, total reflux, and with vapor rates from 370 to 3400 lb./hr.-sq. ft. Details of the column design and operation are available (Rampacek, 1967). BECAUSE
Experimental Results
Three characteristically different ranges of tray operation were observed. One occurred a t vapor rates from 370 to 700 lh./br.-sq. ft. and was considered to be stable because the
Figure 1. Jet tray operation in the lower stable range Vapor flow rote. 694 Ib./hr.-rq. Hole diameter. ‘/&inch Tab angle. 45VOL. 7
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313
- ~ ~ - , Upper renter.
_.
.._I_._
r,
...~
-.~~ .. .
~~
~
,...
Jet troy begins to dump
Upper right. Jet troy continues to dump lower left. Dumping liquid approaches weir Lower center. Blowoff of liquid begins Lover rjghf. Jetting of liquid ~ C T O I I troy, lowest point of unstable cycle
liquid level on rnc rrays did not fluctuate (Figure 1). The second occurred at vapor rates from 700 to 1660 Ib./hr.-sq. ft., and was considered to unstable because the tray liquid and froth built yp and-dumped in a cyclic fashion, causing the bottom tray liquid composition to vary significantly throughout the cyclic period. Under these conditions, a t the point where the liquid height was a t a maximum, the downcomer weir was completely submerged by liquid and froth, and the weir had little, if any, effect in maintaining a froth level on the plate. However, a t the point in the cycle where the liquid height was at the minimum (directly after the liquid had dumped), the weir was not submerged. Figure 2 contains photographs of this range of operation. The third range was stable and occurred a t vapor rates from 1660 to 3400 lb./hr.-sq. ft. Because the liquid froth level was well above the weir, it was concluded that the weir had little or no effect on liquid and froth level. This range of operation is shown in Figure 3. Over-all efficiencies for the jet trays and for one perforated tray were calculated as a function of both F factor (vapor ve314
i & E C PROCESS DESIGN A N D DEVELOPMENT
locity times the square root of the vapor density) and vapor rates (Table I). The velocities were based on the active tray area, which is the cross-sectional area of the tray minus the area of the two downcomers. Vapor densities were calculated hy the method of Pitzer (1955). The efficiency curves in Figures 4 to 6 were fitted from the data by the method of least squares. Discussion
Figures 4 and 5 indicate that efficiency varied somewhat sinusoidally with respect to the independent variables. Using the 45" tab angle, and the 3/8-inch hole diameter tray curve as a reference (Figure 4), the shape of the curves in Figures 4 and 5 can be explained as follows. As the vapor rates are increased from 370 to 925 Ib./hr.-sq. ft., the efficiency decreases, because the liquid is jetted a t an increasins velocity across the tray and into the dawncomer by the increasing vapor rate, resulting in a shorter vapor-liquid contact time. When the vapor rate
Table 1.
G , Lb./ Hr.-Sq. Ft.
Lb./Hr.
H , Inch
663 1845 2035 2600 1721 1410 364 3372 1348 1562 681 1082
110.3 307.5 338.4 433.0 286.5 235 . O 60.7 560.2 224.2 260.3 113.5 180.4
2 10 9 15 7.5
410 745 1350 1916 2262 2633 2861 1078 1552
68.3 124.2 224.9 319 0 377.4 438.5 476.5 179.6 258.4
1941 2693 1240 398 694 952 960 1269 1261 2337 1535 1710
323.5 448.8 206.5 66.2 115.6 158.6 159.9 211.0 210.0 388.6 255.8 284.6
395 1984 2348 2718 2350 1663 1351 1034 689
65.8 330.6 391 . O 452.4 391.5 277.2 224.9 172.2 114.6
1250 1760 2246 2681 778 3418 1062
209.6 294.5 376.0 449.4 130.3 572.0 177.9
GO,
Y,
Mole
70
Vapor Rate and Efficiency Data x,
Mole
Hole Diameter
Unsteady 1.5 17
Unsteady Unsteady 1.5
Unsteady
81.5 81.7 81.2 79.0 84.7 75 .O 80.5 67.2 75 .o 76.3 82.0 81.6
Unsteady Unsteady 8 11 14.5 16 Unsteady Unsteady
87.5 83.7 76.5 84.1 85.2 82.6 72.5 80.5 70.1
Unsteady 1.2 1.5
Unsteady Unsteady Unsteady Unsteady 14 5 7.5
87.4 75.0 74.1 88 .O 75.6 70.0 54.4 62.0 70.5 69.8 85.8 86.1
3/8
33.5 31.2 15.5 13.5 14.0 13.5 6.5 25.8 8.8
Hole Diameter 12 15
3/8
'/a
19.5 12.6 15.5 28.9 21 . o 19.4 13.0 24.9 19.5 6.3 24.4 20.1
Hole Diameter 1.2 7.5 9 17 9
Unsteady Unsteady Unsteady 2
83.2 82.8 82.1 76.4 80.4 73.9 74.5 74.9 82.8
Inch.
17.0 16.1 10.8 15.5 9.1 15.4 23.6 19.3 23.8
(Y
Inch.
7c
A./Sec.
uh,
Ft ./See.
F Factor
65.5 86.5 95.3 92.8 84.6 70.0 81.3 72.1 71.9 78.5 66.1 63.8
17.69 49.12 54.24 69.02 46.02 37.21 9.68 87.91 35.60 41.31 18.20 28.87
1.098 3.052 3.371 4,290 2.860 2.313 0,602 5.455 2.211 2.563 1,130 1.793
0,450 1.249 1.379 1.760 1.170 0.952 0.247 2.259 0,910 1.056 0.462 0.734
11.03 19.91 35.72 51.28 60.59 70.41 75.36 28.61 40.58
0.685 1.238 2.219 3.181 3.763 4.368 4.672 1.779 2,521
0.278 0,505 0.913 1.301 1.539 1.787 1.929 0.732 1.042
50.46 68.52 31.52 10.33 17.67 24.01 23.61 31.58 31.87 58.93 41.37 44.32
3.243 4.410 2.028 0,665 1.138 1.546 1.520 2.032 2.053 3.794 2.659 2.852
1.324 1.817 0.836 0.271 0.468 0.639 0.636 0.845 0.847 1.567 1.085 1.163
10.19 51.09 60.51 69.30 60.44 42.32 34.38 26.36 13.79
0.655 3.284 3.894 4.454 3.881 2.720 2.214 1.693 1.043
0.268 1.343 1.591 1.834 1.587 1.121 0.911 0.698 0.427
17.69 24.80 31.52 37.71 11.03 46.39 14.53
2,072 2.903 3.688 4.410 1.291 5.426 1.701
0.849 1.191 1.520 1.811 0.529 2.268 0.709
E,
u,
Tab Angle 45 2.538 2.719 2.689 2.740 2.574 2.775 2,618 2.860 2.679 2.778 2.538 2.538
27.7 12.7 8.7 8.2 18.4 14 7 15 3 9 .O 15 .O 11.5 28.0 29.2
Hole Diameter 2
70
Tab Angle 60" 2.399 2.459 2.679 2.689 2.621 2.689 2.880 2.559 2,809
75.2 67.4 73.0 89 .O 92.5 86.3 85.9 65.8 77.2
Inch. Tab .Angle 45 2.530 2.778 2.680 2.473 2,640 2.661 2.961 2.711 2.640 2.880 2.473 2.493
90.4 74.3 69.7 79.9 63.2 58.0 47.8 39.9 58.9 83.6 80.9 87.7
Inch. 'Tab .Ingle 60" 2,594 2.594 2.718 2.679 2.740 2.680 2,620 2.661 2,536
83.5 84.5 90.9 72.8 92.1 69.6 58.3 64.5 73.5
Perforated Trays. Hole Diameter 3 / ~ aInch 3.5 5 9 12.5 3 17 3.5
82.2 78.2 76.4 77.3 83.5 57.5 61.9
21 .o 14.3 8.3 7.4 19 .O 5.1 4.3
reaches 925 lb./hr.-sq. ft., the vapor is jetting the liquid and froth across the plate with just enough force either to move it directly into the downcomer or to impinge it onto the wall behind the downcomer and then immediately into the downcomer. Thus, at this particular rate, vapor-liquid contact time i s minimum because the frothy liquid is carried away by the downcomer and is not building up on the plate by recirculation. As the vapor rate increases above 925 lb./hr.-sq. ft., the vapor starts to jet the froth and liquid across the tray and to establish recirculation by impingement on the column wall. The jetted liquid recirculates until a dynamic balance is established between froth height and downcomer capacity. Thus, because the frothy liquid remains on the plate for an increased time and interfacial contact is greater, vapor-liquid contact time is longer and efficiency increases. As the vapor rate con-
2,558 2.711 2.807 2.880 2 576 2.981 2 981
76.0 76.9 86.6 88.7 81 2 73 8 82.1
tinues to increase to 2300 Ib./hr.-sq. ft., the jetting effect increases, causing increased impingement and increased frothy liquid holdup and interfacial area of contact. However, when the vapor rate reaches 2300 lb./hr.-sq. ft., liquid and frothy entrainment increases to a point where the efficiency begins to decrease. At vapor rates beyond 2300 lb./hr.-sq. ft., the efficiency continues to decrease because of increasing entrainment. I n this experiment, it is believed that the abnormally high liquid froth levels on the trays resulted from inadequate downcomer capacity for handling the frothy liquid. I n normal jet tray operation in larger diameter columns, little liquid buildup should occur. However, in this experiment, a very high internal liquid doit nflow resulted from total reflux operation and the introduction of subcooled reflux. In addition, the VOL. 7
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315
Figure 3.
If tray operation could be conducted with a substantial liquid froth level on the trays, the plate efficiencies of both jet and perforated trays could he significantly increased, if pressure drop and capacity limitations were not exceeded. Meaningful scale-up of efficiencies from the 6-inch column used in this experiment to commercial size equipment is not possible. The jetting effect produced by the tabs should he primarily a function of the vapor rate and not of the column diameter. Therefore, as the column diameter increases far a given vapor rate, the trajectory of the jetted liquid will not change, hut the path of flow and therefore the time of contact will increase, and the allowable vapor rate limit will increase. Also, as the column diameter increases, the percentage of material relative to the total tray inventory jetting directly into the downcomer will decrease and the efficiency should increase. Probahly the wide variation in efficiency encountered in this work should he much less for larger diameter columns. Hole Diameter. The plates with the 3/&ch diameter holes give slightly higher efficiencies a t both the high and low peaks of the curves (Figures 4 and 5). The efficiency of the '/t-inch hole diameter trays is higher than that of the 3/~-inch trays at very low vapor rates, hut lower a t very high flow rates. This observation agrees with that for perforated plates (Hellumsctal., 1958). T a b Angle. T h e difference in efficiency between the high and low peaks of the curve for the 60' tab angle trays is not as great as that of the 45' tab angle trays. This indicates that as the tab angle of the jet trays approaches 90°-i.e., perforated trays-the operating flexibility increases somewhat. Perforated Tray. The efficiency curve for the perforated trays does not fluctuate as widely as for the jet trays. This indicates that for the 6-inch diameter column the range of suitable operation of the perforated tray is slightly greater than that of the jet trays, although the comparison of the tray performances is not strictly valid because the hole diameters are different,
Jet tray operation in the higher stable range Vapor flow rate. Hole diameter. Tab angie. 45'
1856 ib./hc.+q. ft. 1/4
inch
froth density was low, resulting in a large froth volume which apparently overloaded the downcomer, and thus the frothy liquid level increased as the vapor rate (and the internal liquid flow) was increased. This buildup also occurred for the perforated trays, hut to a lesser extent. Because of this liquid buildup, the vapor-liquid contact time was greater than for normal operation in which relatively little liquid buildup occurs. Since increased vapor-liquid contact times increase plate efficiencies, the observed high peak efficiencies of approximately 90% for both jet and perforated trays are consistent.
.-
40
-
0
0.25
0.50
0.75
1.00
1.25
F
Figure 4.
FACT,
Comparison of jet tray hole dib
..._._._._. - __.._.
Tab angle. 45' 0 Hole diameter. 3/8 inch A Hole diameter. inch
316
l & E C P R O C E S S D E S I G N A N D DEVELOPMENT
_..=._
100
90
8
80
0
z
w0 E 70 I&
W
60
50
,
40
0
0.25
0.50
1.00
0.75
1.25
F Figure 5.
1.50
1.75
I
2.00
,
,
,
2.25
2.50
FACTOR
Comparison of jet tray hole diameters for a constant tab angle Tab angle. 60" 0 Hole diameter
A
inch Hole diameter l/4inch
z W
50
t
LI
40 o 0
1
025
370
050 740
1
1
,
075
100
1110
1480
'
1
I
l
l
'
~
~
125
150
175
1850
2220
2590
200 2960
225 3330
'
1
1
1
250
3700
F FACTOR VAPOR RATE (Ibs/hr-ft')
Figure 6.
Efficiency as a function of F factor and vapor rate Perforated trays Hole diameter. 3 / 1 6 inch W e i r heighi. 2 inches
VOL. 7
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1968
317
= average relative volatility Vapor rate based on active area of tray
Acknowledgment
a
The authors express their gratitude to the National Science Foundation and to the University of Texas Graduate School for the financial support which made this investigation possible.
Literature Cited
Nomenclature
E = over-all plate efficiency, % F = F factor, (lb.)1/2/sec.-sq.ft. G = vapor rate, Ib./hr.-sq. ft. Go = vapor rate, lb./hr. H = observed froth height, inches u = superficial vapor velocity, ft./sec. uh = hole velocity, ft./sec. x = lowest plate liquid composition, mole % MEK y = top plate vapor composition, mole % M E K
Forgrieve, John, Proceedings of International Symposium on Distillation, Brighton, England, p. 185, Institution of Chemical Engineers, London, 1960. Hellums, J. D., Braulick, C. .J., Lyda, C. D., Van LVinkle, Matthew, A.I.CI1.E. J. 4, 465 (1958). Pitzer, K. S., J . Am. Chem. SOC. 77, 3427 (1955). Rampacek, C. M., thesis, University of Texas, 1967. Todd, LV. G., Van LVinkle, Matthew, IND.ENG.CHEM.PROCESS DESIGN DEVELOP. 6, 95 (1967). C. M. RAMPACEK MATTHEW VAN WINKLE L'niversity of Texas Austin, T e x . RECEIVED for review January 23, 1967 A C C E P T E D November 24, 1967
CORRESPONDENCE SIMULATION OF T H E HEAT TRANSFER PHENOMENA IN A ROTARY K I L N
SIR: A recent paper (Sass, 1967) reported the results of research efforts directed toward modeling the heat transfer which takes place in rotary kilns. Having performed similar work in the past, Lve would like to comment on and enlarge upon the author's findings. Our curiosity was aroused by Figure 3 of the paper, which depicts calculated temperature profiles for both the gases and solids in an ore heating kiln, together with observed gas temperatures at several locations. These calculated profiles, M hile appearing to agree with the observed data, do not agree with either results \ve have obtained for similar simulations or with intuitive reasoning. To aid in the discussion, the author's Figure 3 has been redrawn uith added markings of our o n n . We submit that the true shape of the temperature profiles is more nearly approximated by the dashed lines in the modified Figure 3 than by the himulation results presented in the paper. Our reasoning is as follo\vs: 1. Even with the perfect mixing assumed by the author, one
-0
MEASURED TEMPERATURES COMPUTER RESULTS EXPECTED RESULTS
1600
c
/
OAS
w 12003
iw! Boo-
1
0
Figure 3. (modified) 318
20
40 60 80 PERCENT OF KILN LENGTH
100
Temperature profiles in an ore heating kiln
I&EC PROCESS DESIGN AND DEVELOPMENT
cannot expect the temperature of the solid material to increase by some 1100" F. in 15 feet of kiln length. There are no exothermic Ieactions taking place, and flame radiation is neglected. The heat transferred to the material will: as the material travels down the kiln and the water evaporation nears completion, cause a more gradual increase in material temperature. One must assume that the amount of heat used to evaporate the water will become less as the water content diminishes, and the material will not stay at 212' F. until all the water has been evaporated. At some critical moisture content, depending on the physical characteristics of the material as \vel1 as the degree of mixing, the drying rate will begin to decrease owing to the bound moisture present. At the lower drying rates, more heat \vi11 be transferred to the material than is consumed in the evaporation of the water; the excess heat will increase the temperature of the material. By the time "all" the water is gone, the material temperature may be significantly above 212" F. The lower dashed line in the sketch is therefore more likely to represent the actual conditions than is the author's calculated result. 2. In the author's simulation, all of the heat received by the material is assumed to have been transferred by the gases, either directly by convection and radiation or indirectly through the kiln lining. With no flame radiation, the slope of the gas temperature profile \vi11 continue to increase (reading left to right) as the end of the kiln is reached, as the simulation results show. The data points, however, can be better fit with a straight line than \vith the curved profile resulting from the simulation; the presence of an inflection point indicates that flame radiation exists, and the assumption is not valid. (If a large amount of radiant heat is transferred from the flame, such as in a cement kiln, the solid material ill receive heat directly from the flame, and the gas temperature profile will become flat and asymptotic to the flame temperature.) By ignoring flame radiation, to both the solid material and to the inner kiln wall, the entire gas and solid temperature profiles will be distorted. The differences betLveen the calculated temperature profiles
