Induced voltage electromagnetic flowmeters

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A Thesis Submitted for the Degree of PhD at the University of Warwick

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INDUCED VOLTALE ELECTROMAGMIC FLOWMETERS

by

M. K. PEVIR

A

A Thesis submitted to the University

_of

Warwick for the degree of Doctor of Philosophy

`ý ý' i`ft t r' !týi

ii.

1

.

ter'

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BEST COPY

AVAILABLE

Poor text in the original

thesis.

Some text bound close to

the spine.

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ABSTRACT

This thesis

_{analyses the behaviour}

_{of electromagnetic}

flowmeters

_{in which the flowrate}

_{of a suitably}

_conducting

liquid

is measured by means of the e. m. f. induced between two electrodes

by the motion in a magnetic field. The distortion _{of the field}

by the induced _currents _{is assumed negligible,} _{thus excluding} the

more highly conducting liquid metals, but the method is well-

known to be suitable for _certain liquid _metals, _water-based

liquids _{and blood.} _Attention _{is concentrated} _entirely _{on the}

flowmetering device _itself _{and not on any associated} _electronics.

The idea of the contribution. of each element of fluid

moving with velocity _{v in a magnetic} field B to the total electrode

e. m. f. is first _{put on a sound mathematical} basis by the

introduction

_{of a weight vector}

^Qy

WB

where Vt is a vector entirely determined by the flowmeter and

electrode geometry. The contribution is W. v per unit volume.

The ultimate

aim of the flowmeter

designer

is to make the

meter sensitivity

(the electrode e. m. f. divided by the flowrate)

entirely

independent

of the flow pattern,

i. e. of v.

The

behaviour

_{of a flowmeter-4s}

_specified

_entirely

_{by W and the}

necessary condition

on W to achieve this

sensitivity

independence

(an ideal meter) is found to be curl W-0.

It is _{shown that} there _{is no magnetic} field _that _will _achieve

this if _point _electrodes _{are used.} The condition _{on W weakens}

progressively as more assumptions are made about the flow _pattern

and weight functions suitable for different types _{of velocity"}

profile are defined. The effects of different types _{of electrode}

are also discussed.

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This basic theory is then applied, _first _{to a class} _{of ideal}

uniform field flowmetere with transverse line electrodes. These

meters can, moreover, be used as valves and flowmeters

_{simultaneously.}

Their performance has been experimentally

checked.

Secondly the theory of long flowmeters

(in which all quantities

are assumed invarignt

in the flow direction)

is exhaustively

discussed.

Thirdly _{the practically} important _{case of circular} _meters _with

non-conducting _walls, _point _electrodes _{and short} magnetic fields is

examined. In view of the difficulty of manufacturing a magnetic

field that _{may be mathematically} desirable, this _problem is by-passed

by first _choosing _{a technologically} _simple _{method of produc3rig}

magnetic fields for these _meters using coils specified by certain

parameters _{and then analysing} the relevant _{weight-'function} in terms

of these parameters. Design tables for. -the case of rectilinear

axisymmetric flow _{are presented} (one. specimen), and experimental-

tests described _which _{check the performance} _{of designs} _produced

from these tables.

A different _{method of producing} _{the magnetic} _field'is _suggested

which saves power consumption _{and enables} the field to be more

accurately

designed,

but to which the tables mentioned above still

apply.

Finally _{some outstanding} _problems _{are discussed.}

ý .J

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PREFACE

Since _October _{1965 I have been working} _{as a research} _student

under Professor J. A. Shercliff at the University of Warwick.

During that _{time I have been on leave} from-the English Electric

Company's Mechanical Engineering Laboratories _{at Whetstone,} _near

Leicester _{and been given} _{a grant} by them. I wish to record _my

gratitude to the Company for supporting me over the last few years,

and for giving the University a grant which enabled me to attend

the 6th Symposium on Magnetobydrodynamics at Riga in September 1968.

My main field has been the theory _{of the electromagnetic} flow

measurement of liquids. The flowrate is measured by the voltage

induced between _{two electrodes} _{by motion in a magnetic} field

(Faraday's _{Law) under conditions}

where the magnetic field is not

significantly distorted by the currents induced by the motion, and

the voltage _{is effectively} _measured _{on open circuit.} _{Though much}

of the theory _{may be applied} to any flow. measurement situation I

have concentrated _{on flow} _{in pipes and the improvement} of modern

flowmeter design. _{At one time there were hopes of not only}

measuring the flowrate, but also analysing the velocity profiles.

However, _precisely those _{characteristics} _which _{make an electro-}

magnetic flowmeter good at measuring the"mean velocity (in some

sense) make it bad at picking up local velocities, and this part

of the project was dropped.

Six or nine months were spent. on a. different project, _which is

not reported in this thesis. This was the flow of electrolyte in

a long circular pipe under a pressure gradient and streamwise body

force _varying _{quadratically} _with _radius. The force _{is produced}

electromagnetically by imposing quadrupole current and magnetic

fields (lying in planes perpendicular to the pipe) the conditions

being _{such that} there is no interaction between the fluid _motion

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and imposed fields. The aim is to produce artificial _velocity

profiles in normal fluid mechanic pipe flow by the use of carefully

controlled non-uniform body forces, so that the turbulence in

abnormal shear flow profiles and the stability of the laminar

profiles _{may be examined.} A small amount of elementary _analysis'

of the laminar _profiles has been done, and the experimental

apparatus half-constructed. The problem of getting the necessary

non-uniform _current distribution inside a circular pipe has been

solved by using porous walls (impervious to the flow) and

electrodes _-lying _-outside _,them.

The°flowmetering*work is frilly _reported in'this thesis. The

material has been arranged as a general survey of the theory

followed by the experimental _results, _with three _reports _containing

details _{of the"theory'in} _{the appendix.} _{" This} _enables'the flowmeter

designer _{to get a-good idea of the scope of the mathematical} _work

without having'to"read it unless he particularly wishes to.

Most of the theoretical _results in this thesis _{are in}

analytical form, from which numerical results have been obtained

using a digital computer _{when necessary.} Though there are

situations _{when numerical} analysis _{may be preferable,} the geometry

of many flowmetering _problems is sufficiently _simple to be

amenable to algebraic _analysis.

During the first

_{year the analysis}

_{was restricted}

to long

flowmeters

_{and it was only then that,}

_{spurred on by the work of}

Ketelsen

(6),

_{we started}

tackling

_{the problem of short meters.}

The first _{and third} _reports _{in the appendix} _result from this

work on short meters and formed the basis of papers given in the

autumn of 1968, by Prof. J. A. Shercliff to the 12th International

Congress _{of Applied} Mechanics _{at Stanford} _{and myself} to the 6th

Symposium on Magnetohydrodynamics at the Latvian Academy of

Sciences _{at Riga.} The second report, though it _contains _{the work}

on long flowmeters done before the other two, was written up after

them in a shorter and neater form after the advantages _{of couching}

the theory in terms of the virtual current (defined _{on page 4 of}

the first _report) _{were realised.}

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The reader will notice that to some extent the work of other

people is discussed after the basic theory has been introduced.

The reason is that some of their work has been simplified to ease

analysis and some is inaccurate. Different notations have been

used and it is easier to discuss the work in a common notation and

with an awareness _{of what is being} ignored.

I should like to take this _opportunity _{of thanking} _xry

supervisor, Professor J. A. Shercliff, for his advice and criticism

throughout _{my work,} Dr. G. Rowlands for _convincing _{me that} in

practical situations separable solutions are often the only way of

solving Laplace's equation, Dr. C. J. N. Alty for reading this thesis

in draft, Dr. R. C. Baker, Dr. C. J. Mills, Dr. D. G. Wyatt _{and George}

Kent Ltd for discussions _{on flowmetering} _{and the requirements} _of

practical designs. Finally I should like to thank Mr. A. E. Webb

and Mr. A. C. Ross for their work in the construction of the apparatus

and Miss J. A. Green who had to read my writing when typing this thesis.

" _'r

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CONTENTS PAGE

i

Abstract

i

iii

Preface

vi Contents _-- vi

viii

Nomenclature

viii

Errata

1. Introduction 1

2. Outline Theory 4

2.1. The Basic Theorem 4

2.2. The Design Problem 5ý

2.3. Conditions

_{on the Weight. Vector W}

7

2.4. The Effect

_{of Large Electrodes}

9

2.5. The Effect _{of Various} Flow Patterns 12

3. The Work of Others 18

3.1. H. Kanai

18

3.2. _{Th. Rummel and B. Ketelsen}

21

3.3. V. I. Mezburd

23

3.4. L. M.

-Korsunskii 214,

4. A Short Summary of The Three Reports in the Appendix 26

(4.1., 4.2., 4.3. )

5. Experimental Work _- 29

5.1. Apparatus ₂₉

5.2. Artificial

_Velocity

Profiles

-

33 Table

.

1:

Velocity

Profiles

36 Table 2:

Canvas Pipe Behaviour

₃₇

5.3" Circular Flowmeters _with Point Electrodes ₃₈

Table 3:

Circular

Flowmeter Sensitivities

_43/4lß.

(Predicted _{and Experimental)}

5.4"

Valve/Flowmeter Experiments

45 5.5"

Virtual

Current Measurements

47

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6. _Conclusion

_{and Possible}

_Developments

49

7. References

₅₄

6. _Figures

If

Baker's

_solutions:

Notation

2. _{The Basic Types of Electrode}

3. _{Lines of Constant W for a Transverse Line Electrode}

4. (see figure)

5. Co-ordinate Axes for _{a Circular} Flowmeter

6. _Sketch _{of Flow Circuit} _{and Electronics}

7. Theoretical Effect _{of Canvas Pipe on Sensitivity}

8. _Velocity _Profiles _{used for} _Experiments

9. _{Sketch of "Uniform" Field Magnet, showing Field}

10.

11.

12.

13.

14.

15.

16. 17..

18.

9. Appendix

1.

2.

3. t

Non-uniformity

Photographs _{of Uniform} _Field _{Magnet and General}

Arrangement _{of Apparatus}

Sketches _{of Circular} _Flowmeter _with Point Electrodes

Photographs _{of Circular} _Flowmeters

Sketches _{of Valve/Flowmeter}

Photographs _{of Valve/Flowmeter}

Valve/Flowmeter _Sensitivity _Dependence _{on Stop-}

cock Position.

Virtual

_Current

_{along the Axis of a Circular}

Flowmeter

Alternative

_{Magnetic Field Designs}

Possible _Pole-pieces _for _Flowmoter _with _Uniform _W

consisting _{of three} _separate _reports

Contributions _{to the Theory} _{of Induced} _Voltage

Electromagnetic _Flowmeters.

The Theory of Long Induced Voltage Electromagnetic

Flowmeters.

Some Design _{Data for} _Induced _Voltage _{Electromagnetic}

Flowmeters _with _Non-uniform _Fields.

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NOMENCLATURE

a radius _{of flowmeter}

A _coefficient _{in Fourier} _Series

b _radius _{of iron} _sheath

B

_coefficient

in Fourier

Series,

_{magnetic field}

_component,

B

_{magnetic field}

_vector

Bo Uniform _{or reference} _magnetic field

D Flowmeter internal diameter

d, _{Canvas pipe diameter}

E(k) Complete Elliptic _Integral _{of the second kind,} _modulus k.

F Magnetic field _potential Ba VP

f(z) Magnetic field _potentiaL" _Bx

- i, By n f' (z) _ i= I1

G Virtual _current _potential, _current _VG

g(z) Virtual current _potentials 4

H Manometer _reading

144 _{Bessel Functions}

'lam

zero of J, (p) a0

[of

4-

01 K

Bessel Function,

_{Complete Elliptic}

_Integral

_{of the first}

_kind

L

length

_{of coil}

m, n

integers

p

pitch

of conducting

wall made of alternate

conducting

and

insulating

_strips

Q Flowrate

r

co-ordinate

radius

R

distance

_{from origin}

J$2

_{+ y2 + z2}

S

_sensitivity

t _canvas _pipe thickness, _potential (if it _exists) _{such that}

Vt=BAVG

T

Ratio of iron sheath diameter to flowmeter diameter

U

Electric

_potential

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AV

_Electric

_potential

_difference

V

Velocity

_vector

(V,

_{, Vy , V.}

)

v

Velocity

component, normally

in stream-wise direction

w

Conducting wall thickness

W

_{Weight vector}

B

,,

VG _ (Wx

, Wj ,

WZ )

WWZ

(the only surviving

_{component in long flowmeter}

theory)

W=r _{Wz dz, asymmetric} _rectilinear _profile _weight function

Jz=-

oD

zT

W2r, _axisymmetric _rectilinear _profile _weight function

jwde

[x, _{y, z]} _Cartesian _co-ordinates

z, z

x+ Ly, x-

iy in long flowmeter

theory

oý electrode (or pole piece) half angle, Fourier integral

exponent

Fourier integral _exponent

Y Angle between virtual _current _stream line _{and electrode} _axis

in a long flowmeter

Eý _Polar _co-ordinate

Z _Coil _half _angle

Iý Wall _conductivity

Canvas wall conductivity factor, _variably _conducting _wall

parameter

6 _Fluid

_conductivity

t dt

_{Volume, volume element}

Co-ordinate

in spherical

_polare

(RI9

_p4

)I potential

in plane'

geometries.

Stream function

_{in plane geometries}

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1. INTRODUCTION

The basic _equation for the induced _electric _potential U of a

fluid _{in motion in a magnetic} field, _given for example by Shercliff

(8a, _p12), _is

v 2U

- div

(v ý)

(2.1)

together _with _suitable boundary _conditions _{on any walls} present.

Subsidiary _conditions _available, if _needed, _{are div v=0}

(incompressible flow), div B-0 (always) _{and curl} B-0 (if the

distortion _{of the magnetic} field due to the induced currents is

negligible). Throughout this thesis the applied field B will be.

assumed steady or varying sinusoidally with time, and only the

potential U due to the motion will be considered. The out of

phase potentials due to electromagnetic induction must be otherwise

taken _{care of.}

Various _{methods of obtaining} _solutions _{of (2.1)} have been tried.

In (8a) Shercliff has used separation _{of variables} (e. g. for laminar

flow in an insulating _rectangular _pipe _{and uniform} _magnetic field)

image systems (in the case of the weight function for a circular

flowmeter _with _point _electrodes _{and a uniform} field), and plain

juggling _making _simple _assumptions _about the velocity _profile and

using the fact that the field is Laplacian (in the solutions for

wall velometers with the fluid half-space bounded by an insulating

plane).

Baker has obtained a very elegant set of solutions for _{such wall}

velometers with insulating walls (2a) and also for circular flowmeters

(both _internal _{and external)} (2b), initially by separation _{of variables.}

At first he assumed the velocity _{was in one direction} _only _and

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and that the magnetic field

lay in the plane perpendicular

to

the flow.

(He also considered an axisymmetric magnetic field

for the wall velometer).

The notation

is in figure

1. He

gives the solution

everywhere in the fluid,

but on the wall it

takes the form (U('-*) = B(oo) = 0)

00

half

_{space x>0:}

_U-2f

vz (x)

_{By (2x, y, z) dx}

(2.2)

0

circular

_pipe

:Uä

fvi(r)

BBC

)

rdr

(2.3)

(radius

_a)

with slight

changes in notation.

Disregard

the dependence of By

on z in (2.2)

for the moment.

Although these formulae take account

of ohmic losses,

the total

signal

is found by simply multiplying

the

velocity

at one station

by twice the transverse

component of magnetic

field

_{at a different}

but related

_station

_regardless

_{of the magnetic}

field _anywhere _else.

Because B is a plane solution

of Laplace's

equation,

the magnetic

field

(under certain assumptions about symmetry and singularities)

is

specified

everywhere by its

_{value along a line,}

_{so it is not possible}

to vary the field

_{anywhere without}

_{varying. it along the relevant}

lines.

This follows

_{from a theorem in the theory of complex variables}

_which

may also be used to prove (2.2) and (2.3)

(for the latter

see page 9,

report

2).

The factors

_{2 in (2.2) and (2.3) appear embarrassing,}

_especially

in the case when the moving fluid

_{is confined to a thin layer close}

to the wall and the magnetic field

is fairly

_uniform

there,

_{so that}

Bj (2x, y, z) zBy

(x, y, z).

_{How, one-may ask, do we get twice the}

expected potential?

The answer is that (2.2) applies in its present

form only when B-O

at oa ; half the signal

_{comes from the motion of}

the shear layer in Bý as expected and the other half from the

motion in Bx

for large y, i. e. from the distant

_{motion in the other}

transverse

field

_{component, which is implied}

if

_{the field}

_{goes to}

zero at infinity.

It can be shown that the signal in the (non-

Laplacian)

field

_{created by reversing}

_{the sign of By is zero.}

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Similar _statements _apply to (2.3). _{The potential} _using _the

(non-Laplacian) field (B. _{9 -Be} ) found by reversing _{the sign of BB}

for _{a Laplacian} field is also zero.

(2.2) _also _holds _{even when the magnetic} _field _{B is three}

dimensional. _This _{means that} for _{a wall} _velometer _with _{two point}

electrodes it is necessary only to arrange the difference in By

along the two perpendiculars to the wall through the electrodes to

get a specified response, the field everywhere else can be left to

look _after itself. (2.2) _also holds if the wall can be removed,

though _{the potential} is then halved _{and any motion} _{and field} in the

half _{space x<} _{0 must be taken account} _of. It is more likely that

the motion and field will be symmetrical, in which case (2.2) holds

without alteration.

The generalisation of (2.2), immediately leads one to ask if

(2.3) _{can be similarly} _generalised _because, _if _this _{were the case,}

the problem of designing short flowmeters with rectilinear flow with

an axisymmetric profile would have been solved. The answer is no,

and the reason must be connected with the two facts that., firstly,

circles in two dimensions and half planes, in two and three dimensions,

have simple image systems and

i secondly, the magnetic field is Laplacian.

Circular _pipes in three dimensions _{do not-have} _simple image systems.

It _{is still} _possible _{to attempt} _{the solution} _{of the flowmeter} _equation

(2.1) _{by separating} _{the variables} _{(see page 7 of report} 3) though the

terms _{no longer} _{add up neatly} _{to give} the transverse _component _{of the}

magnetic field at an easily recognisable position.

There are, however, two difficulties _using _separable _solutions.

The first is the need to take account _{of variation} _{of velocity} in two

dimensions

(in the case of a pipe arbitrary

_variation

_{over the pipe}

cross-section)

and the second is the mixed boundary value conditions

relevant

when the electrodes

are not small.

In this thesis _another _approach is adopted, _which is an extension

of the idea of the weight function originally given by Shercliff (e. g.

8a, p. 29) (which has, incidentally, been abused by being applied outside

the context of long flowmeters). The idea indicates the use of Green's

functions. (The term Green's function will henceforth be used generically

implying that the function is suited to the boundary _conditions).

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I

2. OUTLINE THEORY

2.1. The Basic Theorem

Analysis, _given _{on pages 3 and 4 of the first} _report, _{shows that}

the signal öf a flowmeter is

Jx.

W cti

(2-4)

where the volume integral is taken over the whole of the fluid space

of the flowmeter and W is a weight vector, given by

wB

AV 9

(2.5)

Qg is a vector that does not depend on B but only on the shape of

the electrodes _{and the flowmeter} (and the electrical boundary conditions

on the walls). Here it has been given the name "virtual current" since

if _unit _current _{is passed between the electrodes} _{by placing} _{a suitable}

voltage across them, the- resulting current distribution is QýC _.5 is

.

the potential associated _with the virtual _current. This theorem holds

even if the conductivity of the liquid is non-uniform, though the

distribution _of 0ý is of course altered. ' The result _{may also} be

derived by modelling the fluid by a resistance network and using the

reciprocity theorem in the form used by electrical engineers. The

current is called 'virtual' because although there will'in general be

currents circulating (which must of course depend on v_) the virtual

current is fictitious and only a means of thinking about the problem.

In this thesis _{we deal} _{only with} _{cases in which} the magnetic

field is unperturbed by currents induced by the motion thus excluding

high _conductivity liquid _metals. With this assumption BaQF _where

F, as well as 5, in the case of fluid with uniform conductivity, is

a solution of Laplace's equation. -

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It _{can now immediately} be seen why certain _well-known _types _of

flowmeter have desirable _{characteristics.} Consider, _for _example, _a

rectangular flowmeter with large electrodes of high conductivity

covering opposite walls and long in the flow direction. The virtual

current is obviously uniform, even if the other _walls, _{or the fluid,}

have conductivities _varying _with the distance between _{the electrodes,}

though _{not if} _{they vary in the other direction.} _{In a uniform} _magnetic

field _{the weight} _vector _would _{have only one component,} _which _{would be}

constant _{and in the flow} direction.

Axial _current _flowmeters (Shercliff 8b) are another _example.

Suppose the liquid _{passes-through} _{a highly} _conducting _circular _pipe

which forms one of the electrodes, the other being ,a wire placed on

the centre-line of the pipe.

If the flowmeter is long, it _{is clear} that _{the virtual} _current

is radial, OG,

r, in strength, and does not depend on the, conductivity

of the fluid or on contact resistance provided they. are axisymmetric.

It _{is clear} that _{an azimuthal.} _magnetic _field _{pc w will} _{make W uniform}

and in the flow direction. Such a field is provided-by a uniform

current flow along the fluid.

2.2. The Design Problem

We now turn to the basic _problem _which _{is to make a flowmeter}

whose signal

depends only on the flowrate

and is independant

of the

shape of the flow pattern.

From here on such flowmeters are called

"ideal",

though the term is sometimes used in connection with a flow

pattern about which some restrictive assumption has been made.

The first

_{stage of the problem is to work out. a suitable}

condition

on W to ensure that

a flowmeter

is ideal,

or range of

conditions

depending on assumed forms of flow-patterns,

since only W

itself

_{appears explicitly}

in equation (2.1).

The second stage is to choose the magnetic field

_{and virtual}

current so that the condition selected is satisfied; if this is possible.

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The third _stage _{is to produce} _{the required} _magnetic _field _and

virtual current. This is where difficulties can arise. Suppose we

have written down mathematical expressions for the field _{and virtual}

current potentials (for F and G) satisfying Laplace's _equation _{and each}

containing _{a set of unknown coefficients} _which _{we are at liberty} to

choose.

The use of a suitable (as yet-undiscussed) condition on W

would result in relations between the sets of coefficients, that might

either specify the sets completely, or else allow some element of

choice. Put another _{way, the condition} on W would either result in

the specification of the magnetic field and virtual current, or in the

specification of a range of matching fields and currents.

The problem would then arise of producing them practically.

Though any magnetic field _{can be produced} _{by putting} _suitable _coils

around the flowmeter (and surrounding them with high permeability

iron if _{so desired)} there _{is no guarantee} _that _{the coils} _{would be}

easy to make. In general, it _{would also} be too difficult to

manufacture the electrical boundary _conditions required on the wall

/f

. of the flowmeter to realise the stipulated virtual current.

For these reasons we have reversed. the procedure.

The virtual

current

has been analysed first

for electrode

_arrangements

that are

easy to realise

in practice

(and which normally have undesirable

effects

_{on the weight vector W), and secondly for arrangements that}

are-possible

in practice

_{and which have less undesirable}

effects.

In the second report, _{on long} flowmeters, _{where attention} has

been concentrated on what is possible rather than what is practicable,

the magnetic field has been chosen after considering what field is best

suited to a particular virtual current. The method requires the choice

of a mathematical function and the resulting field is not always easy

to reproduce.

On the other hand in the third report, which is an analysis _of

the case of a circulär flowmeter _with diametrically _opposed _electrodes,

assuming rectilinear flow, with special emphasis bn flow with axi-

symmetric velocity profiles, a range of magnetic fields which are easy

to produce are analysed in terms of certain parameters describing the

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1,

range. The behaviour of the relevant weight function resulting

from these fields _{is set out in tables.} The design _{is then} _found

by selecting a suitable number of these basic fields _{so that} the

weight function has the correct behaviour. Normally two, sometimes

three,

_{of these fields}

_{are necessary.}

The main advantage of this method is that while we do not know

(and do not care) what the magnetic field is, _{we know how to make it.}

The corresponding disadvantage is that it _{must be made in the}

prescribed _manner, _{or by using} some other technique that satisfies

the same magnetic boundary _conditions. If these _conditions _are

different _{the analysis} _{does not apply} _at'all.

2.3. Conditions

_{on the Weight Vector W}

The minimum assumption we may'. make about the flow is that it

is incompressible (div _v= 0) for, if it _{is not,} the volumetric flow

rate is different at various cross-sections and is therefore not

uniquely defined. On page 5 of the first _report the necessary and

sufficient condition on W for an ideal _meter (assuming W-, O far from

the electrodes) _{is shown to be curl W=0.} _{On page 12 of the same}

report it is also shown that _whenever _point _electrodes _{are used this}

condition cannot be achieved. Since this is so, and since most

practical flowmeters use point _electrodes, further _conditions must be

imposed on the velocity so that the flowmeter becomes ideal. The

most useful

necessary and sufficient

conditions

are summarised in the

following

table.

Assumption _{on v} _Condition _{on W}

div vm0

_{curl W a-0}

(1)

Z'=

CO, O, V (x, y, z)J

WZa

function

of z

(2)

Lo, o, V(x, y)j

_{w(x, y) .-}

jWdz

constant

(3)

va

ý0,0,

V(r),

2rr

Wi(r)

- 2i

WdO

constant

(4)

(20)

A condition intermediate between (1) and (2) is possible _if, _for

example, v is confined to a plane.

Condition

_{1 is the most desirable;}

_{as mentioned above it}

cannot be achieved _using _point _electrodes but it _{can be satisfied}

using a uniform _magnetic field _{and line} _electrodes in specially

shaped, flowme ter channels (p. 7, first _report). Whether, it _{can be}

achieved in other _{circumstances} is still _{an open question,} though it

is unlikely.

Condition 4 (p"9, _report _{1),. normally} _only _required for

circular flowmeters, _{makes the maximum assumption} _about the flow

(apart _{from slug flow):} _that _it _{is rectilinear} _{and axisymmetric.}

This _assumption _{has been made or implied} _inmost _flowmeter designs

until recently. Condition 4 may be achieved in many ways: for

example with a uniform field _{and any electrode} _system (p. ý, reports )

or with any matching electrode arrangement _{and magnetic} field.

Even for _{one particular} _arrangement, _such _{as point} _electrodes, there'

are many suitable magnetic fields. The justification for this

remark is not merely some existence theorem, (see the remarks on p. 8

of report 3), but also that report 3 describes _{a set of design} tables

which do demonstrate _{many ways of achieving} designs _{of constant} i

The possibility

_{of achieving}

_condition

_{3, for an asymmetric}

rectilinear

velocity

_profile

is not so well established.

Pages 13

and 14 of report

1 suggest that this is possible in principle

_with

point

electIodes

but that in practice

difficulties

_will

be experienced

in-a region whose distance

_{from the electrode}

_{is small compared with the}

flowmeter

_{wall thickness.}

_{The analysis}

_{there is not entirely}

satisfactory

and the question

is discussed

further

on page 13 of this

thesis.

The term matching is used in the sense given in the middle of p. 8

of report 3P which only. suggests the existence _{of matching} _solutions

but does not show how to find them. _{This would involve} _mathematical

inversion _{of equation} _{16 (p. 7 same report)} _which _{may be very difficult}

(as in the classical _problem of moments) _{and would still} leave the

designer _with _{a mathematical} _{specification} _{of the magnetic} _field _once

he had selected the electrode arrangement.

(21)

It is possible

_{to produce a set of tables for this case similar}

to (but much bulkier

than) the design tables

_{for the axisymmetric}

_case.

When report

1 was written

it was thought such tables would be too

large for practical

use, but with drastic

pruning

their

_{size can be}

reduced and tables

for this

case are being prepared.

Conditions (3) and (4) also apply in the theory of long flow-

metersy _{when they} take _simplified forms. In such flowmeters the

magnetic field _{and virtual} _current _{are assumed to be perpendicular} to

and invariant _along the main stream direction, and the weight vector

W has only one component W which is also in that direction. Under

certain conditions (p. 3 report 2) other flowmeters may be analysed as

if _{they were long.} Report 2 deals _almost _en-irely _with long flow-

meters and the mathematical situation-is _clear. To achieve condition

(4) either the magnetic field _{or the virtual} _current _{must be uniform}

and the other may then be arbitrary: to achieve (3) both must be

uniform. The rest of the report is concerned _with _analysing

situations that either approximate to these conditions, or can be

easily realised in practice.

The velocity distribution _{assumed in-condition} _{(2) will} _never

occur in practice, _since the assumption _{of rectilinear} flow and the

continuity _condition _requires _{a 0.} Condition (2). hasp however,

been used as the basis _{of some designs.} Its _shortcoming is that it

is either _{too weak or unnecessarily} _strong. If the flow is recti-

linear then condition (3) _{is all} _that _{is necessary:} _if _{the flow} _is

not rectilinear then condition (2) is not sufficient _since in general

there _will _also be velocity _components transverse _{to the streamwise}

direction _which _will _contribute _{to the output} _voltage; _{we must have}

condition (1) or some variant of it.

2.4. The Effect

_{of Large Electrodes.}

For the purposes of discussion,

the possible

electrode

_systems

can be listed,

in order of increasing

manufacturing

difficulty,

_as

follows:

(22)

a)

Point electrodes

(in practice

_{small circular,}

discs)

b)

_{Line electrodes,}

transverse

_{to the flow}

_{p. 10 report}

₁

p. 12 report

2)

c)

Long electrodes

(report

2), in particular

long quadrant

electrodes.

d) Anything _{more complicated,} _{such as multiple} _electrode

systems, or some of the semes suggested in report 2,

using _walls of variable conductivity.

To get some idea of the strength of the singularity at the edge

of the electrodes, consider the virtual current potential function

when the first three electrode systems are, set in an otherwise non-

conducting plane (or in free space, since the solutions are symmetric),

assuming that the far end of a line electrode _{and far} edge of a long

electrode are sufficiently distant, to be considered at infinity (see

figure 2 for _sketch _{and notation).}

In order to simplify _{the comparison} _{assume that} each of these

electrodes is situated in a uniform transverse _magnetic field _with

the flow _rectilinear _{and invariant} _{in the flow} _direction. _{Then the}

integrated _{form of the virtual} _current _{is relevant,} _{i. e.} ý vdt,

since these are essentially long flowmeter _situations. b

For a point electrode 5ocý _{and for} _{a semi-infinite} line

electrode ýW f _{or log(R} _{+ _),} _{a much weaker singularity.} The

integrated _virtual _current, _represented _using _{the complex variable}

W-

z' =x+

iy as

_-La

d.

_{, has the formz7,}

fora

point

electrode,

log z' for a transverse

line

_electrode

_and

_{for a large electrode}

(which,

_{being already invariant}

_{in the flow direction,}

_{needs no}

integration).

The weakest of these singularities is log _z'. _and

in a uniform field the weight function _{is uniform} _over _{the electrodes}

and always finite.

The other component

_,

Qý

_{= log r',}

_{which must be taken into}

OAG

account if the field is inclined to the electrode; though it tends

to infinity _{at the electrode} _{edge it} does so very weakly. _The

weakness of this logarithmic singularity is the reason why the long

flowmeter _performance _{of the curved} line _electrodes _studied _{on page 12}

(23)

of report 2 is so good. In figure 3 the lines _{of constant} _for

the integrated _virtual _current for _{a transverse} _line _electrode

with two

ends at xs +a are shown. isa log _{and the lines}

are a set of coaxial circles through the electrode _ends, the value of

on a circle

being the angle subtended there by the electrode.

1-

ä

is always finite _{and is uniform} _over _{the electrode.} _{The effect}

of the slight curvature if the electrode is small _{and set in a curved}

wall is remarkably small.

The point electrode has the strongest singularity, for which

3 which is o91 on the perpendicular to the plane through

Ir _Ij

the electrode _{and zero on the plane} itself. The lines _{of constant}

are the limit of figure 3 as the electrode length tends to zero

anä are to be seen in Shercliff's original _weight function (see report

2 figure 2a).

For the large _electrode _{on the electrode} _{and is}

T

zero on the wall. The singularity for this _{case is worse than for}

the transverse line _electrodes. _{In practice} _{the use of long electrodes}

not only lengthens the flowmeter, but also results in reduction _{of output}

signal unless the electrodes _{are sufficiently} _short for the long flow-

meter assumption to be invalid. For these _reasons long large _electrodes

should not be regarded _{as practical} _competitors _{to point} _{and transverse}

line _electrodes.

Point _electrodes, _although _{they have a potentially} _{bad performance,}

are very simple to make. In practice _{of course} they _{are not points} but

circular discs in a virtually flat _plane, _{a well-known} _problem in

potential theory. The virtual _current _{on the disc} o( (r

being _{the radius} from the centre _{of the d? sc, whose edge is at r} ₁₎

with the typical singularity at the edge. On a hemisphere of

radius one disc diameter, the virtual current is within +20% of the

value for an infinitesimal point electrode (it must average out to the

same value since the flux is the same). The integrated virtual

current is within +10% on a similar circle: in fact the integrated

current distribution is the same as for a line electrode one diameter

long. This _remarkable fact is true for _{any elliptically} _shaped

electrode, the equivalent line electrode having the same length _{as the}

axis of the electrode perpendicular to the flow. This _{can easily} be

(24)

seen in the case of the circle

since

(ýý),

_o

f()t

[where

_ý,

yo

for r<1,

_{a0 for r>}

11 is independent

of x, and similar

r

formulae apply for an elliptical

electrode.

Marked curvature

_{of the}

disc would affect

the virtual

current

distribution

both before and after

integration, but _{not affect} the essential logarithmic _nature _{of the}

integrated

_virtual

_current.

Slight

_curvature

_{can be presumed to have}

little

_effect.

In general we may be confident that the point electrode

approximation will be good far from the electrode, compared with its

size. Only within, says one electrode diameter will the departure from

this _{approximation} be marked, and this _will _probably ease the problem of

finding _{a suitable} _magnetic field.

2.5. The Effect _{of Various} Flow Patterns

In this _section _{we discuss,} _{in rather} _general terms, _whether it

is possible to design the magnetic field distribution _{so that} _using

either point or transverse line electrodes the flowmeter sensitivity

is independent _{of specified} _classes _{of flow} _pattern, _namely

a) _axisymmetric _rectilinear flow

b) _asymmetric _rectilinear _flow

c) flow pattern unspecified.

. The conditions on the weight vector W have already been discussed

on page']. On the simplest view the problem is to make the magnetic

field (B _-7 F) small where the virtual _current Q5 is large so that

their _product is constant. It is not in fact as easy as this since

the magnetic field

and virtual

current

are-vectors

with,

in general,

three components;

the product W is a vector product and the relevant

weight functions

are integrals

so that in some cases locally

infinite

values of"a component of W can be tolerated

provided

the integral

is

not infinite.

(25)

Axisymmetric

Rectilinear

Flow

We consider here the case (a) of circular

flowmeters

_with'asi-

symmetric profiles and point electrodes, or equivalent schemes in

other geometries when the profile depends only on one variable (e. g.

flows _past _{a wall} _{where the velocity} depends only on distance from _-

the wall). With a uniform transverse _{magnetic-fieldl} _which is

obviously non-zero at the electrode, the aaisymmetric weight function

is well-known to be uniform, so one would not expect infinite values

of i because of the singularity at the electrodes using a non-uniform

magnetic field. This is clearly shown in report 3 where the effect of

shortening the magnetic field is merely to add correction terms to the

value of W for the uniform field case.

Asymmetric Rectilinear Flow

Making the asymmetric _weight function W _uniform is not nearly _so

easy, using point electrodes. Report 2 shows it is impossible using

a long non-uniform magnetic field. The question is then whether the

extra freedom gained by allowing the magnetic field to vary three

dimensionally _will _{make it} _possible. _{In fact} _{the magnetic} field

cannot be adjusted arbitrarily _{at every} _positions it is a Laplacian

field _{and once it,} _{or its} _potential _{F, is} _specified _{on the surface} _of

the flowmeter, it _{is specified} _everywhere _inside. This fact is the

basis _{of report} _{3 and also the reason why attempts} _{to satisfy} _condition

(2) on page 7 exactly are likely to be unsuccessful as there is not

enough%freedom in B to be able to specify it over a volume. In long

flowmeter theory _specifying the magnetic field _along a radius (i. e.

one dimensionally)

(see Baker (2b) or report 2, p. 9) completely

specifies

it

and the axisymmetric

weight function

V;

we may plausibly

expect the situation

to extend to short flowmeters

and to be able to

arrange the distribution

of the field,

or its

potential

F, over the

surface

to produce a specified

distribution

of the asymmetric weight

function

W. For point electrodes

it

now becomes necessary to make

the magnetic field

zero at the electrodes.

The appendix of report

1 models the conditions

close to a point

electrode

by assuming it lies

on an infinite

flat

plane so that,

in the notation

of report

1, figure

5d

(26)

the potential

due to a point source, so that

w'-

R3

4z

(2.6)

There is only one non-singular magnetic field with the right order _of

smallness at the electrode to make i finite and the necessary _symmetry,

(p. 13 report 1), _{and it} happens to be two dimensional, _causing _{the same}

behaviour _{at the electrodes} _{as shown in report} 2 figure 2a.

If _{we allow} fields _with _{singularities} _{at the electrodes,} the

possibilities are not so obvious. It is clear from (2.6) however

that if is to be independent _of äF _{must be as well} ( is

itself _{a solution} _{of Laplace's} _equation). This _{case was also} _analysed

on p. 13 of report 1 by considering the case Fs 4R2P (cos which

would be the first term in the expansion of such a magnetic field that

was zero at the electrode. It was found-that W was an integral

diverging logarithmically _{as; - -- °°.} This difficulty has been glossed

over in the appendix since the expansion is not valid for large zs it

suggests however that Wv log r for small r (and not constant as stated

in the appendix).

In order to clarify _{the situation} _suppose is the same as

the potential _{due to a ring of radius} _{c and charge} Q' _{whose axis} is

the z-axis, _{and whose centre} _{is the origin} in figure 4. This is an

axisymmetric solution _{of Laplace's} _equation that _{goes to zero at oV ;}

on the z-axis it is _{and may be expanded around} the origin

(for _R< _{c) as} ý1ý

CQc rý_

R2 PýiGuý6)

3 R4 Pif°(cos0) f-...., 1

(2.7)

cL

zc

ýr ' E'+

which may be written

as

r' 2z'' r34

(z4L 3z

g r`ß) +. ý,.

(2.8)

1164 _'G

L

'4 G'' $G

The resulting W (found from (2.6) _and (2.8) by expanding _{the elliptic}

integral) is:

(%0 [1+ (i-In)+

oC jý

_(2.9)

2T1r,

(27)

We may superimpose these fields

for different

_{values of Q_ and c.}

We find

IF

_ýQc

_t

(If 105-,

_{r) 7 6", *tG}

G _G3

(2.10)

We see clearly now that it is not sufficient to suppress the first

term in (2.10) _{by choosing} ? ýG _{a 01 we must also} _choose OC _m0

to suppress the logarithmic _singularitys _{in general} 0: _{111- Ilk 0 so}

that _{W would have the desired} _uniformity _near _{the electrode.} This

behaviour is due to the field far _{away from the electrode} _since _the

order of smallness _{of the marjetic} field _{near the electrode} _{now more}

than cancels out the singularity of 175 V.

Put another _{way, the magnetic} field _component _{on the axis}

is

Taking _=Z

)if we ignore the misrepresentation at the origin,

the resulting _weight function is

.

ea

Ctz

'`ý1i

. 00Z3

and it is seen by expanding

_{as a power series}

for small Z/c

that

CO

and Z

S4

must be zero to make the integral

finite.

Unfortunately _it _{is not possible} _{to suppress} _{the magnetic} _field

at the electrode and make the distant field responsible for W using non-

singular magnetic fields since on the z-axis for these.

The implications _{of this} _piece _{of analysis} (which _{is a rather}

inferior _existence theorem) _{are that} the magnetic field _coils _shown

in report 1 figure 6a will _produce _{an asymmetric} _weight function _{W eG}

log r at a point electrode. The coils necessary to make W uniform

2w

would, from (2.7), be the lines of constant P 6nO), {rn>rz, not

and would have at least five, not three, branches

either side of the z-axis. The conclusions about the possibility

of achieving uniform W with point electrodes are unaltered; it is

(28)

not possible near the electrodes unless the coils _{are so close} that

they approximately produce a magnetic field _with _{a singularity} _along

the z-axis through _{the electrode.} While the suggestion, _{made on page}

13/14 of the first report that W be made 1 on the wall and -1 on the

radius is possible, I am inclined now to think it a bad one since if

we have to assume that the velocity profile depends only on distance

from the wall in the electrode region there are many other possibilities.

1

In practice the departure of S from the formula R assumed in

order to 'examine the singularities of W will have an effect but only

far _{from the electrode.} So will _small _circular _electrodes, _{at which}

the magnetic field-should be small but finite, since under these

conditions the integrated version of the virtual current is relevant

and for a circular electrode we have already seen this is large, but

finite.

There _{is always the possibility,} too, that _{even with} _coils _and

iron _sheath _very _close _{to the electrodes,} _unwanted _magnetic fields

could occur, since we cannot control the normal component locally:

it is controlled by conditions _elsewhere. _{The only harmful} field

(using _{the relevant} _symmetry) _{is the non-singular} _{one we have mentioned}