The proposed methods provide a much speedier way of using the instruments than calculation by the equation. The author knows of no other single-line method, universal for all pressures and temperatures, of correctly recording flow rate data through such instruments, although two incorrect methods are known to be practiced. The first admits its error but fails t o do anything t o remedy it. The meter is calibrated a t one time of year and recalibrated when seasonal changes result in widely different average room temperatures, No effect of pressure variation is ever included. The second method is probably the more insidious, since it involves a supposed “correction” which does not actually effect a true correction. The volume flowing per unit time is measured,corrected to standard conditions, and plotted against the reading of the differential manometer ( 2 ) . This is obviously equivalent to a plot of weight rate (or molal rate) against R. The equations developed here prove that this is not a correct method for weight rate plots in any of the meters discussed. Since the volume rate a t standard conditions, V,, is equal to
Discharge Rates from Gas Flowrneters Proper Temperature
it will be apparent from the equations presented here that this
and Pressure
constitutes an incorrect method for volume flow rate plots as well, for none of the derived plot ordinates is of this form.
Corrections JOHN C. WHITWELL Princeton University, Princeton, N. J.
E
&U.ATIONS, well tested by experiment, are available for flow of gases through primary devices such as the plate or nozzle orifice or the Venturi tube. These involve a difference of pressure across the constriction, usually expressed as differential manometer reading. By the use of the equation it is possible to calculate the flow, as volume (or weight) per unit time, which will be discharged through the instrument a t any given differential pressure manometer reading; the inlet temperature and pressure must be specified. The converse, the manometer reading for a given discharge, is similarly calculable. Less well developed but equally well proved are the equations for discharge from an instrument of the same type in which the pressure-reducing constriction is a long piece of capillary tubing. Calculations of discharge rate, a t specified conditions, may be made for any manometer reading. If it is desired to draw a line of discharge rate us. manometer reading for any of the above instruments, the conditions a t which this line applies must be specified. For any other conditions it is necessary to draw another line. An infinite number of lines is therefore required to provide for all the operating conditions possible. The purpose of this paper is to show that, by a rearrangement of the terms of the known equations, plots of discharge flow rates may be recorded in such a manner that a single line represents all conditions of temperature and pressure. The data for the drawing of this original plot may be calculated a t specified conditions, or measurements of actual discharge rates (i. e., calibration of the instrument) may be employed. Examples of lines drawn by each of these methods are given in Figures 3 and 4.
A new method is proposed of plotting discharge flow rates, for instruments of the Venturi and orifice type, against the differential’ manometer reading. The groups to be plotted are so calculated that one line on such a plot will be universal for varying values of inlet pressure and temperature. The groups are derived by rearrangement of the terms of well-known equations. Types of instruments included in the discussion are plate and nozzle orifices, Venturi tubes, and long-tube capillary orifices. The last are studied in both viscous and turbulent flow ranges for the gas in the capillary. For each of the types of meter, a sample discharge rate plot is shown, as recommended. Alignment charts are also given as a convenient method of calculating the groups found to be necessary in the new type of plot.
If V. is plotted against R , subsequent use of the plot will result in the follo&g errors for temperature change of 10” C. and barometric change of 20 mm. of mercury in the opposite direction (i. e., temperature rise and barometric fall, or vice versa): for plate or nozde orifices or Venturi tubes, 3 per cent on either weight or volume rate plots; for capillary meters, 2.3 per cent on volume rate plots and 12 per cent on weight, rate plots. 1157
a
INDUSTRIAL AND ENGINEERING CHEMISTRY
1158
By rearrangement this becomes :
Assumptions and Limitations I n all subsequent work it is assumed that the simple gas laws apply. No cases in which A p is large as compared to p A (not t o exceed 10 per cent a t the most) were considered. High pressure drops require the use of more complicated equations of discharge and invalidate the simplifying assumptions made in relating AH to R. The situation may be avoided by the proper selection of the size of the line in which the throttling constriction is placed.
Plate Orifices,Nozzle Orifices, and Venturis APPARATUS.Meters considered in this section consist of two parts: (a) the primary device, orifice or Venturi, and (b) the differential manometer connected to show the pressure drop across the constriction of the primary device. Illustrations of plate orifices, nozzle orifices, and Venturis are available in the literature (1,s). The capillary orifice is not included in this section. PRESENT EQUATIONS. Years of research by mechanical and gas engineers have resulted in the following equations for the devices of this group (1):
-
VA(CU.ft./sec. at p a , TA) = 140.6xYADL
4%
VOL. 30, NO. 10
W
a = C*< 'R
(6)
Therefore, the universal plot to be used should be W d m as ordinate against R < as abscissa. The results of Equations 3 and 6 may be obtained by a different method; the equation for liquids flowing through orifices or Venturi tubes @,6) is used as the basis. This equation is given as
-U
~ u B Z
A E ~
C42gH
(71
I n order to employ this equation for gases, i t is necessary to make the same assumptions used in simplifying Equation 1. Then, for any instrument, uBmay be expressed in terms of uA,uA may be expressed in terms of volume flow rate, V , or weight flow rate, W , and the following equations may be derived :
I n Equations 8 and 9,
4 F (1)
where D is in feet, p a in pounds per square foot, and T Ain O K. For the Venturi a preferable equation exists with substitutions for f( and Y A . Since these terms are t o be eliminated, the presentation of both equations is unnecessary. If we consider only one instrument and only one gas a t a time, and if we assume the ratio p , / p A to be nearly unity so that Y Ais essentially unity, and the simple gas laws to apply so that y is unity, considerable simplification will result:
It is apparent that the value of C1' may also include a term t o correct the units of V to cubic feet per minute or per hour. It is also apparent that with small values of Ap it is reasonable to drop the subscripts on V , T, and p . Rearranged, Equation 2 gives (3)
Equation 3 now indicates that a plot of V A d x -as ordinate against C R as abscissa should result in a curve which is universal for all values of p A and T A . If V A alone is a different line will result for every set of plotted against pressure and temperature values, as indicated by Equation 1. It is often more desirable to know a weight rate than a volume rate. The weight rate is obtained by multiplying both sides of Equation 1 or 2 by pA. Then, since
a,
By the simple gas laws, p = pM/BT and Equation 9 may be written
W = C, 2 T
.\/=; Cg = CIM/B
(10)
(11)
Although Equations 8 and 11 may be used to specify universal plots with AH as abscissa, they will be more useful normally if A H is replaced by R:
where dL and dQ are specific gravities referred to the same standard, Equation 12 is simplified by the assumption that one is negligible in comparison with dL/da, which amounts t o neglecting the weight of gas on top of the liquid in the manometer. With an air-water manometer, if the temperature of each phase is 20' C. and the air is a t a pressure as high as 30 pounds per square inch gage, dL/dQ is approximately 276; the assumption that one is negligible in comparison with this figure involves an error of slightly less than 0.4 per cent. For heavier gases or lighter liquids in the manometer arms, the error under the same conditions is magnified. I n general, the error increases with increase of pressure and decrease of temperature. For normal conditions it is therefore reasonable to write AH = R dL/dQ
VAPA W
(13)
(4)
since dolis dependent upon conditions. Since d,, T,, p,, and dL are all constants, Equation 13 becomes:
From the A. S. M. E. report on fluid meters ( I ) , PA
=
0.01042 ~ A G ~ / / T A
(5)
Simplifying by the same assumptions used to produce Equation 2 from l , they p a y be dropped and G regarded as another constant. Then, substituting Equation 5 in 4,
A H = aRT/p; a = dLp./daTa
(14)
By substituting Equation 14 in 8 and 11, the following are obtained :
v W4-p
m
-
=:
CI'G Cs'TR
(15)
(16)
OCTOBER, 1938
INDUSTRIAL AND ENGINEERING CHEMISTRY
9
J'g
8
7
0.6
E p'
/.s /.4 13
.ai?s/??
0.8
XP
4
-
€xump/es 4
a ,
:
8OOmm.
t = 50.C.
Vlor W)= zs f/hd V(p/T)*= / / . O , y fi~uyT;r/p)4c4Z76
1159
manometer reading to be set is then read from the plot of V(p/T)l12 against R. If the converse, the flow rate for a definite recorded value of R, '00 had been desired, V(p/T)1'2 would have been read from the chart a t this known €2, and V would have been calculated by the aid of Figure 1 or by direct calculation, using the recorded 7values of p and T. POSSIBILITY OF VARIABLE VALUEOF C. When the equations on orifice plates and Venturi tubes 600 are plotted on appropriate graph paper, they will produce straight lines only if coefficient C is constant over the range a t which readings are required. This means that the rate of flow in the main tube must be of a Reynolds number greater than 10,000 ( I , 3). Since it is not always feasible to design a low Ap orifice of such high flows, it is entirely possible that a varying C may be involved. The methods recommended for the recording of calibration curves will, in such cases, produce curves instead of straight lines. If desired, this variable C may be removed from the group of constants (e. g., C may be withdrawn from the constant, CI, of Equation 9A) and included in the variables. Equations 15 and 16 would then appear as
F FIGURE 1. ALIGNMENTCHARTFOR OBTAININQFACTORS T r ( p / T ) V ~AND w(T/p)'/g O F EQUATIONS 15 AND 16
These results are the same as those obtained with the A. S. M. E. equation as a starting point (compare Equations 3 and 6). I n the recommended plots the intercept on log-log paper (the slope on coordinate paper) may be calculated if values of C, the discharge coefficient, are known (see subsequent paragraphs on the variable value of C) ; the values of C1'and C,' are known for any instrument and any gas (Equations 8, 9.4, 11, and 14). These should be of principal use in predicting a calibration line as a check on the accuracy of the data taken. ALIGNMENT CHART. A n alignment chart is presented in Figure 1 for the rapid calculation of values of V(p/T)l12 or W ( T / P ) ~ / ~ .The method of use is illustrated in Figure 1. The line ( T / P ) ~ / ~(p/T) , l I 2 is essentially a reference line; the units are not essential to the solution of the groups desired. EXAMP- OF METHOD.Data are available for volume discharge rates IS. manometer readings for a Venturi tube; the pressure and temperature conditions are known a t each rate of discharge. Since it has been assumed that p,/p, approaches 1, outlet conditions may be substituted for inlet T and p if desired, although accuracy is decreased thereby. At each discharge rate, calculate V(p/T) j 2 , utilizing Figure 1 if desired. Plot these values against R on log-log paper. Draw a line through the points. At some subsequent time a definite %owrate, V, is desired. The pressure and temperature a t which the instrument is to operate are determined and V(P/T)'/~is calculated. The
Use of plots would require a knowledge of the variation of C with zc, or Dup/p, which would be obtainable from the calibration. The use of the calibration is complicated by this correction factor and should be avoided whenever possible by design of the orifice to provide the proper range of Reynolds numbers. Since this becomes increasingly difficult with smaller pipe sizes, i t is a point which must not be overlooked. It is probable, atmospheric temperature and pressure being the sole variations, that variation of Reynolds number will have little effect upon C because of the small range covered. For most accurate work, however, the effect should be investigated unless all Reynolds numbers are in the range - of 10.000 and greater when constancy of C is well assured.
Capillary Tube Orifices METERTYPE. The meter consists of a differential pressure manometer connected across a primary device in which the pressure-throttling constriction is a long tube of smaller diameter than the main gas line. This type of device is usually employed in glass equipment, the throttling constriction being a piece of capillary glass tubing. The validity of the following equations is dependent upon the design of the instrument so that Ap from friction shall present the major resistance, Ap from expansion and contraction a t entrance to, and exit from, the capillary being negligibleincomparison. If any doubts occur in a particular case as to the fulfillment of such conditions, the situation may be assured by using small tapered contraction and diffusion tubes a t the inlet and outlet of the capillary. PRESENT EQUATIONS AND THEIR REFORMATION. Viscous Flow in the Capillarg. The equation for this type of meter, with the gas in viscous flow in the capillary, was presented by Benton (2) in 1919. It is Poiseuille's equation for pressure drop for fluids in viscous flow:
If this equation is to be used in a manner similar to that recommended for plate or nozzle orifices and Venturi tubes, p and p must be expressed in terms of pressure and tempera-
.
INDUSTRIAL AND ENGINEERING CHEMISTRY
1160
w
VOL. 30, NO. 10
Eliminating AH in favor of R by Equation 14, yTO.76 = K 1 ' R WT'*7b/p= Kz'R
phmJ 600
/ooo
These two equations suggest the proper plots for this type of meter-namely, VTa.7svs. R or WT1.75/p us. R. Both are to be plotted on coordinate paper. Alignment Chart. Figure 2 is an alignment chart to aid in the calculation of the values of WT1.75/p (Equation 22). No figure is given for VT0.75 as it was believed that, in general, weight rate calibrations would be found more useful than those of volume rates. KOfigures are presented for gases in which p = CYTO.~.All such omitted charts are simple of construction. Complete information on the subject may be obtained from the Chemical Engineer's Handbook ( 3 ) . An example is given of the use of Figure 2 on the chart. The use of the method is similar to that explained for plate and n o ~ z l eorifices and Venturi tubes. Turbulent Flow in the Capillary. It seems well worth while to consider this case, for the maintenance of viscous flow in the capillary limits one to extremely small flows. For a 2-mm. capillary, the critical velocity for air is roughly 50 feet per second, corresponding to 0.1 cubic foot per minute; similar figures for a 4-mm. capillary are 25 feet per second and 0.25 cubic foot per minute. The equations for this situation are not in the literature as applying to capillary meters. They are available for pure frictional drop in straight pipe (3, 6) : fLUZ
where
f
=p
(y) DUP
P
2 0
This function, cp, never assumes the simple form,
),(.=I
Dup
(24)
-L/
FIGURE2. ALIGNMENTCHARTFOR OBTAININGFACTOR but it can be so considered over short ranges of D u p / p . W T 1 . 7 6 / OF p EQUATION 22 The larger the range, the greater the error involved. For the case of D u p / p from 2500 to 20,000, an average value of the slope of the f vs. D u p / p line shows n. = -0.28, from ture. For gases p has already been expressed in terms of p which the following equations result : and T (Equation 10). It may be said that p is independent of pressure over normal ranges; its dependence upon temperature is given by p = aT0.76 (18) Substituting Equations 10 and 18, T 0 . 7 5 T 0.28 Tl.16 0.28 Theoretically this relation will be found expressed as AH = b' u 2 = b' u1.71
(?)
(?)
= a@
The author's investigation of the data on many common gases has led to the conclusion that the use of the 0.75 power of T is consistent with the data without resort to more complicated equations, such as Sutherland's (S), in the range of 0" to 100" C. The gases for which this has been found to be nearly true are air, oxygen, nitrogen, carbon monoxide, nitric oxide, hydrogen, and ethane. Certain others (ammonia, carbon dioxide, and methane) have viscosities which appear to vary with the 0.9 power of the absolute temperature. For these gases the temperature powers in the subsequent equations must, of necessity, be changed. Examples of this will be shown. The data from which these powers of the temperature have been derived were obtained from the Chemical Engineers' Handbook (3). Substituting Equations 10 and 18, 17 becomes
Since AH = aRT/p (Equation 14),
A log-log plot of V(po.42/T0.ao) vs. R should thus produce a slight curve of average slope equal to 0.58. It should be noted that these equations are developed from the assumption that the coefficient of viscosity of the gas varies with the 0.75 power of T . If this is not true for the gas being metered, the powers of T will change throughout the derivation. For those gases for which p = aTO.9, the power o f T i n Equation 25 becomes 0.27. Design of the meter should be such that Ap may not become so high that the relation AH = aRT/p is in danger of in-
OCTOBER, 1938
INDUSTRIAL AND ENGINEERING CHEMISTRY
1161
entia1 manometer connected to a 4 X 2 inch Venturi tube. From these assumed values the discharge weight rate was calculated (Equations 1 and 5). As indicated by Equation 6, these weight rates were multiplied by ( T / P ) ~ /All ~ . values are given in the following table: Manometer Reading, R Inches Hi0 1 2
4 5 10
12 15 17
FIGURE 3. NEWDISCHARGE RATEPLOT Calculated disoharge rates from a 4 X 2 inch Venturi tube, replotted as suggested by Equation 16
validation by widely differing values of p in the two arms of the manometer. It seems quite satisfactory to use Ap values up t o 10 per cent of the inlet pressure.
Experimental Confirmation Since calibration data are usually taken over a short period of time, frequently with constancy of conditions a definite object, it is not easy to find experimental data to check the equations. However, Equation 1, based upon long experience and many data, requires no experimental check. Equations 3 and/or 6 require proof of their applicability under universal temperature and pressure conditions only. For this reason widely divergent values of air pressures and temperatures were assumed to correspond to varying readings of a differ- C
20
tA
C. 20 20 25 60 25 40
100
20 40
PA
Woalcd.
Inches H g 29.92 30 40 40 20 20 30 30 48
Lb./aec. 0.113 0.162 0.256 0.272 0.281 0.302 0.380 0.456 0.606
w
(y
0.678 0.971 1.35 1.50 2.10
2.30 2.57 2.73 2.96
The values of W ( T / p ) / ? are plotted on log-log paper against R in Figure 3. This shows the perfect straight line obtainable a t high Reynolds numbers. Pinkus (4) presented very accurate data for the calibration of a capillary meter, operating in the viscous flow range, with air and oxygen. Conditions with air varied to include pressures from 775 to 830 mm. of mercury and temperatures from 3.9’ to 19.5’ C. These data were recalculated and plotted as suggested by Equation 19; the 1.75 power of T was used, since the coefficients of viscosity of both air and oxygen were found to be dependent upon the 0.75 power of T . This work is presented in Figure 4. “Corrected” points will be noted for the oxygen line. The corrections are theoretical, made by determining the relative values of K , for oxygen and air in Equation 19. The factors involved in the correction are the molecular weights of the two gases and the cy values in the expression: = aT0.76
Since the factor is 0.995, the iower points are inappreciably affected. The single line for the two gases indicates that it should be possible to use an instrument of this type, calibrated with one gas, for any oth.er gas whose coefficient of viscosity varies with the same power of the absolute temperature.
Acknowledgment The author wishes to express his appreciation of the valuable advice and suggestions offered by J. C. Elgin, Princeton University, and T. H. Chilton, the du Pont Company.
Nomenclature
FIGURE 4. NEWDISCHARGE RATEPLOT Pinkus’ data ( 4 ) replotted as suggested by Equation 19
Consistent units are used throughout. a = constant = dlp,/daT, b = constant d = specific gravity, dimensionless f = friction factor, dimensionless g = gravitational constant p = absolute pressure t = normal scale temperature (Centigrade or Fahrenheit) u = mean linear velocity 9 = supercompressibility factor (1) A = cross-sectional area B = gas law constant in p V = nBT, where n is the number of moles of gas in a volume V C = constant in orifice or Venturi equation D = diameter G = specific gravity of any gas referred t o (air = 1) H = static head, units of length = constant in A. S. M. E. orifice-Venturi equation K = constant in capillary meter equation L = length of capillary M = molecular weight of gas R = differential manometer reading R‘ = differential manometer reading, corrected to inches of water at 60” F. 7’ = absolute temperature
INDUSTRIAL AND ENGINEERING CHEMISTRY
1162
V = volume rate of flow W = weight rate of flow Y = expansion faotor ( I ) = constant = difference, applied to AH or A p 9 = function p = coefficient of viscosity p = density Subscripts: A = upstream pressure tap position, differential pressure producing device B = constriction, differential pressure producing device
VOL. 30, NO. 10
s = reference state 1, 2, 3,4 = different values of constants or different conditions of temperature and pressure
a
A
G
L
s f y ~ d b = downstream pressure tap position, differential pressure producing device
THE LOVE FILTRE
Literature Cited Am. SOC.Mech. Engrs., “Fluid Meters-Their Application and Theory,” 4th ed., 1937. (2) Benton, J. IND.ENQ.CREM.,11, 623 (1919). (3) Perry, Chemical Engineers’ Handbook, New York, McGrawHill Book Co., 1934. (4) Pinkus, J . chim. phys., 31,241 (1934). ( 5 ) Walker, Lewis, McAdams, and Gilliland, “Principles of Chemical Engineering,” 3rd ed., New York, McGraw-Hill Book Go., 1937. RECEIVED January 10, 1938.
By Edgar Bundy
h list of Reproductions Nos. lFt,o 60 appeared in our issue
of January, 1936, page 129; the list of Nos. 61 to 72 ap’peared in January 1937 page 74; NOEL 73 to’84 a d listed in January, 1938, page 70, where also is shown No. 85 with details for ob-
With this, No. 94 in the Berolzheimer series of Alchemical and Historical Reproductions, we add a new artist to the series. Edgar Bundy, A. R. A., was born in Brighton, England, in 1862, and spent most of his life in London. His work, mostly historical and realistic genre paintings, was exhibited a t the Royal Academy, commencing in 1881, and from 1907 on a t the Paris Salon. He died in 1922.
This painting depicting, as it were, the extraterritorial activities of the alchemist, was painted in 1894,and is signed and dated in the lower right corner. It was reproduced in Leshe’s Magazine the next year. The present location of the original is not known. D. D. Berolzheimer 50 East 41st Street New York, N. Y.
