4- 4.1 Differential-Pressure Flowmeters

(a) Basic Considerations. All flowmeters are con­sidered to consist of two parts a primary element, which contacts the flowing fluid, and a secondary element, which indicates or otherwise displays the desired information.

In the common differential-pressure or head-type flow­meter, the primary element is an obstruction placed in the pipe to create a pressure drop, and the secondary element is a device to measure this pressure drop and convert it to rate of flow. The secondary element itself may consist of two parts, a transmitter and a receiver, in case the information is to be displayed at some distance from the point of measurement The primary element is usually one of three types orifice plate, flow nozzle, or Venturi tube.

(b) Orifice Plates. The orifice plate is a thin disk clamped between gaskets in a flanged joint, with a usually concentric circular hole smaller than the internal pipe diameter. This is the simplest type of primary element and the most easily reproducible. It can be used without individual calibration with the greatest assurance of accu­racy.

Flow through a sharp-edged orifice plate is character­ized by a change in velocity, which reaches a maximum at a point slightly downstream from the orifice. At this point, called the vena contracta, the flowing stream has its greatest convergence. Beyond this point the flowing stream diverges until it again fills the entire pipe area, the velocity is reduced back to its original value (assuming fluid density and pipe cross section are the same upstream and down-

• •

*MCT CORE MOVES TOWARD PRIMARY A ON MEASURED VARIABLE INCREASE.

ACTUAL DIRECTION OF TRAVEL DEPENDS UPON LINKAGE REQUIRED FOR PARTICULAR INSTRUMENT.

□ CUSTOMER CONNECTIONS

Fig. 4.23—Schematic of a linear voltage differential transformer (LVDT) and demodulator circuit.

■*— IRON — CORE

BAKELITE OR CERAMIC COIL FORM

COILS

LEADS-

ALUMINUM SHIELD

VARIABLE-INDUCTANCE INDUCTANCE-RATIO UNIT ELEMENT

MUTUAL-INDUCTANCE ELEMENT

A-C VOLTMETER 0.1 V
FULL SCALE

0. 10

0. 08

>

l_" 0.06

Э

ь 0.04 О

0. 02

0

PHASE RELATIONSHIP IN MUTUAL-
INDUCTANCE ELEMENT

BOURDON TUBE /

CIRCUIT CONNECTIONS USED IN SIMPLE
PRESSURE-INDICATING SYSTEM EMPLOYING
MUTUAL-INDUCTANCE ELEMENT

Fig. 4.24—Examples of inductance elements for LVDT’s

stream of the orifice), and the pressure increases to a value less than its original value.

Several locations for metering connections are shown in Fig 4 25.

Vena Contractu Tapi, The high-pressure connection is one pipe diameter upstream from the orifice, and the low-pressure connection is at the vena contracta. This tap arrangement is commonly used, particularly in the power industry. It has a slight advantage in that the connections are in regions of nearly constant velocity, thereby offering some tolerance in tap location without a noticeable change in differential pressure

ID and %D Taps These are an approximation of vena contracta taps. The high-pressure connection is one pipe diameter upstream from the orifice, and the low-pressure connection is one-half pipe diameter downstream from the orifice inlet.

Flange Taps These have connections drilled through the edges of the flanges 1 in. from the adjacent orifice surface. Flange taps are widely used in the gas and chemical industries because they are convenient to use and install “Orifice flanges” are readily available with pressure con­nections and jackscrews to spread the flanges to facilitate orifice replacement

Corner Taps These have effective connections im­mediately adjacent to the orifice plate. Corner taps are
commonly used in Europe but are seldom used in this country because they require special flange machining. They are used in some small orifice pipe assemblies furnished by several instrument manufacturers

(c) Flow Nozzles. The flow nozzle, usually of ASME long-radius high-ratio design (Fig 4.26), has an elliptically flared inlet and a cylindrical throat section. There is no vena contracta effect, because maximum velocity takes place in the throat. The flow nozzle passes approximately 60% more flow than an orifice with the same differential pressure and the same ratio of throat diameter to internal pipe diameter. (This latter ratio is the (3 of the flow nozzle, see Sec. (e) below.) A flow nozzle can be installed in welded piping but cannot be used to meter flow in either direction. The flow nozzle can be used successfully in some installations where limited length of straight pipe would not be suited to the use of an orifice.

A throat-tap nozzle, with downstream connection installed directly in the wall of the flow nozzle throat (Fig 4.27), has been developed by turbine engineers and is strongly recommended by them for the performance testing of turbines.

(d) Venturi Tubes. A Venturi tube has conical sections between the cylindrical pipe and throat, with curved transitions (Fig. 4.28) Its capacity is similar to that of a

flow nozzle, but pressure restoration is more complete because of minimized turbulence in the outlet cone. The Venturi tube may have piezometer rings or annular cham­bers surrounding the inlet and throat to average pressures at four or more points. These rings also have the beneficial

effect of permitting an installation to be made with a minimum length of straight pipe.

A nonstandard Venturi tube, with equal angles of taper in the inlet and outlet cones, can be used to meter reversing flow.

R, = 1.375 D t 20% F^= 3.625 d * 0.125 d 5d 5 R3 г I5d a, = 21° ± 1° 7° г oj г 8° or 7° і а г 15°

L, i D or L, i(D/4 +10") z ;D/2i D/4 for 4’і 0*6"

0/40; D/2 for 6"i D г 32"

Lt* d/3 у * d/6

5/32" ї 8 < 25/64" ond
8 < 0.1 D or 013 d

Fig. 4.28—Herschcl or classical Venturi tube. (From Fluid Meters, Fig II III 26, p. 231, American Society of Mechanical Engineers, N Y, 1971.)

359Cd2 FAPyh)54 W= (TW

where C = coefficient of discharge (dimensionless) d = primary-element throat or hole diameter (in.) Fa = thermal expansion factor accounting for change in cross sectional area (dimensionless) h = differential pressure (inches of water at 68°E) W = rate of flow (lb/hr) Y = expansion factor accounting for density decrease at downstream pressure (dimensionless) (3 = ratio of throat diameter to inside pipe diameter (dimensionless) 7 = density (lb/ft3)

Several modified Venturi tubes or flow tubes, shorter than the standard Venturi tube but claiming a lower unrecovered pressure drop, are now available. One such tube is shown in big 4.29 In general, these tubes take advantage of (1) an impact component to increase the pressure felt at the high-pressure connection and (2) a change in direction of the flow in the boundary layer at the low-pressure connection to decrease the pressure at that point. The result is that, for a specified differential pressure and flow rate, the throat diameters of these modified Venturi tubes are larger than those of conventional tubes, thus the overall pressure loss for a given flow rate may be less.

(e) Sizing Primary Elements. The basic equation for a differential-pressure flowmeter, using an orifice, flow noz­zle, or Venturi tube, may be stated

This equation exists in other forms, e. g., for convenient use in gas measurement where flow rate in volumetric units and density at standard or base conditions are involved

The coefficient of discharge, C, may be presented by curves or in tables as a function of pipe Reynolds number, Ro, or of throat Reynolds number, Rq Sometimes C is replaced by’ a combined coefficient, already divided by the denominator of Lq 4.12, then it is labelled the “coefficient К with velocity-of-approach factor included ” Rather than including here sufficient information to permit complete and accurate flow determinations to be made by measuring differential pressure across a primary element of any type and dimensions, this text is limited to presenting a method for determining approximate dimensions. These are suitable for estimating purposes or for assessing the acceptability of a proposed piping installation for flow-measurement pur­poses. ’

The diameter ratio, (3, is determined from Fig 4 30 with a calculation of the capacity factor, I, from the applicable equation For liquids

_ W 1 “ D2(h7f^

or

I = Ql7s

7.48D2 (h7f)1’4

For steam

,______ w___

D2 (h/v)’*

For gases

QgTs

D2(h7f)^

or

Qg [G(T + 460)1*
21 5D2 (hP)^

where D =mternal pipe diameter (in.)

G = specific gravity (dimensionless)

I = capacity factor (dimensions consistent with above equations)

P = pressure at primary-element inlet (psia)

Qq = gas flow rate [ft3 /hr at standard or base con­ditions, often abbreviated “scfh” (standard con­ditions are usually 30 in Hg and 60°F)]

Ql = liquid flow rate [gal/hr at standard or base temperature (usually 60°F for petroleum prod­ucts, but flowing temperature for water)]

T = temperature (°F) v = specific volume (ft3/lb)

7S = density (lb/ft3 at standard or base conditions)

7f = density (lb/ft3 at flowing conditions)

In general, a primary-element /3 ratio of 0 50 to 0 70 may be considered normal. Higher ratios may be expected in high-pressure or high-temperature applications where pipeline economics suggests using the smallest acceptable size with consequent high velocities. An extremely low /3 value might indicate that the pipe size is larger than required or that a meter of lower differential-pressure rating would be preferable. The low /3 ratios of throat-tap nozzles are desirable in turbine testing to provide high differential pressure which may be read with accuracy on a manometer. The high differential pressure can be tolerated during the short duration of a test (after which the nozzle is removed)

The minimum and maximum limits of the diameter ratio, (3, for various primary elements are shown m Table 4.18. These (3 values are important in the selection of maximum meter differential pressure. For compressible fluids (steam, air, and gases), the value of the maximum differential in inches of water should not exceed the operating pressure in psia in order to avoid inaccuracies due to variations in the expansion factor, Y, that are not taken into account.

(f)

Installation. Accurate flow measurements require that the primary element be in a “normally turbulent” flow pattern, with fully developed velocity profile and without spiral motion or velocity stratification. Because of this

straight pipe section of adequate length.

It is generally accepted that the required pipe length depends only on the diameter ratio, 13, and on the structure of the fitting or combination of fittings preceding the straight pipe. On this basis the curves of Fig 4.31, originally published by the American Society of Mechanical Engineers (ASME), have been developed. It should be emphasized that longer lengths are desirable when the arrangement of building and equipment permits, the mini­mum lengths represented by the curves are a compromise and may not result in complete removal of an abnormal inlet turbulence.

(g) Accuracy of Primary Elements. Any meter manu­facturer can be expected to furnish conventional primary elements (orifices, flow nozzles, or Venturi tubes) sized in accordance with the standard equations and discharge coefficients published by ASME These coefficients are average values that may be considered to be correct within tolerances that vary with /3, Reynolds number, and pipe size. The approximate accuracy of these coefficients is as follows

±1.0 for concentric orifices with flange or vena contracta taps, D ^ 2.0 m., 0.20 j3 3? 0.70 ±2.25% for same, (3 = 0.75 ±0.75% for Venturi tubes ±2.0% for flow nozzles ±1.4% for eccentric orifices, D > 4 in.

±2.0% for segmental orifices

When greater accuracy is required, the primary element in its pipe section may be calibrated at a hydraulic laboratory to establish, as nearly as possible, the true value of its individual coefficient of discharge. Ideally the calibration should be performed at Reynolds numbers corresponding to expected operating conditions. This may not be possible, however, because the application may involve the flow of steam, or high-temperature water, at low viscosity (hence a high Reynolds number), and in the
laboratory it is necessary to use low-temperature water with relatively high viscosity, which limits the maximum Reyn­olds number available in the calibration runs If the coefficient curve is stable and flattens out at the high flow rates available in the laboratory, it is considered satisfactory practice to extrapolate to obtain a coefficient at the higher Reynolds number expected m actual service. The uncer­tainty of a coefficient so determined may be approximately

0. 5%. Greater accuracy may be expected, although it is difficult to guarantee, in view of the number of variables that must be measured during the calibration.

(h) Pitot Tubes. The Pitot tube is a primary flow­sensing element. It consists usually of a small-diameter tube pointed upstream with a pressure tap in its side (Fig. 4.32). The second (static) connection may be a trailing connection inside the pipe. Sometimes both connections are in the sides of a straight cylindrical probe. The Pitot tube measures velocity pressure itself, rather than the change in static pressure resulting from a change in cross-sectional area and velocity.

The equation for the Pitot tube may be written

V = CpKp(2gH)* (4.14)

where V = average velocity in the pipe line (ft/sec)

Cp = ratio of average velocity to velocity at the Pitot tube tip

Kp = coefficient based on Pitot tube design H = differential pressure (in feet of flowing fluid) g = acceleration due to gravity (32.17 ft/sec2)

From Eq 4.14 an equation can be derived similar to the basic orifice equation

W= 359CpKpD2 (hw7f)’* (4.15)

where W = rate of flow (lb/hr)

D = internal pipe diameter (in.) hw = differential pressure (inches of water at 68°F)

7f = density at flowing condition (lb/ft3)

If the sensitive tip of the Pitot tube is located at or near the center of the pipe, Cp may be 0.70 to 0.81 The value of Kp is usually supplied by the manufacturer, it may be close to 1.00 for a laboratory Pitot tube such as that shown in Fig. 4 32, or approximately 0.82 for a commercial Pitot tube such as that shown in Fig. 4.33.

The Pitot tube is normally used in temporary installa­tions and for estimating. It is not generally accepted for permanent installation because (1) the small passages tend to plug, which makes the meter insensitive or inoperative,

(2) a traverse must be made across the pipe to establish the point of average velocity required to evaluate the correction factor, Cp, (3) the differential pressure usually encountered is small, and (4) the device cannot be adjusted to provide a given value of flow at a selected differential pressure.

ALL FITTINGS IN SAME PLANE

oo

SO

(i) Secondary Elements. In a nuclear plant, the selec­tion of a secondary element to measure and interpret the differential pressure is limited by the need to avoid the presence of mercury, a common flowmeter sealing fluid, and, in the majority of cases, the requirement for a remote transmission system between the measuring mechanism and the display equipment.

In a remote transmission system, the secondary element is considered to include both the transmitter containing the

IMPACT Fig. 4.33—Commercial Pitot tube. (From L. K. Spink, Prin­ciples and Practice of Flowmeter Engineering, Fig. B-2165, The Foxboro Company, Foxboro, Mass., 1967.)

measuring mechanism and the receiver (i. e., the compo­nents for recording, indicating, integrating, etc.). In effect, the direct mechanical connection of a self-contained secon­dary element is replaced by a pneumatic or electric position-transmitting mechanism.

As noted in Sec. 4-3.2, measuring mechanisms of differential-pressure transmitters are of two general classes: motion balance and force balance. In a motion-balance system, the differential pressure exerted on a bellows or diaphragm displaces it, and the displacement is opposed by a spring, the force exerted by the spring being directly proportional to the applied differential pressure. A typical motion-balance-type electric transmitter is shown in Fig. 4.34.

VENT FILTER Fig. 4.34—Electric motion-balance transmitter. (From In­struction, Sec. E21-17, Fig. 26, Bailey Meter Co., Wickliffe, Ohio.)

In a force-balance system, the output signal is supplied to a pneumatic bellows or electromagnet that opposes the force exerted by the measuring mechanism. In a typical pneumatic force-balance transmitter (Fig. 4.35), a change in differential pressure changes the air gap at a nozzle tip, thereby changing the nozzle pressure, which is then amplified to provide the output signal. The changed output pressure changes the force opposing the measuring force, restoring equilibrium at a new value of the opposing forces with only an imperceptible change in nozzle air gap.

The receiver of a flowmeter may contain two features peculiar to flow measurement, a square-root extractor and an integrator. The square-root extractor is required if the pointer motion is to be directly proportional to flow rate

instead of to differential pressure. The integrator is required to convert flow rate to total flow for accounting purposes.

Because the rate of flow is proportional to the square root of differential pressure, a flow scale on a differential-pressure meter is not uniform, the divisions at the low end of this scale are compressed close together and difficult to read. The square-root extractor interposes a variable gain between the measuring mechanism and the display, magnifying the motion significantly at lower flow rates to make the scale divisions uniform. The extractor may be in the transmitter of a transmitting system, in the receiver, or in a separate unit installed between transmitter and receiver. A simple mechanism that performs this function pneumatically is shown in Fig. 4.36 In the figure, the rise d is proportional to the differential pressure, and the pointer motion indicating flow is proportional to the angle a A cam follower operates through a vane and nozzle to maintain a light contact with the cam. For low values of a. (i. e., d is very small compared to the beam length R) the value of a is almost exactly proportional to the square root of d.

The integrator is a counter that is usually driven by a constant-speed motor through a friction clutch and cam — operated escape mechanism in such a way that the counter rotates during a portion of cam rotation, the duration being proportional to rate of flow or point of position. The amount registered during any time period therefore is the total flow m that period. A diagram of a typical integrator is shown in Fig. 4.37.

COLLAR Fig. 4.36—Square-root extraction. (From Instruction, Sec. P22-8, Fig. 5, Bailey Meter Co., Wickliffe, Ohio.)

(j) Accuracy of Secondary Elements. Secondary — element accuracy may be represented as a single tolerance or as an individual tolerance for each component, such as transmitter, square-root extractor, receiver, and integrator.

For several tolerances to be combined, they must be expressed on the same basis, i. e, in terms of differential pressure or of flow, and as percent of maximum or of actual reading. The square root of percent maximum differential is percent maximum flow, and a tolerance in percent of maximum flow is divided by percent of scale to determine the percent of actual flow.

For instance, at mid-scale on a differential-pressure transmitter, a tolerance of ±1% of maximum would represent an actual flow tolerance of 1% since the square root of (0.50 — 0.01)/0.50 = 0.99, but, at 25% scale, the flow tolerance is 2% since the square root of (0.25 — 0.01)/0.25 = 0 98 The addition of individual toler­ances may be made on a root-mean-square basis and stated as such.

Because the high gam action of the square-root extrac­tor at low flow rates magnifies uncertainties as well as the flow signal itself, the rangeability, or turn-down ratio, of a differential-pressure meter is practically limited to about 25% of maximum flow (6.25% of maximum differential) When operation at wider range is required, multiple primary elements or multiple transmitters can be used with auto­matic switching, which simultaneously changes meter ca­pacity and integrator speed. Alternately, a meter with linear output might be specified.