790
ANALYTICAL CHEMISTRY
tube at -78” C. and warmed t o 0’ C. the solid phase persists and an unbalance about 10 mv. smaller than the calibration value is observed. This indicates a vapor pressure about 0.1 mm. lower than that present during calibration, about the expected difference between solid and liquid deuterium oxide a t 0’ C. It is concluded that a liquid condensate is always present in normal operation. APPLICATION
The above analytical technique was used in a study of hydrogen eyrhange between water vapor and cracking catalyst. It proved particularly useful for the in situ determination of catalyst water content. These applications will be described in later publications. The method should be applicable to numerous problems n-here small samples of water must be analyzed with an accuracy within ahout 0.5% over the entire range of composition. ACKNOWLEDGMENT
The author is indebted to P. H. Emmett for his participation in ninny helpful discussions of this problem.
LITERATURE CITED (1) Clemo, G. R., and Swan, G.8.,J . Chem. SOC.(London), 1942,
370. (2) Farkas, A., “Orthohydrogen, Parahydrogen, and Heavy Hydrogen,” p. 172, London, Cambridge University Press, 1935. (3) Friedman, L., and Irsa, A. P., ASAL. CHEM.,24, 876 (1952). (4) Harteck, P., Z. Elektrochem., 44, 3 (1938). (5) Kirshenbaum, I., ”Physical Properties and Analysis of Heavy Water,” h-ew York, McGraw-Hill Book Co., 1951. ( 6 ) Lange, N. -4,,“Handbook of Chemistry,” 7th ed., p. 1483, Sandusky, Ohio, Handbook Publishers, Ino., 1949. (7) Ibid., p. 1487. (8) Miles, F. T., and Mensies, A. W. C., J . Am. Chem. Soc., 58, 1067 (1936). (9) Niwa, K., and Shimaraki, E., J . Chem. SOC.J a p a n , 60, 985 (1939). (10) Orchin, M.,TT’ender, I., and Friedel, R. A., ANAL.CHEY.,21, 1072 (1949). (11) Thomas, B. W., Ibid., 22, 1476 (1950). (12) .Thornton, V., and Condon, F. E., Ibid., 22, 690 (1950). (13) Korthing, A. G., and Geffner, J., “Treatment of Experimental Data,” 2nd ed., p. 205, New York, John Wiley BE Sons, 1943. (14) Worthing, A. G., and Halliday, D., “Heat,” pp. 43-6, Sew York, John Wiley & Sons, 1948. RECEIVED for review December 3 , 1952. Accepted February 9, 1953
Precision Measurements of Gas Flow Rates Mathematical Procedure in the Calibration of Flowmeters SABRI ERGUN Coal Research Laboratory, Carnegie Institute of Technology, Pittsburgh, Pa.
Precision in the measurements of rates of gas flow is essential in studying rates of reaction, diffusion, sorption, and heat transfer in flow processes and in the determination of geometric surface area and particle density of crushed porous solids. A method is outlined for precision measurements of gas flow rates and calibration of capillary and orifice type flowmeters. A meter is identified by two constants, a and b , representing viscous and kinetic factors in the pressure loss. These constants are determined by calibrating the meter with a selected gas-for example, nitrogen. They permit the use of the flowmeter for all gases with known physical properties under any temperature and pressure conditions without additional calibration. The method is not restricted to laminar nor to turbulent flow regions. I t is statistical and simple. It enables calculation of the precision of measurements made and serves as a guide in the design of laboratory flowmeters. With the method proposed, a flowmeter can be calibrated with a suitable gas at a convenient temperature and pressure and can be used for any gas under any pressure and temperature conditions.
T
HE problems confronted in the measurements of rates of gas flor are: selection of the meter, its calibration, and the precision of the measurements. Numerous types of flowmeters are available commercially. It has long been recognized that capillary and orifice-type flowmeters are among the most suitable to be used in the laboratory. The formulation of flow through these meters, therefore, has been the subject of numerous investigations (3, 4,7 , 10-13, 16, 20-22, 25, 26, 29, 51-33, $5, 38). Theories advanced or empirical curves proposed are not gen: erally intended, hoTyever, to be used in place of actual calibra-
tion. The problems arising from the range and flexibility of floameters have led to various modifications in the design of capillary flowmeters (1,4-6,8,9, 17, 19, 23, 28, SO, 31, 56). Generally, the literature on the methods of calibration of flowmeters deals with the apparatus used and the modifications thereof (b,18, 27, 54,37,
S9). A mathematical procedure to be folloTTed in the calibration and in the formulation of an equation of flow both in orifice and capillary type of flowmeters has been wanting. Only by the development of the f l o equation ~ can the precision of the measurements be calculated. THEORETICAL
The pressure loss for isothermal gas flow through a capillary or an orifice flowmeter can be expressed as the sum of viscous and kinetic energy losses:
AP =
apQm
f bpmQ2
(1)
n-here AP is the pressure drop in dynes per square centimeter, p is the viscosity of the gas in dynes per second per square centimeter, p is its denPity in grams per cubic centimeter, Q is the rate of gas flow in cubic centimeters per second and a and b are constants. The subscript m refers to conditions a t the temperature, T, a t the average of inlet, P I ,and exit, P,, pressures. The coefficients a and b are functions of the dimensions of the meter. They are found t o be independent of the properties of the gas. For circular channels, for example, a = 128L/aD4. The kinetic energy term includes entrance, exit, and head losses due to eddies in the channel. It is not intended here to formulate either a or b, but rather to determine them for the flowmeter to be calibrated. Since p0Q0 = p m Q m , where the subscript 0 corresponds to standard temperature and pressure conditions, a transformation of Equation 1 leads to:
APIQm =
(UP)
+ (bpo)&o
(2)
791
V O L U M E 25, NO. 5, M A Y 1 9 5 3
errors, as calculated by the method of least squares, in the values of intercept, u p , and slope, (Sitrogen flow) bpo, respectively, are zk0.06 and 1P" Volumeb, Time, Qh. QmC, Qo x AP/Qm = Y 0.29%. Since the viscous conAd Cni. H ~ O CC. Sec. Cc./Sec. Cc./Sec. Cc./Sec. Cm.HtO/(Cc./Sec.) tributions are dominant in the -0 001 1.844 2.956 2.074 480.8 2.080 6.13 1000 pressure loss for flow rates rang3.006 +O 003 3.273 2.915 304.0 3.289 9.84 1000 ing up to 24 cc. per second 3.069 4 0 001 4.945 4.414 200.8 4.980 15.17 1000 a t standard temperature and -0 002 3.123 6.405 5.733 154.6 6.468 20.00 1000 3.184 +O 006 7.773 6.974 127.1 7.868 24.75 1000 pressure, standard errors in the values of Qo corresponding to -0 007 3.229 9.245 8.315 106.6 9.381 29.85 1000 -0 002 3.287 10.61 9.568 92.7 10,79 34.87 1000 a value of AP are less than -0 004 3.339 11.95 10.80 82.1 12.18 39.90 1000 &O.l%. The standard devi+O 004 11.89 3.393 13.12 74.5 13.42 44.52 1000 ations of a and b can be oh-0 003 13.15 3.441 14.84 14.48 67.4 49.82 1000 tained by dividing uap and ubpo 3.494 15.72 14.31 +o 001 92.9 16.15 54.92 1500 by p and PO, respectively. The 3.548 +o 002 17.01 15.53 85.6 17.52 60.35 1500 3.587 -0 003 18.08 16.54 80.4 18.66 64.85 1500 standard deviation for the pa3.646 19.2.5 17.66 +O 008 75.3 19.92 70.18 1500 rameters of another fluid wit! 3.679 20.23 18.59 +O 001 71.5 20.98 74.43 1500 viscoaitv and density, say ( p -0 004 19.76 3.732 22.30 21.44 89.7 80.02 2000 and p i ) , ,would \e uap' = p'ua -0 005 20.78 3.767 22.50 83. 3 23.45 84.75 2000 and abPO = p,,Ub. Similarly, -0 004 3.813 23.58 21.83 81.2 24.63 89.90 2000 3.858 24.50 22.73 +o 002 78.0 25.64 94.53 2000 the flowmeter factors, a and h, -0 001 3.892 25.36 23.58 75.2 26.60 98.70 2000 can be obtained by dividing ap zxy - 4zy and bpo by the values of visbpa = z z 2 = 0.04303 ~ i. 0.000123 cosity (at 7') and density a t standard temperature and presu p = 9 - ( b p o ) 4 = 2.878 i 0.0018 sure of nitrogen. Experiments Se = d X A > / ( . V - 2) = 0.0039 with carbon dioxide and hydrogen, respectively, gave 2.394 o a p = sc d / 2 2 2 / 9 ( 2 2 2 - f S 2 ) = 0.0018 and 1.433 for the values of mbpo = S e / d Z m = 0.000123 np. The ratios of these values a Distilled water a t 26.2' C. to that obtained with nitrogen, b A t 28.2' C. and 738.2 inn?. of mercury. where ap = 2.878, are 0.832 c A- defined following Equation 4. d 1 = Y - ( u p + bpor) and 0.498, which compare favorably with the respective ratios of viscosity a t the same temperature-i.e., 0.833 to 0.843 and 0.498 to 0.503-which are obtained from the International where Critical Tables (24). Similarly, the ratios of b p o , obtained v i t h carbon dioxide and hydrogen, to that obtained with nitrogen (3) are within +0.3% of the ratios of the standard temperature and pressure densities of the respective gases. By combining Equations 2 and 3 and solving for Qo Table I.
Calibration of a Capillary Flowmeter 5
DISCUSSIOY
EXPERIMENTAL
The meter constants, a and b, can be calculated by means of Equation 2 from the measurements oY pressure drop as a function of the rate of flow of any gas. When AP/Q, is plotted against Qo,the intercept of the resulting line is ( a p ) and the slope, f b p o ) . Once a and b are determined, the meter can be used for any gas of known density and viscosity without calibration by the use of Equation 4.
Q., cc/scc.
Figure 1. Determination of the Coefficients of a Laminar-Stable Flowmeter Calibration of a capillary meter (diameter X 1.07 mm., length X 50 cm.) is shown in Table I. B constant head distilled xater manometer, 120 em. long, was used to measure the pressure drop. Experimental conditions, calculations, and precision determinations are included in Table I. The values of AP/Qm are plotted against Qo in Figure 1. The straight line is drawn according to the method of least squares as indicated in the table. Standard
When repeated measurements of rates of gas floiv are required, calculation of each rate by Equation 4 will be time-consuniing. Flow rate us. pressure drop charts, therefore, should be constructed by the use of Equation 4. If a constant temperature and a constant downstream pressure, P,, can be maintained, a single curve can serve the purpose. A comprehensive chart should contain two parameters-viz., T and PD-and the construction of such a chart may be tedious. However, a constant, P,, can easily be maintained by means of an open-end manometer and a needle valve to be placed a t the downstream end of the flowmeter. By adjusting the needle valve, a desired PDcan be maintained. This is essential in the use of the flowmeter if the meter precedes a unit-e.g., reaction vessel, packed bed, stopcocks, etc. The dolvnstream pressure then will change with the change in flow rate if not adjusted. A pressure drop reading corresponds to a different flow rate with different downstrekm pressures. The constant pressure, P,, to be maintained should be slightly higher than the maximum pressure the downstream end can attain. A chart for QO us. AP consequently needs to be drawn with only one variable parameter-Le., 2'. For precision the charts should be large enough-viz., comparable to the length of the manometer arms-in order to minimize the errors in reading. Expressions for pressure drop in the form of polynomials of the rate of fluid flom- are not novel ( 7 ) . Moreover, general equations for orifice meters implicitly recognize the possibility of calibration with one gas and use with another. The use of capillaries for estimation of fluid viscosity has the same implication. Severtheless, interchangeability of gas when both viscous and kinetic effects are significant within limits of precise measurement of flow has not been demonstrated nor has the application of a tn-oterm expression t o the calibration and t o identification of a flowmeter by means of two coefficients been shown. I t is not
792
ANALYTICAL CHEMISTRY
implied here that a two-term pressure drop equation covers an unlimited range of flow rate. Indeed, it will be shown that coefficients do change following transition regions; yet Equation 2 has been found to be applicable both to the laminar-stable and to the turbulent-stable regions.
plots-at higher flow rates for the meter of Figure 1 and a t the lower rates for the meter of Figure 3-but they lie outside the useful range of the meters. Therefore, in order to avoid two pairs of coefficients, as in the case of Figure 2, meters should be designed SO that the transition regions lie outside their useful range. If this is unavoidable, the meters should have long approaches to the capillary-Le., calming sections about 40 diameter lengths-in order to avoid fluctuations in the transition region. The mathematical procedure outlined has been successfully applied to calibration and measurements of gas flow rates up to 21,000 cm.3 per minute (maximum pressure drop 120 cm. of water) (14, 15). The method has been found not to be limited to any pattern of fluid stability. It leads to mathematical formulation of flow, to determination of precision, and to adaptation of flowmeters for use with any gas. LITERATURE CITED
(1) Alyea, H. S . ,IND.ENG.CHEM.,BNAL.ED.,12, 686 (1940). ( 2 ) Anderson, J. W.,and Friedman, R., Rev. Sci. Znstr., 2 0 , 61
(1949).
P.,
cc./sce.
Figure 2. Determination of Coefficients of Flowmeter Showing Trsnsition Region
(3) Bean, H. S.,Buckingham, E., and Murphy, P. S., J . Research A-atZ. Bur. Standards, 2 , 561 (1929). (4) Benton, A. F., I n d . Eng. Chem., 11, 623 (1919). ( 5 ) Bordos, F., and Pouplain, F., Bull. soc. emour. ind. natl., 1929, 257-65. (6) Brady, L. J., and Corson, B. B., IXD.ENG.CHEY.,ANAL.ED.,14, 656 (1942). (7) Brillonin, RI., “Lecons sur la viscosite des liquides et des gaz,”
Paris, Gauthier-Villars, 1907. (8) Bruun, J. H . , IND.ENG.CHEY.,A x a ~Eo., . 11, 655 (1939). (9) Croxton, F. C., Ihid., 14, 69 (1942). (IO) Davey, 77’. J. G., Gas World, 125, 598, 760 (1946). Diebief, J., J . phys. mdium. 7 , 402 (1926). Dorsey, N. E., Phys. Rev., 28, 833 (1926). Elliott, hf., I n d . Eng. Chem., 20, 923 (1928). Ergun, S.,A N A L . CHEIf., 23, 151 (1951). Ibid., 24, 388 (1952). Erk, S., J . Rheol, 2, 205 (1931). Fabris, R. J., and Peacock, K.AI., Can. Chem. Process Inds., 31,
Q.,
Figure 3.
cc./sec.
Determination of Coefficients of TurbulentStable Flowmeter
The magnitudes of the meter coefficients are dependent upon the dimensions of the floBTmeter. Whereas the dimensions nped not be kno-ivn for the purpose of calibration and use of the flon metpr, they are of interest in their design. For true Poiseuille flow, except for end effects, b = 0 and consequently the plot of Equation 2 should have a zero slope. It should be stressed her? that the existence of a slope does not necessarily mean that the flox has lost its laminar-stable pattern. The region corresponding to a change in the stable flow pattern-Le., transition regionresults in a curve in the plot. This is shown in Figure 2. An important point is that Equation 2 is applicable to both the laminarstable and turbulent-stable regions which have different coefficientsas shown in Figure 2. This is further illustrated by comparing Figures 1 and 3. Figure 1 is an example of laminar-stable flow. Viscous losses constitute the major portion of the total pressure drop, yet the kinetic effects are not negligible (13% a t 10 cc. per second). The data of Figure 3 were obtained m-ith a glass orifice and are an example of turbulent-stable flow. Kinetic energy losses are dominant. Transition regions exist for both
153 (1947). Gooderham, W, J., J . SOC.Chem. Ind., 63, 351 (1944). L. CHEX,21, 1154 (1949). Haller, W,, and Trakas, V.,KoZZoid-Z., 47, 301 (1929). Heiss, J. F., and Coull, J., Chem. Eng., 56, N o . 4, 104 (1949). Herbs, Cl., Bull. soc. chim. Belges, 51, 133 (1912). Hofsiiss, M.,Gas- u. Wasserfach. 70, 293 (1927). International Critical Tahles, Vol. I-,p. 1, Sew York, McCrawHill Book Co., 1928. Linden, H. R., and Othmer, D. F., Trans. Am. SOC. Mech. Engrs., 71,765 (1949). llaercks, J., Gliicknuf, 6 5 , 1234 (1929). Meuron, H. J., ISD. ESG. CHEY.,ANAL. ED.,13, 114 (1941). Nashan, P., Chem. Fahrik, 1940, 471. Perry, J. d.,Jr., Trans. Am. Soc. Mech. Engrs., 71, 757 (1949). Piazza, J., Anales inst. inrest. cient. y technol. (Univ. nac. litoral, Santa FB, A r g . ) , 10-11, 97 (1940-1).
Pinkus, A , , J . chim. phys., 31, 211 (1934). Radins, E. L., Oi2 Gas J . , 26, S o . 51, 111. 125, 129 (1928). Sips, R., -4nn. chim. anal. chim. nppl., 15, 97 (1933). ED.,4 , 244 (1932). Smith, G. IT.,TND.Eso. CHEX.,.IK.LL. Swift, H. W., Phil. Mag. 2 , 851 (1926). Williams, S.F., Chem. Eng., 53, S o . 10, 123 (1946). JTingard,R. E., and Williamson. H. P., Jr., J . Southern Research, 1, s o . 2, 7 (1949).
Zipperer, L., Gas- u. Fasserfach, 71, 1033 (1928). Ibid., 74, 27 (1931). R E C E I V EfD o r review October 15 1952.
Accepted February 9 , 1933.
