SeDtember. 1928
ILVDUSTRIAL A V D ENGIiVEERING CHEMISTRY
923
Friction Coefficient for Gas Flow through Small Glass Tubes’ Marshall Elliott UNIVERSITYOF TEXAS,AUSTIX, TEXAS
H E continuous measurement of small rates of gas flow is a frequent problem in chemical engineering research. Benton2 has comprehensively discussed the types of apparatus suitable for such work and has presented the results of a large amount of experimental work done in that connection by the Chemical Warfare Service. He sets forth a number of empirical formulas, each fitting, quite arcuratelp, gas flow under a given set of conditions. He concludes that the most generally suitable type of flowmeter is simply a capillary glass tube connected to T’s a t both ends. With proper functional relations or with exact calibration, the rate of gas flow through such a tube can be determined from the observed pressure drop between the two T’s. Benton presents a set of curves from which suitable sizes of tube may be selected, but he does not claim that these curves may be used in place of actual calibration. He states that in all cases where accuracy better than 10 or 1.5 per cent is desired the apparatus should be calibrated against some sort of standard meter. Such calibration requires a lot of time, is always troublesome and, where a standard meter is not available, is very inconvenient. Therefore some means of making accurate calculations from known tube sizes would be eminently worth while. Benton has quoted or determined a number of exact formulas for this purpose but no one of them is generally applicable. In the belief that the wellknown Fannin equation for fluid flow could be used for such a purpose provided only that proper constants were available, the present work was undertaken.
T
Experimental
The experimental procedure consisted essentially in the calibration of several capillary flowmeters of the type recommended by Benton. A glass tube of the desired bore was connected to glass T’s a t both ends and, through these, to manometers. Various gases were then passed through the tube and the pressure drop was recorded. The rate of flow was measured by a a-et-type displacement meter timed by a stop watch. The wet meter had previously been checked against a gasometer. The glass tubes used were of two diameters, 1.139 and 3.014 mm.; so chosen as to have as nearly uniform a bore as possible and that the variation in bore was uniform from end to end, as observed by measuring the length of a “slug” of clean mercury a t various positions along the tube. In no case did the variation in bore exceed 3 per cent of the total diameter of the tube. Tubes of various lengths were used a t first, but owing t o the possibility of inaccuracy resulting from the so-called “end effects,” losses of head due to sudden contraction or enlargement, only those tubes of greatest length (120 cm.) were used as a basis for the data plotted. The diameter of each tube was determined by the usual method using mercury. Gases varying in density and viscosity from hydrogen to carbon dioxide, with air and methane in between, were used. The velocities of flow ranged from 5 to 121 feet per second. The time required for the passage of 0.1 cubic foot of gas a t constant pressure was recorded. The pressure drop was determined by means of a differential manometer, 1
Received May 21, 1928. J. I N D . E N D . CHEM.,11, 623 (1919).
and the total pressure on the gas at the upstream end of the capillary tube was also noted. The area of the tube being known, the velocity, u, could be calculated from the amount of gas passing in a given time. Since the pressure drop in the Fannin equation is expressed in feet of the fluid flowing and it was observed in this case in millimeters of water, the latter must be converted to the former by means of the relation : Ah =
pressure drop in mm. of HzO X density of water (62.4) 304.8Xdensity of gas under flowing conditions(1bs.per cu. ft.) Calculations
The data thus obtained were substituted in the Fannin equation where Ah = loss of head between two ends of capillary expressed in feet of the fluid actually flowing, due account being taken of effect of temperature and pressure L = length of tube in feet u = velocity of flow in feet per second d = diameter of tube in feet g = constant for acceleration of gravity (32.2 feet per second per second) f = so-called “friction factor” and is the only arbitrary constant involved in the equation
L and d were of course directly measured and g is a constant, thus leaving f as the only unknown term. By substituting the experimentally determined data in the equation, values of f were obtained for various velocities, tube diameters, densities of gas, kinds of gas, lengths of tube, and temperature of flow. This term f has been shown to be a t all times a function of several variables-viz., the velocity of flow, the density of the fluid flowing, the viscosity of the fluid, the diameter of the conduit, the physical state of the fluid (gas or liquid), and the material from which the conduit is made. When the last two variables are kept constant, as is the case when a gas is flowing through a capillary glass tube, the value of f may be expressed
I’=
d
(%)
where D = diameter of capillary in inches u = rate of flow in feet per second p = density of gas a t upstream temperature and pressure M = viscosity in English units (0.0672 X c. g. s. units) of gas a t temperature of flow. This may be obtained from standard physical-chemical tables such as Landolt-Bornstein
The term ( D u p l p ) is known as the “modulus.” There are a number of tables, curves, and equations available in the literature and standard texts from which the value o f f can be obtained when the value of (Dup/p) is known. These are, however, all confined to those problems encountered in hydraulics in which a liquid is flowing through metal pipes and are worthless for calculations under other conditions of flow-. The values o f f obtained as above described were plotted on log-log paper against the corresponding values of (Dup/ p ) , S o effort was made to determine the functional relation
Vol. 20, No. 9
INDUSTRIAL AND ENGINEERING CHEMISTRY
924
which exists between f and (Dup/p)-that is, the relation which will fit the curve thus obtained-because there is no real need for such a relation; the curve is far more usable. The individual measurements from which the curve was made are tabulated below. Each point on the curve, which corresponds to a line in the table, is the result of four closely agreeing measurements. D
U
P/P
Fl./sec. L b s . / c u . f l . / v k a 10.87 ' 6i70' 22.50 6570 29,40 6570 44.34 6570 61.22 6550 10.22 951 16.56 903 31.67 925 44.85 947 62.50 951 90.50 946 956 121.00 5.20 11280 21.50 11430 35.60 11720 11870 43.80 9.76 6150 29.10 6150 42.32 6150 58.60 6150 6860 0.1184 6.18 6860 2 2 , SO 6860 36.42 6850 47,24 20.18 945 945 44.60 945 72.60 945 112.40 a Viscosity in English units.
Inch 0.0449
DW/P
f
3220 6660 8690 13020 18020 436 67(! 1310 1905 2660 3855 5200 2630 11080 18720 23300 2690 8050 11680 16150 504 1858 2962 3830 2260 5000 8140 12580
0.0479 0.0232 0.0183 0.0123 0.0088 0.4190 0.2730 0.1390 0.0900 0.0720 0.0470 0.0290 0.0610 0.0152 0,0093 0.0079 0.0609 0.0245 0.0136 0.0103 0.3120 0.0967 0.0643 0,0442 0.0644 0.0308 0.0168 0.0162
0.
r
0.4
03
I
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lYlllll XIIII
I
I I
I I I Ill1
I IIIIII
I
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I
I I l l II
GAS
FLOWING Air
Hz
CO? CH4
Air
H 2
Discussion
Since the curve was obtained by the calibration of a large number of flowmeters, the process can be reversed and the values from the curve used instead of the calibration when a flowmeter is desired. This method of calculation is not a general substitute for calibration, but in those very numerous cases where the approximate velocity at which the gas is to flow is known and where the total pressure on the gas and the temperature are fairly closely known, a series of calculations requiring about 30 minutes may be substituted for a calibration requiring 3 or 4 hours with no sacrifice in accuracy. This method of calibrating a flowmeter would consist in
the selection of a capillary tube of about 1 meter length and of suitable diameter. The diameter must then be determined accurately. Since the density and viscosity of the gas are known, several values of velocity covering the range expected may be assumed and corresponding values of f read from the curve. By substituting these in the Fannin equation, values of Ah corresponding to each of the assumed velocities may be obtained and a curve of pressure drop us. velocity may be drawn. The close approach to a straight line obtained in the curve justifies the assumption that it could be extrapolated on either end if necessary. This assumption is further justified by the fact that other similar curves, made for different conditions of flow, do not show any sharp inflection except when the critical velocity of the fluid is reached. It is very rare that gases are handled at such low velocity. However, it is hardly likely that in ordinary work there will be any need to extrapolate the curve here given. The sizes of the tube and velocities used cover pretty generally the range likely to be met in practice. It will be noted that the minimum rate of flow is less than 100 cc. per minute while the maximum is nearly 25 liters per minute.
Treatment of Missouri River Water for Locomotive Use' H. H. Richardson MISSOURIPACIFICRAILROAD Co.,ST. LOUIS,Mo.
ISSOURI River drains wholly or in part ten statesColorado, Iowa, Kansas, MinnesotJa,Missouri, Montana, North Dakota, South Dakota, and Wyoming. The drainage area of 528,850 square miles, which is approximately one-sixth of the area of the United States, is larger than that of any other of the tributaries of the Mississippi; in fact it comprises 43 per cent of the total of the Mississippi basin. For a considerable part of the distance from Omaha to Kansas City, Mo., and from Kansas City to St. Louis, the Missouri Pacific Railroad parallels the course of the Missouri River, taking advantage of the water-grade line. It is to be expected that on this rail distance of 491 miles the major 1 Presented before the Division of Water, Sewage, and Sanitation Chemistry at the 75th Meeting of the American Chemical Society, St. Louis, Mo., April 16 t o 19, 1928.
source of water supply both for the cities and the railroad would be the Missouri River. Although not subject to the rapid fluctuations in quality that are common to smaller rivers nor to serious industrial trade waste pollution, the quality of the supply is unsatisfactory for steam-making purposes on account of the scaleforming and suspended matter present. The total hardness averages approximately 14 grains per gallon and total dissolved solids, 20 grains. The seasonal variations in quality are more or less regular, the maximum dissolved solids and minimum suspended matter occurring in December and January, and the minimum dissolved solids and maximum suspended matter in the late spring and early summer months. The spring floods of 1927 caused an unusual increase in hardness during May and June. There was also an un-
