Baker R.C. Flow Measurement Handbook: Industrial Designs, Operating Principles, Performance, and Applications

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8.A.1 SOME HISTORY 177

will continue to have a place. Several of the designs described have been introduced

relatively recently.

If a manufacturer is considering the introduction of new designs, the user will

be dependent on manufacturer's data for most of the designs that fall outside a stan-

dard's specification. The quality of the manufacturer's total operation will, therefore,

be a key consideration. The effect of installation should be plausible compared with

similar devices from elsewhere. The biggest uncertainty will be in the special condi-

tions that are unforeseen: small traces of gas in a liquid becoming trapped in flow

vortices, deposition of small quantities of a solid or liquid film, and materials prob-

lems.

Increasingly, however, the question will need to be asked, "Is there a linear

meter that would be more appropriate?"

APPENDIX

8.A

History, Equations, and Accuracy Classes

for the VA Meter

8.A.1 SOME HISTORY

In 1873, G. F. Deacon produced a design that resembled a variable area meter. This

was very similar to an earlier device patented in 1868 and made by

A. Chamberoy

of Paris (Coleman 1956). Something similar was devised by Sir J. Alfred Ewing

(1924-5) in about 1876 and was initially known as the Ewing flowmeter. Further

experiments were carried out by Awbery and Griffiths (1926-7). They found that

the float, if a ball, started chattering at high flows and caused an overreading. Three

solutions were suggested:

i. Inclining the tube so that the ball rolled on the wall. This was Ewing's preferred

solution,

ii.

Using a float of a different shape. Awbery and Griffiths (1926-7) mentioned that

the meter used a cylindrical float with helical channels to cause slow rotation,

iii.

Stretching a fine wire along the tube close to the wall, to prevent the chattering

motion (Advisory Committee on Aeronautics 1916-17). Awbery and Griffiths

verified this in their observations.

One possible reason for this instability (Leder 1996) is a saddle point (free stag-

nation point) that forms in the wake downstream of the ball and off-axis and that

rotates and induces a large scale rotating helical vortex together with small scale

turbulence motions.

Ewing suggested that the range of the instrument could be extended by incor-

porating more than one float. The largest would be on top and would record low

flows,

and the smaller ones underneath would rise when the flow rate was sufficient.

Ewing also used a dye trace to show the nature of the flow. An alternative could be

to have two floats of similar size but with the top one of lower density.

178 OTHER MOMENTUM-SENSING METERS

Awbery and Griffiths showed that their experimental results collapsed onto

single curve for

given diameter ratio

D/d, where D is the diameter

the tube

at a certain height, and d is the diameter of the ball. They plotted log[q

D/(D

—

)]

against \og[d

(pf

—

p)/v

p], where

the volumetric flow rate,

p is the

density

of the fluid,

pt is the

density

the float,

and v is the

kinematic viscosity

the

fluid.

British Patent 2428 granted to

Joslin of Colchester (1879) covered a device for

ascertaining rate

consumption of gas, which resembles

meter, and this was

followed

1908 by

patent granted

K. Kuppers

Aachen for

tube and float

very similar to

Joslin's.

Felix Meyer of Aachen obtained the patent and started man-

ufacturing the tubes.

1921 Trost Brothers Ltd. of England took up Meyer's agency

for the United Kingdom, and in 1931 Schutte and Koerting secured the manufactur-

ing rights for the United States. Coleman (1956) gave a very useful historical review

from which some of these notes are taken.

Head (1946) refers

to a

downdraught instrument

which

the

float

lighter

than the fluid.

8.A.2 EQUATIONS

The basic theory

the flowmeter depends

the upthrust

the float resulting

from the pressure loss across

where

and p

are the upstream and downstream pressures,

is a loss coefficient,

is the density and V is the velocity in the annular passage past the float. Replacing

velocity by volumetric flow rate q

such that

V =

(8.A.2)

where A

is the annular area around the float, we obtain

Pi-p

(8.A.3)

This equation assumes that the dynamic head

inlet

negligible, and that

the

pressure difference across the float is caused by the loss of the dynamic head down-

stream of the float. Dijstelbergen (1964) uses a contraction coefficient C

for the ratio

where

is the area of the tapering tube at height

The immersed weight of the float is given by

W=V

(

-p)g

(8.A.4)

where V

is the volume of the float, p

is the density of the material of the float,

and

g is the acceleration due to gravity. Thus, the balance is given by

- p)g (8.A.5)

where A

is the maximum cross-sectional area of the float.

8.A.2 EQUATIONS

179

This is usually rewritten as

(pt-p)g

(8.A.6)

with l/y/K replaced by

(e.g., Martin 1949,

Coleman 1956; cf. Head 1946-7).

Coleman (1956) gave values of C for

three float shapes (Figure 8.A.I), which sug-

gested that at full flow the value of

ranged

from 0.6 for a float of low Re sensitivity

to about unity. The plots also showed a

decrease in C at low flow rates, and this

would broadly reflect the fact that the loss

coefficient for an orifice increases with re-

duced Re (Miller 1990). Coleman also plot-

ted the effect of viscosity on the various

float shapes and confirmed the implications

of Figure 8.A.I, that the plumb-bob is the

most viscosity dependent.

Schoenborn and Colburn (1939) gave a

curve of C against Re/C, where Re is the

Reynolds number based on D

. They called

it the equivalent diameter but defined it

as the difference between D and d. This

curve is reproduced in Figure 8.A.2. It was

obtained for a large high pressure design

using a steel tube with extension rod ob-

servable through a graduated glass window.

This curve could be used as a first approxi-

mation if no other is available. They used a

balance of forces of the weight of the float in

liquid to the upward force due to the pres-

sure reduction across the float to provide

the basis for their experimental correlation.

They also noted that the curve for large

meters appeared to be less fluid-dependent

than for smaller instruments.

Although it may be difficult to predict

flow rates, it is possible to convert from

one fluid to another at the same read-

ing. Schoenborn and Colburn (1939) gave

an equation similar to that given by ISA

(1961b):

(8.A.7)

1.0

0.5

10°

1.0

0.5

Re 10

1.0

0.5

10°

Re 10

Figure 8.A.I. Effect of float shape on flow coefficient

(after Coleman 1956).

1.0

0.5

Re/C 10

Figure 8.A.2. Plot of C against Re/C for

typical vari-

able area meter (after Schoenborn and Colburn 1939).

180

OTHER MOMENTUM-SENSING METERS

I.UO

1.02

1.01

1.00

0.99

0.98

0P7

/(P

-P

)p2

-\(P,-PI>PI

-—-^^^

—^^^

float

s.g.

7.9^/^

^^^^^—^^^ 1.44

SPECIFIC

GRAVITY

METERED

FLUID

i i i

0.68

0.69 0.70

0.71

0.72 0.73 0.74 0.75 0.76

s.g. FOR

CALIBRATION

FLUID

Figure 8.A.3. Density compensation float comparison curves (after Coleman

1956).

The difference is that ISA (1961b) omits

C2/C1,

thereby making the assumption

that the ratio is unity. Coleman (1956) also used this equation with C2/C1 equal

to unity, noting that variation of viscosity is ignored. He referred to the fact that if

Equation (8.A.6) is rewritten for mass flow rate and differentiated for variation in

fluid density, it can be shown that a float density, of twice the fluid density gives a

zero change and, therefore, is least sensitive to variation in density (cf. Head 1946).

Figure 8.A.3 shows values of the density correction factor with change of fluid density

and for various float densities, based on calibration for aviation gasoline of specific

gravity

0.72.

Head (1946) also raised the possibility that, with a downdraught design,

the requirement from differentiating the volume flow equation of a float of zero

density could be achieved approximately. Head also suggested that float automatic

compensation might be used to allow for temperature variation and other variables.

Several authors have pointed out that the value of C is constant if /x/y/pipf

—

p) is

constant.

8.A.3 ACCURACY CLASSES

To define the accuracy of the variable area flowmeter, VDI/VDE 3513 (1978) recom-

mended an accuracy class and divided the reading into two parts:

• First part is three-quarters of the actual flow rate M;

• Second part is one-quarter of full-scale or upper range value E.

The summation error F in flow rate, in actual flow rate units, is then given by

8.A.3 ACCURACY CLASSES 181

where K is the accuracy class number selected from a range from 0.4 to 10. For

example, taking a full-range flow of 100 units, the actual error will be

at full scale f | x 100 + i x 100^ -^- = K units

at 1/10 full scale (- x 10 + - x lOo) — = 0.325 K units

\4 4 J 100

If we take a value for K of 2.5, then the error at full flow will be 2.5 units in 100

units (2.5%), whereas at 1/10 full flow it will be 0.812 units in 10 units (8%).

CHAPTER

Positive Displacement Flowmeters

9.1 INTRODUCTION

The material in this chapter is based on the paper by Baker and Morris (1985) on

positive displacement (PD) meters, which in turn has been updated with more re-

cent industrial and published material and extended to gases. P.D. Baker's (1983)

paper provided some additional useful information on these meters. The main types

of liquid meter were given by Barnes (1982), Hendrix (1982), Henke (1955), and

Gerrard (1979) (cf. Mankin 1955). The reader should refer to API (1992) and similar

documents.

At least four of the meter designs to be discussed have been around for over 100

years.

The nutating disk flowmeter for liquids was developed in 1850. The rotary

piston meter appeared in the late nineteenth century (Baker 1998).

The measurement of gas has depended, from an early date, on two types of

positive displacement meter: the wet gas meter of high accuracy and credited to

Samuel Clegg (1815), and the diaphragm meter of lower performance but greater

range for which William Richards (1843) should take the credit.

9.1.1 BACKGROUND

The concept of carrying known volumes of fluid through a flowmeter is a short step

from the use of a discrete measure such as a bucket or measuring flask. Thus in each

of the designs described later, the flow enters a compartment that is as tightly sealed

as is compatible with relative movement of adjacent components. A knowledge of

how many of these compartments move through the flowmeter in one rotation of

the shaft leads to a knowledge of the flow rate for a certain rotational speed. A sim-

ple theory is developed in Appendix 9.A.I. Clearly every leakage path will reduce

or increase the amount carried and cause an uncertainty in the measurement. We

can make an approximation of the leakage flows in the simple model. In addition,

pressure and temperature will distort the volume and may cause small errors, and

we may need to develop compensation for these. Compensation for pressure can

be provided in the form of a double-walled meter in which the precision-measuring

chamber is unpressurized since the pressure is the same inside and out, and the pres-

sure drop to the outside world is taken across the second and outer wall, the shape of

which will not affect the accuracy. The temperature compensation, if necessary, can

be incorporated into the calibration adjustment by a temperature-sensitive element.

Very often these devices provide a direct sales ticket that is produced by the

rotation of the meter and the starting and stopping of the flow. On the other hand,

182

9.1 INTRODUCTION 183

it is very difficult to make small adjustments as needed in calibration where the

signal is from a mechanical rotation. We, therefore, need to introduce complicated

mechanical calibrators where a frequency output is not acceptable. The combination

of mechanical counter and ticket machine can cause a substantial drag on the meter

and may cause slippage errors in certain types of calibrators.

Many of the points made in this chapter relate primarily to liquid measurement,

have some relevance to gases, and should be noted by those mainly interested in gas

measurement.

9.1.2 QUALITATIVE DESCRIPTION OF OPERATION

The essential of any positive displacement meter, as indicated earlier, is that a discrete

and well-defined portion of the fluid is carried from inlet to outlet without loss or

mixing with the remainder of the fluid. This applies to all the fluid that passes

through the meter. The skill and ingenuity of the designer is shown, therefore, in

achieving this as simply and precisely as possible. If, for instance, we know the

volume of each compartment of a reciprocating pump, then knowing how many

cylinder-full portions are transmitted in a revolution, we can relate the amount

passed to the speed or revolutions of the pump. This is essentially the principle

of a metering pump. Taking this a step farther, we could reverse the reciprocating

pump. Instead of providing power externally to drive the pump, the fluid moves the

pistons, and the valves ensure that each portion is transmitted, without loss, to the

outlet. Of course there will be some loss due to imperfect sealing between the pistons

and the cylinders, but provided this can be kept very small, we have the basis of a

very accurate instrument. The reciprocating principle may not be satisfactory for

high throughput meters, and so other designs have been devised. It will be easiest

to explain the working of each in turn with respect to diagrams.

The theory of the device is worked out in Appendix 9.A.I. Here we need only

note that the simplest equation of flow through such a device is

= Volume per revolution x Rotational speed

—

Leakage (9.1)

Leakage is unlikely to be a constant value for one revolution and is more likely to

be speed-dependent. The output from all meters is a rotating shaft, and this needs to

be related to the flow rate and total passed. If the shaft passes through the pressure

containment of the meter, then a rotary-shaft-sealing arrangement will be needed.

In one type of meter, an externally lubricated packing gland is used to isolate the

dynamic shaft seal from the product (Baker 1983), thereby increasing the life of the

packing gland. The external lubricant must, of course, be chemically compatible

with the product being metered. In some cases, the problem is avoided by using a

magnetic drive coupling.

In the past, calibration and transmission methods have tended to be by mecha-

nical linkage. Such a linkage loads the rotor and may alter the characteristic of the

meter. It is also difficult to design a mechanical system to adjust the calibration of

the meter. Some ingenious methods have been adopted and will be described later in

the chapter, largely for historical interest and for those who have dealings with such

devices that remain in service. Today the possibility of using electrical or optical

184

POSITIVE DISPLACEMENT FLOWMETERS

Inlet

Outlet

Figure 9.1. Nutating disk meter (reproduced from

Baker and Morris 1985; with permission of the

Institute of Measurement and Control).

transmission and still retaining intrinsic

safety means that an output pulse train can

be used, and signals can be manipulated

electronically.

9.2 PRINCIPAL DESIGNS OF

LIQUID METERS

This section reviews the main designs

and their special operational features and,

where possible, reviews the published information that is available about them.

Inlet

9.2.1 NUTATING DISK METER

In the nutating disk meter (Figure 9.1), the disk is constrained by the central spheri-

cal bearing and by the transmission to nutate. Nutation is a movement rather like a

spinning top that is slowing down. The disk

prevented from rotating by a partition

separating the inlet and outlet streams. The incoming flow causes the disk to nutate

and in doing so the disk forms a closed compartment trapping a fixed volume of liq-

uid. There are effectively two such compartments, one above and one below the disk.

The disk element has the shape of the planet Saturn. The important difference is

that the disk has a slit in its "rings/' which makes a close fit with a radial partition.

Flow enters, say, under the disk on one side of the partition and causes the disk to

rise and fill the region under the disk, and so to nutate. When it has nutated half

a turn, the entry port becomes blocked by the disk, and the other entry port above

the disk has started to allow fluid above the disk. At the same time, the exit port

for the liquid below the disk opens to discharge. Thus in terms of Equation (9.1),

in one rotation, the cavity on one side of the disk is both filled and discharged. It

is unlikely that the leakage can be kept very small, and the overall accuracy is not

expected to be very high. Although rather difficult to describe, it is a simple meter

of low cost, rugged and capable of uncertainty within

±1.5%.

(Henke 1955).

Typical industrial designs, some of which may be for water flow measurement,

may have a built-in filter. Claims for measurement uncertainty may be of order

for flow rates of 10-100 1/min with minimum pressure of 0.1 bar, viscosity up

to 1,000 mPa, and temperature up to 80°C.

Materials may be stainless steel, polypropy-

lene,

or ethylene-tetra-fluor-ethylene. Mod-

ern designs may include a microcomputer

to convert flow rate into the correct units.

9.2.2 OSCILLATING CIRCULAR

PISTON METER

The oscillating circular piston meter

(Figure 9.2) is similar to the nutating disk

meter in that the rotation of the piston is

constrained by a partition, so that the pis-

ton, in fact, rocks. Its center is constrained

Outlet

Figure 9.2. Oscillating circular piston meter (repro-

duced from Baker and Morris 1985; with permission

of the Institute of Measurement and Control).

9.2 PRINCIPAL DESIGNS OF LIQUID METERS 185

Fluted

Rotors

Flow

(a)

Main

Rotor

Vanes

Stationary

Block

Flow

(b)

Second

sealing

rotor

(C)

Figure 9.3. Multirotor meters: (a) Two rotor (re-

produced from Baker and Morris

1985;

with per-

mission of the Institute of Measurement and

Control); (b)

Two

rotor, main rotor has four vanes

(reproduced from Baker and Morris 1985; with

permission of the Institute of Measurement and

Control); (c) Multirotor: large rotor with two

small rotors (after Baker 1983 and Baker 1989).

to move in a circle by the transmission so that the radius of the cavity is essentially

the sum of the radii of the rotary piston and the circle on which its center moves.

When the center of the rotary piston is at the top of its travel, the piston forms a

closed compartment with the cavity. One rotation of the shaft will cause the rotary

piston to return to its starting place and so to discharge the volume of one compart-

ment. In some designs, the leakage may not be negligible. As well as use with water,

these meters may be made in designs suitable for liquid foods and other nutritional

fluids (Anon. 1966a). Hamblett (1970) suggests that they can achieve a measurement

uncertainty within ±0.25% for flow rates from 0.018 to 110 m

/n (4-25,000 gal/h).

This accuracy is rather higher than would be expected. He also lists the various types

of registers and controls that are suitable for a range of applications.

9.2.3 MULTIROTOR METERS

There are several designs of meter with multiple rotors. Figure 9.3(a) has two rotors

that seal against each other; each rotor carries liquid through the meter as it rotates.

Figure 9.3(b) shows another two-rotor meter, but in this case the main rotor consists

of four vanes that form metering compartments. The second rotor is a sealing rotor

that returns the vanes to the inlet side of the meter. The second rotor may be two-

or three-lobed, and its rotation will be precisely linked to that of the main rotor. It

transmits a net fluid flow across the meter equal to the volume of the vane that it

returns to the inlet. The mechanism, its design and engineering, is complex and,

unless manufactured to a very high standard of precision, may lead to high pressure

loss and interference between rotors and housing. Figure 9.3(c) shows a design with

a large rotor and two small rotors.

9.2.4 OVAL GEAR METER

The oval gear meter (Figure 9.4) is a special form of a multiple rotor meter in which

each oval rotor is toothed, and sealing between the rotors is enhanced by the

186

POSITIVE DISPLACEMENT FLOWMETERS

Percent error

+0.6

+0.3

-0.3

-0.6

(b)

Pressure

drop

0.5

0.4

bar 0.3

0,2

0.1

50%

100%

Figure 9.4. Oval gear meter: (a)

Working principle (reproduced

with permission of Bopp and

Reuther); (b) Error curves for a

range of viscosity values (after

Haar Messtechnik); (c) Typical

pressure drop for operating range

(after Haar Messtechnik).

flowrate

(c)

50%

100%

flowrate

resulting labyrinth. Each rotor transmits fluid from inlet to outlet and forms a closed

compartment when its major axis is aligned with the flow direction. The volume

passed per revolution of each rotor is four times the volume between the rotor and

the oval housing when the rotor is confining liquid [Figure 9.4(a)]. In place of the

leakage paths between the rotors of multirotor meters, there will be, for this meter,

extremely small leakage where the rotors mesh, and the tolerances for the other

surfaces are likely to be to a high standard, giving a very small value for the overall

leakage.

Small examples of this device are found in oil-flow lines in hydraulic sys-

tems,

whereas larger high precision stainless steel versions are also available.

They are claimed to have a low pressure drop, to have low maintenance, to be

of simple construction, and to achieve accuracies within ±0.25% (Henke 1955).

Meters are available with a stress-free measuring chamber for high pressure appli-

cations.

The meter can be subject to viscosity variation. Figure 9.4(b) shows the error, and

Figure 9.4(c) shows the pressure drop for a typical commercial design. Measurement

uncertainty is in the range ±0.1-2%. Sizes are from 6 to 400 mm, with flow ranges

from 0.1 1/m or less up to 1,200 m

/h, but many are for smaller flows. Turndown

is of order 10:1 (Endress and Hauser 1989), and operation is possible with, and

largely unaffected by, viscosity in the range 0.4-20 cP. However, the possibility of

a calibration shift (of order 0.5%) should be checked with the manufacturer. One

manufacturer suggests that operation is possible up to very high viscosities. Pressure