Heat and Vapor Transfer in a
Wetted-Wall Tower WALTER I. BARNETl AND KENNETH A. KOBE University of Washington, Seattle, Wash.
-4pparatus and Method
H e a t and vapor transfer data were obtained from the operation of a wetted-wall tower with countercurrent flow of liquid and air. Three series of runs were made with different liquids-two with light paraffin-base oils to obtain data on heat transfer from gas to liquid without simultaneous vapor transfer, and the third series with water so that a study might be made of simultaneous heat and vapor transfer. In all cases the gas film was found to control the rate of heat transfer. Correlation of the data b y means of the general Nusselt-type equations gave:
The apparatus consisted of a wetted-wall tower (Figure I), a steam heat exchanger for preheating the air, an air lift for lifting the oil to the top of the tower, and a small air pump. The main section, A , of the tower was a 4-foot length of seamless steel pipe (standard 1-inch iron pipe size), and the lower calming section, B, was a 5-foot length of the same material. The upper calming section, C, was standard I-inch pipe, 3 feet long for the oil runs and 1 foot long for the water runs. The sections were connected by two standard 3-inch iron couplings, equipped with bushings and drilled to allow for flow of liquid into the top coupling, D, and out of the lower one, E. The top end of the main section was turned to a sharp edge on the inner side, and the top calming section was set in place so that only sufficient space was allowed for passage of liquid into the tower.
h e d / k = 0.028 (R0)O.S hvd/pDv = 0.016 (R0)O.W According to the Chilton-Colburn theory: jH =
0.031 (Re)-O.2
j D = 0.025 (Re)-O.2 According to the A r n o l d theory:
dr/BH = 0.021 (Re)O.82 dr/Bo = 0.022 (Re)O.82 The ratio of
hH/h, i s in the range of 0.25 to 0.27.
HE concept of a fluid film or laminar layer existing beT tween a solid body and a fluid flowing turbulently past it is well supported by numerous investigations. In the instances of heat and mass transfer between the phases, this laminar layer offers a t least part of the resistance to the transfer process. The remainder of the resistance occurs in the main body of the fluid. The total resistance through the fluid to its interface is generally attributed to a hypothetical film having an effective film thickness proportional to the total resistance. Unless otherwise designated in this article mention of fluid film refers to the "effective" film. When two fluids are moving relative to each other and are separated by a definite interface, two fluid films will offer resistance to heat transfer. I n the special case in which the fluids are a volatile liquid and a gas, only the gas film is concerned in a transfer of the vapor. B u t for heat transfer one of two possibilities must be considered, either both films offer appreciable resistance or one film has a negligible resistance as compared to that in the other film. I n this latter case the effect of fluid velocity upon the high resistance or controlling film may be studied directly. The object of this investigation was to obtain data on the rates of heat transfer from a gas to a liquid, with and without the simultaneous occurrence of mass transfer, and to plot these data by the methods given in the literature. Because of the scarcity of information on heat transfer between a gas and a liquid without simultaneous mass transfer, this case is of particular interest, especially as it was thought that only the gas film might control the rate. 1
FIGURE 1. DIAGRAM OF WETTBDWALLTOWER
The lower end of the main section was turned to a sharp edge on the outer side, and a cone of light sheet metal, F , was affixed to it to carry the liquid out and away from the air stream. The air line, leading from the preheater t o the tower, and t'he tower and calming sections were insulated with 1-inch-thick corrugated asbestos. Portions of the tower such as the couplings and the liquid outlet line, G, were covered with 1 inch of asbestos cement. The top 8 inches of the lower calming section consisted of 1-
Present address, Federal Power Commiesion, Washington, D. C .
436
April, 1941
INDUSTRIAL AND ENGINEERING CHEMISTRY
inch hard-rubber tubing to keep heat from flowing up the lower calming section and into the outgoing liquid. A constant-head device (not shown) ensured a constant rate of liquid feed to the top of the tower throughout the run, the amount of the feed being controlled by needle valve H. A liquid seal on the liquid outlet line kept air from blowing out at this point. For the runs in which oil was used, an air lift raised the oil from the reservoir a t the bottom of the tower to the constant-head device at the top. Thus the oil was recirculated continuously during the operation of the tower. A constant ingoing oil temperature was made possible by means of a cold water cooling coil in the reservoir. For the water runs city water was piped directly to the constant-head device, the water temperature being regulated by the introduction of steam into the line. Liquid discharge rates were determined as the liquid came from the tower by direct measurement with a graduate and stop watch. The air was supplied by a small rotary pump. From the pump the air passed through a &foot calming section to the flowmeter and then through the reheater to the tower. The air flow was measured by a 3/le-incg Venturi meter for the low velocities and a */s-inch sharp-edged orifice for the high velocities. They were previously calibrated against a dry gas meter which had been checked by a wet test meter. Short-stem mercury thermometers, accurate to 0.1 C., were used to measure the temperatures of both liquid and air. The ingoing liquid temperature was obtained by inserting a thermometer, J , in the upper coupling which was always partly filled with liquid passing directly into the tower. The outgoingliquid thermometer, K , was inserted into the lagged outlet pipe leading from the lower cou ling. Air temperatures were obtained by placing thermometers directly in the air stream below the lower and above the up er couplin s. Due care was taken to keep the thermometers f r y during &e course of a run. Errors in air temperature measurement due to radiation were considered negligible, as the inside wall temperatures were close to the temperatures of the air a t the points at which the thermometers were inserted. For each run the liquid and air rates were maintained constant, and the tower was operated until temperature readings were constant for a 10- or 15-minute period. For the water runs the humidity of the ingoing air was determined by a sling psychrometer, and the humidity of the outgoing air by means of the wet- and dry-bulb thermometers in the top of the tower. The wet-bulb temperature was obtained by wetting the cloth bulb cover with water several degrees higher than the wet-bulb temperature of the air, placing the thermometer in the air stream just above the dry-bulb thermometer, and noting the minimum reading; care was taken that the wick did not dry out in the meantime.
437
the top and bottom of the tower was used, except when the ratio of the difference was less than two, in which case the arithmetic mean is sufficiently accurate. For the calculation of the vapor transfer coefficient h, i n the general equation, (2)
h, = w / A ( A H ) , , .
a procedure was followed similar to that used for the calculation of hx. The same value of A was used as before. The rate of vapor transfer, w , was obtained as the product of the weight rate of air flow and the change in absolute humidity of the air on passing through the tower, the absolute humidity being determined from the psychrometric chart with the aid of the adiabatic cooling lines. The barometric pressure was close to 760 mm. so t h a t no correction for its effect upon the
O
%
Two grades of light paraffin-base oil were used for the oil runs. Oil 1 was a mineral seal oil and oil 2 was a half-andhalf mixture of mineral seal oil and of SAE 30 lubricating oil. The physical properties of oil 1 are shown in Figure 2 and in the following table: Density
Temp.
c.
Gram/cc.
22.5 0.861 0.850 40.0 0.840 56.0 70.8 0.830 73.0 o:iis 90.2 92.0 Specific heat, 0.45 cal./(gram)(' C . )
...
Viscosity Centipoises
11.39 6.25 4.01
... ...
2.72 1.94
Data and Calculations For the calculation of the over-all heat transfer coefficient, h,, the general equation for heat transfer was used: hx
n/A(At),v.
(1)
The rate of heat transfer, 4, was obtained as the product of the weight rate of air flow, the specific heat, and the change in temperature of the air on passing through the tower. The interfacial area, A , for the tower was calculated using the nominal inside diameter of the pipe (1 inch) rather than the actual diameter (1.049 inches) ; thus allowance was made for the thickness of the layer of falling liquid. For determining the value of the average temperature difference, the log mean of the difference of air and liquid temperatures at
0.801 0
20
FIQURE 2.
40
60
80
0 100
DENSITYAND VISCOSITY FOR OIL1 CURVES
absolute humidity was necessary. For the calculation of the humidity gradients, the absolute humidity of the air in equilibrium with water at the particular water temperature was read from the psychrometric chart and taken as the humidity on the water side of the film. T h e method of obtaining the average humidity difference, (AH),,, was t h e same as t h a t employed in obtaining the average temperature difference . A sample calculation (for water-air run 21) is as follows:
d (diameter of tower) = (1.00)/(12) = 0.0533 ft. A (interfacial area in tower) = (4.0)(3.14)(0.0533) = 1.05 sq. f t , q (rate of heat transfer baaed on temperature change of air) = (0.24)(1.8)(14.3)(22.9) = 141 B. t. u./hr. ( At),Y. (log mean temperature difference across interface) = (31.8 - 8.0)(1.5)/(ln 31.8/8.0) = 31.1" F. hH (beat transfer coefficient = q/A( At),,. = (141)/(1.05)(31.1) = 4.32 B. t. u./(sq. ft.)(hrj('F.) p, (viscosit of air a t average temperature of film) = 0.0465 lb./(hr.)&.) at 43.3" C . k (thermal conductivity of air a t avera e temperature of main body of air) = 0.0145 (B. t. u.)(ft.y/(sq. ft.)(hr.)(' F.) at 53.2" C. hHd/k (Nusselt number) = (4.32) (0.0533)/(0.0145) = 24.3 p j / p / D y = 0.60 (for all temperatures for water vapor and air) c,uf/rC, = 0.76 (an average value for air) G' (mass velocity of air per unit cross section of tower) = 4G/ad2 = (4) (14.3)/(3.14)(0.0S33)2 = 2625 lb./(hr.)(sq. ft.) j , (Colburn's heat transfer number) = (hx/cG')(cpj/IC/)2/s = (4.32)(0.76)*/3/(0.24)(2625) = 0.00571 +x = 1 - r r(cp,/kf) = 0.854 (values of + H were obtained h-om a plot of + H us. Re) dr/BH (dimensionless heat transfer number by Arnold's theory) = dhH+x/Cpf = (0.0833) (4.32) (0.854)/(0.24) (0.0465) = 27.5 w rate of va or diffusion based on change of humidity of air) = [14.3)(0.02%l - 0.0111) = 0.227 lb. water/hr.
+
INDUSTRIAL AND ENGINEERING CHEMISTRY
438
Vol. 33, No. 4
ascertaining of whether or not the gas film controlled the rate of heat transfer, was carried out in the following manner: Expecting that final results would agree with the experience of other l\-orkers as to the effect of gas velocity upon the rate
(log mean humidity difference across interface) = (0,0212 - 0.0070)/(ln 0.0212/0.0070) = 0.0128 Ib. water/lb. bone-dry air h, (vapor transfer coefficient) = ~ / A ( A H ) B Y=. (0.227)/(1*05) (0,0128) = 16.9 lb. water/(sq. ft.)(hr.) p (density of air a t average temperature of main body of air) = 0.0675 lb./cu. ft. D, (diffusivity of water vapor at average temperature of main body of air) = (0.853) ((53.2 273)/273]1.76= 1.164 sq. ft./hr. h,d/pD, (dimensionless diffusion number equal to d / x ) = (16.9)(0.0833)/(0.0675)(1.164) = 17.9 j , (Chilton and Colburn's vapor diffusion number) = (h,/G') ( p / / p / ~ , ) ~ /= * (16.9)(0.60)2/s/(2625) = 0.00487 +D = 1 r r ( p / / p , D v ) = 0.758 (values of +D were obtained from a plot of + D us. Re) (dimensionless vapor diffusion number by Arnold's theory) = d h , , + ~ / p f = (0.0833)(16.9)(0.758)/(0.0465) = 22.9 Re = dG'/pf = (0.0833)(2625)/(0.0465) = 4700 hH/h, = 4.32/16.9 0.256 T = 1.75 (Re)-'/8 = (1.75)(4700)-1/8 = 0.608 B H = (0.0833)(0.608)(12)/(27.5) = 0.022 inch BD = (0.0833)(0.608)(12)/(22.9) = 0.027 inch (AH),,
+
- +
The various heat and vapor transfer numbers, as calculated above, were plotted against the corresponding Reynolds numbers on a log-log scale (Figures 4 to 7), and the equations for the best straight lines representing the data were calculated by the method of least squares (Tables I and 11).
Discussion of Results
ooo
The data were plotted by several of the methods suggested by previous investigators. -4 prerequisite to this study, the
TABLEI. DATAAND Watez Temp., R , , ~ Air Temp., ' F. F. No. Bottom Top Bottom T op 170.4 110.3 79.3 75.6 169.3 109.4 77.7 75.2 167.0 107.8 78.1 75.2 164.8 104.9 77.0 73.4 157.1 100.4 76.5 74.3 176.9 114.8 82.0 75.2
018
a24
FIGURE 3. PLOT OF l/hH l/Ga.a
CALCULATED
Humidity, Lb. H20/Lb. Bone-Dry Air Rate, Lb./Min. Bottom Top Air Water 0 0112 0 0163 0.610 1 . 3 1 0 0110 0 0159 0.558 1.31 0 0110 0 0161 0.484 1 . 3 1 0.0110 0 0161 0.434 1 . 3 2 0.313 1.29 0 0109 0.0160 0.735 0.87
012
006
1
1
hvd
hu&
pDv
i~
jD
0,116 0.127 0.143 0.154 0.198 0.101
47.5 43.5 38.8 36.0 28.2 54.2
32.2 28.2 25.9 30.4 19.8
0.00444 0.00445 0.00456 0.00472 0.00506 0.00423
0,00325 0.00311 0.00330 0.00429 0.00385
0.053
0,129 0.124 0.156 0.178 0.229 0.109
43.0 44.5 35.7 31.3 24.4 50.2
46:i 33.6 33.4 17.7 38.6
0.00436 0.00453 0,00468 0,00486 0.00493 0.00443
0.068
0.074 0.095 0.04s
.. . . . 0.00507
dr BH
dr BD
56.0 51.3 45.3 42.0 32.4 64.6
43.3 37.9 34.6 40.2 25.8
..
12,050 11,060 9,570 8,600 6.230 14;460 11,190 11,040 8,530 7,240 5,520 12,650
168.8 iio.4 167.0 162.5 156.9 168.8
103.5 iii.z 108.5 105.8 104.0 122.0
70.2 81.3 80.2 79.9 80.2 92.8
63.7 75.1 77.7 77.9 78.8 97.3
0.0112 0.0134 o.oii30.0i90 0.0105 0.0182 0.0105 0,0187 0.0113 o.oisi 0.0122 0.0282
1.26 0.5601.26 0.432 1 . 1 9 0.365 1.06 0.278 0.88 0.650 1.01
58.7
6i:g 44.8 43.9 22.9 51.6
13 14 15 16 17 18
171.5 172.4 167.0 162.5 153.5 152.6
121.1 119.3 120.4 118.8 113.9 112.1
91.8 91.4 92.1 93.9 94.3 93.9
96.1 95.5 95.9 97.9 96.3 96.3
0.0103 0.0105 0.0107 0.0110 0.0110 0.0107
0.0277 0,0281 0.0277 0,0290 0,0295 0,0296
0.724 0.770 0.523 0.406 0.320 0.276
1.01 0.97 0.97 0.99 1.65 1.30
0.049 0.047 0.064 0.078 0.094 0,106
0.094 0.083 0.135 0,165 0.198 0.213
5s.1 65.8 40.6 33.5 28.0 26.2
47.0 54.9 34.6 25.9 23.2 20.8
0,00462 0.00491 0.00446 0.00469 0.00497 0.00534
0.00403 0.00442 0.00409 0.00392 0.00444 0.00459
68.2 77.4 47.4 38.6 32.2 29.9
63.0 73.8 45.9 33.8 30.1 26.8
14,080 13,020 10,210 7,930 6,280 5,430
19 20 21 22 23 24
162.5 151.5 148.5 143.6 137.8 131.4
117.3 108.5 107.2 105.1 103.1 100.8
93.2 90.3 91.2 91.2 91.6 91.9
96.4 92.1 92.8 93.0 93.2 93.4
0.0106 0.0110 0.0111 0.0111 0.0111 0,0110
0.0282 0.0263 0.0270 0.0278 0,0284 0,0289
0.416 0.274 0.238 0.202 0,167 0.134
1.32 1.28 1.28 1.28 1.28 1.28
0.076
0.156 0.211 0.231 0.257 0.297 0.340
35.3 26.5 24.3 21.9 19.1 16.8
27.2 20.3 17.9 16.6 14.3 12.1
0.00484 0.00345 0.00571 0,00607 0,00637 0.00691
0.00402 0.00460 0,00457 0,00496 0,00519 0,00544
40.9 30.2 27.5 24.8 21.4 18.7
35.7 26.1 22.9 20.9 18.0 15.1
8,150 5,360 4,700 3,980 3,310 2,660
25 26 27 28 29 30
125.4 119.3 105.6 169.0 175.3 176.9
98.6 97.7 94.6 122.9 123.1 124.7
91.9 92.1 92.5 91.8 92.1 90.7
93.2 93.4 93.2 98.4 99.3 99.3
0.0105 0.0291 0.0108 0.0286 0.0103 0.0293 0.0105 0,0273 0.0107 0.0279 0.0106 0.0264
0.102 0.075 0.051
1.28 1.28 1.28 0.77 0.73 0.47
0.234 0.301 0.409 0,054 0.043
0.410 0.559 0.685 0.112 0,084 0.081
14.0 10.3 8.5 48.9 64.8 67.5
9.8 6.6 5.0 36.9 44.7 44.3
0,00753 0.00752 0.00902 0,00433 0.00474 0.00453
0.00972 0.00520 0.00573 0,00352 0,00352 0.00320
15.4 11.2 9.1 57.2 76.5 79.9
11.9 7.9 5.8 49.1 60.1 59.9
2,030 1,500 1,030 12,620 15,340 16,740
31 32 33 34 35 36 37
177.4 128.5 177.8 128.3 177.8 125.6 177.8 122.0 172.4 122.7 158.0 116.6 164.3 118.4
91.8 91.9 92.5 93.2 95.6 93.9 96.4
97.0 98.2 99.7 100.2 99.7 99.5 99.5
0.0110 0.0115 0.0124 0.0132 0.0118 0,0109 0.0110
0.915 Or16 0.920 0 . 2 5 0.920 0.41 0.920 0 . 5 5 0.645 1.28 0.312 0 . 6 8 0.415 1 . 5 0
0.041 0.040 0.040 0.040 0.054 0.095 0.076
0 088 0.085 0.076 0,066 0,102 0 200 0.147
62.1 64.3 71.9 82.8 53.8 27.8 37.6
36.3 39.7 47.6 55.5 41.2 27.7 29.6
0.00392 0,00404 0.00452 0.00520 0.00479 0.00505 0.00617
0.00249 0.00271 0.00325
73.3 75.9 84.9 98.0 62.9 31.8 43.4
49.4 54.0 64.7 75.0 55.1 28.2 38.8
17,740 17,830 17,830 17,870 12,530 6,090 8,100
0.0235 0.0250 0,0280 0.0303 0.0309 0,0309 0.0320
0.648
0.790 0.862
0.060
0.074 0.085 0.105
0.107 0.119 0.136 0.158 0.189
0.046
0,00480 0.00560 0.00387 0.00369
0.00375
0.00398 0,00428 0.00440
50.4 52.4 41.5 36.3
Re
7 8 9 10 11 12
0.560
0,060
AGAINST
RESULTB FOR JI-ATER
k 0.056 0.060
030
28.0
INDUSTRIAL AND ENGINEERING CHEMISTRY
April, 1941
TABLE11. DATA AND CALCWLATED RESULTS FOR OILS 1 Run No.
Air Temp., F. Bottom Top
Oil Temp., O F. Bottom Top
Rate, Lb./Min. Air Oil ’
1 G0.S
hHd k
1 hH
Oil 1 (Mineral Seal Oil) 0.926 0.093 0.155 0.099 0.171 0.785 0.720 0.117 0.209 0.136 0.249 0.692 0.673 0.172 0.310
AND
439
2 dr iH
ir;;
Re
18.1
0.00624 0.00601 0.00619 0.00626 0.00678
0.095 0.100 0.097 0.09% 0.094
0.166 0.180 0.187 0,146 0.200
32.3 29.8 28.8 38.6 28.0
0.00598 0.00586 0.00544 0.00686 0.00495
38.7 35.7 34.5 44.2 32.2
6,270 5,910 6,140 6,240 6,300
0.078 0.063 0.054 0.072 0.072
0.149 0.114 0.056 0.118 0.136
37.3 48.5 98.5 47.3 40.6
0.00522 0.00527 0.00874 0.00596 0.00514
43.4 57.1 116.4 55.1 47.7
8,020 10,400 12,740 8,910 8,910
0.107 0.139 0.187 0,212 0.212 0.111
51.5 39.7 29.7 26.2 26.2 49.7
0.00548 0.00485 0.00462 0.00476 0.00530 0.00476
60.7 46.6 34.5 30.3 30.3 58.6
10,630 9,240 7,230 6,170 6,170 10,590
0.063 0.084, 0.099 0.109 0.122 0.135
0.111 0.187 0.194 0,240 0.270 0.299
49.6 29.8 28.8 23.4 21.0 18.9
0,00539 0.00454 0,00534 0,00491 0.00503 0.00517
58.4 34.6 32.8 26.7 23.7 21.3
10,400 7,360 5,970 5,280 4,590 4,030
0.524 0.514 0.486 0.500 0.500 0.477
0.164 0.207 0.076 0.072 0.066 0.071
0.357 0.474 0.157 0.147 0.114 0.136
16.0 12.2 35.3 37.8 48.6 40.6
0.00555 0.00558 0.00482 0.00480 0.00561 0.00509
17.8 13.3 41.1 44.1 57.1 47.6
3,150 2,360 8,220 8,840 9,800 9,010
0.524 0.645 0.281 0.252 0.243
0.085 0.071 0.060 0.057 0.055
0 182 0.143 0.110 0.088 0.081
32.2 41.7 49.8 82.1 67.7
0.00474 0.00483 0.00514 0.00592 0.00624
35.4 45.4 59.1 73.8 80.4
7,230 9,040 11,020 11,950 12,340
4 5 7 9 11
208.4 208.4 193.1 185.0 163.0
120.2 122.0 112.6 111.6 101.3
105.1 106.2 100.4 97.5 90.9
88.2 88.0 85.8 86.0 83.3
0.325 0.307 0.243 0.202 0.150
12 13 15 18 23
199.4 118.8 199.4 119.8 196.7 125.2 155.1 9 9 . 5 156.4 109.4
107.6 106.3 120.9 95.0 99.5
85.6 86.4 84.6 81.7 81.9
0.316 0.298 0.310 0.315 0.318
27 29 30 31 32
159.6 165.2 169.3 161.8 161.8
109.4 112.3 109.4 108.7 115.3
99.9 108.3 136.0 105.8 117.0
82.4 80.6 83.5 83.7 82.0
0.405 0.608 0.525 0.561 0.644 0.365 0.450 0.673 0.450 0.290
33 34 35 36 37 38
166.3 117.3 165.0 118.9 160.0 115.9 156.7 113.0 156.2 109.9 166.1 119.7
124.2 118.9 110.8 108.0 99.5 119.1
82.8 83.5 83.1 82.9 83.5 82.2
0.536 0.466 0.365 0.312 0.312 0.535
0.327 0.346 0.327 0.327 0.505 0.281
0.062 0.070 0.085 0.096 0.096 0.062
40 42 43 44 45 46
165.2 158.0 149.0 147.6 143.8 140.4
115.3 113.5 100.9 103.3 100.8 98.8
113.9 104.2 88.0 90.3 88.2 86.7
81.7 80.6 78.4 78.4 78.1 78.4
0.525 0.375 0.302 0.267 0.232 0.203
0.374 0.346 0.813 0.533 0.524 0.524
47 48 49 50 51 52
136.4 123.8 159.1 161.8 164.8 163.2
96.1 91.9 111.2 113.7 113.0 113.5
85.1 82.8 99.7 103.6 108.5 104.4
78.6 78.8 80.6 82.9 84.4 82.6
0.159 0.119 0.415 0.446 0.495 0.455
53 54 55 56 57
83.3 76.1 165.6 168.8 167.5
105.8 99.0 123.8 124.7 125.6
110.5 103.6 122.0 129.2 134.2
120.9 113.4 87.8 88.9 89.6
0.365 0.457 0.556 0,603 0.624
0.626 0.645 0.280 0.720 0.439
34.4 31.3 25.9 21.8
41.5 37.7 30.7 25.6 20.5
6,440 6,070 4,820 3,990 2,970
Oil 2 (Mineral Seal-SAE 30 Oil)
of heat transfer through the film, l / h H was plotted against l/GO.* to determine whether there was any appreciable resistance to the transfer of heat through the liquid to the gas-liquid interface. Figure 3 shows that the liquid film resistance was either zero or negligibly small, since extrapolation t o infinite gas velocity, a t which velocity the gas film resistance was theoretically zero, caused the lines for over-all resistance to pass through the origin. The term hH, while calculated from the data as an over-all coefficient of heat transfer through both the liquid and the gas films, can now be considered as the coefficient for the gas film only, and as such may be compared with the vapor transfer coefficient. One method of correlating the heat transfer data was by means of the commonly used Nusselt-type equation: hed/k = a(Re)n(cp/k)m
(3)
Since the Prandtl member for gases is practically a constant a t ordinary temperatures and the investigation did not include a study of the effect of varying that number, the value of (cp/k)” is ipcluded in the coefficient of the equation. The equation obtained by plotting the heat transfer data as in Figure 4 is : h&/k
= 0.028
(4)
Dittus and Boelter (4), Shemood and Petrie (9),McAdams (6),and others report the following: h&/k
= a(Re)0.S(c,~/k)O-~
(5)
in which a lies in the range between 0.022 and 0.024. Taking the value of k as 0.76, (cp/k)O.4 is 0.90,and Equation 5 becomes: , hHd/lc = 0.021 (Re)0.8
(6)
Comparison of Equation 4 with this form of the standard equation for heat transfer to a dry wall shows that, a t a given Reynolds number, the present data lie 33 per cent higher. The dimensionless term d/zD is used by Gilliland and Sherwood (6)for the correlation of vapor transfer data in a Nusselt-type equation. The use in this investigation of a vapor transfer coefficient h, similar to the heat transfer coefficient h H was deemed preferable because of the close relation in the mechanisms of heat and vapor transfer, as pointed out by Chilton and Colburn (2) and as is evident from comparison of the equations representing the two processes. The vapor transfer data were correlated using the dimensionless term h,d/pD, which is identical with d/xD. The equation representing the data plotted in Figure 4 is: hvd/pD, = 0.016 (Re)0.83
(7)
Gilliland and Sherwood (6)report the following: d / x D = 0.023 (Re)O.s8(r/pDv)0.44
(8)
Taking the value of p/pD,, for the system water-air as 0.60, ( p / ~ D , , ) o .is~ ~0.80, and this equation becomes: d/xD = h,d/pD. = 0.018
(Re)0.83
(9)
Comparison of Equations 7 and 9 shows that, at a given Reynolds number, the present data lie 11 per cent lower than the data of Gilliland and Sherwood. With reference to Figure 4 the line representing the- vapor transfer data nearly coincides with McAdams’ line for heat transfer. A variation of the conventional Nusselt-type equation is used by Chilton and Colburn for diffusional processes. Colburn (8) treets ..heat transfer data by plotting (hH/cG’)( ~ p J k , ) ~ / swhich , is here designated as j,, against the Reyn-
I N D U S T R I A L A N D E N G I N E E R I N G CHEMISTRY
440
Vol. 33, No. 4
DATAC O M P A R E D ON FIGURE 4. H E A T AND V A P O R TRANSFER FIQCRB 5 . HEATAND VAPOR TRANSFER D A T A COMPARED BASISOF MCADAMS~ K DMODIFIEDGILLILANDEQWATIONS ON THE BASISOF COLBURK’S THEORY WITH MCADAVS’ LINEFOR HEATTRANSFER
THE
d d s number. I n an analogous manner Chilton and Colburn (2) correlated vapor transfer data by plotting against the Reynolds number the term: Under the conditions prevailing in the apparatus during this investigation, the value of p,, remained nearly 1 atmosphere, allowing the replacement of KDp,,M,/G‘ by the equivalent group h,/G‘, so that j D =
(hc/G’)( P ~ I P ~ D ~ ) ~ ’ ~
(11)
The above method makes use of different exponents for cpl/]Gr and p f / p f D , from those used in the Nusselt-type equations, and also base the viscosity, density, thermal conductivity, and vapor diffusivity upon the film temperature rather than upon the bulk temperature. I n Figure 5 the values of j H and j D as calculated from the data are plotted against the Reynolds number. “Colburn’s Line” represents Colburn’s heat transfer data and also Chilton and Colburn’s vapor transfer data for a wetted-wall tower. The equation for this line is within 10 per cent of j
= 0
023
Re-0.2
(12)
which is given by Colburn as approximately representing his analysis of other investigators’ heat transfer data. Equation 12 is stoichiometrically equivalent to the Nusselt-type heat transfer Equation 6 but not t o the corresponding vapor transfer Equation 9. The equations for the data plotted in Figure 5 are: j~
=
0.031 Re-02
j~ = 0.025 Re-0,2
(13)
(14) They are stoichiometrically equivalent t o Equations 4 and 7, respectively. Equation 13 is 35 per cent higher and Equation 14 is 9 per cent higher than Equation 12. The method of correlation suggested by Arnold ( I ) also was used, since it is considerably different from the two methods previously described. Arnold employs the actual thickness of the laminar layer rather than the effective thickness of a fictive film for the correlation of data because, as he states, the laminar layer actually exists whereas the effective film is an empiricism. As originally set forth by
Prandtl’s theoretical modification of the Reynolds analogy, the ratio of (1 - r ) / r t o cp& is the ratio of the resistance t o heat transfer in the turbulent zone to the resistance to heat transfer in the laminar layer. From the definition of B H , zH, and [(I - r ) / ~ ] / ( c p ~ /the k ~ )folloming relation may be derived: BIZ = ZHTCPj/JCf+H TCPf/h&H (14 Rearrangement of Equation 15 and introduction of the characteristic linear dimension d yields the dimensionless heat transfer term: dr/Bx dh.drr/CPf (16) From the ratio for vapor diffusion, [(I - r ) / ~ ] / ( p ~ / p ~ D J which is similar t o the heat transfer ratio, the vapor diffusion term is obtained: dT/BD =
dhtdD/&f =
dPjD&D/ZDPf
(17)
The values for d r / B H and d r / B D were calculated and plotted against the Reynolds number as shown in Figures 6 and 7 , respectively. The equations expressing the results are: d r / B H = 0.027
(Re)0.82
d r / B D = 0.022 (Re)Oa2
(18) (19)
Equations 18 and 19 show that the value of BH is 23 per cent less than the value of B D . According t o Arnold’s theory the thickness of the laminar layer should be the same for heat transfer as for diffusion of any vapor. Gilliland and Shermood (5) plotted their own vapor transfer data and McAdams’ heat transfer data, using instead of dr/Ba and d r / B D the respectively equivalent terms, dhH$Jcp, and dpfD,$,/zp1. Their results show the value of BH to be apparently 30 t o 35 per cent greater than the value of B,. Sherwood (7) reports an analysis of Mark’s wet-bulb measurements for various liquids as having a deviation of 15 to 25 per cent in the opposite direction-that is, in the same direction and t o about the same extent as reported above for this investigation. Arnold suggests that the apparent inequality of B H and BD may be due to a considerable error in the calculated value for r. After analyzing the question mathematically, it would
INDUSTRIAL AND ENGINEERING CHEMISTRY
April, 1941
441
FIGURE 6. COMPARISON ACCORDINGTO ARNOLD’STHEORY OF FIGURE 7. COMPARISON ACCORDING T O ARNOLD’S THEORY HEATTRANSFERO F THE VAPORTRANSFER THE HEATTRANSFER DATAWITH MCADAMS’ DATAWITH THE HEATTRANSFER LINE AND WITH GILLILAND’S LINE FOR VAPORTRANSFER LINE AS PLOTTED BY GILLILAND AND SHERWOOD AS PLOTTED BY GILLILAND A N D SHERWOOD
seem that the error is not merely in the value of r , but also in assuming that r has the same value for vapor transfer as for heat transfer. Setting up Equations 16 and 17 as a proportion:
Assuming the experimentally well-verified equality for the system water-air of hH/h, =
8
=
nc
(21)
where n is equal to or greater than unity, the requirement for T H being equal to rD is for + J H / + J D to be equal t o or less than unity. These two conditions can be met simultaneously only when rH and rD are both approaching zero, in which case n approaches unity. The physical interpretation that might be given to this in respect t o a wetted-wall tower would be air entering dry and flowing a t a very high velocity so that the resistance to heat and vapor transfer through the laminar layer was negligible and the humidity of the air was practically zero throughout the tower. Another example might be the wet-bulb thermometer in a sling psychrometer being whirled a t a rapid rate through dry air. Under ordinary circumstances in the operation of apparatus such as a wetted-wall tower, the values of r for heat transfer and for vapor transfer must be different if the equality of BH and BD is to be possible. The ratio, hH/hv,for the system water-air is of particular interest since previous investigations have shown that it is fortuitously nearly equal to the humid heat of the moist air. The assumption of this relation is important in the use of the adiabatic cooling lines on psychrometric charts. Values of h,/h, calculated from data obtained in this work follow: Re = 1000 From data when correlated by Nusselt-type equations Chilton and Colburn’s theory Arnold’s theory Erom other sources McAdams and Gilliland’s equations Chilton and Colburn’s theory Arnold’s method (Gilliland’s and MoAdams’ data) Arnold’s theory
Re = 10,000
0.25 0.25 0.26
0.25 0.26 0.27
0.17 0.21
0.16 0.21
0.13 0.21
0.16 0.22
Although these values compare closely with the humid heat calculated for the average conditions existing in the tower during its operation, they cannot be taken as offering full proof that the relation of h H / h , = s is correct, since the calculation of h, was dependent in part on the use of the adiabatic cooling lines of the psychrometric chart. Also given in the above table are the values of hH/h, calculated from Chilton and Colburn’s and from Arnold’s theoretical equations, which are from 8 to 12 per cent less than the lowest possible value for humid heat. Values deviating still more in the same direction are obtained when McAdams’ heat transfer equation and Gilliland’s vapor transfer equation are combined and when the heat and vapor transfer data plotted by Gilliland and Sherwood (6) according t o the Arnold theory are used. A tabulation by Sherwood (8) of hx/h, calculated from wet-bulb determinations by several investigators give values that are greater than the specific heat of dry air and within the range of the humid heat.
Practical Applica Lion The heat transfer coefficients for direct contact of gas and liquid, as determined in this investigation, are considerably larger than those normally obtained when the fluids are separated by a metal wall; for which reason heat transfer between gas and liquid should be carried out whenever possible by direct contact. The results of the above work with a wetted-wall tower show that i t is reasonable to expect the gas film to control the rate of heat transfer when a gas and a liquid are in direct contact. For the case of vapor transfer when hot gases are being used to vaporize a liquid, no effective liquid film exists, so that again the gas film controls. Thus, for an apparatus in which heat and vapor transfer is taking place, the most important factor to consider with respect to obtaining better efficiency of operation and greater capacity is a decrease in the gas film resistance by a n increase in gas velocity. The best method to be used to increase the gas velocity would depend upon the type and construction of the apparatus; however, the initial velocity of the gas might be raised by means of a blower, or the equipment might be so constructed that the gas would flow through narrower passageways and
442
INDUSTRIAL AND ENGINEERING CHEMISTRY
thus at higher speed. Another factor entering into the capacity of the apparatus is the amount of interfacial area of gas and liquid. Increased capacity resulting from increased area of interface is obtained by breaking u p the gas stream with baffles so as to provide greater mixing of gas and liquid. The advantages of heat transfer by direct gas-liquid contact are made use of in the submerged combustion burner developed by K. A. Kobe and J. H. Jensen. The burner is designed to operate under the surface of the liquid being heated and evaporated, and the combustion gases pass upward through the liquid. Tests showing the rapidity with which the hot combustion gases reach a temperature near that of the liquid are a practical illustration of the value of direct gas-liquid contact. The chief contributing factors to this high rate of heat transfer are the high velocity with which the gases issue from the burner and the considerable turbulence resulting from the passage of the gases upward through the liquid. A mathematical procedure could hardly be applied to the design of this apparatus, but the principles discussed above could be applied in the designing for increased capacity on the basis of actual test data. Although the previous discussion has dealt only with the heating of liquids, the same principles apply to their cooling by direct contact with gases. This latter, in fact, is a process of considerable usage in such equipment as atmospheric water-cooling towers and other water-cooling devices. The cooling of liquids other than water by direct contact with a gas in heat transfer apparatus is of small present use but worthy of consideration, particularly for liquids having low vapor pressure so that loss of vapor to the cooling gas is negligible. I n special cases even volatile liquids might be advantageously cooled by recooling and recycling the vaporrich cooling gas in a closed system.
Conclusions 1. The rate of heat transfer in the wetted-wall tower is controlled b y the gas film whether or not simultaneous vapor transfer occurs. 2. The ratios of dr/BHto dr/B, = 1.23 and of jH/jD = 1.24 for the Arnold and Colburn theories are better correlation between heat and vapor transfer than the ratio of hHd/k to h,d/pD, = 1.33. 3. The value of the ratio hH/h,was found to be in the range of 0.25 to 0.27 and thus nearly equal to the average humid heat for the conditions existing in the tower during its operation.
Vol. 33, No. 4
Nomenclature A B c
d
D,
G G’ hx h,
j~
JD
12 KD M, p,, 4
r
Re S
area of film interface, sq. ft. thickness of laminar layer, ft. specific heat of air at constant pressure, B. t. u./(lb.) ( ” F.) = diameter of tower, ft. = vapor diffusivity of water vapor in air, sq. ft./hr. = air mass rate, lb./hr. = specific air mass rate, lb./(hr.)(sq. ft.) = heat transfer coefficient, B. t. u./(sq. ft.)(hr.)(’ F.) = vapor transfer coefficient, lb./(sq. ft.)(hr.) = ( h ~ / C G ’ ) ( C p j / k f ) ~= / ~Colburn’s heat transfer number = (k,/G’)(pf/~jD~)~/3 = Chilton and Colburn’s vapor transfer number = thermal conductivity of air (B. t. u.)(ft.)/(sq. ft.) (hr.)(’ F.) = molar mass transfer coefficient, lb. moles/(hr.)(sq. ft.) (atm.) = mean molecular weight of diffusing vapor = log mean pressure of inert component in film, atm. = heat transferred in unit time. B. t. u./hr. = 1.75 (Re)-”S = dG‘/p, = Reynolds number = humid heat of moist air, B. t. u./(lb. bone-dry air) = = =
le
I
n \ ‘.I
water vapor transferred in unit time, lb./hr. 2 = effective film thickness of laminar layer and turbulent zone, ft. d r / B = Srnold’s transfer number (At),,, = log mean temperature difference, F. (AH)&”.= log mean humidity difference, lb./lb. = viscosity of air, lb./(hr.)(ft.) P = density of air, lb./cu. ft. P @H = 1-r r(c!-V/b) +D = 1 7 r(w/mDV) Subscripts: D = vapor transfer f = term computed at average conditions of laminar layer H = heat transfer ?I$
=
O
-
++
Literature Cited Arnold, J. H., Physics, 4, 255-62 (1933). Chilton, T.H., and Colburn, A. P., IND. ENG.CHEM., 26, 1183-7 (1934). Colburn, A. P., Trans. Am. Inst. Chem. Engrs., 29, 174 (1933). Dittus and Boelter, Cniu. Calif. Pub. Eng., 2, 443 (1930). Gilliland, E. R.,and Sherwood, T. K., IND.ENG.CHEM.,26, 516-23 (1934). McAdams, W. H., “Heat Transmission”, p. 165, New York, McGraw-Hill Book Co., 1933. Sherwood, T. K., “Absorption and Extraction”, p. 48, New York, McGraw-Hill Book Co.,1937. Ibid.,pp. 54, 55. Sherwood, T. K., and Petrie, J. M.,IND.ENG.CHEM., 24, 736-45 (1932).
COOLIXG TOWER, SOCONY
- VACUUM
OIL COMPANY AT PAULSBORO, N. J. The Fluor C0r;loration
Courtesy
