space between the tubes at the same height. The stopper was positioned at five different heights in the tube. Each height corresponded to a different flow area ...
Chemical Engineering Science, 1965, Vol. 20, pp. 425429. Pergamon Press Ltd., Oxford. Printed in Great Britain.
Constant cross-section, variable area flowmeter E. KEHAT Department of Chemical Engineering, Technion-Israel Institute of Technology, Haifa, Israel (Receiued 7 September 1964; in revised form 22 October 1964) Abe&act-A cm&ant cross-section, variable area flowmeter is described. It consists of a rotameter type float fitting closely inside a slitted tube. For a properly designed flowmeter the calibration curve is linear over a wide range of flow rates. Design data for this flowmeter are given. Discharge coetBcients through the slits and past the float were determined for water and air from calibration curves of tubes 2.0 and 3-O cm in diameter and floats with a range of weights of 324-256-8 g.
THE object of this work was the development of a cheap, easy to fabricate flowmeter that will operate over a wide range of flow rates, and that will give linear calibration curves of flow rates against an indicator position. DANCKWERTSand SIKDER[l] have developed a constant cross-section, variable area flowmeter, using a rotameter type bob inside a perforated tube. This work extends their work to a higher flow range, and to linear calibration curves.
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akminum fh?lgQ
-Q!aQt
DEWXPTION OF FLOWMETER
stit
The
flowmeter consists of three main parts: An external transparent tube, a brass inner tube in which two parallel longitudinal slits are machined and a rotameter, type Aoat inside the inner tube. The fluid enters at the bottom of the inner tube. Most of the fluid edts through the slits to the external tube. A small “leak” flow, approximately 10 per cent of the maximum ffow rate flows past the float in the open internal tube. Both flows leave through the top of the external tube. The position of the float indicates the flow rate. A schematic drawing of the flowmeters used in the experimental work is shown in Fig. 1. The external Perspex tube was tightened against rubber gaskets, in the ahuninium flanges, by four stretching wires. The brass tube was sealed to the bottom flange by a rubber “0” ring. This arrangement withstood operating pressures up to 30 lb/in2. A simpler seal made from two rubber stoppers is recommended for atmospheric pressure work [l]. C
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float
bmshba
FIG. 1. Schematic drawing of flowmeter.
The position of the top of the float was read on a scale stuck to the side of the front slit. This position was read more easily for slits 05 mm in width, if a thin needle was inserted through the slits and the top centre of the float, where it was held by means of a slotted headless screw and a nut. The needle also served to keep light floats within the tube. Heavier floats were kept in the tubes by coiled spring stops. THEQRETXALTREATMENT The total flow rate (Q) consists of the flow rate through the slits (Q’) and the flow rate past the float (Q,). The theoretical equations for the flow through the slits are similar to those derived by DANCKWERTS and SIKDER[l] for a perforated tube. The final expression for the flow through the slits is J(2k)H]
Q’ = gJ(y)sin[$
(1)
where W is the weight of the float in the fluid, p the density of the liquid, d the internal diameter of the slitted tube, D the maximum diameter of the float, b the width of the slit, H the height of the float above the bottom of the slit, C the discharge coefficient through the slits and k the pressure recovery coefficient. Equation (1) indicates that the flow rate through the slits becomes linear with the height of the float if 8Cb ITd2 J(2k)H
I 0.245
(2)
for a maximum deviation of 1 per cent from linearity. In the linear operating range equation (1) becomes (3)
In this range the flow is independent of k. The maximum flow through a given flowmeter
(4) 426
and the maximum position of the float is H
1*5711d2 max= 8Cb,/(2k)
(5)
The “leak” flow rate past the float Q, is evaluated by the rotameter equation
Qo=co.
d2-D2
JJW& Jc -S-
(6)
)
where C, is the discharge coefficient for flow past the float. The slope of the linear part of the calibration curve is (7)
This value can be used to calculate C from calibration curves. An average value of C for entire length of the slit can be calculated using orifice equation, if the pressure drop across slit, Ap, is measured experimentally
the the the the
EXPERIMENTAL The geometries of the three inner tubes are described in Fig. 1 and Table 1. Fifteen floats of the Table 1.
Tube No.
Characteristics of the experimental flowmeter tubes 1. D. (mm)
Slit width (mm)
Slit height (mm)
Max. H. Linear* (mm)
350 350 350
48 96 216
__1
20
1.0
2 3
20 30
0.50 0.56
* Calculated for air from equations (2) and (10).
plumb bob type, ranging in weight from 3.24 to 3707 g were used in tubes 1 and 2. Nine floats, cylindrical in the upper part and conical in the lower part, ranging in weight from 19.48 to 256.8 g, were used in tube 3. The fluids used were air and water. The range of flow rates was 36-450 l/min
Constant cross-section, variable area flowmeter
for air and 3-1-43 l/min for water. The pressure upstream of the flowmeter was varied from 5 to 20 lb/in’. The tested flowmeter was placed in series with a precalibrated rotameter of the same range. The total flow rate at each float position was measured three to five times, not successively, and averaged. This also served as a check of the reproducibility of the readings of the flowmeter. The “leak” flow rate was measured for tubes 1 and 2 by placing the floats used in these tubes in a similar tube without slits, and measuring the flow rate. Since the flow in a slitted tube may differ from the flow in a non-slitted tube, the “leak” flow rate for tube 3 was measured by covering the slit lightly with an adhesive tape, placing a float inside the tube and measuring the flow rate. The discharge coefficient for the slits was measured in tube 3 by measuring the pressure difference as a function of flow rate between two 4 in. pressure taps, one at the centre of a rubber stopper inserted inside the brass tube and one in the space between the tubes at the same height. The stopper was positioned at five different heights in the tube. Each height corresponded to a different flow area through the slits. The rubber stopper caused a slight distortion of the slit. The slit width along the tube was measured with a sensitive gauge and an average value was used in equation
*..
(8). The pressure was read by a U-tube manometer, or a micromanometer. EXPERIMENTAL &SUiTS A typical calibration curve for air flowing in tube 3 with float No. 22 is given in Fig. 2. Two lines are shown, one for a float with an indicator needle in the slit and one for a float without a needle. The needle has two effects. It decreases the leak area and increases the weight of the float. The decrease in leak area displaces the calibration line to give a lower flow rate at the same float position. The increase in weight should increase slightly the slope of the calibration line, but this effect is too slight to be seen in Fig. 2. The calibration of floats without needles were highly reproducible. The addition of a needle made the readings easier, but resulted in a 2 per cent loss of reproducibility. LINEARITY
OF
FLDW-
The float position, which corresponds to the maximum linear portion of the calibrations of the flowmeters was independent of the float used. Figure 2 shows that the calibration curve is linear up to a float position of about 225 mm, which fits the range calculated and presented in Table 1.
L
:
100
i 200 ) P- TOTAL FLOW RATE IIl/m it
400
FIG. 2. Typical calibration curves of a linear flowmeter.
427
E. KEHAT
Direct measurements of the discharge coefficients in my-five runs with water at flow rates of lO40 l/min, and forty-seven runs with air at flow rates of 76 - 375 l/min in tube 3, without floats, at five stopper positions, gave an average C for water of O-67 and for air of O-79. The spread of the results was less than 10 per cent for each fluid. For design purpose and the recommended floats of the plumb bob type a C of O-7 for water and O-8 for air can be used.
PRESSURF! RECOVERYcoEFFIcIENI-k
FIG.3. Discharge coefficients for flow past float.
In general the linear portions of the calibration curves, for all tubes, were close to the values in Table 1, predicted by equation (2) which proves the validity of equation (2).
RFJ~~MMIZNDED TYPE OF FLOAT
For tube 1 and various floats a maximum flow rate was obtained at float positions of 33-34 cm for water. Introducing H = 335 mm and C = O-7 into equation (5) gave k = 0.51. DANCKWERTS and SIKDER[l] obtained an approximate value of k = 0.56 for a perforated tube flowmeter. DISCHARGECOEZFFICIE~S, C,, FOR THE “LEAK” FLOW RATE PAST THE FLOAT
The recommended type of float is a float with a low centre of gravity. It can be made from a combination of two materials with the higher density material at the bottom, glued together by an Epoxy resin. The optimum value of d - D is 1 mm. Larger space between float and tube results in high values of Q, which is independent of float position. Floats D of 05 mm were tested. These floats withddid not move freely inside the tube.
The leak flow rates were calculated by extrapolating the calibration curves to H = 0. Comparison of the measured and calculated “leak” flow rates showed good agreement for tube 3. The discharge coefficients, C,,, were calculated from the measured values of the leak flow rates and equation (6). These coefficients are plotted against the Reynolds number in Fig. 3. The Reynolds number in this case is defined by Re=
SLIT DISCHARGEC~~CIENTS The slit discharge coefficients, C, were calculated
4pQ0 Wd + 0)
(9
where p is the viscosity of the fluid. The coefficient Co increases with increased Reyfrom the linear portion of the calibration curves and equation (7). The spread of the results was nolds number from O-7 to 1, for tubes 1 and 2. Almost identical curves are available in the literasmall. The discharge coefficient is approximately O-7 ture for the discharge coefficients against the for water and O-8 for air for the plumb bob type Reynolds number, for rotameters with plumb bob floats of tubes 1 and 2. The cylindrical floats of type floats [2, 31. The discharge coefficients were in the range of 0.5-0.7 with a great scatter of the tube 3 resulted in higher coefficients, probably because the assumption that the weight of the float data, for the floats of tube 3. These low values of is centered in a flat disc at the top of the tube [l], CO are reasonable in view of the higher friction in the flow past the cylindrical floats of tube 3. is not valid for this type of float. 428
Constant cross-section,
variable area flowmeter
THE DESIGN OF LINEAR,CONSTANTCROSS-SECTION, linear flowmeter of this type. The maximum total flow rate, including the leak flow rate past the float, VARIABLEAlWAFLo~ is approximately 10 per cent higher than the maxiThe dimensions of these flowmeters are dictated mum flow through the slits. by equation (2). In order to obtain high sensitivity expressed by high values of H, b should be small and d should be large. The smallest practical slit Acknowleclgement-The experimental work of this study was conducted by 0. AUERBACH, H. BARLES, and Y. BLAM. width was found to be 0.5 mm. Therefore, for Thanks are due to Dr. A. ORELLfor editorial advice, floats of the plumb bob type and taking k = O-5, and C = 0.7 and C = 0.8 for water and air reNOTATION spectively, the following relations are obtained from equation (2) : b Width of slit H max= 2.4d2 for air
(10)
H max= 2*75d2 for water
(11)
A range of H of 20 to 30 cm is needed for a reasonable sensitivity. This requires tube diameters of 335 cm. For H,,,, from equation (3)
Equations (10-12) can be used for the design of
c co D d H k AP i
L
Discharge coe&ient through the slits Discharge coefficient for flow past the float Maximum diameter of float Internal diameter of tube Position of float above the bottom of the slit Pmssure recovery coefficient Pressure difference across the slit Total flow rate in flowmeter Flow rate through the slits “Leak” flow rate past float Reynolds number Weight of float in fluid Density of fluid Viscosity of fluid
~FERENCES
D~~cxwaars P. V. and SIKDERA. K., C/rem. Engng. Sci. 1960 13 34. POLIZ~Z L. M., Znstrum. Control Systems 196134 1048. J~SCHER and PORTER,Theory of the Flowmeter. 1947 Catalogue Section 98A. Hatboro,
Pa.
R&surn&Un tel debitmetre est d&it. 11consiste en un rotametre a flotteur etroitement ajuste ii l’interieur dun tube pourvu de fentes. La courbe de calibration est lin&dre pour une large marge de debits. Sont don& les r&sultats correspondants. Les coeEcients de decharge ii travers les fentes et l’espace annulaire autour du flotteur sont determines pour l’eau et l’air a partir des courbes de calibration correspondant 21des tubes de 2 a 3 cm de diametre et des flotteurs de poids compris entre 3,24 et 256,8 g.
429
