SEPTEMBER 15, 1935
ANALYTICAL EDITION
aniline and sodium carbonate. It has also been used in solutions of nitric, perchloric, maleic, and fumaric acids where the hydrogen electrode does not work satisfactorily. The electrode was positive to a saturated calomel electrode in a 0.05 1V solution of chromic acid, and as the acid was titrated with sodium hydroxide it passed through gero and finally became negative. The germanium titration curves generally show a greater slope on the alkaline side than on the acid side. The results of t,he titrations compare favorably with those
355
obtained with the antimony electrode. The data for a titration curve of a monobasic acid can be obtained in 2.25 hours.
Literature Cited (1) (2) (3) (4)
Barnes and Simon, J. Am. SOC.Agron., 24, 156 (1932). Britton and Robinson, J . Chem. SOC., 1931, 45% Kolthoff and Hartong, Rec. trav. chim., 44, 113 (1925). Nichols and Cooper, IND. ENG. CHEM.,Anal. Ed., 7, 350 (1935).
RECEIVED April
20, 1936. Based upon the thesis presented t o the Faculty
of the Graduate School of Correll University by S. R. Cooper in partial fulfillment of the requirements for the degree of doctor of philosophy.
Constant-Flow Orifice Meters of Low Capacity R. T. PAGE, Department of Industrial Hygiene, Harvard School of Piiblic Health, Boston, Mass.
M
AXY devices, differing widely in principle of opera-
tion and in design, are employed for measuring the rate of flow of gases. The advantages and limitations of each device are well defined. The usual laboratory methods possess features which render them unsuitable for the metering of small amounts of gases, particularly when a constant rate of flow is required. The displacement type of meter, using the receiver method of measuring gas (typified by the ‘ldry” meter used in the commercial metering of illuminating gas, the more accurate “wet” meter, and the spirometer) doesnot give an instantaneous reading of the rate of flow, but requires the measurement of the time interval during which a definite volume is passed. Anemometers and float meters are not only delicate but also inaccurate a t low rates of flow. The method of measuring change in temperature of the gas due to the addition to it of a known quantity of heat has the disadvantage of not giving a direct reading of the rate of flow without complicated apparatus. Pitot tubes and venturi meters, while excellent in principle because of the comparatively small resistance they introduce into the line, are difficult and expensive to make, and unsuitable for measuring gases a t rates of the order of 50 liters per minute or less. The resistance-tube meters developed by Muster ( I $ ) , and described by Guye and Schneider (Q),were extensively used by the Chemical Warfare Service. In these meters the loss of head due to passing the gas through a capillary tube is a function of the rate of flow. Benton ( 2 ) gives detailed specifications for the calculation of dimensions and the construction of meters ranging from 500 cc. to 200 liters per minute in capacity. These meters are difficult to duplicate accurately and each one must be calibrated separately. They are fragile and not ideal for field use, but are unexcelled for metering pure gases a t rates lower than 4 liters per minute. When the gas is contaminated with particulate matter, an appreciable error may be caused by the deposition of dust a t the mouth of the capillary tube. Frequent cleaning becomes necessary and often proves difficult. The orifice compares favorably with the resistance-tube meter. Orifices can be constructed in noncorrosive materials, are easily cleaned, and are satisfactory at fairly low rates of flow. At low pressure differentials, the rate of flow of gas through an orifice is a function of both the pressure above and the pressure below the orifice. This is also true in the case of the resistance-tube meter and the venturi meter. Both pressures must remain constant, or the rate of change of one must be a definite function of the rate of change of the other, to obtain a constant rate of flow. This result is very difficult to obtain without complicated and expensive compensating apparatus.
Constant-Flow Orifices The problem of securing small constant rates of flow of gas has been solved practicably in this laboratory by the use of small orifices operating a t pressure differentials greater than the critical, the pressure above the orifice being equal to or less than atmospheric pressure, and the downstream pressure being less than 0.53 times the upstream pressure. When an orifice is operated a t a pressure differential equal to or greater than this definite critical value, the pressure below the orifice has no effect on the rate of flow (11); and for constant upstream pressures, a constant flow can be obtained through the orifice irrespective of wide variations in the downstream pressure. When a suction pump or ejector is used to create a less than critical pressure below the orifice, the only factors affecting the constancy of flow will be the atmospheric pressure and the resistance in the line above the orifice. In practically all cases these will remain constant within the allowable limits of error of the problem, but for accurate work a resistance compensator can easily be located above the orifice to insure constant rates of flow. A constant-flow orifice was developed in this laboratory (IO) which passed approximately 28.3 liters per minute (1 cubic foot per minute) of air when it was placed in the suction line below a modified Greenburg-Smith impinger, a dustsampling instrument. The satisfactory performance of this and similar orifices which have been put to a wide variety of uses leads the author to believe that a description of this useful laboratory tool will be of interest.
Orifice Characteristics The rate of flow of a gas through an orifice is a function of both the pressure above the orifice, P I , and the pressure in the vena contracta, P. For an orifice with a smoothly rounded approach, P is the pressure in the throat. Coefficients for use with the theoretical discharge formulas for certain types of orifice operating under certain definite conditions have been determined accurately enough to allow the calculation of the rate of gas flow with high precision, without experimental calibration of each orifice. This holds for large orifices, which, if properly designed and installed with properly located pressure taps, serve as primary measuring instruments for low pressure differentials. The value of the coefficient decreases as the size of the orifice decreases, since the effects of skin friction and any irregularities due to construction vary inversely as the hydraulic radius. Unless very small orifices are made with extreme precision, the coefficient for each orifice should be calculated from an experimental calibration if the orifice is t o be used for precise measurements.
INDUSTRIAL AND ENGINEERING CHEMISTRY
356
VOL. 7 , NO. 5
If a gas flows adiabatically through a n orifice, and if it is assumed to act as a perfect gas under the condition of entrance to the orifice, the weight of gas discharged through the orifice (8) is given by the following formula:
where M A
= = g = y = PI =
P
=
Pp = Vi =
TI
=
grams of gas per second area of throat of orifice in sq. cm. gravity constant ratio of specific heats of the gas = Cp/Cv abpolute pressure on the entrance side of the orifice in grams per sq. cm. absolute pressure in the throat of the orifice absolute pressure below the orifice specific volume of the gas on the entrance side of the orifice in cc. per gram absolute temperature of the gas on the entrance side of the orifice in degrees Centigrade
a. b. c.
d.
e. f.
600-liter spirometer Psychrometer for measuring humidity of air entering orifice Gate valve to control upstream pressure Straight approach tube 1 inch (2.5 cm.) in diameter and over 20 diameters long Orifice holder, orifice, and pressure taps Psychrometer for measuring humidity of air leaving orifice No. 2 Hancock ejector connected t o compressed air line (over 2.4 kg. per s q . cm.)
g. The general gas law for a perfect gas is expressed by the for- mula : FIGURE 2. APPARATUS FOR CALIBRaTING ORIFICE FLOWMETERS PV = BT or P / V = P2/BT (2)
If the gas being measured is air, then B = 2930, and y = 1.4, so that by letting g = 981.5 em. per sq. second and substituting in Formula 1 one obtains the equation:
or
1M =
PiA
1.531 -J-J ( K ) 1
From Formula 3 it is evident that the weight of air discharged depends upon the two pressures, PIand P,and varies directly as K . The maximum discharge will occur when K reaches its maximum value-i. e., when PIPi= 0.53. This also is the minimum possible value for P and is called the
vacuum with the acoustic velocity corresponding to the state of the air in the orifice, no matter what resistance exists during the flow toward the orifice.” The velocity of efflux is constant as long as TI is constant and is independent of Pi, although the quantity of gas delivered depends upon this pressure owing to its effect on the density. If the velocity of the gas passing through the orifice is calculated, using Formula 3 as the basis for calculations, one finds that the velocity of air becomes equal to the velocity of sound, in air a t the same pressure and temperature, when Pz = 0.53 PI approximately. This is the maximum discharge velocity, equal to the velocity of discharge into a vacuum, and beyond this point P2ceases to exert a back pressure on the air coming through the orifice. Substituting 0.53 for P/Pl in 3a one obtains the formula: M
= 0.812
PIA/.\/T
(4)
which is the same as Fliegner’s formula ( 7 ) (expressed in c. g. R units) when PZis less than the critical pressure.
Design and Construction of Orifices
FIGURE 1. SKETCH OF ORIFICE
critical pressure Pm. It is an orifice characteristic that if Pz2 Pm,then P = Pz, but if Pz< Pm,then P = Pm (11). Consequently, for any value of PZequal to or less than 0.53 PI,the weight of gas discharged will depend only on Pi. This was shown t o be true by Napier in 1866 when he proved experimentally that the weight of steam discharged from a n orifice was independent of the pressure of the medium into which efflux took place at pressure differentials greater than the critical (6). This phenomenon was explained by Zeuner (14) on the hypothesis that “the air flows (through an orifice) into a
Careful consideration was given to the form of orifice to be used. There are very few data in the literature on the characteristics of small orifices of any type operating a t the critical pressure differential. Circular orifices with curved approach were adopted, since they were considered to be the easiest type to duplicate, they have a relatively high coefficient, and this coefficient is only slightly altered by wearing of the orifice edge. Figure 1 is a sketch of the type of orifice tested. Eighteen orifices, made with standard twist (tap) drills varying from No. 40 to No. 60, were calibrated. In making an orifice, a brass blank of the required thickness was soldered to the end of a 2.5-cm. brass rod centered in the lathe, and the blank was turned down to 3.3 cm. diameter. The orifice was then drilled in the blank with a standard tap drill held in the lathe foot-stock. The rounded approach was cut by hand with a three-cornered scraper and the curvature was checked with a template. The rod was then removed from the lathe and heated until the orifice plate dropped off, the solder being wiped off while hot. All burrs and rough places were removed from the lower rim of the orifice until it would pass a microscopic inspection. This method did not allow accurate duplication of orifices, but can be followed easily by an amateur mechanic. It
SEPTEMBER 15, 1935
ANALYTICAL EDITION
allows a rough prediction of the area of the finished orifice, but the true area must be accurately measured with a microscope. Orifice numbers denote the drill used in making them, but measured areas were used in all calculations of coefficients. Table I lists areas, diameters, and thicknesses of orifices.
TABLEI. DI&fENsIoNsO F
ORIFICES
Orifice No.
Theoretical Area for Tap Drill Used
40 40a 42 44 46a 46 4Sb 4Sa 48
0.04865 0,04865 0.04381 0,03748 0.03323 4.03325 0,02929 0,02929 0,02929
0.05171 0.04997 0,04672 0.03949 0.03581 0,03488 0.03125 0.03039 0.03029
0.2568 0,2522 0,2438 0.2243 0.2136 0,2108 0.1994 0,1966 0,1963
0.310 0.277 0.310 0.310 0.239 0.310 0,246 0.312 0.310
50a 50 52 52a 54a 54 56 60 58
0.02484 0,02484 0.02013 0.02013 0.01536 0.01536 0,01071 0.00813 0.00897
0,02541 0,02530 0.02136 0.02029 0.01616 0.01562 0.01143 0,01014 0,00928
0.1798 0.1796 0,1649 0.1608 0.1435 0.1410 0.1207 0.1135 0,1087
0.310 0.310 0.310 0.239 0.310 0.320 0,320 0.310 0.310
Sq. cm.
hleasured Diameter Thickness Area Orifice of Plate ( A ) (D = ,/4A/=) (T) sq. cm. Cm. Cm.
____ Calibration of Orifices The apparatus used in calibrating the orifice flowmeter is shown in Figure 2. All joints were sealed and tested for tightness. The spirometer had been previously calibrated by a waterdisplacement method. The air-filled spirometer was connected to a sealed tank filled with water, which had a valved outlet opening into a weighing tank. As water was drawn from the tank, an equal volume of air was drawn from the spirometer. The air in the spirometer was completely saturated with water vapor and temperatures were kept constant. The volume of the spirometer between the 500- and the 300liter marks, and between the 300- and 100-liter marks, was calculated from these data. I n conducting the tests, most of the runs were made for the volume of air contained between the 500- and the 100liter marks on the spirometer, although a few runs were made between the 500- and 300-liter marks. Air was drawn through the orifice from the room until the upstream pressure had been regulated and then the source of air supply was switched to the spirometer. The stopwatch used in timing was started as the 500-liter mark passed the indicator and was stopped when the 100-liter mark passed. The following data were secured for each point on the calibration curve:
357
M
11.03 P1'A
(5)
The measured volume of air discharged was converted to equivalent volumes a t 1 atmosphere (76.0 cm. of mercury) and 21.11" C. This corrected volume divided by the time gave the flow in liters per minute. These values are plotted against the pressure, PI, in Figure 3. The volumes per minute were corrected for moisture content and converted to actual discharge, M', in grams per second. Wet-bulb thermometer readings taken in the upstream supply line were corrected for error in depression due to the velocity of the air. Corrections were calculated from charts given by Carrier and Lindsay (b), and were very small in all cases. After the air passed the psychrometer, the vapor content could not increase; and, since the pressure was falling while the temperature remained nearly constant, the vapor in the expanding mixture was becoming more dilute and farther from saturation, so that there was no tendency for water to be precipitated. The composition of the moist air remained unchanged along the line and was the same a t the orifice as a t the psychrometer in the supply pipe. This could be checked by the readings from the psychrometer below the orifice (S). The partial pressure of water vapor in the air was calculated from the corrected psychrometer readings by means of Carrier's psychrometric chart (4). I n calculating the theoretical discharge, the upstream pressure, P1,was calculated on the basis of a mercury column a t 21.11" C. In the tests the temperature of the column was never over 8" C. warmer than this and the error in manometer readings introduced by this temperature variation was negligible. The actual discharge, M', when divided by the theoretical discharge, M, as given by Fliegner's Formula 5 gave the coefficient of discharge, C: M' = ClV
(6)
Wet- and dry-bulb temperatures of the room Barometric pressure and temperature Dry-bulb temperature of spirometer air Wet- and dry-bulb temperature of air above orifice Wet- and dry-bulb temperature of air below orifice Pressure above orifice Pressure below orifice Time for measured volume of air to pass orifice All these factors were kept as constant as possible during any one run. The upstream pressure was varied from atmospheric pressure to atmospheric minus 30 em. of water, and from atmospheric minus 2.5 cm. of mercury to atmospheric minus 25.4 em. of mercury. Theoretical discharge quantities were computed a t 21.11" C. (70" F.) (T1= 294.2"A) and a t the absolute test pressure, PI, expressed in terms of centimeters of mercury. By substituting in Formula 4 letting 1 gram/sq. em. = 0.0736 em. of mercury-i. e., PI' = PI '0.0i36-one obtains the working formula: where PI' = the pressure in em. of mercury.
OF AIR TEIROUGH ORIFICESAT 21.11' C. FIGURE3. DISCHARGE AND 85 PER CENTRELATIVE HUMIDITY
INDUSTRIAL AND ENGINEERING CHEMISTRY
358
The values of the coefficient of discharge as determined in this investigation include the factors of velocity of approach, viscosity, irregularities of construction, and differences between actual and assumed conditions. Because of the effect of the last two factors and the limited number of observations on each orifice, there is a large probable error in any equation for the discharge coefficient formulated from these
0960 0950
variation in the value of C, with respect to the value of A when the value of PI is stated, were calculated by application of the method of least squares. These lines are shown graphically in Figure 5. These orifices were made by two people using the same technic so far as possible. Individual differences were not evident. If the area of an approximately circular orifice is accurately measured, the value of C is given approximately by the following equations. If the flow must be measured with a high degree of accuracy, each individual orifice should be calibrated for a t least three points on the curve, and the value of C can then be calculated.
L, a940 0930
09.70 OF DISCHARGE FIGURE 4. VALUESOF COEFFICIENT
For use with Fliegner’a formula for stated pre~suresabove orifice 44.
data. The data show a trend in the value of the coefficient which appears to indicate that it is a direct function of both orifice area and upstream pressure. This may be only an apparent effect with irregularity of construction acting as the controlling factor, but similar results have been reported by other experimenters. Polson, Lowther, and Wilson (IS) determined coefficients for orifices of 0.6 to 2.5 em. diameters when air under pressure discharged into the atmosphere with pressure drops in the critical range, and found that values of C for use with Fliegner’s formula varied directly with area and upstream pressure. Similar results with respect to area were reported by Bean et al. for impact nozzles of similar shape (1). Each value of C was plotted against PI for each orifice. Equations for the straight line of best fit were determined by means of the method of least squares. A typical curve for one orifice is shown in Figure 4. The values of C were calculated from these equations for each orifice a t pressures of 50.8, 63.5, and 76.2 em. of mercury, and these values of C were plotted against area of orifice A . Orifice 58 was dropped from the calculations a t this point, since it was markedly irregular in form. The equations of the straight lines of best fit for the
VOL. 7, NO. 5
Equations for C The following equations for C were calculated from the experimental data with respect to orifice area (em.’) a t the given upstream pressures : -
Cm. Hg
I
(Inches H g )
50.8 63.5 76.2
(20)
K
Equation for C C = 0.897 C = 0.899 C = 0.904
- 48.128 - 50.83.4 - 49,158
The probable error of the above equations was h0.6 per cent-values of C as determined from the straight line of best fit from the C versus PI graph for any one orifice were as likely to deviate less than *0.6 per cent from the values obtained by solving the above equations as they were likely to deviate more than *0.6 per cent. Individual observations for the relation between C and PI on any individual orifice showed a probable error of approximately *l.O per cent. The assumption of straight-line relationship is allowable within the range in pressures and orifice areas investigated.
Comments and Conclusions The object of this investigation was twofold: first, to develop a set of constant flow orifices for laboratory use, and second, to study the characteristics of these orifices. In accord with laboratory practice, results have been expressed in metric units. The author’s showed that small circular orifices to give constant flows of gas under critical pressure differentials could be easily made for laboratory use; that the rate of flow for any given upstream pressure could be calculated with a probable error of approximately * 1.0 per cent for an orifice of measured throat area, but that for more accurate determination of rates of flow, the orifice should be calibrated against a satisfactory primary standard.
Literature Cited
/w a95
0
0.90
OS 0.80
A FIGURE 5. VARIATIONOF COEFFICIENT WITH
RESPECTTO AREAAT DIFFEREKTPRESSURES ABOVE ORIFICE
Bean, H. S., Buckingham, E., and Murphy, P . S., Bur. Stahdards Research Paper 49 (1929). Benton, A. F., J. IKD. ENG.CHEX.,11, 623-7 (1918). Brooks, D. B., Bur. Standards, Miscellaneous Pub., M . 146 (1935). Carrier, W. H., Trans. Am. SOC.Mech. Engrs., 46,736 (1925); 33, 1005 (1911). Carrier, W. H., and Lindsay, D. C., Ibid., 46, 739 (1925). Church, I. P., “Mechanics of Engineering,” New York, John Wiley & Sons, 1913. Fliegner, il., Proc. Inst. Civil Engrs. London, 39,370 (1874). Goodenough, G. A., “Principles of Thermodynamics,” 3rd ed., Henry Holt & Co., 1929. Guye and Schneider, Helv. Chim. Acta, 1, 35 (1918). Hatch, T., Warren, H., and Drinker, P., J . Ind. H y g . , 14, 301-11 (1932). Marks, L. S.,”Mechanical Engineers’ Handbook,” New York, McGraw-Hill Book Co., 1930. (12) Muster, Thesis, Geneva, 1907. (13) Polson, I. A., Lowther, J. G., and Wilson, B. J., Cniv. Ill. Bull. 27, No. 39 (1930). (14) Zeuner, G., “Technical Thermodynamics,” Arthur Felix, 1874; tr. by Klein, J. F., New York, D. Van Nostrand Co., 1907. RECEIVED June 27, 1935.
