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INDUSTRIAL AND ENGINEERING CHEMISTRY
3. Receiving operator, having received his requirement, steps off foot treadle; this cuts off air supply to diaphragm valve, which closes slowly as air leaks from diaphragm. 4. Restriction of flow at initial movement of diaphragm valve starts relisure wave through pipe t o pump. 5. %hen pressure wave reaches pump control, slight increase in pressure causes control to shut off steam valve. 6 . As valve stem reaches closed position, arm opens air block, admits air to auxiliary diaphragm, and locks steam valve closed.
If an attempt is made to run the pump without following the above sequence, the control system automatically shuts the pump down. Installations of the type described have been made for a number of long lines. Since these protective measures have been taken, over two years of satisfactory service has been given. Aclmowledgment The authors are indebted to C. K. Flint, Eastman Kodak Company, for permission to publish this investigation.
451
Thanks are due the Babcock and Wilcox Tube Company for the photographs of Figures 1 and 2 made in that company’s metallurgical laboratories. The cooperation of J. K. Hill of the Taylor Instrument Company was valuable in the comparison of diaphragm valves and in the development of the pump control.
Literature Cited Badger and McCabe, “Elements of Chemical Engineering,” 2nd ed., p. 76, New York, McGraw-Hill Book Co., 1936. Enger, M. L., private printing of Symposium on Water Hammer, pp. 100-6, Am. 900. Mech. Engrs., 1933. Joukovsky (tr. by 0. Simin), Proc. Am. Water Works Assoc., 24, 335-424 (1904). Moore, H. F., Metals & Alloys, 4 , 3 9 4 0 (1933). Oshiba, F., Kinzoku-no-Kenkyu, 11, 328-43 (1934): English abstract in Metals & Alloys, 6 , MA161 (1935). Perry, J. H., Chemical Engineers’ Handbook, 1st ed., p. 1652, New York, McGraw-Hill Book Co., 1934. Russell, G . E., Textbook on Hydraulics, 4th ed., p. 295, New York, Henry Holt and Co., 1934.. Symposium on Water Hammer, private printing, Am. SOC. Mech. Engrs., 1933. Trans. Am. SOC.iMech. Engrs., 59, 651-713 (1937).
CORRELATION OF ROTAMETER FLOW RATES JOHN C. WHITWELL DAVID S . PLUMB
AND
Princeton University, Princeton, h.J.
A method is proposed for the correlation, through one rotameter, of flow rates of liquids of varying physical properties. Using Bernoulli’s equation as the basis for theoretical study of the instrument, a new method of plotting calibration data is developed. From the slopes of lines on this new plot, a value of a constant for each liquid is obtained. If this constant is plotted against the proper function of fluid proplerties, a single correlation is obtained for all ten of the fluids studied. The constant is not a single power function of fluid properties, as shown by the two distinct portions of the correlation line. The correlation is applied only above certain critical values of the Reynolds number; this fact, combined with experimental observations, indicates the induction of turbulence by irregularities of the float and by entrance conditions. Variation of the float density is included in the correlation. Use of the method is outlined with examples.
T
H E rotameter is a recently exploited flow-measuring device whose mechanical development has far surpassed any theoretical knowledge of its action. It has always been desirable but impossible to predict calibration lines (discharge rates us. readings) for liquids of known physical properties. The perfection of interchangeable precisiontapered glass tubes for such instruments has made this desirability a practical necessity. With such tubes and a satisfactory method of prediction, experimental study of one meter type with a few standard liquids would make possible the ’ calibration, with any liquid, of all instruments of the same size and type without further experimental work. Without such a method experimental calibration of every meter is required with every liquid for whose use it is proposed. The authors’ investigations have been performed on one such standard rotameter using various liquids and floats of different compositions. A method has been developed whereby all available data for the meter may be correlated. Theoretical considerations were used to develop a working equation, which, when plotted as in Figure 3, produced a single line for each fluid. The slopes of these lines may be determined from a knowledge of the physical properties of the liquids and the float material.
Apparatus Figure 1 represents the essential portions of the standard commercial rotameter used in the investigation. The tapered glass tube is graduated in millimeters. The reading, R, at any flow is
INDUSTRIAL AND ENGINEERING CHEMISTRY
452
the vertical distance measured by this arbitrary scale to the top of the float or rotor, F , of length L. The position of the float at a given flow is determined by the net effect of the buoyant and dynamic forces upward and of gravity downward. The dimensions of the meter studied are as follows: Graduated length of tube, om. (in.) 25 (9.8) Diameter at tube top, mm. (in.) 9.14 (0.36) Diameter a t tube bottom, mm. (in,) 6.55 (0.258) Overall length of tube, om. (in.) 31.656 (12.463) Distance from tube bottom t o zero reading, om. (in.) 3.018 (1.188) Diameter of floats (top), mm. (in.) 6.35 (0.250) Length of floats, mm. (In.) 13.894 (0.547) Volume of floats, oc. 0.225 Specific gravity of floats: Rubber 1.168 Aluminum 2.79 Steel (stainless) 7.66 Lead 10.25
Theoretical Considerations The basis of the correlation is Bernoulli’s equation taken across the float itself, from point 1to point 2 (Figure 1). For this analysis the meter may be treated as an orifice of annular cross section; the size of the annulus varies with the position of the float. In order to derive a satisfactory working equation, the following assumptions were made: 1. The rotameter is a constant pressure drop device. 2. All the ressure drop in the apparatus takes place across the float itself fluid friction along the walls of the tube being negligible. 3. The pressure drop across the float is independent of “skin” (or wall) friction.
With the aid of these assumptions, the Bernoulli equation may be rearranged to express volume rate of flow in terms of certain constants and of a function of the reading of the meter. With no work added, the classic equation for liquids is:
VOL. 31, NO. 4
be the term through which eventual correlation of different liquid flow rates is obtained. Conversion of the equation to a form more suitable for use may be obtained by expressing UI and ua in terms of q, the volumetric flow rate, and the cross-sectional areas a t points 1 and 2, respectively, variables which involve the position of the float. Thus since q = ulA1 = uzAt,
Finally, areas A1 and A, may be expressed in terms of R. If the taper per unit length of tube is a, and the diameter at zero reading is DO, and from which
DZ = DO+ aR DI = DO+ a(R - L) 7r Az = -(Do + aR)’ - Ap 4
and
A1 =
x
+
[Do a(R - L)]2
The final equation may be written as
where 4 ( R )is a function of the reading obtained by substituting the above expressions involving R into the term (A12 -
Az’)/AI’A:*. Experimental Results The experimental data were taken with one meter equipped with a constant head device, four floats of identical size but of different materials, and ten fluids involving a twofold variation of density and a ninety fold variation of viscosity. Laboratory work consisted of a complete calibration of the meter with each of the fluids and each of the floats. The data are given in Tables I, 11, and 111, and a typical calibration curve is shown in Figure 2.
where density is written as PO, since in subsequent work p will be used for density in the metric system (i. e., specific gravity). Rearranging, UZ2
2g
- xi - xz+ (Pl - P Z ) / P O - F
(2)
(pl - p z ) and g are constants, the former by assumption; (XI - X,)= - L , and F has been assumed to be zero. Equation 2 then becomes:
up
FIQURE 1. DIAQRAM OF ROTAMETER
- UlZ
=
c - 2gL
(3)
where C for convenience includes fluid density po and the constant 29. For a given fluid, C is a constant but must of necessity vary between fluids. It is the only quantity in Equation 3 which can be dependent upon fluid properties and must be expected to
FIGURE2. TYPICAL CALIBRATION CURVEFOR SUGARSOLUTION D WITH A STEBLFLOAT The floats were lead, steel, aluminum, and rubber. Acetone and carbon tetrachloride were used with the three metal floats, and water was used with all four floats. Seven sucrose solutions, ranging in specific gravity from 1.138 to 1.260 and in viscosity,from 6 to 27 centipoises, were employed with the lead and steel floats, since the rubber and aluminum were too light to give an appreciable flow even at high readings. These fluids were arbitrarily grouped into two classes, thick and thin; the viscosity of the former was greater than 5 centipoises; that of the latter was 1 centipoise or less. Figures
APRIL, 1939
INDUSTRIAL AND ENGINEERING CHEMISTRY
453
4 and 5 seem to show that a closer definition of t h e classes m a y be made when more data are available. T h e experimental data were calculated and plotted as shown in Figure 3. Thus a fundamental check of the equation is a f f o r d e d since, according to Equation 4, this plot should give a straight line of slope (C - 2gL), where C, g, and L are constants for any one liquid and float. For the thin liquids, water, acetone, and carbon tetrachloride, a straight line is obtained for practically the entire range of flow of the meter on a l l f l o a t s . However, t h e sucrose solution lines show a tendency to c u r v e a t low flows, the curvature becoming more pronounced with the heavier and more viscous solutions. Courtesy, Schutts & Koerting Company In all cases the UNIVERSAL ROTAMETER WITH LUCITE lines are straight PROTECTION TUBE over more than half the range of flow of the meter, and for each fluid a break point may be located below which the curvature begins. No attempt was made to analyze the conditions below this break point. The lines for acetone and water have several points out of the range of the ordinate in Figure 3. Although the figures extend only to 9000, the maximum value of q2 in these cases is 12,000, increasing the number of points for determination of the proper line. Although Equation 4 is that of a line with no intercept, none of the experimental lines passes through the origin. Considering only the straight portions, all may be extended to a “common point” on the vertical axis. The slope of each CORRELATION OF FLUID PROPERTIES. line is the entire quantity (C - 2gL). By adding the value of 2gL (in consistent units) to the slope, quantity C was determined, a constant characteristic of any fluid. It was found that C depends upon the viscosity and density of the fluid, and that it is a power function of both variables. It was also found that there are two distinct ranges of correlation of C,
TABLEI. WATERCALIBRATION DATA Reading,
Rubber float
Cm. 3 5 7 9 11 13 15 17 19 21 23 25
...
0.90 1.71 2.73 3.77 4.90 5.95 7.15 8.26 9.52 10.55 11.74
Capaaity, Cu. In./Min. Aluminum Steel float float 3.30 8.96 6.85 16.3 10.80 24.2 14.70 33.0 19.15 41.8 23.4 51.1 27.4 58.6 32.0 67.7 36.2 77.5 40.5 84.0 45.2 93.6 49.7 103.5
Lead float
9.95 17.85 27.80 37.70 48.30 58.3 68.5 78.7 88.5 99.2 109.4
TABLE11. ACETONE AND CARBON TETRACHLORIDE CALIBRATION DATA Reading, Aluminum Cm. float
..
3 4 5 6 7 8 9
10:66 15:90 2i:3
lo 11 12 13 14 15 16 17
26:s
19 20 21 22 23 24 25
4915
32:4 38:l 43:4 5510 60:s 66:4
-
Capacity, Cu. In./Min. -Aceton-CCIdSteel Lead Steel float float float
13.0 19.4 22.0 27.1 31.0 36.9 42.1 46.9 50.7 58.2 63.4 68.3 73.3 79.0 84.1 89.0 94.4 97.6
..
.. *. ..
15.4 2i:o 3i:3 49:7 62:l 75:O 8612 9+:5 104.6
.. ...... .. ..
Lead float
6.92 9.70 12.31 15.35 18.50 21.5 25.2 28.6 31.7 34.7 37.8 41.5 44.4 47.7 51.5 54.5 58.5 61.0 64.4 68.5 73.0 76.1
8.86 12.1 15.4 19.4 23.0 27.1 30.9 34.7 38.7 42.8 46.8 50.5 55.0 58.8 61.5 67.2 71.0 74.8 78.8 83.3
TABLE111. CALIBRATION DATAFOR SUCROSE SOLUTIONS Reading,
Cm. 3 4 5 6 7 S
9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25
.A
B
Capacity, Cu. In./Min. C D E Lead Float
4.85 7.82 11.20 14.60 18.66 23.0 27.2 31.3 35.5 39.7 44.0 48.0 52.6 56.6 60.6 65.5 68.2
3.46 5.71 8.45 11.75 15.28 18.75 22.4 26.2 30.1 33.8 37.9 42.0 46.0 49.7 53.2 57.5 61.5 65.6 69.4 73.7 ’77.8 82.3 85.0
3.80 6.04 8.60 11.84 14.89 18.08 21.7 24.7 28.3 32.4 35.1 38.6 42.1 45.6 49.4 52.9 56.2 59.4 62.9 66.7 69.8 72.0 74.5
2.54 2.03 4.12 4:45 6.47 8.81 6.60 11.70 8.51 14.38 11.00 17.35 13.34 20.6 16.00 23.5 18.61 21.4 26.5 24.1 29.8 33.0 27.0 35.8 29.6 32.9 39.0 42.6 35.7 45.9 .38.5 41.2 49.0 44.2 52.5 47.0 55.4 58.7 50.0 62.3 53.0 56.3 66.0 59.4 68.8
... ... ... ...
2.32 4.04 6.32 8.97 12.15 15.06 18.46 21.7 25.2 28.9 31.9 35.9 39.4 42.6 46.8 50.3 53.5 57.6 60.9 65.0 68.3 71.7 75.6
1.56 2.61 4.10 6.40 8.18 10.56 13.40 16.00 18.82 21.8 24.9 27.4 31.3 34.5 37.7 40.8 43.6 47.0 51.8 54.7 59.5 61.4 64.5
1.42 2.30 3.60 5.14 7.03 9.24 11.66 14.38 17.10 19.66 22.4 25.4 28.6 31.3 34.4 37.8 40.6 43.8 46.5 49.3 53.3 56.3 59.5
F
cf
1.17 1.89 3.08 4.53 6.16 8.07 10.28 12.50 15.02 17.58 19.80 22.9 25.9 28.1 31.2 33.8 36.7 39.6 42.2 45.3 48.4 51.3 54.3
0.85 1.39 2.03 3.16 4.34 5.58 7.05 8.66 10.04 12.10 14.60 16.25 18.98 21.1 23.0 25.8 27.7 30.4 32.7 35.3 37.9 40.4 42.5
0.98 0.71 1.17 1.50 2.27 3.20 2.73 4.41 3.85 5.88 4.90 7.70 6.40 9.30 8.12 9.45 11.30 12.95 10.19 15.1 13.20 17.4 15.0 19.4 16.8 19.1 21.7 21.1 23.8 26.4 23.0 28.6 25.1 27.3 30.7 33.0 29.5 35.4 31.7 37.7 34.1 40.2 36.4 42.9 38.4
0.64 1.05 1.56 2.24 3.00 3.99 5.08 6.45 7.85 9.35 10.52 12.35 14.10 15.58 17.62 19.38 21.2 23.1 24.8 26.6 27.6
Steel Float
3 4 5 6
7 8 9 10
11 12 13 14 15 16 17 18 19 20 21 22 23 24 25
1.33 2.16 3.36 4.64 6.38 8.20 10.34 12.42 14.76 17.15 19.58 21.8 24.5 26.9 29.6 32.4 34.8 37.4 40.1’ 42.5 45.4 47.9 50.9
...
30.6
32.7
454
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TW-YR
VOL. 31, NO. 4
but that both may be recorded on one graph, since the ratio of powers of viscosity to density is the same in both ranges. To obtain the correlation, it is necessary to use a new constant, C’ (= C p ) . This quantity is 2gAp in the Bernoulli equation. The first range treats the thin fluids, water, acetone, and carbon tetrachloride, all of which have a viscosity of one centipoise or less but vary widely in specific gravity. It is found that for a given float,
,8000
-7000t f /” or
where k , the empirical constant, and the values of the powers were obtained from the intercept and slope of C‘ vs. p 2 / z loglog plot. The thick range was investigated with sucrose solutions, all of which had a specific gravity greater than one and viscosities in the range of 6 to 27 centipoises. By similar analysis, or
4
.io00
i
I6
I2
20
24
where p0J4is considered to be negligible. The curves from which these results were obtained are shown in Figure 4. CORRELATION OF FLOAT DENSITY. Figure 5 shows C ’ / P ~plotted . ~ against p 2 / z . C’ vs. p 2 / z gave a separate line for each float calibration. The lines could be made one by the indicated use of the 0.7 power of float density. In Figure 5 the point for the water calibration with a rubber float would fall directly on the other points and was omitted for clarity. The points for the aluminum float, used with thin range liquids, also correlate well. The final complete expressions for C may be written to include fluid properties and float densities. The powers are written to the nearest 0.05, since the data do not justify more accurate recording:
28
9000
thick range: thin range:
C b‘ pF00.70/ z 0.60 C = b pp0.70/(p0.76
(5)
(6)
,0.16)
Limitations and Observations
000
4
8
12
16
20
24
The correlation, as completed to date, has certain limitations. First, no work has been done on the curved portions of the lines of Figure 3. This difficulty might be minimized by a proper selection of meter sizes intended to provide a straight section over the desired flow range. Secondly, the break point below which this curvature begins must be known. On Figure 3, lines X Y are drawn through these points. Figure 6 represents a possible solution of this difficulty. The Reynolds number for the break point is calculated a t the annular space at the top of the float. The slope of the two parallel lines is 0.75. These two lines may
28
FIGURE 3. MASTERPLOTOF CALIBRATION DATA FOR STEELAND LEADFLOATS REPLOTTED AS SUGGESTED BY EQUATION 4
I
I 2
FIQURE 4. CORRELATION OB C’ PROPERTIES
WITH
FLUID
I
2
I
FIGURE 5. INCLUSION OF FLOAT DENSITYIN CORRELATION OF C‘ WITH FLUID PROPERTIES
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APRIL, 1939
be made one by again dividing the values of C by the 0.7 power of the float density. The same relation does not hold for the thin fluids, although nothing definite is indicated by this fact since the break points are barely distinguishable.
TABLETV. DATAFOR CORRELATION OF SLOPES FROM +(R)PLOT
Acetone Water CClr Sugar A
B C D E F G
q2
point. While operating in thin liquids, the float was constantly vibrating with extreme rapidity in horizontal planes. All readings for these liquids are above the break point. In thick fluids the float was stationary until the reading corresponding to the break point was reached, when it would again
vs.
0.330 0.893 0.980
0.79 1.00 1.59
1,890 1,120 2.640
2688 1968 1407
2110 1938 2280
414 380 448
2161 1571 1087
1767 1606 1785
425 386 429
6.0
1.138 1.170 1.200 1.220 1.236 1.244 1.260
0.216 0.169 0,157 0.106 0.089 0.077 0.061
1404 1218 1046 919 840 770 651
1596 1412 1228 1100 1045 954 821
313 277 241 216 205 187 161
1062 944 817 723 638 603 551
1210 1130 997 898 794 777 723
291 272 240 216 191 187 174
8.1 9.2 14.1 17.1 20.0 27.0
455
Thirdly, the proper correlation must be selected for the fluid to be metered. The change from thin to thick range correlation lies between 1 and 5 centipoises. T o determine whether this change is in a critical range or at one absolute point will be the subject of future work. Figure 5 indicates that the change occurs at a value of p 2 / z of approximately 0.22, by consideration of the intersection of the two straight lines. Throughout the laboratory work a change was observed in behavior of the float which coincided precisely with the break
FIGURE6. CORRELATION OF BREAK POINTS FOR SUCROSE SOLUTIONS
begin to vibrate, However, the inertia of the thicker fluids, even a t high flows, was always sufficient to produce a slower and more readily visible vibration than that in thin liquids. The period and magnitude of vibration were such that the action would be termed a “wobble.” At the break point it was found that Reynolds numbers were always much lower than those of the standard smooth-pipe critical range (2100-5000). The behavior of the float therefore indicates an induced turbulence which is produced in the tube by the obstruction to flow offered by the float itself. Since the lines are curved below the points of initial wobble, it is apparent that such artificial turbulence is essential to the successful application of Equation 4.
Method of Use Six steps are required in the application of the correlation to use with liquids for which the meter has not been calibrated: 1. Using Figure 5 and the properties of the fluid, calculation of C ’ j p p O J ; from this, calculation of C‘. 2. Calculation of C from the relation C = C’lP. 3. Calculation of (C - 2gL). 4. Location of the common point on the:@ os. $ ( E ) plot (from previous work on the same
meter with the same float). 5. Drawing of a line of slope (C through the common point.
ROTAMETERS USED IN THE PLANT OF CALVERTDISTILLERS CORPORATION AT ST. DENNIS,MD.; THE PANEL WAS INSTALLED BY E, B. BADGER& SONS COMPANY Courtesy, Fischer & Porter Company
- 2gL)
456
INDUSTRIAL AND ENGINEERING CHEMISTRY 6. Estimation (by Figure 6) of the lower limit of applicability of the straight line if a thick range fluid is employed. W i t h a thin range fluid, the upper 90 per cent of the line may be considered satisfactory.
,
Sugar Solution
-Reynold8 Steel float
The work at present represents a correlation of data available for one meter only. If it is possible to duplicate this work on other meters, by taking the square root of Equation 4, a simplification of the method will probably result. This change will allow the reading of q direct from a new master plot (i. e., p vs.l/Q(R)).
I
It is advisable to have two additional c u r v e s t o simplify use: (1) 4(R) us. R (Figure 7); (2) q 2 vs. q. It is possible to calculate $ ( R ) from any R, and q2 from any q or vice versa, but it has been found more satisfactory to employ the two suggested graphs.
EXAMPLE1, THIN LIQUIDS.It is desired to have a calibration line for the fluid toluene, using a stainless steel float in this particular meter. The density and viscosity of toluene are known; from the correlation plot (Figure 5 ) the quantity c ' / p F Q ' 7 may be determined. from this and the densities of the float and liquid, C may be calculated. Knowing the length of the float, the quantity 2gL is calculated and subtracted from C to give the value (C 2gL), representing the slope on the p2-+(R) plot; as the common point for the stainless steel float has been determined, the line may be drawn. The break point for toluene is at such a low flow that it may be ignored. Courtesy, Schutte & KOeVtinQ C o m p a n y For a given reading, +(R)is read from the SPECIALTWIN ROTAMETER FOR +(R)-Rline (Figure 7), GASOLINE and the square of the volume flowing is read from the toluene line just drawn. EXAMPLE 2, THICKLIQUIDS. Consider a solution with a viscosity of 18 centipoises and a specific gravity of 1.2. From Figure 5 and the fluid properties C' and C are determined as in the previous example, the slope (C - 2gL) is calculated, and the +$(E) line is drawn through the common point. I n this case, however, the line is applicable only above the break point which will occur for this fluid at less than one fourth of the full reading of the meter. Below this point the straight line is incorrect. Above this oint the line is used the same as for the thin liquids. The breaz point may be estimated from Figure 6, data for which are as ollows: No.Lead float
VOL. 31, NO. 4
W i cm,
FIGURE 7. +(R)us. R
The limitations previously discussed prescribe at present a relatively small field of application for the correlation. However, such limitations appear to represent only gaps which must be filled in. Future work in this laboratory will include tests of the method with different sizes of rotameters and an attempt at standardization of the common point.
Acknowledgment The authors are greatly indebted to Fischer and Porter Company, who supplied the meters used in this investigation and the additional investigations now in progress, and have shown a continued interest in the work.
Nomenclature A I = cross-sectional area of tube at low tip of float, sq. in. A2 = cross-sectional area of annular space at top of float, sq. in. AF = area of top face of float, sq. in. a = taper of tube, diameter increase per unit of length b' = empirical constant in Equation 5 b = empirical constant in Equation 6 C = 2 ( P I - p z ) / p o , Bernoulli's equation C' = DO = diameter of rotameter tube at zero reading, in. D, = equivalent diameter, annulus at top of float F = frictional head, inches, Bernoulli's equation = gravitational constant, in./min./min. = empirical constant, determined in thin range correlation IC' = empirical constant, determined in thick range - correlation L = length of float, in. P = pressure, Ib./sq. in. = volume of flow per unit time, cu. in./min. = reading from graduation on tube, cm. [in + ( R ) , R ia converted to inches] U = velocity, in./min. X = static head, in. 2 = viscosity, centipoises = viscosity, English units P = density of liquid, metric units (i. e., sp. gr.) P = density, English units PO PF = density of float, metric units (i. e., sp. gr.) D.21po = Reynolds number, consistent units U suiiscript 1 = point at low tip of float subscript 2 = point at top Wce of float
cap
B
