NOTES
Pressure Change Effects on Rotameter Air Flow Rates Paul Urone*l and Richard C. Ross National Enforcement Investigations Center, Environmental Protection Agency, Denver, Colo. 80225
For a given float height, the ambient volumetric air flow rate through a rotameter increased with drop in pressure. This was due to changes in the drag forces of the air moving past the float. The volumetric air flow rates of three rotameters having spherical steel floats and covering flow rates from 1to 15 L per min were studied at pressures from 630 to 430 mmHg. The slopes of plots of flow rates vs. pressure varied from -1.2 to -6.6 X L/mmHg at fixed float heights of 30,60,90, and 120 scale divisions. In general the slopes were less than predicted by theory, and the more negative slopes were associated with the higher flow rates. Dividing the slopes by the “calibration” flow rate measured at the laboratory pressure of 630 mm gave fractional flow rate factors ( E ) , which varied over a remarkably short range of -0.5 to -1.2 X 10-3/mmHg. It is proposed that for those instances where recalibration is impractical, the fractional air flow rate correction factor of 0.08 f 0.02% (or roughly 0.1%) per mmHg pressure change be used to estimate the field air flow rate. Rotameters are used extensively to measure rates of flow of gases and liquids. They are a form of a variable orifice meter and generally consist of a vertical, transparent tube having a tapered inside diameter and containing a float of a predetermined shape and density. A scale attached to, or etched on, the tube gives directly, or can be calibrated to give, the flow rate for the level indicated by the float. Rotameters range in size from 1/16 to 12 in. i.d. and can measure flow rates from a few milliliters to more than 300 ft3per min. A wide variety of float shapes and densities are used, depending on the type of fluid and flow being measured. A second float of greater density than the first is often included in the tube to extend the range of the rotameter. Not surprisingly, rotameters are commonly used for air and gas flow rate measurements in a wide variety of air pollution activities. Among these are their use for measuring air sampling flow rates and rates of flow of reagent or diluting gases. Because of the wide geographical range of the environmental problems investigated by the National Investigations Center, it became necessary to determine the effect of relatively large ambient air pressure (altitude) changes on the flow rates indicated by rotameters. Whenever possible, rotameters are calibrated near the site of their use. However, in many instances this is not always easily acomplished. In particular this is true when sampling at remote sites, in air planes, in shortterm, spot-checking projects, or where a sampling device causes a large pressure drop in the air being measured by the rotameter. The theory of the rotameter has been studied extensively (1-6). The velocity of the fluid (gas or liquid) as it moves up the tube decreases due to the increasing inside diameter of the rotameter. Within the flow rate range of the rotameter, its float will rise until the drag forces of the fluid moving past the Present address, Environmental Engineering Sciences, University of Florida, Gainesville, Fla. 32611. 732
Environmental Science & Technology
float equal the gravitational forces pulling it down. When the float is in an equilibrium position ( I , 2):
CpAu2 = Vfbf - p ) g (1) The left side of Equation 1represents the drag forces, while the right represents the gravitational forces. C is the resistance coefficient; p is the density of the moving fluid; A is the maximum cross-sectional area of the float perpendicular to the direction of flow; u is the linear velocity; Vf and pf are the volume and density of the float, respectively; and g represents the force of gravity. When the rotameter is used to measure air at ambient pressures or less, the density of the air is negligibly small compared to the density of the float, and since the area of the float is a constant, Equation 1becomes: u =K
’
a
(2)
where K’ represents the square root of the weight of the float divided by its maximum cross-sectional area. For a given position of the float, the annular area between the float and the tube wall is fixed. By multiplying both sides of Equation 2 by the annular area, the volumetric flow rate, Q, may be obtained:
Q = K m
(3)
If the resistance coefficient, C, remains constant as the pressure of the air changes, then: (4)
and at constant temperature: where Qt is the theoretical volumetric flow rate a t ambient conditions. The subscript a indicates the actual or ambient conditions and c indicates the calibration laboratory conditions of density ( p ) , pressure ( P ) ,and absolute temperature ( T ) respectively. , Equations 4 or 5 may be used in situations where the resistance coefficient, C, does not change or changes but slightly. However,the resistance coefficient to a large extent depends ), can change with updn the Reynolds number ( N R ~ which pressure and velocity changes in the air stream: P N R =~Du W
where D, in this case, represents the diameter of the float, w the viscosity, and the other symbols are as defined above. If the resistance coefficient does change with changes in pressure, Equation 1 becomes difficult to resolve theoretically and is more expeditiously solved by experiment. Experimenta 1 Figure 1 schematically shows the apparatus used for measuring the flow rates of air through three selected rotameters, each containing spherical steel balls as floats. The rotameters were selected for the air flow rates covered in the 0-15 L/min
This article not subject to U S . Copyright. Published 1979 American Chemical Society
Table 1. Comparison of Measured Volumetric Flow Rate Data ( 0 , ) with Theoretical Calculated Data ( 0 , )and Fractional Factor Corrected Data ( Q ) , Liters per Minute, as Functions of Float Height ( H ) and Pressure ( P ) , Millimeters of Mercury Matheson No. 603
H 30
P
63 1 581 53 1 48 1 43 1 60 63 1 58 1 53 1 48 1 43 1 90 63 1 581 531 48 1 43 1 63 1 120 58 1 53 1 48 1 43 1 a By extrapolation
Oa
01
Of
P
1.40 1.50 1.58 1.60 1.65 2.5 1 2.67 2.88 3.04 3.10 3.50 3.67 3.96 4.00 4.13 4.10 4.25 4.54 4.76 4.99
-
-
1.46 1.53 1.60 1.69
1.46 1.51 1.57 1.62
630 580 530 480 430 630 580 530 480 430 630 580 530 480 430 630 580 530 480 430
-
-
2.62 2.74 2.84 3.03
2.61 2.7 1 2.81 2.91
-
-
3.65 3.82 4.01 4.23
3.64 3.78 3.92 4.06
-
-
4.27 4.47 4.70 4.96
4.26 4.43 4.59 4.76
I
Ace No. 4-15-2
Matheson No. 604 Qa
3.16 3.20 3.33 3.52 3.69 6.74 6.84 7.13 7.29 7.65 9.89 9.99 10.34 10.71 10.90 12.23 12.55 12.78 13.13a 13.62a
pr
Qt
Of
P
Oa
-
-
628 578 528 478 428 628 578 528 478 428 628 578 528 478 428 628 578 528 478 428
1.34 1.38 1.47 1.50 1.53 2.49 2.52 2.70 2.72 2.82 3.93 4.11 4.25 4.44 4.45 5.36 5.45 5.93 6.02 6.44
3.29 3.45 3.62 3.82
3.29 3.41 3.54 3.67
7.02 7.35 7.72 8.16
7.01 7.28 7.55 7.82
-
-
-
-
10.30 10.77 11.32 11.96
10.28 10.67 11.07 1 1.46
-
-
12.75 13.33 14.01 14.80
12.72 13.21 13.70 14.18
-
ot
Of
-
-
1.40 1.46 1.54 1.62
1.39 1.45 1.50 1.55
2.60 2.72 2.85 3.02
2.59 2.69 2.79 2.89
-
-
-
-
4.10 4.29 4.50 4.76
4.09 4.24 4.40 4.56
5.59 5.85 6.14 6.49
5.57 5.79 6.00 6.22
-
-
Table II. Slopes and Fractional Ambient Air Flow Rate Changes with Changes in Air Pressure Slopes 01 Amblent Air Flow Rates ( Oa) wlth Decreasing Pressure LlmmHg X Ha Mathesgn 603 Matheson 604 Ace 4-15-2
B Drv
Test
30 60 90 120 Vacuum
Precision Valve 1
Valve
2
Meter
Figure 1. Schematic of apparatus used to measure air flow rates at
decreasing pressure increments overall range. They were a Matheson No. 603, a Matheson No. 604, and Ace No. 4-15-2. The flow rates of air passing through the rotameters were measured with the float at the 30,60,90, and 120 scale unit markers, respectively. At each setting, the upstream pressure (PI-also called Pa,the ambient pressure) was varied, so that measurements could be taken at each of the predetermined float heights. This was made possible by carefully adjusting precision valves 1and 2 (Figure 1).The experiment was conducted a t room temperature, which was relatively constant a t 25 f 2 O C . The downstream pressure, P2, varied from 1to 20 mmHg less than P1 depending on the air flow rate through the rotameter. It gave a measure of the pressure drop through the meter but was not used in any of the calculations. The volumetric flow rate through the dry test meter (Qm) was obtained by dividing the dry test meter volume (V,) by the elapsed time ( t ) : Qm
= Vmlt
(7)
Since at pressures lower than the laboratory, the air expands after passing through precision valve 1 (Figure I), the volumetric flow rate at P1 (Qa) was obtained by multiplying the
276 4.54 5.52 6.64
1.20 3.10 3.18 4.58
1.oo 1.72 2.74 5.46
Fractlonal Changes of Amblent Alr Flow Rate ( Oc ) wlth Decreaslng Pressure per mmHg x 10-3 Ace 4-15-2 H= Matheson 603 Matheson 604
30 60 90 120
av
0.89 1.24 0.91 1.12 1.04
overall av a
0.87 0.67 0.56 0.54 0.66 0.83 f 0.22 X
0.75 0.69 0.70 1.02 0.79
H = steel ball height, rotameter scale units
dry test meter volumetric flow rate by the pressure expansion ratio: Qa
= Qm(pm/pl)
(8)
where P , is the pressure of the air in the meter.
Results Table I lists the measured air flow rates as well as related calculated values. The measurements are given as a function of the rotameter float height ( H ) and the pressure of air passing through the rotameter. The “ambient” air flow rates ( Qa) are those calculated from Equations 7 and 8. The theoretically calculated values (Qt) were obtained from Equation 5 using the flow rate measured at 630 mmHg pressure as the calibration flow rate (QJ. The flow rates obtained by use of Volume 13, Number 6, June 1979 733
and
+
Qc = Q c ( l C A P ) (11) where dQ/dP represents the slope or rate of change of the flow rate with pressure and A P is the drop from the calibration pressure to the ambient pressure in millimeters of mercury. A P = P, - P,
P r e s s u r e . mm Hg
Figure 2. Theoretical and measured air flow rates as functions of air pressure and steel ball height (H) for Matheson No. 604 rotameter
the empirical fractional change factor ( e ) were calculated with Equation 11 (see below). Although the ambient volumetric flow rate (Qa) increases as the pressure drops, it should be noted that the volumetric flow rate adjusted to standard temperature and pressure (25 “C or 70 O F and 1atm) (Qstd) decreases with drop in pressure. I t is calculated by: Qstd = Q a ( p a / p s t d ) ( T a t d / T a )
(9)
All flow rates for a given height of the float (where the annular areas are the same) were plotted vs. the air pressure of the measurement as both the measured and the theoretically calculated values (Figure 2). The slopes of the measured flow rates vs. pressure lines were calculated (Table 11). All the slopes differed but were within one order of magnitude, L/mmHg. The higher slope varying from -1.2 to -6.6 X values were associated with the higher air flow rates. Dividing the slopes by the flow rates at calibration conditions (Qc) gave the fractional change of the volumetric flow rate per mmHg pressure drop (e): e=- dQ/dP
(10)
Qc
734
Environmental Science & Technology
(12)
The volumetric flow rate fractional change factor ( t ) as shown in Table I1 varies over a remarkably short range of values considering the pressure and flow rate changes covered. The average and standard deviation of t for the four float positions in each of the three rotameters was 0.83 f 0.22 X 10-3/mmHg. This is roughly equivalent to an increase of 0.08 f 0.02% of the calibrated volumetric flow rate for every millimeter mercury drop in the pressure of the air being measured by the rotameter. In a related study, data published by Craig for a Fischer and Porter rotameter with ‘/d-in. stainless steel float also gave an average t of 0.08 f 0.02%/mm for the upper two-thirds of the rotameter scale range, but lower t values for the lower third of the scale. Using the simplified value of 0.08% for t , the fractional change corrected volumetric flow rates ( Q e ) were calculated using Equation 11. In some instances the theoretical calculated flow rates (Qt, Equation 5) were closer to the measured flow rates. However, the maximum difference between the theoretically calculated and the measured flow rate was 9.7%, and the overall average difference was 3.2%. The maximum difference between the fractional factor corrected flow rates (Q,) and the measured flow rates was 7.6%, with an overall average difference of 2.6%. It seems reasonable, therefore, to conclude that as a close approximation, or under circumstances where it is not convenient to recalibrate a rotameter in the field, a correction flow rate factor of 0.08 f 0.02% (or approximately 0.1%) increase in ambient air flow rates per mmHg pressure drop can be used with low probability of large error. For an increase in pressure, a decrease in air flow rate would be calculated (Equations 11 and 12). At this point, however, the correction factor applies only to rotameters having steel ball floats and flow rates between l and 15 L/min. Literature Cited (1) “Fluid Meters: Their Theory and Application”, 6th ed., ASME Research Committee on Fluid Meters, H. S. Bean, Ed., American Society of Mechanical Engineers, New York, 1971. (2) Ower, E., Pankhurst, R. C., “The Measurement of Air Flow”, Pergamon, New York, 1966, pp 251-5. (3) Martin, J. J., Chem. Eng. Progr., 45,338 (1949). (4) Dijstelbergen, H. H., Chem. Eng. Scz., 19,853 (1964). ( 5 ) Craig, D. K., Health Phys., 21,328 (1971). (6) Wehster, H. L., “Gas Flow Modeling of Variable Area Flowmeters”, US.Atomic Energy Commission, Publication No. SLA73-1006, Sandia Laboratories, Albuquerque, N.M., 1973.
Received for review June 26, 1978. Accepted December 11, 1978. This study was made possible by a n Intergovernmental Personnel Act Fellowship granted t o P. Urone.
