Richard G. Gann Naval Research Laboratory Washington, D. c. 20375
An Inexpensive Wide Range Capillary Flowmeter
There are numerous experimental situations in which it is desirable to admit a gas over a wide dynamic range of flow rates. Some examples are gas phase titration, fuel admission to flames, and mass spectrometer calibration. This paper describes a versatile but simple capillary type flowmeter which satisfies these needs while remaining inexpensive. At a source pressure of 1 atm, deliveries of 10-3 to 10 NFT cm3/s have been achieved with high reproducibility. These limits are due only to the experimental necessity a t the time, and wider ranges should be possible. A capillary flowmeter operates on the principle that a pressure drop occurs when a gas undergoes laminar flow through a long straight tube. For such a tube of circular cross section, the Poiseuille flow formula gives the relationship between the flow rate and the pressure drop
where F is the gas flow in molecules/s, r and 1 are the tube radius and length, in cm, respectively, 7 is the coefficient of viscosity of the gas in poise (dyne-s/cm2), k is the Boltzmann constant (erg/"K), T is the absolute temperature ('K), AP is the pressure drop P, - Pd between the upstream and downstream pressures, (dynes/cmz), respectively, and P is the arithmetic mean pressure, (P, Pd)/2 (dynes/cm2). Two equivalent, but more convenient, forms are
+
where the pressures are now in torr. Dushman and Lafferty2 note that upon entering the capillary, the gas requires some distance to establish a fully developed laminar flow. To correct for this, the following dimensionless factor should appear in the denominator of eqns. (1)-(3)
where M - i s the gram molecular weight of the flowing gas. The equations then would read
Simple substitution demonstrates that nitrogen delivered a t NTP and flowing through a capillary of 1-mm radius and 5-cm length, will produce a 1 ton. pressure drop if the flow rate is -24 NTP cm3Jsec. Under these conditions, the correction factor is negligible only for flows less than -0.2 NTP cm3/s. The essential components, then, of a capillary flowme-
Figure 1. (A) Capillary. (Bj Swageiak3 crosses. (C) Connections to system. (Dj Relief stopcock. (El Overflow bulbs, ( F ) Manometer column. (0) Fluid drain. ( H ) ~ w a g & k ~ c a p .
ter are the capillary itself, inlet and outlet connections for the gas flow, and a differential pressure measuring device across the length of the capillary. This latter need is well filled by the use of one of several commercially available capacitance manometers. However, they are quite expensive relative to the cost of the rest of the instrument, and a simple differential manometer is more commonly used. This introduces a constraint in the use of a single flowmeter for introducing a gas a t widely varying rates, since a manometer of reasonable length is only readable over 2 orders of magnitude. This is easily overcome by making the ca~illariesinterchanceable. As can be seen upon examinaiion of eqns. (1)-(3), small changes in the capillary radius have a pronounced effect on the pressure drop for a given flow rate: There are many different designs of capillary flowmeters in use. One variation, used by the author and shown in Figure 1, combines versatility, low cost, and ease of construction, and will be discussed as an example. The U-tube manometer fF) is constructed of 8-mm medium wall tubing with a %-in. 0.d. filling spout at the bottom. The length of the tubing can be made short to meet spa'Part of this work was performed at the University of Pittsburgh in collaboration with Dr. Frederick Kaufman under the sponsorship of the Advanced Research Projects Agency. Department of Defense, and monitored by US. Army Research OfficeDurham, Box CM, Duke Station, Durham, N.C. 21706, under Contract No. DA-31-124-ARO-D-440. Dushman. S., and Lafferty,J. M., "Scientific Foundations of Vacuum Technique," 2nd Ed., John Wiley & Sons, Ine., New York, 1962, pp. 80-7. Volurne51. Number 11. November 1974
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tial limitations or long to provide a wider dynamic range. Generally a length of 20-50 cm has proved successful. The spacing between the manometer columns was reduced to within the width of a mirrored metric scale (Scientific Glass Apparatus Co.) for elimination of parallax. The drainspout a t the bottom (G), sealed with a Teflon-ferruled Swagelok3 cap or a Teflon stopcock (H), simplifies the removal and refilling of the fluid. A low vapor pressure silicone oil, DC-702 or 704 (Dow-Corning Co.), is generally used as this fluid since its density is a factor of 12.7 lower than mercury, thus increasing the manometer sensitivity. (In some situations, it is also undesirable to admit a mercury-saturated gas to the system.) The overflow hulbs (E)must be large enough to contain the fluid in the event of a $as surge. Kjeldahl hulhs are preferred since they can prevent spurting of the oil into the capillary. The bulbs are topped with %-in. 0.d. glass tubing. The relief stopcock (D)is of a standard glass-Teflon or metal bellows variety. I t is closed during normal gas metering, but is opened as a capillary bypass in the event of too large a flow or for faster evacuation of the flowmeter. The crosses (B) used to connect the components may be blown of Y4in. glass if desired, but the use of inexpensive brass Swagelok3 crosses reduces construction time as well as cost. By reversing the front ferrule and inserting an O-ring between it and the cross body, a tight seal with some flexibility can be obtained. The separation of the crosses is determined only by the width of the valve. The fitting of the capillaries (A) is not difficult. They consist of the propcr length of capillary tubing, bent into a "U" for insertion into the Swagelok crosses. For relatively large flows, capillary tubing of an appropriate i.d. may be commercially available. For smaller flows, it is necessary to reduce the tubing bore. There are two simple methods for achieving this. The first consists of inserting a length of metal wire through the length of the capillary, heating the tube till i t collapses on the wire, and then removing the wire. Performing this while the capillary is mounted on the flowmeter proper will prevent distortion. The second method involves mounting the capillary, flowing gas through i t a t about % the desired maximum flow, and heat-collapsing the tube till the desired manometer reading (with a cool capillary) is achieved. In both cases, the collapsing must be reasonably uniform to avoid sudden constrictions which result in non-laminar flow or even total closing of the capillary. Both techniques require only a little practice to achieve a desired flow range. A schematic of a typical flow-delivery system is shown in Figure 2. If the gas is to be delivered a t a source pressure of 1 atm, a bubbler 1 0 will maintain the reference pressure, P,. Otherwise, a pressure gauge can be installed. A relief valve (D),set a t -0.1 atm positive, will serve to prevent damage to the system in the event of an overpressure. A metering valve (F) downstream of the flowmeter (E)provides the actual control of the flow rate. The flowmeters can be simply calibrated either (1) by measuring the pressure rise in a bulb of known volume during a known time interval or (2) by measuring the time taken by the gas to transport the mercury-sealed piston in one of the commercially available "volumeters" (e.g., from George Porter Co.). Figure 3 illustrates calibration curves for Nz and COz for a low range flowmeter in which the capillary was partially collapsed from -0.5-mm
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Figure 2. Schematic of a Typical Gas Delivery System. (A) Gas source. (Bj stopcocks or valves. (C) Bubbler or pressure gauge, (Dl Relief valve, (Ej Capillary flowmeter. (Fj Gar metering valve. (GJ To apparatus. (Hi To calibration device.
Figure 3. Typical calibration curves for an interchangeable capillary flowmeter with a 25-cm manometer column filled with DC-702 silicone oil. O-COz: O-Ns. A few points in the very low flow range have been omitted far clarity.
i.d. tubing. The ratio pf the flows for a given measured pressure drop is equal to the inverse ratio of the viscosities as expected from the earlier discussion. (Due to even small irregularities in the capillary radius, it is not possible to precisely calculate the absolute flow of a function of pressure drop. Hence experimental calibration is necessary.) These plots are generally reproducible to within the reading error of the manometer. Swagelok is a trademark of the Crawford Fitting Co., Cleveland, Ohio.
