March, 1931
INDUSTRIAL A N D ENGINEERING CHEMISTRY
301
Heat Transmission to Water Flowing in Pipes' Effect of Tube Length A. E. Lawrence and T. K. Sherwood* WORCESTER POLYTECHNIC INSTITUTE. WORCESTER,
SCASS,
Experiments were carried out on heating water flowbe greater than in a long ing at 0.62 to 22.4 feet per second through a clean pipe, where the effect would affecting the rate of copper tube, 0.593 inch inside diameter, by means of be relatively small. heat transmission besteam condensing from the surrounding steam jacket. I n 1910 Nusselt (11) d e tween the walls of a pipe and To determine the effect of pipe length on the film rived the expression (obtained a fluid flowing through it are coefficient of heat flow from pipe to water, the pipe earlier by Graetz, 3) for the ordinarily assumed to be the length was varied from 59 to 224 diameters, the entire heat conduction into a fluid fluid velocity, the diameter length of each tube being heated. The average deviaflowing in a round pipe, based of the pipe, the length of the tion of the heat balance was less than 7 per cent. on the assumption of a parapipe, the nature of the fluid, Thermocouples were used to determine the pipe surface bolic velocity d i s t r i b u t i o n and the temperature conditemperatures, and the film coefficients for the water side across the pipe. Such a vetions in the fluid from pipe calculated and correlated in several different ways. locity distribution is not found wall to center line. NumerWhen plotted, the water-side coefficients for all four even in viscous flow for, as ous investigators have shown pipe lengths fall in narrow bands, but not in order of Keevil a n d McAdams (4) that for b o t h gases a n d the pipe lengths. For heating water in turhulent flow have pointed out, the effect of liquids in the region of turit is concluded that the effect of pipe length on the change of viscosity with tembulent flow the film coeffifilm coefficient of heat transmission is negligible. p e r a t u r e , neglected in the cient of heat flow between pipe A graphical analysis of the data on the over-all derivation of the Graetz equawall and fluid varies directly coefficients of heat flow from steam to water, not retion, is to distort greatly the as the nth power of the mass quiring the use of the thermocouple readings, led to parabolic distribution of vevelocity of the fluid, the value the same conclusion. of n being in the vicinity of locities. However, the equsThe film coefficients for condensing steam on the 0.8. Few investigators delibtion serred to focus attention outside of the horizontal pipes were found to agree erately varied other than the on the possible effect of tube well with the Nusselt theoretical equation for this case. fluid velocity and tempcralength, and Xusselt later (12) introduced the dimensionless ture, so conclusions regarding the influence of the pipe diameter and length, and of' the nature group (d/L)into his original equation derived by dimensional of the fluid, have of necessity been based on a comparison analysis, The new equation, proposed as bettcr fitting his of data of several investigators, working with somewhat early data on heating gases with two different lengths of different apparatus and experimental technic. The present heating surfaces, took the form article describes research on the effect of the tube length in Ek = constant x (1) which similar series of tests were repeated using four different lengths of the same tube, the conditions a t the fluid entrance in which, for gases, n = m = 0.786, and p = 0.054. Although and exit being held constant. this equation was later modified, the value of p of 0.054 was Two entirely different aspects of the possible effect of retained. I n 1925 Nusselt (14) found that Ptender's data on tube lengths have been discussed by various writers. Kus- heating water in pipes of two different diameters indicated selt (11) and other German writers have pointed out the the value of p to be 0.2. Very recently Stender (18) has effect of the changing temperature gradient across the pipe pointed out that Equation 1 indicates the coefficient to as the fluid flows through it, and have developed theoretical approach zero for very long pipes, and has proposed an equaequations for heat conduction into the core of the fluid. For tion of the form example, as the fluid flows through the pipe and is heated, the temperature gradient in the fluid near the pipe wall, and consequently the rate of heat flow, is shown to decrease faster than the average temperature difference, and the film for gases, and a similar form to represent his own and Stancoefficient therefore decreases. The relations derived, how- ton's data on water. McAdams and Frost (6) considered the extra turbulence ever, are believed to apply only approximately to heat transfer to fluids flowing in viscous flow, and probably not a t all a t the entrance to the pipe to be the important factor in to turbulent flow, where it would seem to be impossible for connection with the effect of tube length when the fluid is a n appreciable temperature gradient to be set up except in flowing in turbulent motion, and introduced a factor of the the film near the wall. On the other hand, McAdams and form 1 C) d/L, which is analogous to that sometimes used Frost (6) have pointed out that a thinning of the film and an to multiply the actual pipe length in order to obtain the improvement in the film coefficient might be expected near equivalent frictional length. These authors studied data the tube entrance because of abnormal turbulerice a t that of several investigators and determined the value of CS as point. The average coefficient with a short pipe would then 50, their comparison of one set of data with another being made by plotting hd/k vs. dup/z,. They proposed the equa1 Received January 5, 1931. Based on a thesis submitted by A. E. tion Lawrence in partial fulfilment of the requirements for the degree of master
HE important factors
T
(y)"(y)" (g)'
+
of science in chemical engineering, Worcester Polytechnic Institute, June, 1930. Special lecturer in chemical engineering, Worcester Polytechnic Institute, Worcester. Mass. Present address, Massachusetts Institute of Technology, Cambridge, Mass.
-k=
14.5 (1
+ 7)
(3)
as well representing the data studied. The difficulty was encountered, however, that each apparatus had not been
INDUSTRIAL A N D ENGINEERING CHEMISTRY
302
carefully described by the original investigator, so it was necessary to take L as the length of heating surface, whereas in some cases i t was not known whether there had been an unheated straight approach section or whether the fluid had entered the pipe through an ell immediately a t the inlet t o the heating section. Furthermore, the results of the one investigator who had used more than one pipe (Stanton) did not fall in order of the values of d/L, and one of the two other sets of data considered had to be extrapolated for purposes of comparison. McAdams and Frost fully appreciated these diEculties, clearly stating the limitations to their correlation.
3 / B in. Cover ? l i r e
a p p r o a c h Tube 1.44 in. 1.6.
Figure 1-Tube
Sheet Assembly at Water-Inlet End
The purpose of the present investigations was to obtain data on heat transmission to water flowing in turbulent motion through several lengths of the same pipe, the conditions being as nearly as possible identical for each series of runs. The water entered and left the pipe immediately a t the ends of the heated section, the conditions being very similar to those a t the ends of a condenser tube in a large surface condenser. The water supplied was preheated to various temperatures, making it possible to distinguish between the effects of temperature and of tube length, for any one water velocity. Arrangement of Apparatus
The apparatus employed consisted essentially of an experimental single-tube condenser together with a doublepipe cooler, standard orifice, measuring tanks, and pump for recirculating the cooling water. The condenser consisted of a 12 foot 4 inch length of 4inch pipe lagged with ll/rinch magnesia covering and fitted with "tube sheets" supporting a single S/4-inch copper condenser tube concentrically within the 4-inch jacket. Cooling water was passed through the copper tube and steam supplied to the 4-inch jacket, the whole apparatus being inclined slightly from the horizontal allowing the condensate to be drained from the end a t which the cooling water was removed. At the cooling-water-exit end of the condenser, the jacket was fitted with a flange to which was bolted an iron tube sheet 1 inch thick drilled in the center and tapped for a standard 3/4-inch brass ferrule. One end of the 3/riach 0. d. copper condenser tube was held by this plate, being packed with asbestos and graphite packing pressed in place by the brass ferrule. Downstream from the plate was a 6-inch section of 4-inch pipe serving as an outlet chamber, and into which two thermometers were inserted to measure the temperature of the water leaving the apparatus. A 4 by 2 inch galvanized sheet-iron baffle was placed across this outlet chamber a t right angles to the direction of flow a t a distance of
Vol. 23, Nd. 3
slightly less than '/z inch from the end of the ferrule. This served to mix the water well before its temperature was measured, and to prevent erroneous readings due to stratification of hot and cold layers of water, especially a t low water velocities. Heat conduction through the plate or tube sheet to the water in the outlet chamber was minimized by a '/$-inch sheet cork insulation held flush with the outlet chamber side of the plate. The water leaving the outlet chamber entered a standard 1-inch line leading to the cooler. At the cooling-water entrance end of the condenser the copper tube was supported in a specially designed movable tube sheet illustrated in Figure 1. This was made 4 inches 0 . d., fitting snugly in the 4-inch steam jacket, and could be located a t any point within the jacket depending on the length of the copper condenser tube used. With this arrangement various tube lengths could be tested without changing the length of the 4-inch steam jacket. It was originally intended that the water entering the apparatus should fill the 4-inch pipe up to this tube sheet, but on trial it was found that the water entering the tube had been heated to a considerable extent by heat conducted through the metal of the 4-inch pipe from the steam side to the entrance-chamber side of the tube sheet. As it was difficult to correct for this effect or to measure accurately the true average temperature of the water as it actually entered the condenser tube, the arrangement illustrated in Figure 1 was adopted for all the tests here reported. The tube sheet was made in two parts, a 1-inch plate and a 3 / ~ i n c hcover plate. The plate was drilled in the center and tapped for a standard J/4-inch brass ferrule which served to hold the cooling water entrance end of the condenser tube under test. The outer edge of this plate was cut away for a width of 7/32 inch and a depth of l/2 inch, and a ring made just fitting this groove. Packing in this groove made a tight joint between tube sheet and the jacket, being held in place with the ring pressed down by the 3/a-inch cover plate which was bolted to the 1-inch plate by 3/8-inch stud bolts. The cover plate was drilled and tapped to hold the end of a 6inch length of l'/rinch hard-rubber pipe. A groove was cut as indicated in the 1-inch plate, holding packing pressed in place by the end of the hard-rubber pipe. With this arrangement there was no possibility of the water being heated as it approached the entrance ferrule. Furthermore, the construction was such that the conditions a t the tube entrance were similar to those a t the cooling-water-entrance end of a condenser tube in a large steam condenser, or multitubular heat exchanger. The temperature of the tube a t various points along the length was determined by means of thermocouples soldered in grooves or holes in the outer surface. An attempt was made to use No. 36 copper and constantan wires for these couples, but these were found to break easily because of their small size and the tendency of the copper wire to corrode in the steam, so couples of No. 24 constantan and No. 26 copper wire were used in the tests reported. The couples were first made by soldering the copper wire in a short groove cut a t right angles to the tube length, the excess solder being filed away to make a continuous smooth surface. A '/&nch hole about 3/64 inch deep was drilled in the opposite side of the tube, the hole filled with molten solder, and the carefully cleaned constantan wire inserted vertically. This type of junction was used in the runs of Series I and in the first runs of Series 11. Since the constantan wires tended to pull out of the small holes, a change was made to a couple soldered in a short groove. About 3/32 inch of the end of the constantan wire was bent a t a sharp right angle and this short end soldered into the short groove, the wire then leading away vertically from the pipe surface. The external appearance of
IAVDVSTRIALA N D ENGIiVEERISG CHEMISTRY
March. 1931
this type of junction was similar to the former constantan wire junction soldered in the shallow hole, and no difference was apparent in the results with the two types. I n both types it was found extremely important to scrape the wire clean exactly down to the pipe surface, as the least amount of adhering solder caused the actual thermocouple junction to occur a small fraction of an inch up the wire from the pipe surface, and the couple would read abnormally high. Table I shows the positions of the couples for the various tests. The constantan wires were attached to the top surface of the tube, except in runs 6 to 26 of Series 111,when they were attached a t the bottom, and all runs of Series IV, when they were attached a t the side. Couples were also used to check the entering and exit cooling water temperatures, but the readings were not used in calculating the results. One couple was inserted in the water in the hard-rubber nipple a t the cooling-water entrance, and found to read the same as the thermometers in the tees in the I-inch line leading to the apparatus, showing heat conduction back along the feed line to be eliminated. 'Table I-Thermocouple COUPLE
4
Positions. Distance from Cooling-WaterEntrance End S E R I E IS SERIESI1 SERIES 111 SERIESIV Inches Inches Inches Inches
73
The most satisfactory way of calibrating the thermocouples used for measuring the temperature of the pipe surface was found to be by cutting off a section of the copper pipe about 1 inch long, with the thermocouple leads in place, and suspending the whole in a beaker of water beside a thermometer. The water was heated very slowly and stirred while simultaneous readings of both thermometer and thermocouple were made. A Leeds and Northrup portable potentiometer indicator was used with the couples, and was believed to be accurate to somewhat less than 1O F. Steam was supplied from a 140-pound line, and was sufficiently wet so that when reduced to atmospheric pressure it was in general superheated only a few degrees. I n Series I the steam separator was not used, and the steam was wet even after passing the reducing valve. I n Series I the steam entered the steam jacket from a manifold through three standard I/c-inch pipes located 3.0, 6.87, and 8.77 feet, respectively, from the entrance end. The steam velocity through these small pipes was so great, however, that the pipe-surface temperature opposite these inlets was abnormally high and thermocouples located near these points did not indicate the average pipe-surface temperatures. For Series 11, 111, and IV a 4-inch tee was inserted between the exit tube sheet and the downstream end of the 4-inch jacket. Steam was then supplied a t a low velocity through this tee and flowed into the steam jacket in a direction countercurrent to the flow of cooling water. The steam space mas maintained under 2 to 3 pounds pressure, and pet cocks attached to the jacket were cracked slightly to provide for continuous removal of air. The pressure in the steam chest was measured by means of a mercury manometer. A condensate receiver supported under the condensate outlet was provided with a marked gage glass to indicate the level of the condensate which was collected and weighed during each run. The system as a whole was arranged so that the cooling water could be cooled and recirculated or, if desired, city water could be passed once through the apparatus and run to waste. The water leaving the condenser passed through a 3 by 1 inch double-pipe cooler 10 feet long to a sharp-edged thin-plate orifice used to measure the rate of flow when the cooling water was being recirculated. This orifice consisted of a 3/64-in~h
303
plate having a 'ls-inch hole drilled concentrically with the 2-inch chambers. Rubber-tubing manometer leads were connected to a single tap 6 inches upstream from the plate, and to a piezometer ring with four taps 1.4 inch downstream from the plate. The orifice was calibrated over a wide range of velocities with both cold and hot water. The water leaving the orifice flowed to waste or to a movable outlet emptying into either of two measuring tanks of 150 pounds capacity. The water was pumped from these tanks either to the sewer or back through the system by a small Goulds centrifugal pump. Procedure and Method of Calculation Collection of data for a run involved several repeated series of readings of manometers, thermometers, and thermocouples, and one or more measurements of the rate of flow of condensate and of cooling water into the measuring tanks. Before each run the apparatus was allowed to run steadily for 15 or 20 minutes to allow time for the conditions to become constant. During the run particular care was taken to insure a constant steam pressure of about 2.4 pounds. A p proximately 3 minutes were required for two men to take a complete set of readings, and as these were repeated several times the runs usually lasted from 10 to 15 minutes. All thermometers were graduated to 0.1" C. and calibrated by reference to a standard, and the average corrected readings are tabulated in the tables of data. The inside of the copper tube was cleaned thoroughly with a stiff brush a t intervals of from 1 to 3 days.
SERIES
LENGTH, F T
-- M U D A M S Q FROST EQ.
,
D US Zf
60 80 la 20 Figure 2-Comparison of Results w i t h Equation of McAdams a n d Frost 06
OB I O
20
30 40
The heat picked up by the water was calculated directly from its rate of flow and temperature rise, taking the specific heat and specific gravity as unity in all cases. The heat given up by the steam was calculated by multiplying the weight of condensate by a constant latent heat of 970 B. t. u. per pound. I n experiments on heat transmission of this type the method of calculating the mean temperature difference is always somewhat arbitrary, and should be stated in detail. I n these tests the initial and final temperature differences were obtained by subtracting the inlet and outlet water temperatures from the arithmetic mean average temperature of the inner surface of the pipe. The latter value was obtained by subtracting the calculated small temperature drop through the coppeppipe from the arithmetic mean of the temperatures indicated by the several thermocouples attached to the outer surface of the pipe. I n calculating the film coefficients the rate of heat flow as figured from the tem-
I : Length 11.09 Feet
Table 11-Series R~~
4
6 7 8 9 10 11 12 13 14 15 16 17 18 20 21 22 23 24 25 26 27 28 29 30 31 32 33
F.
F.
41.0 41.5 40.4 39.9 39.35 38.8 38 8 38.6 38.5 38.3 38.3 42.4 45.6 42.2 43.4 40.9 156.7 166.6 39.9 39.6 39.0 40.4 41.6 161.1 163.8 164.8 160.0 161.9 160 7 41.05 40.05 39.6
154.2 166.0 145.1 130.4 119.1 112.4 106.6 102.3 97.3 92.9 89.2 167,6 196.6 175.3 187.2 141,9 191.2 194.8 124.4 130.0 120.0 189.0 192.0 188.2 186.2 183.8 178 9 181.0 183.7 111.6 99.9 88.3
VELOCITY
Ft./sec. 2.52 1.87 2.91 4.05 4.96 5.98 6.97 7 99 9.36 10.24 11.57 1.76 0.63 1.50 1.05 3.26 3.57 3.50 4.04 3.72 4.87 0.88 0.60 5.05 7.46 9.92 11.00 10.65 7.95 6.10 8.00 11.50
PICKED UPBY WATER
TEMP.
F. 210.7 210.6 210.6 210.2 209.9 210.7 210.4 210.4 210.7 209.8 210.4 211.4 210.9 211.1 210.9 210.9 209.9 209.9 210.4 210.4 210.2 210.7 210.6 209.9 210.9 210.2 210.4 210.9 210.6 211.4 210.6 211.3
HEAT GIVEN
HEAT
INLET- OUTLET- Av. wATER wATERwATER S T E A M TEMP,
1 2 3
UPBY STEAM
B . 1. u./ht 122,500 100,100 131,700 157,300 170,000 188,800 204,000 218,800 236,400 240,000 253,000 94,500 40,900 84,300 66,000 141,800 52,800 42,500 147,000 144,700 170,200 56,300 39,000 59,200 72,400 81,900 89,400 87,700 78,200 184,700 206,000 241,000
B. t . u . / h r . 135,000 98,000 132,700 163,000 183,300 188,000 197,000 2 13,000 2 18,500 238,000 247,000 103,000 56,100 96,700 69,500 133,500 63,000 49,600 170,600 157,000 170,200 77,200 60,100 65,200 79,000 87,200 175,000 142,100 211,000 270,000 347,000
MEAN
INLET- OUTLET- Av.
STEAM
WATER
WATER
TEMP.
TEMP. VELOCITY TExP'
Q
F.
39.5 39.5 42.7 43.1 43.8 43.9 45.3 41.0 41.2 41.7 41.4 43.2 42.4 45.8 42.1 41.9 44.5 45.6 46.1 164.1 143.6 63.4 154.4 155.8 158.1 159.0 163.2 160.9 136.0 114.9
e
F.
82.5 76 4 80.8 84.0 91.8 89.8 96.7 115.2 142.0 180.2 158.9 129.3 121.5 208.0 104.8 98,4 167.4 201.0 202.0 197.2 188.6 150.8 187.9 185.9 186.1 184.3 184.8 184.5 195.6 192.2
WATER
HEAT
Ft./sec. 15.30 20.30 17.50 15.80 11.85 12.90 10.00 6.00 2.80 1.23 1.90 3.77 4.70 0.62 7.87 9.32 1.66 0.77 0.765 3.21 3.22 2.925 5.54 6.900 7.81 8.89 11.27 10.10 1.51 1.01
F. 244.3 247.8 247.1 248.6 244, a 247,8 244.1 240.1 234.7 226.7 227.7 226.3 238.0 222.6 236.0 241 0 227,4 226.3 223.6 222.4 227 226 225.2 223.4 226.5 229.5 229.5 228.8 223.2 222.2
'ICKED WATER UPBY
'm
Dus/zm
2
1
F. 0 697 0 655 0 671 0 74 0 81 0 872 0 913 0 948 0 976 1 010 1 043 0 6.7 0 55 0 625 0 575 0 751 0 360 0 344 0 84 0 816 0 868 0 590 0 587 0 360 0 359 0 360 0 374 0 368 0 365 0 905 0 982 1 069
2 1 2 3 3 4 4 5 5 6 6 1 0 1 1
2 5 6 2 2 3 0 0 8 12 16 17
14 69 57 24 63 06 53 00 69 02 56 61 68 42 08 57 89 04 85 70 33 88 61 31 30 3 4
17 1
12 9 4 0 4 82
6 78
F. 166 6 176 0 169.7 150.8 122.8 137 0 127.3 111.6 103 9 89.4 97.8 184.8 181.1 179.8 183.0 165.0 191.8 191.6 145.0 158.1 138.9 163.4 163.4 175.8 177.8 174.2 176.5 178.3 125.3 103.9 90 8
O
F.
178.7 186.4 176.5 165 4 161 0 157.4 154.1 150.0 147.7 145.2 143.2 186.4 197.6 188.0 192.5 170.4 197.6 198.5 166.6 167,9 165.2 196.8 203.5 196.1 193.1 190.6 187.3 187,5 190.0 154.3 149.0 146.1
3
4
3
6
F. 178.7 188.6 171.7 163.4 l52,9 147 0 136.5 131.6 123.1 118.2 108.5
' F. 177.5 182 8 169.0 161.0 156 5 148 1 144.0 147,4 135.2 131.4 126.2 188.0 208.4 190 9 199.2 166.3 200.3 203.7 159.6 162.5 155.9 204.0 209.3 197.6 194.0 190.4 185.0 185.9 188.7 147.7 140.0 126 6
' F.
F.
184.1 189 3 180.9 174.9 171,5 165 7 165 0 161.0 156.5 155.8 151.5 194.5 209 1 196 8 200.5 180.9 198.1 202.1 174.0 176.0 173,3 204.6 209.0 197.4 195 8 194.0 189.5 191.3 196.3 165.4
194.1 200.3 191.1 180.7 172.4 168.3 165.6 163.4 161.2 155.0 154.5 203.7 210.8 203.2 205.7 190.2 201.9
203.0 206.6 202.0 197.6 198.5 200.6 170.6 160.5 147.5
11: Length 9.03 Feet
HEAT CP BY
THERMOCOI PLB
TEMPERATURE ha
181.1 188.2 177.2 167.3 157.6 154.8 149,9 145.6 140.0 134.4 132,O 191,5 203.2 192.0 197.6 175 0 198.0 200.3 165.8 170,2 162 7 197.2 201.8 196.7 195.0 192.3 188,2 189.0 192.2 154.1 143.8 129.4
AVERAGETHERMOCOUPLE TEMPERATURE
MEAN 'IPESURFACE
STEAM TEMP.
B.1. u . / h r . B . 1. u . / h r . 219,000 299,000 266,000 257,000 221,000 233,000 206,000 177,500 115,000 72,000 94,500 108,000 152,300 35,400 195,000 210,000 85,500
50,900 47.7nn 60:900 112,800 77,000 81,200 88,000 90,900 95,800 98,500 37,720 40,000
perature rise of the cooling water was used, and the area was that of the inner surface of the 0.593-inch copper tube. The coefficient h, was calculated using the arithmetic mean of the initial and final temperature differences found as stated above, and hL was calculated in a similar manner, but using the logarithmic mean of these values. I n plotting the results zm is the viscosity of water in centipoises taken a t the arithmetic mean of the inlet and outlet water temperatures, and e! is the viscosity of the film, the temperature of which is taken to be a t the arithmetic mean of the above main body average temperature and of the average inner pipe surface temperature. Results
AVERAGE
PIPESURFACE TEMP.
Table 111-Series RUN
Vol. 23, KO.3
INDUSTRIAL AND EiVGINEERING CHEXISTRY
304
I
The principal data are shown tabulated in Tables I1 to V. It may be seen that the ranges covered for the main variables were: tube length, 2.91 to 11.09 feet; water velocity 0.62
ha
'*
Dus/zm
F. 1517 145 138.8 145.8 153.6 150.4 158.1 177.1 192.7 205.7 201.2 188.7 185.4 212.8 171 165.4 205.1 210.2 210.6 211.1 208.1 195.8 206.3 205.3 205.9 203.0 202.3 202.8 215.6 214.5
1 11 1 15 1 10 1 07 1 01 1 03 0 97 0 89 0 75 0 61 0 68 0 798 0 843 0 525 0 939 0 98 0 642 0 543 0 537 0 345 0 383 0 635 0 368 0 37 0 365 0 367 0 362 0 365 0 382 0 421
8.19 10.44 9.44 8.76 6.96 7.44 6.11 4.0 2.21 1.20 1.65 2.8 3.31 0.70 4.97 5 64 1 53 0.84 0.85 5 51 4.99 2.73 8.92 11.10 12.7 14.3 16.4 16.4 2.34 1 42
2
3
4
O F . ~. 122.9 121.7 116.1 118.4 125.7 129.8 134.4 157.0 167.9 185.9 185.0 165.7 163.7 201.0 146,3 136.4 192.1 191.6 196.1 196 6 192.4 169.1 190.9 188.6 193.1 179.5 184.1 186.8 210.4 206 6
F. 150 6 144 5 132 6 144 1 150 9 149 0 156 5 175 1 192 7 203 2 199 5 190 4 185 9 210 2 176 3 169 4 202 0 207 5 207 3 211 6 204 8 195 2 205 1 204 0 203 0 202 0 201 0 201 3 214 7 213 8
158 8 153 1 149,O 152.2 160.5 157.0 166.1 181.9 196,3 208.6 203,9 194.1 191,3 214.9 176 0 171.8 206.1 213 6 212 9 214 2 211,s 201.1 209,3 208.2 209.1 208.4 206.6 207.0 216.5 214.9
O
F.
5
6
F. 158,5 147 7 132.8 141.1 149.3 147.2 153.1 177.3 196.6 211.5 205.7 191.8 187.5 216.7 173 0 168.3 209.7 215.9 215 9 215.3 113.5 204.5 211.1 210.7 210.4 210.0 20s 1 207.7 217.4 217 2
F. 160.7 153.2 154,s 161.6 168.8 161.0 169.9 187.3 201.3 214.5 208.0 195.8 192.5 217.4 178 0 174.8 210.8 216.5 216.1 215.3 213.5 202,i 211,l 209,3 209.3 209,3 207.0 205,Q 217.4 217.4
to 22.4 feet per second; inlet water temperature, 38.3' to 164.8' F.; and the main body average viscosity, 0.36 to 1.25 centipoises. The rates of heat flow, calculated from both the temperature rise of the cooling water and from the weight of condensate collected, are also tabulated, so that a comparison of these quantities may easily be made. The runs omitted from these tables were rejected because the apparatus had not come to equilibrium and the conditions were not constant during the run. In the majority of cases, however, the points representing these runs not tabulated fell in well with the others of the same series. Inspection of the heat balances shows very fair agreement between the rate of heat flow as calculated from the rise in temperature of the cooling water and that corresponding to the latent heat of the steam condensed. The average deviation is less than 7 per cent, which is satisfactory for this type of work. This comparison, together with the fact
ISDCSTRIA L A-YD ESGI-YEERISG CHEMISTRY
hlarrli. 1931
Table IV-Series
305
111: Length 6.03 Feet AVERAGETHERMOCOUPLE TEMPERATURE
6
k
10 11 12 13 14 1.J
16 17 18 19
;: 22
__
23 24 56 30 32 33 34 35 36 37 38 39 41 42 43
F. 47 4 47 2 48 1 46 7 46 0 45 8 46 0 45 8 45 8 45 8 45 8 45 8 45 6 46 1 46 0 46 0 46 0 45 8 45 8 53 3 128 6 150 2 62 3 49 9 I57 6 133 2 136 3 137 7 140 8 144 2 144 9
F. 107.1 98.9 123.3 96.9 91.3 87 0 88 4 82.0 80.2 78 8 76.9 75 0 74.1 72.0 io 9 70 4 69.7 68.z 67 , 170 0 168 182 i 126 113 9 183.7 167.9 168.6 166 3 166 8 167 5 166 5
2
Fl./sec. 3.61 4.84 2.35 5.27 6 90 8.05 8.93 9.97 11 00 12.00 13.10 14.00 15.00 15.90 17 10 18.00 19.00 20.00 22.40 0.912 3.08 3.08 3.20 3 17 4 02 4 02 4.85 5 80 6.81 7.86 8 97
F. 219.6 220.9 220 222.2 222.7 224.1 224.1 224.5 225.6 227.2 227 222, ,5 222.7 223.4 227.4 229 229.5 228.8 230.9 219.3 219.3 222.0 223.4 219.5
B. 1. u . / h r 94,500 107,200 75,350 114,000 134,200 142,000 151,000 154,200 163,000 170,700 175,600 175,200 184,000 177,800 184,000 188,000 194,800 196,300 210,600 46,800 52,700 43,000 87,800 87,400 45,000 59,900 67,500 71,300 79.300 76,300 83.600
B . t . u. / h r . 92,500 106,400 75,200 136,700 135,300 142,600 147,200 166,700 166,700 160,000 163,700 166,500 163,000 165,700 172,000 175,200 349,000 47,900 52,500 40,900 86,500 88,400 45,000 61,400 64,700 64,700 77,100 77,530 79.100
Table V-Series R~~
IXLET\vATsR TEMP,
F. 1 2 3
1 6
SI
9 10 11 12 13 I4
15
16
17 18 19 20 51 21
23 54
23 26 27 29 30 31 32 33 34 35
49.2 49 0 48.8 48. 8 48 6 48 6 48 3 48 a 48.d 48.3 48.1 48.1 48 48 48.8 49.2 18.6 02.1 31 2 33 $1 50.4 114 9 106, i 194 5 136 h 136 i
117 4 122 h 123.5 123.3 123 b 122 1 122 4 n2 6
OUTLET\\,ATER
Av. \vlrER
TEMP.
TEMP,
O
F
68 7 67 7 66 6 65 3 65 0 64 1 63 2 62 3 61 4 61 0 60 1 59 4 58 7 5s 4 72 9 77 9 82 9 98 4 95 1 128 0 83 s 136 3 130 4 143 6 156 0 152 4 135 7 138 3 136 3 137 2 136 , 134 .I 134 1 80 3
STEAM
Fl./sec. 7.10 7.95 9.11 10 22 11 00 12 10 13 00 14 00 15.00 16.00 17 00 18.00 19.90 21.40 5.11 3.72 2 835 1.35 1.49 0 678 2.65 3.44 3 44 3.41 3.45 4 93 4 91 f, 00
, .oo
7 97 9 00 10.00 11 10 3 97
F. 219 222. 9 222 6 221.6 221 6 222 223 2 220 222.4 220 218.4 218.6 219 219 6 218.6 218 6 216 4 218 218.2 218.2 218 0 219 219 218.6 222 2 218 218 4 219 3 219 3 218 6 218 6 122 3 221 3 220 6
HEAT PICKED
UP B Y \\'AIER
O
F.
178.7 169.7 190.9 170.8 163 154.5 154.5 153.1 149.7 147.0 141.7 137.9 136.5 132.2 129.3 130 130 128.4 124.4 208.1 204.6 209 192,9 192.7 208.4 203 201,9 199.4 198.5 197 198.5
2.39 3.05 1.73 3.25 4.09 4.63 5.09 5 47 6 05 6.53 7.08 7.43 7.94 8.26 8.75 9.21 9.61 10.1 11.1 0.89 4.17 4.78 2.61 2.23 6.45 5.53 6.76 8.04 9.61 11.2 12.8
-
2
3
F. 171,l 157.2 184,8 157.0 146.6 140.9 139.1 134.3 130.1 125.7 121 3 116.1 113.8 120.3 114.0 117.0 110.4 105.4 106.2 201.2 201 9 206.6 191.1 194.3 207.3 199.5 199.0 195.8 196.6 195,9 199 2
183,4 176.5 192.0 175.8 168.6 164.7 161,s 163.6 159.8 153.1 149.9 146.4 144.6 136.5 134.3 136.3 138.0 137.2 127.7 206.6 203.3 208.4 194.0 191 6 208.2 203.0 202.0 198.3 199.2 198.1 197.9
' F.
4
5
F. 178.9 170.2 194.3 178.1 171.8 165.7 163.8 153.4 150 8 148.1 14i,4 144.5 144.3 136.5 134.3 134.6 132.3 131.8 129.7 211 3 204,o 209.1 190 2 189.9 208.4 203.0 102.0 199.4 197.6 196,7 196.7
180.4 171.8 191.6 168.1 160.7 152.7 149.9 152.0 149.0 146.7 144.1 140.7 139.1 136.5 135.2 135.5 135.9 134.6 134.2 212.9 207.1 210.7 195. 4 194 0 209,3 205.2 203,4 200.9 199.2 196.5 198,s
' F.
IV: Length 2.91 Feet
HEAT GIVEW UPBY STEAM
B . 1. u . l k r . B . 1. u.l h v 59750 50200 64150 62700 69600 67900 ,1100 I 3300 74200 77600 75600 80800 76400 82600 17500 83100 ,7500 84200 81000 87800 83100 88700 82400 88600 92400 95600 88100 53300 58200 48500 46100 44500 40700 26800 33000 28100 33000 21600 23800 40200 42000 31700 34800 3J550 31100 33200 28400 29100 33600 33900 38700 38200 39700 40700 4 1500 40500 42900 43700 49'200 46100 52700 50900 55200 50700 47700 47500
that when properly correlated the points representing the data of any one series of runs fall very close to a smoclth curve, indicates that the data are reliable. The film coefficients tabulated are those calculated using the arithmetic mean temperature difference, obtained as descrilied above. These are the values used in the plots discussed below, as it wah found that hetter correlations resulted than Tvhen using the coefficients h ~ calculated , using the logarithmic mean temperature differences. Actually the ~ a l u e sof ha and h~ are almost identical for all four pipes a t the higher water velocities, and it i s only in case of the runs a t less than about 2 feet per second that the values of ha are appreciably less than those of ILL. ,4s a justification for the use of the arithmetic mean temperature difference it may be pointed out t h a t the derivation of the logarithmic
htE.4N PIPE-
SURFACE TEMP.
' F. 158.7 155 2 153 6 150 8 146.7 143.6 142 6 139 7 136.3 137, 3 137.2 134 1 129.1 131.1 17d. 3 180.9 189.9 201.7 200.5 208 0 193.8 197 9 197,7 200. 8 200 8 198 6 191 3 19.5 0 187 9 186 8 184 8 181 8 180 3 185 4
ha
zl,,
Dus/z?,,
AVERAGETHERMOCOUPLE TEMPERATURE 2
3
' F. 1 15 1 16
1 17
1 18 1 18
1 19 1 '0
1 21 1 22 1.22 1.23 1.24 1.24 1.25 1.11 1.07 1.03 0.92 0.945 0.755 1 01 0 33 0 567 0.488 0 446 0.453 0 527 0 308 0 51 0 309 0 309 0 518 0 517 1 03
3.66 4.06 4.61 5 14 5 53 6 04 6 42 6.86 7.29 7.77 8.19 8.61 9.53 10.1 2.72 2.06 1,63 0 87 0.935 0.53 1.55 3.84 3.59 4 14 4.58 6 4*5 ii 51 7 00 8 14 9.29 10 5 11 4 12.7 2 28
158.0 l34,4 153.4 153.8 139 1 143.6 142 9 138 2 133.7 137.9 141.1 137.5 130.1 134.4 177. 178., 191.6 204.6 203.9 208 0 199 0 197.6 198 4
f7
200 6
199.8 197 9 190.0 193.6 181 6 188.6 178 7 174 4 177.3 184 6
4
F. 164.1 162.8 160 0 156.0 155 0 150 8 147. 7 146 3 143.2 144.8 142 139., 135.3 138.2 178.7 187.2 194. I 201.9 201.3 209.0 196.6 202 6 202.0 203 2 207 5 103.0 196 6 200 2 197.6 197 6 195 2 194 3 192 0 190 9
2
F. 156.1 151.6 149,9 144.5 146.3 138.2 138.6 136.2 133.2 131.5 129.6 127.3 122.9 122.4 170.6 178.1 184.6 199.0 196.6 207.6 187.7 195.7 194.3 197.7 197.3 196,3 188.9 192 7 188 6 182 9 184.1 181 0 178.0 183 0
mean assumes a constant film coefficient from one end of pipe to the other; actually the film coefficient increases as the fluid is heated. The true mean temperature difference for heating a liquid is probably somewhere between the arithmetic and logarithmic means, and in some cases may lie nearer the arithmetic mean. It will be of interest to compare the results both with the equation proposed by JIcAdams and Frost for water and with the curve representing the correlation suggested by Morris and Whitman (8) for heating both oils and water. Figure 2 s h o m the data on the four pipe lengths plotted as h,Dlk ~ sD.u s / z / ,together with lines representing Equation 3 for the four values of r used. As mag be seen, the points fall in a band lying between the lilies for the two extreme pipe lengths, but there is no apparent tendency for
INDUSTRIAL AiVD ENGINEERING CHEMISTRY
306
the points to segregate on four distinct lines falling in the order of the pipe lengths. Close inspection of the plot reveals the fact that the low points for any one series are for those runs in which the water entered considerably preheated, and in which both film and main body viscosities were relatively low, showing that for these tests the viscosity effect was not so great as indicated by Equation 3. Although there is this scattering of the points because of variations in the water viscosity, it may be seen that the best straight line through the points would correspond to a line representing Equation 3 for a tube length of 130 to 140 diameters.
04
0.6
as
Figure 3-Comparison
1.0
21)
4.0
6.0
OD ia
20.
30.
of Results w i t h Morris a n d W h i t m a n Curve
Figure 3 shows data replotted as
y/(7)0.31 D ~ s vs.
zm
as suggested by Morris and Whitman. Since the only variables are h,, u, and zm, plotting the data in this way i s es~ ~ u/z,. . ~ ~ The sentially the same as plotting h , / ~ vs. result is to bring the mass of points closer to a single smooth curve, and close inspection reveals that the points for any one pipe length fall very close to a single straight line. Although the points for the shortest pipe fall lower as a group than those for any other length, the highest group represents the data on the next to the shortest pipe and the data do not, in general, fall in order of the pipe lengths. The solid curve is that proposed by Morris and Whitman, and the data are seen to be in remarkable agreement with the upper end of this curve. Since the straight line shown drawn through the points on this plot (using logarithmic coordinates) has a slope of 0.68, the net effect of viscosity is that of z, raised to the 0.37 - 0.68 or the -0.31 power. This again brings out the fact that the influence of viscosity is less than indicated by Equation 3. A somewhat similar method of plotting is shown in Figure 4, in which the data are shown replotted with the exponent on the c?,,,/k group changed to 0.50. The coordinates are the same as those used by Rice (15), who, however, used the physical properties of the fluid a t the mean film temperature. The data fall slightly closer to a single curve than when the 0.37 exponent is used, and again no definite variation with tube length is apparent. The slope of the straight line shown is 0.70, indicating the fJm coefficient to vary inversely as the 0.20 power of the viscosity. The equation of the line shown is (4)
which is recommended as the form of the original Nusselt equation best correlating the present data.
Vol. 23, No. 3
Although the present data show a slightly better correlation using the exponent of 0.5 rather than 0.37 on the group crm/k, it is known that an exponent of 0.33 better correlates data on both gases and liquids. This has been pointed out by McAdams ( 5 ) and also by Cox (a). Any general equation for both gases and liquids may consequently be expected to show a smaller effect of the cr/k group than indicated by the exponent of 0.50. Since k is taken as a constant and c, D, and s were not varied, it is not necessary to include these quantities in order to obtain a good correlation of the data of the present investigation. For example, Figure 5 shows the data plotted as ~ ~ , it may be seen that the straight line h, vs. u / z , , , ~ .and shown represents the point quite well. The equation of this line is
indicating h, t o vary as the -0.24 power of the viscosity z,. Although this equation well represents the present data, it cannot be expected to apply outside the range of variables covered in the present investigation and listed above. I n view of the good correlation obtained when plotting the data as shown in Figure 4, it is of interest to compare the result with the classic data of Stanton ( l 7 ) , one of the few reporting data on heat transfer to water using more than one pipe. Figure 6 shows Stanton’s data plotted in this way, and it may be seen that for the higher range of Duslz, the data fall on two well-defined curves, the upper line representing the data on the pipe 33.8 diameters long, and the lower curve the data on the three pipes of lengths 31.6, 41.6, and 62.4 diameters long. Thus, although one pipe is out of line, the data on the other three indicate no effect of pipe
a4
as a8
0.1
Figure 4-Correlation
29
31) 4~
so a0
io.
2.0
34
Using Coordinates Suggested by Rice
length, when correlated in this way. The dotted curve represents the data of the present investigation, taken from Figure 4. Stanton’s data are seen to fall in reasonably well with the present results, except in the case of the data on the 0.547-inch i. d. pipe, which are approximately 40 per cent higher. With reference to these data of Stanton, it is particularly significant that the five points representing data obtained when heat was being transferred from water to pipe, fall on a line with the results for heating water in the same pipe. This throws light on the much-discussed discrepancy between Stanton’s data on heating and cooling of water in the same pipe, and suggests that if the data are properly correlated the apparent effect of the direction of heat flow disappears.
IND7JSTRIA.L AND ENGl ‘NEERING CHEMISTRY
March, 1931
307
Analysis of Over-All Coefficients
04
LO
0 6 08 LO
10 4.0
Figure 5-Simplified
60 81)I Q
3a
20.
Method of Correlation
Many other data on heat transmission to water in turbulent flow in pipes hare been collected and plotted as haD/(?)O.$
ys
&s zm
for purposes of comparison with Figure 4. Although the resulting plots will not be reproduced here, good correlations resulted in practically every case, the slopes varying from 0.69 to 0.80. As an arbitrary basis for comp:trison, the values of the ordinates at an abscissa (Duslz.,) of 2.0 were read from these curves and tabulated. Table T’I shows the results, arranged in order of the values of the ratios of length to diameter. Table VI-Comparison
The important conclusion from the above analyses of the data obtained on the film coefficients is obviously that for these data the effect of tube length on the coefficients was negligible. This conclusion is a t such variance with those of Nusselt and of McAdams and Frost that a check method of analysis of the data is highly desirable. I n particular, it is preferable40 have a method of analysis of the data which does not involve the use of the pipe-surface temperature measurements, as it is probable that criticism of the above analysis n-ould be directed a t the arbitrary definition of mean temperature drop through the film, or a t the use of thermocouples for obtaining accurate pipe-surface temperatures. Fortunately, such a method exists in that proposed by E. E. Kilson (&I), and later used by hlcildams, Sherwood, and Turner (7), for the analysis of data on over-all coefficients obtained by a number of investigators. Using this method, Figure 7 shows the reciprocals of the over-all coefficients L’ plotted vs. zm0’24/u0‘73 for the tests where the water was not preheated. The slopes of the lines, and not the elevation of the points, determine the values of the film coefficients inside the pipes. The slopes of the four lines are all practically the same, although the slight differences in the slopes of the lines shown are in inverse order of the pipe lengths. This substantiates the above conclusions based on the film-coefficient data, regarding the effect of pipe length. 6000 I
I
1
I
l
l
DATA OF S T A N T O N
4000
I
I
l
l
I
I
I
I
of Data w i t h Those’ of Previous Investigators
INVESTIGATOR DIAMETER LENGTH
Lawrence and Sherwood Lawrence and Sherwood Morns and Whitman ( 9 ) , water only Lawrence and Sherwood Soennecken ( 1 6 ) Baldwin and Sherwood (7) Stanton (17) Webster ( 1 9 ) Lawrence and ShPrwood Stanton Stanton Stanton Burbach ( I )
Inches 0.593 0.593
Inches 133 108
0.62 0.593 0.669 0.494 0.290 0.50 0.593 0.421 0.547 1.125 0.197
121 72.5 75.5 49.5 18.1 30 34.9 17.5 18.5 35.6 9.8-78.8
D 224 183
I:
970 920
196 122 113 100 62.4 60 59 41.6 33.8 31.6 50-400
Except for two high values, the results fall surprisingly close to a mean (not including Burbach) of about 1010. This table serves as a basis for a quantitative comparison of the present data with data taken from the literature. I n a series of tests with a given pipe length, there are in reality only three variables-namely, the film coefficient, water velocity, and the water temperature. Since temperature distribution, both length~iseand across the section of the pipe, is doubtless important, it is obvious that a single term representing the influence of temperature can never be entirely successful in correlating all the data where the temperatures are varied. It has been shown that z?m-0.20 or 7m-0,31 allows for temperature variations quite satisfactorily in the case of the present data, but it mal be noted that when the data are correlated in this way zm with a suitable exponent must alone bear the burden of allowing for the complicated temperature effect. The thermal conductivity of the water increases with temperature and it is possible to have some function of k instead of zm represent the temperature effect. It is probable that allowance for the variation of both k and ?,n (or z r ) , using suitable functions of these variables, would result in equally good correlations of the data.
5
400
b.’
A
7’
x
/2
p(
200
318
DIAMETER,IN. L E N G T H ) I N
0-7
Ia5
-
0421 12s S A M E , W A T E R BEING COOLED 8 62A 0290 18.1 031.6 1.125 35.6 e
416
/ I 02
DUS 2,
IO0
OA
0.6
OB LO
20
Figure 6-Comparison
40
60
80 IO.
20.
w i t h Data of S t a n t o n
The average slope is approximately 0.00305, corresponding to a value of 415 for the constant in Equation 5. This is 12 per cent higher than the value of 370 obtained from the film-coefficient data, but the discrepancy is no doubt due to the fact that the vapor side resistance is in general larger for the high-velocity runs, and the observed slopes of the lines on Figure 7 are consequently slightly low for purposes of comparison with this constant of Equation 5. Limitation of Present Data
I n concluding that tube length has no appreciable effect on the film coefficient of heat transmission, it should be kept in mind that the present data cover only water flowing in turbulent motion and one pipe diameter. Preliminary data obtained by the authors indicate a pronounced effect of tube length for oils flowing in the viscous and lower turbulent ranges, and it seems possible that the Graetz theoretical equation may be found to hold a t least approximately for the case of oils or other liquids having low thermal conductivities, when flowing in viscous motion. Burbach (1) has recently published the results of experiments on the cooling of water in a 0.197-inch i. d. tube, from
I-VDC’STRIAL il.ITU EAVGISEERI.YG CHEMISTRY
308
which he deduces a considerable effect of tube length. Hot water -was forced through the small tube surrounded with ice and the tube-mall temperature near the outlet end measured by means of an imbedded thermocouple. By repeating experiments using different tube lengths and measuring the temperature of the well-mixed water leaving the tube in each case, Burbach was able to construct curves of tube and water temperatures us. distance from the hot-water inlet. He then calculated the film coefficientsof heat flow from water to pipe a t various points along the tube, obtaining the rate of heat flow a t any point from the carefully measured slope of the water-temperature curve. On plotting hD/k vs. Dupc/k he found the curves for the points near the water inlet to
Vol. 23, S o . 3
taken as equal to the heat picked up by the water, and the temperature of the outer pipe surface as determined by the thermocouples, and the temperahre of saturated steam a t the pressure prevailing in the steam chest. The resulting coefficients varied from 1500 to 3600 B. t. u. per hour per square foot per O F. and were noticeably less when the temperature difference was large. The data are shown plotted in Figure 8 -_ as h, +‘Qk3/z vs. the temperature difference At,, in order that they may be compared easily with the Nusselt theoretical equation (23) for condensation of vapors outside single horizontal tubes:
The constant 0.725 applies when the units are consistent; conversion to the units shown in the table of nomenclature, and substitution of the diameter of the pipe (0.75 inch 0. d.) result in
0.002
0.0015
(7)
which is represented on Figure 8 by the dotted line. I n -on this figure, the physical variables in plotting the results the group
