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

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2.7 COMPRESSIBLE FLOW 35

by the nozzle. A point is reached where the mass flow becomes constant, and the

noise disappears irrespective of the downstream pressure.

What is happening? The normal information transfer mechanisms between the

changes downstream and the upstream flow appear to have ceased. The loss of sound

is clearly linked to this loss of information transfer, and it is intuitively reasonable to

link the information transfer to the very small pressure waves that constitute sound

and to suggest that, when the velocity is as fast or faster than sound speed, it is

impossible for the waves to move upstream, and so impossible also to communicate

upstream any flow change that happens downstream. In this case, sound speed has

been reached at the throat of the nozzle.

It is beyond the scope of this book to derive the full equations for compressible

flow, but we state three equations that will be useful in later discussions. These are

for the special case of flow when conditions are adiabatic, isentropic, and reversible.

Although an idealization, the flow for the inlet to the nozzle is a good approximation

to such a flow.

(2.15)

(2.16)

They give the value of T, p, and q

/A for a given y and M, where

Units

A = area m

c = local sound speed m/s

specific heat at constant volume J/kgK

= specific heat at constant pressure J/kgK

M = Mach number V/c dimensionless

p = pressure Pa

= stagnation pressure Pa

= mass flow rate kg/s

—

temperature K

= stagnation temperature K

gas velocity m/s

y = ratio of specific heats c

(isentropic exponent) dimensionless

Note that, in an isentropic flow without heat transfer, the stagnation temperature

and pressure do not change and represent the values if the gas were to be brought

to rest. However, stagnation pressure will change where irreversibility occurs (e.g.,

through a shock wave or in a flow with friction), and stagnation temperature will

change where heat transfer takes place.

36 FLUID MECHANICS ESSENTIALS

Figure 2.6(b) gives the plot for pressure variation through the convergent-diver-

gent nozzle. As gas flows through the convergent-divergent nozzle in Figure 2.6(a)

;

the velocity of the gas increases toward the narrowest point in the duct, the throat.

The flow is created by a low pressure downstream of the nozzle. As the back pressure

is reduced, so the pressures through the nozzle fall, and the velocity increases [a

and b in Figure 2.6(a)]. In the sonic or critical condition, the pressure downstream is

reduced until the velocity of the gas at the throat has increased to the speed of sound

when the Mach number M is unity. Figure 2.6(b) shows that this condition can be

achieved with the flow downstream of the throat subsonic (c), partly supersonic and

partly subsonic with shock waves (d and e), or with the flow all supersonic (f, g, h,

and i with various types of flow in the plenum). Had the nozzle stopped at the

throat, it would still be possible to obtain sonic conditions there, but there would

be no possibility of the pressure recovery that occurs in the diffuser in c, d, and e.

Equations (2.16) and (2.17) give the variation of pressure and area of the duct as

we move toward the throat and M increases. At the throat where M = 1, we obtain

two important relationships: the critical pressure ratio

-)'

V 2

and the mass flow rate at choked or critical conditions

(2.18)

where * as a subscript indicates throat conditions. For air, with y = 1.4, the critical

pressure ratio for sound speed at the throat is

p*/po

= 0.528. It can be shown that

the pressure recovery in the diffuser allows the exit/inlet pressure ratio to be much

higher than this while still achieving critical conditions (sonic at throat).

The importance of choked or critical conditions is twofold.

i. The flow rate is unaffected by downstream variations because the sonic throat

condition acts as a block to downstream changes (see Baker 1996 for a fuller

description).

ii.

The mass flow of gas may be obtained from a knowledge of y for the gas, the

throat area A*, and the upstream stagnation values p

and T

We shall consider the practical version of Equation (2.19) in Chapter 7.

2.8 MULTIPHASE FLOW

The term multiphase flow is somewhat misleading because it covers both multicom-

ponent and multiphase. Thus dirty gas, air-in-water, cavitational, and steam flows

may all be referred to as multiphase flows within this broad use of the term. Because

of the dearth of data, the fluid engineer attempts to learn from data from different

sources.

2.8 MULTIPHASE FLOW 37

OLD

WELL

YOUNG

WELL

GAS SLUGS

AND

WATER DROPLETS

LARGE GAS

BUBBLES AND

WATER DROPLETS

GAS BUBBLES

OIL ONLY

Figure 2.7. An example of three-phase

vertical flow from an oil well.

We may consider first the flow from

an oil well, making the assumption that

this is vertical. This is not strictly true, but

it will give us some basic concepts. This

was described by Baker and Hayes (1985).

The crude oil will reach the well-head,

having flowed up a pipe of about 100-mm

bore for distances of several kilometers or

about 30,000 pipe diameters. The flow will

therefore, presumably, be fully developed.

Initially, the flow will be single-phase, es-

sentially oil only.

oil is removed from the

well, the pressure in the reservoir will de-

crease, and the gas fraction in the well flow

line will increase, and appear as gas bubbles.

This is known as bubbly flow (Figure 2.7).

With further aging, the gas bubbles will become larger. An equilibrium size dis-

tribution will result from breakup due to turbulence and from coalescence due to

the breakdown of the liquid film on close approach of two bubbles. In addition,

the water content is likely to increase. Water will be present as droplets and will

form a third phase. Yet further aging will result in slugs of gas that travel up the

center of the pipe leaving a slower layer of moving liquid on the wall (which, at

times,

may even reverse in direction during the passage of the slugs). These slugs

tend to overtake each other forming larger slugs many meters long. These slugs may

be up to 20 m long and may form an equilibrium size distribution giving a bal-

ance between coalescence due to bubbles overtaking each other and breakup due

to instabilities when they become too large. As a further complication, the oil flow

may contain waxy deposits and sand. Even though separation of components is a

standard process, the development of subsea systems may require multicomponent

handling.

We next consider horizontal two-phase flows (Figure 2.8). The most obvious

effect is the loss of axisymmetry. Gravity now causes the less-dense phase to migrate

to the top of the pipe. Thus, in a gas-liquid flow, the gas will move to the top of

the pipe as bubbles. If these are allowed to become large, plugs of gas result, and as

these coalesce, slugs of gas take up regions against the top of the pipe. Eventually, a

sufficient number of these will lead to stratified flow.

Alternatively, the mixtures may be of two liquids (e.g., water in oil). The droplets

of water will sink toward the bottom of the pipe, mirroring the behavior of air

SINGLE PHASE BUBBLY

PLUG

'• •

(

SLUG

•p-vr:

•

STRATIFIED

• ...

Figure 2.8. Horizontal two-phase flow.

38 FLUID MECHANICS ESSENTIALS

bubbles, and will eventually drop out onto the bottom of the pipe causing a contin-

uous layer of water.

With a sufficient length of straight pipe (100D or more), a fully developed flow

may be achieved. However, in most applications we shall not be able to predict the

resulting flow or how it will affect equipment in the line. This presents a considerable

challenge to the development of instrumentation for handling oil well flows in sub-

sea installations. It may be possible to mix the fluid to create a more homogeneous

fluid for flow measurement, but this will cause severe turbulence and a changing

profile, conditions generally considered unsuitable for such a measurement, not to

mention the pressure loss which may result.

Liquids may contain gases in solution. For water, the maximum amount is about

by volume. The gas in solution does not increase the volume of the liquid by

an amount equal to its volume since the gas molecules "fit'' in the "gaps" in the

liquid molecular structure. For hydrocarbons, the amount of gas that can be held in

solution is very large, and the GOR (Gas-to-oil ratio), which is the volume of gas at

standard conditions to the volume of liquid, can range up to 100 or more. In either

case,

but particularly the latter, changes in flow conditions (e.g., a pressure drop)

can cause the gas to come out of solution causing a two-phase flow with, possibly,

severe effects on instrument precision.

In low head flows (e.g., in a sewage works), the flowing stream may entrain air,

and a dispersion of air in water may result.

2.9 CAVITATION, HUMIDITY, DROPLETS, AND PARTICLES

Cavitation may occur in certain liquid flows at pressures around ambient. Cavitation

is the creation of vapor cavities within the liquid caused by localized boiling at low

pressure. It can cause damage (e.g., in pumps, propellers, valves, and other flow

components) because the cavities can collapse very quickly with large, although

localized, impacts and can erode solid surfaces. It can also cause errors in flowmeter

readings because it results in a larger volume than for the liquid alone.

High humidity may also create problems if it results in a consequent large amount

of water vapor changing to liquid droplets in the gas (e.g., in critical flow venturi

nozzles).

Particulate matter can, in addition, cause wear and may need to be removed with

a fine filter.

Much can be learned by computing the trajectories of droplets and particles (cf.

Ahmad et al. 1986, Hayes 1988), and such an approach may be usefully applied to

flowmeters. Droplets and particles will affect their performance in various ways.

a. They will cause flowmeters to read incorrectly, and most often this will be due

to the flowmeter responding to the volume flow of the gas phase and not fully

accounting for the greater mass carried in solid and liquid.

They may cause erosion of the flowmeter body and sensing element.

c. They may give false pulse readings in vortex flowmeters by hitting the pressure

sensor.

d. They may become trapped in vortex structures around both stationary and mov-

ing parts of the meter.

2.11 STEAM 39

The basic flow around most meter internals is now predictable from standard com-

puter programs, and a bubble/droplet/particle trajectory model should be combined

with such a prediction. From such studies, it may be possible to design internals less

susceptible to bubble/droplet/particle effects or to understand better the effects on,

and to optimize the design of, meters such as the turbines described by Mark et al.

(1990a, 1990b).

For electromagnetic and ultrasonic meters, further work on the interaction of the

field and bubble/droplet/particle may lead to more sophisticated sensors. Further

study should also address the effect of large bubbles on the performance of such

meters and on ways of flow-pattern mapping in general.

2.10 GAS ENTRAPMENT

Gas entrapment appears to occur in some flow geometries with important effects.

Thomas et al. (1983) observed that 'Transient large eddies (vortices) in turbulent

free shear flows entrap and transport large quantities of bubbles, and may also force

the coalescence of bubbles/' At the conference where this paper was presented, the

appearance of this phenomenon was discussed in relation to two papers in particular,

which suggested the presence of the phenomenon. In one of

these,

Baker and Deacon

(1983) had tested a turbine meter that appeared to exhibit hysteresis in its response

to increasing and decreasing fractions of air in water. This may have been due to the

particular meter or the flow circuit, but it raised speculation as to whether vortex

structures in the vicinity of the ball bearing fluid access path could be entrapping air

and holding it in the bearing after the external air content in the flow had dropped.

Presumably such entrapment could occur in the vortex downstream of an orifice

plate [Figure 2.9(a)].

Hulin and Foussat (1983) observed the entrapment of a second phase in the shed

vortices behind a meter bluff

body.

Figure 2.9(b) shows a diagram of this mechanism.

I have also observed, in some field data from an ultrasonic flowmeter, a behavior

that could result from entrapment of air. A meter in a low head flow, where air was

entrained with the water, periodically failed. The possibility that, in such a flow, the

transducer cavities could cause small local vortices which could entrap the air and

block the ultrasonic beam offered a possible explanation [Figure 2.9(c)].

There is considerable scope for investigating in other applications the appearance

of such entrapment and the likely effect. In flow measurement, entrapment may also

occur around the throat of a Dall tube and behind an averaging pitot tube or other

probes inserted in the flow. In all these cases, it is likely to affect the reading of the

flowmeter. Entrapment may also occur in pumps, where it may reduce performance,

and in pipework where abrupt area changes exist.

2.11 STEAM

One truly two-phase fluid is steam. Superheated steam may be treated as a gas, and

its properties are well tabulated. However, it is increasingly important to measure

the flow of wet steam made up, say, of about 95% (by mass) vapor and about 5%

40 FLUID MECHANICS ESSENTIALS

(a)

VORTICES WHICH MAY

ENTRAIN A SECOND

PHASE

MEDIUM VOID FRACTION

LOW VOID FRACTION

HIGH VOID FRACTION

(b)

VORTICES WHICH

MAY ENTRAIN A

SECOND PHASE

Figure 2.9. Possible mechanisms for gas entrapment: (a) Upstream and downstream of an orifice

plate; (b) in vortices; (c) in an ultrasonic transducer cavity.

2.12 CHAPTER CONCLUSIONS 41

liquid. The droplets of the liquid are carried by the vapor but will not follow the

vapor stream precisely. As with water droplets in oil, the liquid will drop through

the vapor to land on the pipe wall, and we may obtain an annular flow regime until

sufficient turbulence is created to reentrain this liquid. The measurement of such a

flow causes major problems because the pressure and temperature remain constant

while the dryness fraction changes. It is therefore not possible to deduce the dryness

fraction or density from the pressure and temperature, and, in addition, the droplets

may cause an error when we attempt to measure the flow.

Some pointers from the literature will be referred to in relation to some of the

types of flowmeter. However, the best and safest advice is to use a water separator

upstream of the flowmeter and to measure the resulting flow of dry saturated steam.

2.12 CHAPTER CONCLUSIONS

I have been highly selective in choosing topics for inclusion in this chapter. Once

again the reader is referred to fuller texts on fluid mechanics. Baker (1996) was

written, particularly, with the industrial flowmeter user in mind and enlarges on

some of the points made in this chapter.

The development of computational fluid dynamics provides a tool for analyzing

flowmeter behavior, and references will be mentioned in the other chapters. I con-

sider that perturbation methods should be explored. These methods would allow a

greater precision in defining performance change between the ideal flow through a

meter and the disturbed flow due to turbulence change, profile distortion, or swirl.

The effects of small amounts of

second phase offer possibilities for investigating

the behavior of particle/bubble/droplet in flows through meters by developing, for

instance, methods such as those of Ahmad et al. (1986) or Hayes (1988).

Work on flow conditioning is at present subject to thorough investigation by

Laws and others as indicated, and Mottram and others continue to provide important

insights into the effect of pulsating flows.

The critical flow venturi nozzle, as we shall see, appears to be capable of higher

precision than has been accepted. There may

value in fluid behavior investigations

in this and other areas to define ultimate limits of accuracy.

CHAPTER

Specification, Selection, and Audit

3.1 INTRODUCTION

The object of this chapter is to assist the reader in specifying a proposed flowmeter

;

operational requirements as fully as possible and hence to enable the reader to select

the right flowmeter. The reason for wishing to select a flowmeter may be obvious:

a flow rate to be measured. However, the reason in some cases may be less obvious.

If you are a flowmeter manufacturer, or prospective manufacturer, you may wish

to identify a type and design that would meet a particular market niche. If you

are a member of an R&D team, you may be exploring measurement areas that are

inadequately covered at present.

In preparing this chapter, I have developed earlier ideas (Baker 1988/9, Baker

and Smith 1990), but I have also benefited from the ideas of others such as Endress

et al. (1989). Baker (1988/9) was the best distillation I could offer in the late 1980s.

There is much there that I would still endorse. In Baker and Smith (1990), we took a

different line. We provided the means to specify the user's needs in as much detail

as possible. We then provided a form for communication with the manufacturer.

Many people have attempted to provide a means of selecting a flowmeter. A

glance at only a few of the many books on flow measurement in the bibliography will

indicate this. I used to feel that the way forward was to write an expert system to do

the

job.

am less convinced about that now. One reason for this is that manufacturers

appear to be extending the flowmeter types they offer and their applications, so that

keeping the expert system up-to-date is a major task, and the decision on the best

meter for a particular job may rest on more subtle considerations than an expert

system is likely to include. This suggests that the expert system should narrow the

field pending the final choice in consultation with manufacturers or consultants.

I have picked out a few examples from the literature of application to specialist

areas.

This selection is not intended to be exhaustive but rather to complement the

applications identified for each flowmeter type in other chapters.

3.2 SPECIFYING THE APPLICATION

The change that I perceive over the last 10 years is that the main manufacturers

have recognized the need to have within their armory a meter for every occasion. In

some cases, this has led to existing meters being applied to new and more demanding

applications. When

wrote Baker (1988/9),

was fairly skeptical as to the extent that

3.3 NOTES ON THE SPECIFICATION FORM 43

flowmeter types could be eliminated for particular applications. I am now more so.

On a regular course at Cranfield University, we divide up into working groups in

order to select flowmeters for particular applications. Almost invariably there are

three or four meters suitable to a lesser or greater extent for each application.

I have, therefore, developed the theme of Baker and Smith (1990) in this chapter.

We start with a form and discuss the implications of each item for user and manu-

facturer. Before the reader turns to the form and concludes that it is unrealistic for

a user or manufacturer to complete or respond to such a form, I would hasten to

defend its inclusion because

i. it provides a check list of features that might otherwise be forgotten in specifying

the meter, and

ii.

it is there to be modified, simplified, and reduced in length and complexity, until

it suits individual needs.

The form is set out in Appendix 3.A.I under eight headings, which are labeled

alphabetically. It can be photocopied and/or adapted. It is aimed at the user and

current manufacturer, but it should provide a starting point for the prospective man-

ufacturer, who may wish to identify a type and design for a particular market niche,

or for a member of an R&D team, which is exploring new development areas. In

summary the eight headings are:

A. General information about the meter manufacturer

General description of the meter user's business and specific application

C. Details of fluid to be measured

Site details

E. Electrical and control details

Other user information such as preferred accuracy, materials, maintenance, and

cleaning

G. Manufacturer's additional information and requirements such as warranty,

filtration requirements, and price

H. Other considerations and final conclusion

The audit of a flowmeter should link closely with the specification and selection.

To ensure consistency, the final section on audit reverts to the specification table

and reviews the correctness of the original assessment (if there was one) or the

appropriateness of the selection.

3.3 NOTES ON THE SPECIFICATION FORM

A INFORMATION ABOUT THE MANUFACTURER

In approaching a manufacturer, it is important to know whether it will want to sell

you its one and only flowmeter type - however appropriate or inappropriate for

your application - or whether it has a range of meters, which will ensure that one

is appropriate to your needs. It is also useful to know how many the manufacturer

sells and the satisfaction of other customers.

manufacturer may be wise to include

44 SPECIFICATION, SELECTION, AND AUDIT

illustrations of a high quality production line in its literature as well as indications

of its international sales and servicing structure.

In making this point, I have no intention of excluding the smaller high quality

manufacturer whose products will be sold on their special features and adaptability.

Such a manufacturer will have its own strengths. Indeed, the manufacturer should

be clear as to what they are and base its selling strategy on them. In fact, many, if not

most, manufacturers' outlets in a particular country will fall in the category of small

and medium-sized enterprises (SME) and are likely to give a personalized service.

B USER'S BUSINESS AND SPECIFIC APPLICATION

The object of this section is to provide the manufacturer with a clear overview of

the application, site, and essential background to maintenance and safety on site.

The relevant safety documents and those dealing with other relevant standards

and legislation on electromagnetic compatibility (EMC) should be referred to (note

the

marking requirement in Europe). Endress et

al.

(1989) provides a useful digest

of the safety requirements for Europe and the United States.

C FLUID DETAILS

The user should endeavor to provide the manufacturer with the fullest information

possible to avoid any incompatibility between fluid and meter. Of those properties

requested, conductivity only applies to liquids where an electromagnetic flowmeter

is to be considered. Opacity would be relevant only for meters that use optical sensing

through the fluid. Compressibility may relate to the behavior of differential pressure

and ultrasonic flowmeters, and it may be useful for the manufacturer in advising on

such applications.

Lubricity has not been included in the list. On the basis of definitions in dic-

tionaries and encyclopedias,

lubricity

may be taken to mean boundary lubricating

properties.* It is a term frequently used in the flowmeter business, where the mea-

sured fluid is often expected to act as lubricant for the flowmeter rotor. Lubricity

generally is used to indicate the ability of the metered fluid to lubricate the bearings

of a rotating meter insert.

Lubricity does not appear to be a term used by lubrication experts. Its meaning

would seem to be related to the high friction coefficient in a lubricated bearing at

low viscosity. This is best illustrated by the Stribeck diagram, Figure 3.1 (a) (from

Summers-Smith 1994). This shows that as the viscosity n or rotational speed n of the

bearing decreases or the load p increases, the friction coefficient

falls to a certain

point. In this region, hydrodynamic lubrication occurs, and the curve has a positive

and fairly constant

slope.

Following this for smaller values of viscosity, the curve has a

sharp negative slope. Where the curve meets the vertical axis, boundary lubrication

takes place as the surfaces come into contact. This is matched by a curve with a

similar shape [Figure 3.1(b)] which shows the friction coefficient

as a function of

the specific film thickness

the ratio of the lubrication film thickness h

min

, and the

combined roughness of the two contacting surfaces a. The minimum occurs (region

Private communication from

D. Summers-Smith.