Electromagnetic flowmeters for fast reactors

Progress in Nuclear Energy, Vol. 1, pp. 41 to 61.

Pergamon Press 1977.

Printed in Great Britain

ELECTROMAGNETIC FLOWMETERS FOR FAST REACTORS Roo~

C. B ~

Mechanical Engineering Department, Imperial College of Science and Technology, London, U.K. (Received 30 November 1976) Almtract--This paper is in four parts. Section I explains the theory of the induced-voltage electromagnetic flowrneter and then considers various types which have been used. For the primary circuit of fast reactors both flow-through type and probe type have been proposed, although obtaining magnets which operate satisfactorily at high temperatures has been a problem. In the secondary circuit the high magnetic Reynolds numbers cause the field to be swept out of the magnet gap and this has led to the use of the long saddle-coil fiowmeter. In Section 2 flux-distortion flowmeters are described. These have been proposed mainly for monitoring the primary circuit flow and again both flow-through and probe types have been tested. Sections 3 and 4 continue the discussion of the flux-distortion fiowmeter by introducing two methods of analysing its performance. The first is a finite difference method which solves the non-linear problem by using a time marching method. It is shown that a linear approximation is adequate for the likely levels of flow encountered in the fast reactor and consequently two linearised solutions are used. The first method is a finite difference one and allows the instantaneous response of a step change in velocity to be observed as well as the effect of bubbles. In Part 4 the second linearized method uses current rings to divide up the conducting material. By considering the interaction of all the rings, it is possible to obtain the current distribution and hence the magnetic field. In conclusion it is suggested that further development would be useful of the devices which are most suited to the liquid metal fast breeder reactor.

A a

B b C D d E E F

I i Jo

J

j~ K k k' ??! tip Ms

R R= r

S t

U AUz~

NOMENCLATURE magnetic vector potential typical length scale tube internal diameter (ID) probe radius coil radius ring radius magnetic flux density tube wall outside diameter (OD) coefficient in eq. (39) ff coefficient in eq. (48) ff diameter of tube electric field elliptic integral coefficient in eq. (56) defined by eq. (28) electric current currents in core model primary coil current current density virtual current density elliptic integral as defined in eq. (35) as defined in eq. (38) as defined in eq. (38) prin~ry coil turns per metre secondary coil turns per metre right hand side of eq. (44) ff magnetic Reynolds number radial coordinate sensitivity coefficient in eq. (53) ff time coordinate electric potential potential differences between electrodes

V ¥ ~m

W W Z

0 u

p o

go

voltages generated in secondary coils velocity of fluid mean velocity weight function weight vector axial coordinate magnetic field function cylindrical coordinate pipe wall conductivity permeability o variable of integration fluid conductivity variable of integration for flowmeter volume contact resistance per unit area magnetic field scalar potential function excitation angular frequency INTRODUCTION

The development of the fast reactor with sodium cooling circuits gave incentive in several countries to the development of new instruments for monitoring the sodium. The problems of designing such instruments are severe: high temperature; radiation; restricted access; hydraulic shock; reliability. However the use of sodium presented the possibility of applying electromagnetic devices. (Duncombe and Thomasson, 1970; Evans et al., 1966). In this paper we are interested in the application of electromagnetic flowmeters to the primary and secondary circuits. The primary circuit items are submerged in a sodium pool at about 430°C. The coolant flow to 41

42

Magnetic ~Electrodes

ROGER C. BAKER

the core and breeder is provided by a number of flow loops through the core ducts. Temperatures as high as 700°C may be experienced and must be withstood. In the British Prototype Fast Reactor (PFR), flux distortion (eddy-current) flow monitors were used to obtain pump flow by insertion into a containment pocket leading, round various curves, 40 ft down into the reactor and they had to perform at 450°C. They were also used to monitor core flows in temperatures for normal operation in the range 350--475°C but they were also able to operate up to 700°C (Dean et al., 1970). The problem of monitoring the primary flow behaviour in future liquid-metal fast breeder reactor (LMFBR) designs is more complex still and the final methods for doing this are still under consideration. One obvious candidate for this is the flux-distortion flowmeter but necessary instrument redundancy for safety reasons would require large numbers of these instruments. The difficulties of accurate measurement of flow in the primary circuit make accurate measurement of prime importance for heat balance purposes in the more accessible secondary circuit (Duncombe and Thomasson, 1970). The design which has been favoured for the PFR is the saddle-coil flowrneter, an electromagnetic induced-voltage flowmeter with a field long in the axial direction. This is satisfactory in performance, but its long length is rather restrictive for siting in a secondary circuit and there appears to be a need for more accurate short flowmeters for incorporation in future fast reactors. In this paper the main divisions are made by flowmeter type and the features which are peculiar to primary or secondary flow circuits have been indicated. In Section 1 the induced-voltage flowmeter is described in theory and then the various types which have been developed are discussed. In Section 2 the same pattern is followed for fluxdistortion flowmeters. In Sections 3 and 4 analyses are presented in some detail of the behaviour of the eddy-current flowmeter, one type of flux-distortion flowmeter. 1. INDUCED-VOLTAGE ELECTROMAGNETIC FLOWMETERS Introduction In his book, Shercliff (1962) gives the basis of operation of these flowmeters which depend on the measurement of motion-induced voltage resulting from the flow of fluid through a magnetic field (Fig. I). The theory may be understood most simply if the

Fig. I. Induced voltage electromagnetic flowmeter. equations are set down for low conductivity fluids. The equations leading to the electromagnetic flowmeter equation are: Ohm's Law

j = ~(E + v × B); V x E=

I

Field equations

- B;

(1) (2)

V x B ffi#uj;

(3)

V . B -- 0.

(4)

where j is current density, o is conductivity, E is electric field, v is fluid velocity, B is magnetic flux density and ~ is permeability. For a steady magnetic field, I~ will be zero, and equation (2) will lead to E =

- VU

(5)

where U is the electricpotential. If wc assume that the currents in the fluid are small enough so that their effect on the magnetic field m a y be neglected we can obtain the magnetic fieldwithin the fluidby solving . V2,/, = 0

(6)

B -- V~b.

(7)

where This condition is known as one of very low magnetic Reynold's number, R., where (if a is a typical length scale)

Rm= / ~ v , . a .

(8)

a dimensionless number which compares the magnetic field resulting from motion induced currents with the applied magnetic field for a mean velocity v=. If we n o w take the divergence of equation (I) wc obtain the flowmeter equation by substituting for E from equation (5) V2U = B.V x v

(9)

where equation (7) has been used to simplify the right hand side.

Electromagnetic flowmeters for fast reactors One simple solution of this equation results from introducing an axisymmetric velocity profile v(r). Equation (9) becomes V~U

=

-

B0 dv ( r ) dr

43

) *_.£._, °.":':=,o

(10)

For a non-conducting pipe wall the boundary condition is

~u c~'-'r-= -- v (r)Bs

(11)

and it is shown by Baker (1968) that a solution of this is U = - ½S[{v(x/pr) + v ( a , . / p / r ) } Bs(p,O)d p (12) for a very long field (Fig. 2 gives the geometry). If

• Non

/X

Magnet(c Flux

1 I__I A

Fig. 3. Shercliff weight function for uniform field point electrode flowmeter. function to a weight vector W where the output signal of the flowmeter was given by AUEE -- Sv.WdT

(I 5)

and where the integration is over the whole volume of the flowmeter. He showed (Bevir, 1970) that the weight vector is given by

W = B ×L

(16)

where j,, is the current density which would pertain in the fluid if unit current entered by one electrode and left by the other. It is known as the virtual current. Bevir showed that for an ideal flowrneter the curl of the weight vector must be zero and that as a result no point electrode circular pipe flowmeter Fig. 2. Long ttowmeter geometry. with insulating walls is ideal in the sense of providing a signal which is always proportional to the mean Be (r, 1)/2) is uniform the potential difference between flow. One important design of flowmeter which is ideal and which is frequently used for liquid metal two electrodes separated by diameter d is flow measurement is the rectangular section flowAUE~ = dB)'= (13) meter with large electrodes and uniform magnetic field. This flowmeter measures the flow correctly and a flowmeter sensitivity is defined as whatever the velocity profile. Shercliff (1962) also discussed the effect of a con5" = A U ~ (14) ducting tube on the flowmeter signal. This has the dBv,, " effect of shorting out some of the signal and the The deceptively simple results of equation (13), sensitivity is reduced for a uniform field flowmeter which seems to imply that the flowmeter is ideal in to the sense of measuring mean velocity directly, has 2a' led to a mistaken view of theflowmeter's performance S -- (a 2 + b2 ) + (k'/o')(1 + uz/a)(b 2 - a 2) (17) and to balance this Shercliff0954, 1962) derived his weight function plot to show the extent to which a where ,~ is wall conductivity, , is fluid conductivity, flowmeter with uniform field is prone to error due T is the contact resistance Ixtween fluid and wall, to non-axisymmetric profiles(Fig. 3). This plot gives 2a is the ID of tube, and 2b is the OD of the tube a weight function by which any velocity in the tube wall. must IX multiplied. Thus flow near electrodes will This effect is important in sodium flowmeters be overweighted and away from electrodes, under- where the wall is usually of conducting material and valued. Bevir extended this concept of a weighting is not insulated from the fluid (see Fig. 4). Equation

ROGER C. BAKER

Unatfogrnet.i I I ~M

characteristic to droop at high velocities and the linearity of the flowmeter is lost.

[ J Flux

Core flow-through type

~ ' \

i

~'~7"~

reslsta nce

Fig. 4. Flowrneters with conducting pipe. 17 is valid when the voltage is measured between points on the outside of the tube wall, a common design for this situation where the fluid has a conductivity high compared with the wall. In aqueous liquids an insulating liner on the wall is necessary to avoid signal shorting. Early experiments by Elrod and Fowse (1952) which allowed for wall shorting and took some care to avoid contact resistances generally confirmed this theory. However they could not account for a 4 ~o low signal. Pfister and Dunham (1957) observed the effects of thermoelectric emf's which caused a small error. They also discussed the problem of field sweeping. This problem in liquid metal applications is a result of the high currents generated and the consequent distortion of the applied magnetic field. This effect leads to a non-linear signal response. The extent of the non-linearity is related to the magnetic Reynolds number. If this is very much less than unity then the applied field will not be disturbed, but if it is of order unity or greater than unity we can no more assume that the applied field is unchanged. The effect of these high values of Rm is often referred to as field sweeping. The magnetic field is apparently swept in the flow direction. Since the electrodes are normally situated at a position of maximum strength the effect of field sweeping is to cause the signal

For flow monitoring within the core of a fast reactor the values of R, are generally low enough to allow field sweeping to be neglected. It has been possible to design heads which may be inserted into a duct to form either a modification of the conventional flow-through meter, or a probe type meter situated centrally in which flow passes around it in an annulus. A device of the flow-through type is described by Popper and Glass (1967) and the calibration and testing are discussed in Glass and Popper (1968). A typical form of this flowmeter is shown in Fig. 5. It is basically the well-known electromagnetic flowmeter using permanent magnets, but designed for insertion in a pipe. The magnetic field is produced by permanent magnets which are positioned behind the electrodes while the pole-pieces of soft iron span the gap between these magnets. In order to fit the small space, the pole-pieces and magnets may be curved (the latter in the direction of magnetization) and the electrodes make contact with the conducting wall of the tube. As indicated by Popper et al. (1967) this flowmeter was at the time favoured for the in-core application. The advantages were those common to the electromagnetic flowmeter. Despite its attractions it did have clear disadvantages for this application. All these could be traced back in various forms to uncertainty in the calibration. The change in temperature could affect dimensions, resistivity of flow tube and liquid (wetting is also important) and magnet properties. Of these the most important was the uncertainty of obtaining a magnet with stability up to 1200°F (Forster, 1973). A further problem with permanent magnet flowmeters is strong neighbouring fields which can alter the field strength in the pole gap.

MAGNETS~-ELEC TRODE LEADS

ELECTRODES

Fig. 5. Flow-through type core flowmeter.

Electromagnetic flowraeters for fast reactors The tests described by Glass and Popper (1968) showed agreement with prediction to within 4% but a change of sensitivity of 30 ~ was experienced after some flow rig modifications. The change was put down to the effect of accidental and large welding currents which had passed through the magnet. Having adjusted the predictions to allow for the change in magnetic field, tests at temperatures up to 1100°F were promising. The use of Alnico V magnets was based on considerations of temperature, metallurgical change, impact, nuclear radiation (Popper et al., 1967) and its ability to be cast and machined to the required dimensions (Forster, 1971). The flowmeters were of all stainless steel construction apart from magnet and soft iron polepieces, and signal cable insulation was of high purity alumina (99.8~ Ala 0~). The flowrneters were filled with helium to aid in initial leak detection and to help remove gamma heat from the magnets. The field of these flowmeters was fairly uniform and about 5 diameters long so that the small degree of field sweeping would be negligible, but signals might have been affected by variation in flow profiles even though the profiles were axisymmetric. In test the signal appeared to have been lower than calculated by as much as 6 ~o. The cause of this is not given. Turbulence noise appears to have been easily observable. Magnet variation could account for discrepancies as its drift was substantial and random. Yada (1970) has discussed a similar instrument and he has also used an equation for wall shorting. This type of meter is also mentioned in Akiyama and Yada (1969). In both references the effect of temperature is illustrated by the change in the calibration curve. This change was about a 6 ~o reduction in output for a temperature increase from 250 to 450°C. Both Yada (1970) and Akiyama and Yada (1969) also show a more conventional design of flowmeter for flow measurement in a liquid sodium loop. This consists of a horse-shoe type magnet with a circular pipe between its pole-pieces. The whole is encased in a perforated box, presumably to reduce stray field effects. The use of coils to generate the magnetic field (thus overcoming some of the problems associated with permanent magnets) was rejected by Popper et al. (1967) for three main reasons: (1) a large amount of power and current is needed to produce a modest flux; (2) the power is not easily transmitted to the meter, and (3) the large coils cannot easily be designed to fit within the limited space available.

45

Popper and Glass (1967) discuss the various calibration factors required for their design. They give the equation for wall shorting obtained from Elrod and Fouse (1952). This is similar to the one given by Shercliff (1962) (equation (17)) except that he included contact resistance, an unknown which nevertheless is likely to be significant in this application. Other correction factors discussed by Popper and Glass (1967) are for end-shorting for which a graph is given and for magnet property variation. They did not observe any thermoelectric effects. It would be useful to know more of the effect of bubbles on the response. Non-uniform distribution of bubbles would presumably behave like nonuniform conductivity in the flow head, the effects of which have been discussed by Baker (1970) and Bevir (1971a) and shown to be small. The effect of a conducting wall would modify this theory and would also cause the wall shorting to increase when bubbles appeared. The effect of sweeping will be small, and so the design of these meters should benefit from the extensive theory built up by Bevit (1970). With this it should be possible to develop a design which had low sensitivity to flow profile.

In-core probe-type flow sensor An interesting variation on this design is given by Verber et al. (1971). The magnet and electrodes are built into a probe which can then be inserted in the centre of the ducts leaving an annulus for the sodium to flow in (Fig. 6). Many of the constructional methods are common to the flow-through type discussed above. Cast Alnico XIII was selected for the magnet because of its high coercive force. The performance was satisfactory and the response linear. The disadvantage of this type of probe is that its response is heavily dependent on the velocity in the vicinity of the probe and little dependent on the velocities away from the probe. It does not give a signal proportional to mean flow rate. This is shown by the equation for the potential difference between the electrodes (Baker, 1968) A UE~ = 2 ~ v ( ~/ar)Be(r,O)dr

(18)

(see Fig. 7 for geometry). 0 would be ,T/= for the usual electrode positions. Bevir (1971b) gives a distribution of the weight function from which it can be seen that the response is primarily due to the flow in the vicinity of the probe and particularly in the vicinity of the electrodes.

46

ROOER C. BAKER

DUCT

Fig. 6. Probe type core flowmeter. MAGNETIC

!

1

¢r

~ ELECTRODE

SURFACE

Fig. 7. Probe geometry. Secondary circuit saddle-coil flowmeters In the secondary circuit of the fast reactor the magnetic Reynolds numbers are up to 5 and field sweeping becomes an important consideration. Correct calibration is also important since a knowledge of the reactor coolant flow rate is essential for safe operation (Meshii and Ford, 1969). These authors, using a flowmeter with a length-to-diameter ratio of about 0.7 observed considerable nonlinearity due to field sweeping. The purpose of the saddle-coil flowmeter is to overcome the problem of field sweeping. Thatcher et al. (1970) have described a meter of this sort which was claimed would give an output

ELECT.ODE /L CO,LSU

T

proportional to flow within an absolute accuracy of 4-3 ~ and requiring no calibration. This flowmeter which is basically an induced voltage type has a conducting wall thus allowing signal shorting. Figure 8 shows the layout of the coils which are laid on to a tube concentric with the flow tube of diameter greater than 2.5 times that of the flow tube. The dimensions used for pipe and coils were approximately: pipe ID coil diameter coil length field strength at axis

0.30 m; 0.75 m; 2.34 m; 5 roT.

The coil was mounted outside the pipe lagging. The resulting field was found to be approximately uniform across the pipe and remained uniform for about half the length of the coil. A number of electrode pairs were welded to the stainless steel flowtube to allow the saddle coils to be moved to assess the effect of a bend upstream. The test results were in good agreement with the predictions when these were corrected for wall shorting (using Shercliff's (1962) equation) and field sweeping (a small effect and tabulated in the paper). The highest value of R . was 5. For a pipe bend in the magnetic

SADDL CO,l \

DUCT Fig. 8. Saddle-coil flowmeter.

Electromagnetic flowmeters for fast reactors field plane five diameters upstream from the measuring electrodes the change in output signal of the flowmeter was negligible. The effect on the fiowmeter was also negligible when a venturi orifice was placed six diameters away. Thatcher (1971) discusses the derivation of the field sweeping correction factor. The assumptions made about the magnetic field are apparently acceptable for the small amount of field distortion which is experienced by the saddle-coil flowmeter. But, as Thatcher points out, they are inadequate for gross field distortion where substantial departures from linearity are expected. Komori et al. (1974) have also discussed the performance of a flowmeter of a similar design to the one just described. They give calculated values of magnetic field distribution for the saddle-coil type of winding. Their results using dry calibration (deducing the velocity-voltage relationship from measurement of magnetic field only) do not appear to have given high accuracy. They also discuss the use of more than one pair of electrodes as a means of obtaining an integrated signal less sensitive to flow profile distortion and they consider Shercliff's (1962) two-dimensional weight function plot. The use of a modified distribution to give a more uniform weight function has been discussed at length by Bevir (1971c). He has also considered the effect of conducting walls. Since the saddle-coil flowmeter approximates a long flowmeter it would be an interesting exercise to optimize the field distribution using Bevir's approach. The use of multiple electrodes or large electrodes are possibilities which could be explored further. Another interesting direction for future development is in shaping the magnetic field in such a way that with a short field a linear signal is retained at high Rm values by positioning the electrodes downstream. This idea is hinted at by Meshii and Ford (1969) and the possibility is further indicated by a diagram in Thatcher's (1971) paper showing the change in induced voltage for an axial array of electrodes. Turner (1960) had earlier tried out this idea with some success and it is rather surprising that no developments have taken place so far. Further work might also be useful on the response time of large flowmeters (Griffin, 1971). This author describes experiments using an aluminium cylinder. The results indicated a fast response. In a recent report (Raptis and Forster, 1975) experiments are described in which a cross-correlation technique was used with a permanent magnet flowrneter and two pairs of electrodes to provide an in-service meter calibration. This technique looks

47

promising but initial accuracies would need to be improved. 2. FLUX-DISTORTION FLOWMETERS Introduction

The basic principle of this type of meter is that the distortion of the magnetic field due to sweeping by a moving conductor may be sensed by a search coil or other means and used as a flow signal. Although a device of this kind was originally patented by Lehde and Lang (1948), and in one example proposed for ship speed measurement, its early use was mainly in ionized gases. The magnetic Reynolds number must be of order 0.1 or greater to obtain adequate signals. This occurs in the high speed flow of ionized gas and was exploited for this purpose by Lin et al. (1958), Pain and Smy (1960) and Cowley (1961). One feature is that the output signal has a similar dependence on both velocity and conductivity and these authors were mainly interested in measuring the latter. Meyer (1961) also discussed its use for conductivity measurement at the skin of a missile as it re-enters the Earth's atmosphere. Fuhs in a number of papers (1964) developed the idea of a conductivity/velocity probe. Mayer (1964) patented a flowmeter which used the change of inductance of the applied field windings resulting from the flow. Cowley (1965) analysed a device for flow measurement either for installation in the wall of a duct or across the duct in an aerofoilsection capsule. He examined the effects of high values of Rm and a skin effect parameter uo~a z where ~ is the fluid permeability, a the fluid conductivity, ,o the excitation frequency for the field and a is a typical dimension. Baker (1969) analysed a design in whichthe field was produced by two concentric opposed coils spaced axially along a circular pipe, external to the pipe, and field sweeping was sensed by an axial Hall effect probe midway between these coils. At low values of R,, the response was found to be linear and for a certain spacing of the coils it could be shown to be almost insensitive to flow profile. This work was extended (Baker, 1970b) to allow for higher values of R,, although it was shown there that linearity persists for the optimum design, to values of R,, .--, 5. This work was primarily concerned with d.c. fields. Apart from the magnetometer flowmeter to be discussed later, interest for reactor applications has been directed towards the a.c. version, the eddycurrent flow sensor. Popper et al. (1967) saw a number of advantages in using this flowraeter in the Fast Flux Test Facility

48

Rc~

C. BAg~a

and additional advantages if further developed. Apart from those common to electromagnetic flowmeters there was the compact size; the satisfactory construction; large a.c. signals; sufficient transient response, and no necessity for ferromagnetic materials. However, calibration was likely to be sensitive to velocity profile and temperature (because of conductivity changes). The electronics were also considered complex.

Eddy-current flow-through type fiowmeter Wiegand (1976) discussed a version in which the coils are outside the tube, the sodium flowing through, rather than around, the coils (Fig. 9). It was

DUCT

Fig. 9. Eddy-current flow-through type flowrneter. expected to give better coil/fluid coupling. Wiegand's analysis of this device made the following assumptions. (1) The axial components of the magnetic field within the fluid at the ends of the primary coil is one-half of that at the centre of a long thin solenoid with equivalent current-turns per unit length and equivalent fluid conditions. (2) The secondary coils are sufficiently long and sufficiently close to the primary that essentially all the flux in the fluid, which issues from the ends of the primary coil, threads through the surfaces of the fluid within the lengths of the secondary coils. That is, a negligible amount of the primary flux reaches the far ends of the secondary coils. (3) If an a.c. current is made to flow in the secondary coils, the resulting eddy-currents in the fluid within the secondary coils are one half of those within a long thin solenoid of equivalent currentturns per unit length. (4) The velocity profile in the fluid is axisymmetric. With these assumptions Wiegand obtained the force on the fluid when alternating currents flow in

both primary and secondary coils. It is then possible to deduce from the reciprocity principle the secondary voltages resulting from fluid motion. As a result of his analysis (described in more detail in Wiegand (1969)) he obtained operating frequencies which gave peak signals. By operating at such a peak the maximum signal would be obtained and temperature errors reduced. He also gave curves showing the variation of signal with Reynolds number. This is a result of the changing profile. It was found that signals increase with Reynolds number which is intuitively reasonable since the signal will be more affected by velocity near to the pipe wall than elsewhere, and this is increased proportionally with increased Reynolds number. Wiegand chose to analyse a five-coil version of the flowmeter, the two additional end coils being primary coils. This device was tested (Wiegand and Michels, 1969) first by using an alumininm rod. The agreement was very good. The errors in the sodium loop tests were higher (about 10~) but peak response position was fairly well confirmed. The errors seem to have been adequately explained by test and design problems, but could also have resulted from anomalous velocity profiles. One major problem experienced in this work was due to imbalance in the secondary coils. This was a particular problem at low flow rates. If the dry test agreement which they achieved is an indication of the prediction accuracy, then the authors suggest that calibration may not be required. Feng et al. (1975) describe a finite difference relaxation method for the analysis of this meter using complex notation for quhdrature signals. They also conclude that if the velocity across the pipe is to be measured with reasonable accuracy a low frequency should be used (5 kHz).

Eddy-current probe-type flow sensor Popper et al. (1967) in their review of flow measurement techniques discussed a device of this type. One possible design is illustrated by Fig. 10. In this form it comprises one central and two outer field or primary coils which excite an alternating magnetic field around the probe. The probe is inserted coaxially into a duct and the sodium flowing through the annulus between the probe and the duct wall sweeps the field downstream. Coaxial with and between the primary coils are two balanced search or secondary coils. With no flow their induced voltages exactly balance. But when flow distorts the magnetic field, the induced signals in the search coils are unbalanced and the imbalance is used as a flow signal.

Electromagnetic flowmetcrs for fast reactors

SECONDARY

FLOW ~

COILS

/

DUCT Fig. 10. Eddy.current probe-type flowmeter.

Libby and Jensen (1969) appear to have built a range of designs of probe: two-coil, in which both coils produce the field and sense the distortion; three-coil in which the centre coil produces the field and the outer coils are balanced against each other; a multiple-coil probe (three primaries and three secondaries). The test procedure was to move the probe inside a solid tube. As a result of these tests, one of the two-coil probes was tested by moving it vertically in a cylinder of Wood's metal. The authors saw potential in this type of flow sensor, but suggested that the three-coil system might be preferable. Brewer et al. (1971) tested further designs. Threecoil versions were tested using an oscillating aluminium tube around the probe to simulate a uniform profile. Three- and five-coil probes were tested in a sodium loop. They gave a number of results for their latest flow sensor, a three-coil type. Linearity and stability were good. Costello et al. (1972) have shown the change in sensitivity resulting if a sensor of this type is unpocketed (in contact with sodium) or inserted in a pocket. A 58 % drop in sensitivity for the latter case was observed. Ohgushi et al. (1972) also report results of tests on a five-coil probe. Dean et al. (1970) have described three-coil probes for use in the British PFR as pump flowmeters and core flowmeters. The pump flowmeter has stainless steel main body construction and is surrounded by a stainless steel tube. A stainless steel straining wire is strapped together with the connecting cables to give flexibility to negotiate the path, and strength for safe withdrawal. Operating frequency was 700 Hz. Sensitivity variation of 1.2% was found between sensors, and the pocket affects the calibration. For these reasons it was decided to calibrate the probes in position in the reactor. Diagrams of drive and amplifier units are given. Careful cable twisting reduced pick-up (primary at 1 turn/it and secondary at 1.6 turns/it).

49

The core flowmeter of robust design will operate up to 500°C and can be inserted and withdrawn from a containment pocket which extends 40 ft down into a reactor. The excitation frequency was 600 Hz. The response was linear in the required range. Thatcher (1971) derives an equation to analyse the behaviour of this device and also gives some preliminary results. Sheff and Lessor (1972) have also developed analytical methods to describe the performance of this meter. Two further methods of analysing this device numerically will be discussed in Sections 3 and 4 of this paper.

Magnetometer flow sensor

Recently Wiegand (1972) has proposed a related device in which the applied magnetic field is produced by two opposed permanent magnets and a fluxgate magnetometer is set between them at the null point of the applied field to sense the induced field due to motion (Fig. 11). One problem with such a device

MAGNETS / ~,,..%M/AGNETOMETE R

CAPS~/UIE

POC~KET

~'~ SHEATHED CABLE

Fig. l 1. Magnetometer flowsensor. is the effect of stray fields, and to avoid these Wiegand has suggested a variant consisting of one permanent magnet with magnetometers positioned at each end, so that under no flow or strong fields their signals cancel, but for flow-induced fields their signals add. This device has had less development than those mentioned earlier, and would seem to suffer from the problems of inadequate magnet stability. However the device appears to have advantages in being less susceptible to design imbalance (and this can be corrected with the fluxgate magnetometer), in having higher signal levels, and, since it is a d.c. device, penetration problems which result from the skin effect are avoided.

Other flow-sensing devices using field ch'stortion

A number of devices have been suggested and tested in Riga, Latvian S.S.R. and have been reported in the literature. One family of these is

50

ROGER C. B~d~R

dependent on travelling magnetic fields, the relative velocity of field and fluid being sensed (Ulmanis, 1962; Kalnin et al., 1966). Another type depends on a pulsed magnetic field. The rapidly changing field induces a current ring in the fluid which is convected with the fluid. The time of transit of this ring can then be sensed by a search coil downstream (Zepgir and Sermons, 1965). These and other devices are covered by Kisis (1968) and Tsivkunov etal. (1973). Although these devices could well yield useful flowmeters for the fast reactor circuits they do not appear to have been exploited for this purpose either in the United Kingdom or in the U.S.A.

3. FINITE DIFFERENCE ANALYSIS OF THE EDDY-CURRENT FLOWMETER WITH TIME MARCHING Introduction In Section 2 we discussed the application of the eddy-current flowrneter to the fast reactor core flow. In this section and the following one we analyse the behaviour of this device. The particular geometry of interest is shown in Fig. 12. The probe is positioned axially in the centre of a duct, so that the

PRIMARY ,/COIL

FLOW

/

--

CE

Under these conditions the governing equation for the magnetic field function ), is given by r~r ~r~r)

=

(0, 0,)

/1~ ~- + vN

(21)

Or = - rB,

(22)

dy Oz

(23)

where

and

rB,.

This equation is valid everywhere except in the vicinity of the primary coils, where the current is h. For these regions we have f l dy'~

\\

i

i

i

7o - 7eL,, 6t

/

St- CONDARY COIL

Fig. 12. Geometry of flux-distortion-flow meter. sodium flows around it in an annular passage. The central coil produces an a.c. field, and as the flow increases, so this is distorted. The outer coils are balanced at no flow and produce no signal. When the flow increases and distorts the field, a signal is produced by the imbalance. Combining equations (1) and (3)

r),_,

[ 1

/zav'~

6r2(r - 6r/2) + ~ z 2 + "~-z/~'-=--

+ 6rl------~2+

+

1 6Z 2

=

le ~r 6r6z

(20)

If however we can assume that the geometry is axisymmetric which is a reasonable assumption we can simplify the equation.

(25)

where yo and Y0-ASTare the values of ~, for the centre of a finite-difference molecule for two successive time steps and 8t is the length of the time step. The values of y at the other four points of the molecule are indicated by y+,, y_,, y+., v-,. Equation 21 may now be rewritten in finite-difference form using this notation

(19)

and the curl of this, using equation (2), gives V x V x B = wrG7 x (v x B) - B).

(24)

Finite-difference method for full solution A time-marching solution is used which follows the time varying field. Consider first the time derivative. This may be written in approximate finitedifference form as

i

V x B =/z(r(E + v u B)

IzrJs.

d27

These equations may be rewritten in finite-difference form making use of the conditions that ~, ----0 on all boundaries.

CORE i

+ ~z2

+

+

Yo

laav~ r7 +, 6Z ] y += + &2(r + 6r/2)

/aaron,,. 6t

(26)

h is the current at lattice points which coincide with the primary coils. Elsewhere h will be zero.

Electromagnetic flowmeters for fast reactors The ¢ocffichmt matrix resulting from the solution of this set o f finite-differenoc equations is solved for each time step substituting from the previous step in the term containing ~'O~sT. Although the solution takes place at each time step, the matrix m a y be set up once only and part o f the Gaussian elimination procedure also need be done once only. The values of ~, are calculated for a series of times and they are found to converge to a sinusoidal repeating pattern. The values may then be used to obtain the secondary signal. The time derivative may be obtained as t ~ = t~?°c°t(t°<~t) - tO?o~..cosec(t~t)

(27)

where w is the angular frequency o f excitation. This form of the derivation was applied to the stepping process first, but was found to cause instability.

Tests for accuracy The computer program that resulted from this approach to the solution was first tested against the exact theory for an infinitely long core*. The duct geometry is shown in Fig. 13 with a typical lattice spacing. The response of the secondary coils to a flow of 2 m s - 1 when there is no conducting pocket surrounding them is shown in Fig. 14 for a range of frequencies. The values are in terms of F which is given by

V

=

8npnpn~alF

(28)

where V is the voltage generated by the flow, n~ and n, are turns per metre for primary and secondary coils, a is the coil radius and I is the primary current.

*Thatcher, G. Private communication.

There is a discrepancy between the two sets of values, numerical and exact, of as much as 20~o. Some likely reasons for this arc the coarseness of the lattice, the finite extent of the lattice (increasing the error at low frequencies) and the simple time stepping procedure. The lattice is not easily adjusted as the actual duct geometry must fall at acceptable lattice positions and computer storage adds a further restraint. Shorter time steps will increase computing time. In the values shown in Fig. 14 the lattice dimensions were 20 × 35 and there were 32 time steps per cycle.

Applications of the numerical solution We shall discuss later a further development of this solution. Here we confine our discussion to applications which are relevant to this form of the solution only. The first feature of this solution is that it allows variation over the whole length of the flowmeter. It is of interest to examine the effect on the signal o f the passage of a bubble through the device. F o r this purpose we have chosen an excitation frequency of 150 Hz and we use a flow rate of 2 ms-1. The axisymmetry requires that wc consider a ring bubble which would be an extreme case. The effect of a small bubble moving along the probe wall is shown in Fig. 15. The bubble is a ring of section 3.1 mm radially by 8.0 mm axially (Fig. 13 gives the duct dimensions). Signal variation of up to about 20% is apparent from Fig. 15. In comparison the effect of a bubble which fills the annulus radially is very severe. Figure 16 shows the effect o f the passage of such a bubble which fills the annulus radially and has an axial length of 20 mm. The signal rises to three or four times the normal size. A distribution of bubbles with spacing approximating to the overall coil length could cause an apparent large increase in

,"0

, l PM~Dh~E

i

i

44. ii;i l/

I

NON-CONDUCTING NON-MAGNETIC MATERIAL

SO£Xu~ FLOW Vs 2 m / s 0".4 545~10 L

II 15mfT~ 4m.~ i

[I

H

I i i I ~I I ]!oCJCKET iru~eO ) 0;1042xt0'

II l l l l II I I L .,t,c I I I I I I I I I I T~r e gsmm t t I I I I tO"=I042xI0 ~:r I J 1 Ill IIIZ4I $ 12 16 "[-- -

4 4 0 mm

5]

~ 14 n ~ m L 10mm

I | I ~3s~m/ ' I ] } I " I=r..,.,~ ~7mm I [ t~--,,m "1L I, [ ;* Z 34m~. SHORmT

C)ORE END

Fig. 13. Dimensions and lattice for finite difference solutions.

52

ROGER C. BAKER

p"

Making use of this linearity, which had previously been noted by Thatcher* and, in a somewhat different geometry of flowmeter, by the author (Baker, 19701)) the solution described above was moditicd, as described below, thus avoiding the need to obtain the flow signal as the difference of two larl~ numbers.

0'3

x 10"2

02

Modified method for linear solution If it is assumed that the flow will have only a small effect on the magnetic field, it is possible first to obtain the magnetic field distribution for zero flow which will have reverse symmetry about the midplane and then to use this distribution to find the distribution of the flow-induced perturbation field which will be symmetrical about the mid-plane. One benefit of this symmetry is in the reduction of lattice points which require separate consideration. Thus in the first part of the solution the velocity term in equation (21) is omitted while in the second part of the solution it becomes a part of the excitation t¢rrn. In the original solution the value of ~ for the core could be specified, but the finite-difference boundary condition was not entirely satisfactory. In the modified program an infinite value of ~ was taken as being an adequate approximation.

0 Exact Theory • Nurri@f-/cal Values

0.1

I

100

I

I

~

I

200 300 400 500 E x c i t a t i o n Frequency. Hz

Fig. 14. Full finite difference solution compared with exact theory.

the flow signal which if responded to by the reactor controls would reduce the flow and possibly accentuate the problem. It is of interest to examine the effect on the signal of the meter being half filled (axially) with sodium. The program was run with 150 Hz excitation frequency at zero velocity and with sodium covering the lattice up to and including the central plane lattice points. The resulting signal was F = 0.017 or six times the signal for a flow of 2 ms- ~. The second test for which this program was used gave the response against magnetic Reynolds number (R,) for 150 Hz excitation (Fig. 17). Note the approximate linearity of the signal up to a value of R , of 0.7 at which the velocity is over 10 ms -z, well in excess of the likely velocities in a fast reactor.

Test of accuracy The computer program resulting from this linearized solution was also tested against the exact theory for an infinitely long magnetic core. The results for the geometry of Fig. 13, without a conducting pocket, are shown in Fig. 18. The lattice dimensions for the values at ~00, 200 and 300 Hz were 33 radially by 20 axially with the same spacing as for the original program. This indicates a larger

*Thatcher, G. Private communication.

-0.4

x F'2

, -.o...~__~

Flow slqnal tnthe absence ot bubbles

~° -oli

-01

I -~

l

l

l

t

l

I

-40

I

I

I

I

I

I

I

I 40

I

I

i

i

I oO

Distance of bubble f r o m nnJd plane, m m

Fig. 15. Signal variation for the passage of a small bubble past the probe.

53

Electromagnetic flowmeters for fast reactors X IF'0-21 .4

~

-8(:~

~owbgoal,nme

-40 0 40 Distanceof bol:~tefrom mid I~ane~mm

80

Fig. 16. Signal variation for the passage of a large bubble past the probe.

F

F~

xlO"= O3

0.3

OP 02 • ExaCt Theory 0 Numerical Values

"0~

0.1

I

1

I

2

t

3

I

4

I

5

I

6

I

Rm

7

Fig. 17. Response of probe to increasing magnetic Reynolds number.

lattice in both radial and axial directions than was possible using the original solution. F o r the value at 500 Hz the lattice size was 20 × 18, considerably smaller to match the lower field penetration at the higher frequency. In all cases there were 32 steps per cycle. The agreement is satisfactory considering the coarseness of lattice and time marching. Effect o f a short core and a pocket In Fig. 19 we show three curves. The first of these is a repeat o f that shown in Figure 18, and gives the numerical values for the response of the eddy-current

I

100

I

t

I

I

I

2OO 30O aOO50O Excitation Frequency, Hz

Fig, 18. Linearized finite difference solution compared with exact theory.

flowmeter with no pocket and an infinitely long core of very high relative permeability. The next curve plotted shows the change o f response when the magnetic core is short (see Fig. 13). It is interesting to note how small the changes in the signal size are. The third curve in Fig. 19 gives the flowmeter response when a conducting pocket surrounds the coils and the core is o f short length. All these results arc for a 33 x 20 lattice except the one value at 500 Hz and in all oases 32 steps per cycle were used in the time marching system.

ROGER C. BAKER

54

F

F x 10"2

x I(3-:

0 -3

0-3

02--

02 •~ Uniform profile O Long core pocketles$

V Boundary layer !profile

A Short core Docketless 17 ShOrt core w=thpOckets

0.1

0.1

i, 100

t

200 Excitation

100

300

400 500 Frequency, Hz

Fig. 19. Effect of core length and pocket on probe response.

Effect of a boundary layer The linear approximation used in this modified solution implies that we can treat the movement of each layer of the fluid separately since the magnetic field perturbation resulting from the movement will be negligible compared to the zero flow field. Thus from a knowledge of the response of individual layers we can build up the response to any profile. In Fig. 20 two curves are shown. In one the velocity is uniform at 2 ms-1 (the response is for a short-core unpocketed flowmeter). In the other a step velocity profile is used to simulate the effect of a boundary layer. The inside annulus with inner and outer radii of 11.6 mm and 13.2 mm has a velocity of I ms -1. The middle annulus with radii 13.2 mm and 14.7 mm has a velocity of 1.667 m s - 1 and the remaining fluid with annulus radii of 1.47 mm and 25.6 mm has a velocity of 2 m s - 1. The mean velocity in the boundary layer profile is 1.90 ms-1, a 5 % reduction over the uniform profile. However the reduction in signal is about 20~. and as for the pocketed flowmeter provides an indication of the sensitivity of the flowmeter to velocity changes close to the coils.

Response to flow changes The time marching solution provides an indication of the signal change following a sudden flow change from rest to 1 ms-1. The change takes place at a point in time when the primary coil current is at a peak and the subsequent signal change is shown in Fig. 21 for a uniform velocity, short core, un-

I

I

I

I

200 300 400 500 E x c i t a t i o n Frequency. HZ

Fig. 20. Effect of boundary layer on probe response.

pocketed flowmeter with 100 Hz excitation. The response under these conditions can be seen to settle down very quickly. 4. CURRENT RING ANALYSIS OF THE EDDY-CURRENT FLOWMETER

Introduction In this section we discuss another approach to the solution of the eddy-current flowrneter. The flowmeter geometry is axisymmetric and all currents generated flow in circular paths. It is therefore possible to divide up the flowmeter material into a number of current loops. In each of these a current flows generated by the motion and the a.c. field, and these currents generate their own field. It is possible to set up a matrix of in-phase and quadrature currents in these loops and by solving it, to obtain the current in each and the induced flow signal. The current induced in one member of this array of current rings is due to the sum of the fields of all the Hngs inducing a current in the one ring either due to frequency or due to motion. Thus we can set up a matrix which gives the effect of one ring on another. If this effect is due to frequency we have that Jo = ~Eo (29) and

VxE---B---V

xD,

(30)

giving Jo = - #"~o

(31)

55

Electromagnetic flowmeters for fast reactors

0.6 × 1 0 -2 0.3

-06f Fig. 21. Probe response for a step change of velocity. where jo, Eo and .40 are components of current density, electric field and vector potential in the circumferential direction. Thus for the interaction of one ring on another due to frequency we need to compute Ao. If the effect is due to flow then

Jo = ~rvB,.

(32)

In this ease we need to know the value of B, the radial component of the magnetic field. These two functions are given by Smythe (1968) as:

Ao=rtk,r] [(1 - ' ~ ) K - E 1 #1 z

(33)

B, = 2"n r[(a + r) 2 + z2] ~

[

a2+r2+z2E] -- K +

(a-

r) 2 + z 2

(34)

where K and E are complete elliptic integrals of the first and second kind, I is the loop current, a is the ring radius, r and z give the position of field measurement relative to the ring and

4ar k2 = (a + r) 2 d- 22.

(35)

The elliptic integrals may be calculated from infinite series given by Dwight (1961) in the form

7~ + m) K=..~(I I

12

12.3 2

l + ~m" + ~-~m"

E= 2(1 - -+

where

m = (1 - k')/(l + k') /

and

k'

x/(l - k2).

/

(38)

With this information we may now set up the field of any array of current rings. First however we consider a simplification of the solution which reduces the amount of computer storage required.

Perturbationapproxhnation In Section 3 it was shown that for velocities of the magnitude to be expected in fast reactor flows the approximation may be made that the signal is directly proportional to velocity. If we make use of that approximation here, we obtain a considerable simplification of the problem. It now becomes possible to solve for the field without flow first, and then to use this solution to obtain the small perturbation due to flow. The advantages of this are that both fields individually have a kind of symmetry while the combined field does not. We are therefore enabled to solve for one half of the axis only and thereby reduce the computer storage. We also obtain the output signal directly rather than as the small difference of two large quantities. We now set up the form of the current ring interaction as a series of equations.

The unperturbedfield 12"32"52

+ 22.42.62 m6 + . . .

1 (36)

m)

m2 12 12"32 1 I + - ~ + ~-~.42 m " + ~ rn6 + . . .

(37)

For the following discussion reference should be made to Fig. 22 which illustrates the current ring interaction. In this diagram two fluid current rings P and Q are shown and their symmetrical counterparts P' and Q'. R and R ' are symmetrical primary field coil rings and S and S' symmetrical secondary search coil rings. We consider first the current generated in Q by all the current rings P and P', R

Roo~a C. BAgvat

56

,~, = C~ D

p, O

0

p []

.

=

(4o)

where (?or and Coy, give the effects of induction due to equations (31) and (33) of P' on Q. F o r this purpose the cross-sectional area of the ring Q is also used. Since the coefficients Cot and CQr, always appear together we may replace their sum by C'op, and with C'op we can rewrite equations (39) and (40) as

S__ C~IL$

CORE

~.(co,+ co,.)n"

- ~(co, + co,.)z~/',

D

SODIUM 8

-

.-z

Fig. 22. Diagram of current ring interaction.

1~/2 = ZC~pI ° + EC~, I u, and R'. I f we use superscripts o to indicate in-phase components o f the current in these rings and s/2 to indicate quadrature components, we obtain:

I~'~ = E ( c o ~ + Co,.)~ + P

(Cox + C o , . ) ~ ,

(39)

P

I~ =

- ~Copl~/2

~Co,I~/'.

-

(42)

These form a set of simultaneous equations in unknowns I ° and I "/2 which may be written in matrix form as:

1

o

...

o

cl,

Clz...

c;.

o

l

...

o

c;,

c;,...

c;,.

o

0...

,

(41)

R

I0

-- ECI R l~/2 R

-

io

C...

Zc;, I~ :2

- y c ' , z ~ :2 R

c;,

C12...

C',.

- l

0

. • .

-Y.cl, 10

11 'z

0

R

C'2...

C~.

0

-l

- Z c h 10

...

R

c:,

c;,..,

c;o

o

o

...

-Xc;,,

-1

°

R a

(43) and it is apparent that the square matrix is made up o f two identical quadrants and two unit bands, one positive and one negative. Replacing the vector matrices on the right by - - R ° and - - R "/" we obtain

1 f ,o 1 --

I

R° R,/2

It is now possible to treat this matrix equation as a pair of simultaneous equations in matrices, the solution of which permits us to compute the currents from a matrix one quarter the original size and consequently to re4uce the required computer storage. We achieve this as follows: I "/2

+

C'l ° =

C'P/" --

(44)

I°

=

R °,

(45)

- R "/z

(46)

-

Premultiplying equation (46) by C' and adding the result to equation (45) gives (CC'

+

I)1=12 = - (R ° + C'R"/2).

(47)

Electromagnetic flowmeters for fast reactors Having obtained I"/2 by the solution of this equation it is then possible to substitute back into equation (46) to obtain I °. By this procedure the values of 1° and I ~/2 for the unperturbed field have been found. It should be noted that these values are symmetrical with respect to the mid-plane, so that current rings positioned at symmetrical points upstream and downstream will have current circulation on the same sense. We now proceed to apply the same procedure to obtain the flow perturbation. But in this case, although symmetrically placed points will have current rings of the same size, they will now be of opposite sign. The reason for thb difference is that the symmetry of the unperturbed rings stemmed from the symmetry of the primary coil currents, whereas the excitation for the perturbed field is current rings generated by the interaction of the flow with the radial component of the unperturbed magnetic field, and it is this which caused the sign change.

The flow-perturbed field For the perturbation currents we replace equations (39) and (40) by

I~/2 = +

~(Cop-

y'

P and R

(Do +

I~ = - ~ ( C o p -

+

~,

Cop.)l ° (48)

The solution of this follows the same course as used in equations (44), (45) and (46). Thus we obtain the set of currents due to the flow perturbation and we can compute the signal induced in the secondary coils due to them as:

V "/" = 4 n a ~ ~ ( S s p -

Ssr)I °

(53)

V°

-- Ss~,)I~,/2

(54)

= -- 4 t t a ~ ( $ s ~ $

P

where J"ttt/2 and V ° are the induced voltages, a is the secondary coil radius, and S is obtained from equations (30) and (33). The 4 includes a factor of two to allow for the two secondary coils. This concludes the discussion of the theory and we now present some results which show the accuracy of this method.

Test of accuracy The output signal was computed in terms of F which is defined by

V = 8npn~aIF.

(28)

The array of current rings used in the computation is shown in Fig. 23. Although conductivity and velocity could be varied for each ring, it was assumed that the velocity in the fluid was uniform for these

CQr)I~,/2

(Doe + DQp,)I°^

(49)

Tr

NON-M&GNETIC TUBE (FOR OOCKETLESS CASE) v • 2m/$ 6 Cr , 4.545 • 10 (FOR POCKETED CASE ) * i

PandR

the main differences being that the two terms with C coefficients are subtracted due to the symmetry change, that the D coefficients give the effects of flow due to equations (32) and (34), and that the summation of the D terms is now taken over both the primary field coil currents and also the induced currents calculated already in the absence of a flow perturbation. We therefore write:

I~/2 = E C o p I + pa~.d D'QpI~,/2 .z?,

57

(50)

, tl!ilt~!i

F

,

soo,uM

] i i i i J I [ r ] I I I r I ]

iii!

~OCKETt,FVSED1

I,..

[ I

co,s; ] T i]

I i } I~

--. L~_~10~m$4mm

/

~ 34t~m

o.l.o4v,'~0~

, ' ! .q,-M.G.E,T,C CO~E

I

i

i

"

'

8 !

II |J ~ Z

Fig. 23. Typical current loop arrangement with dimensions.

= P

+

E D,pIO

(51)

Pandg

giving

I I

(52)

tests and that the conductivity was uniform within each region with the values given in Fig. 23, since these were the conditions for Thatcher's values. One hundred rings were used, of axial length 4 ram. The results of this computation are shown by the circles in Fig. 24 and 25 which provide a comparison with Thatcher's exact theory shown by the lines. Fig. 24 gives the response for varying frequency when the oofls are contained in a conducting pocket,

58

ROGER C. BAKER

while Fig. 25 shows the same but for a flowmeter without a pocket and with fluid surrounding the coils in to a radius equal to the radius of the inside of the pocket. The radial dimensions of current rings for the results in Fig. 24 was 1.55 mm for core rings, 1.8 mm for pocket rings and 2.0 mm thick for fluid rings. For Fig. 25 all rings had a radial dimension of 1.55 mm. The agreement in Fig. 25 is very satisfactory, but that in Fig. 24 is less so. The explanation of this • 0"3

F x

Exact Theory

ld: O

Coml~JteO Values

02

O 0"1

500

1000

Excitation Frequency,

2000 HZ

Fig. 24. Cm'rent ring model compared with exact theory for pocketed flowmeter (non-magnetic core).

discrepancy which seems most likely is that the restricted region of conducting material which has been used, while adequate at higher frequencies at which the field has less penetration, is inadequate in Fig. 24 at low frequencies since for this graph the fluid in Thatcher's solution was supposed to extend to infinity. Notes on the solution

The extent of the current rings in the z-direction is quite limited. In order to test the effect of this, a fluid RnnB]us USing only four layers radially, again of 1.55 ram thickness, was analysed first with nine axial rings and then with fourteen axial rings and it was found that the signal change was only about

0.3%. The program was found to be highly sensitive to the arrangement of the current rings. Large crosssectional area rings caused the program to give spurious results. A similar problem also occurred under certain conditions when some coils with low conductivity were used. The reason for this appeared to lie in the pivotting procedure used in the Gaussian elimination when solving the matrix equation. Since in any row of the matrix the elements around the leading diagonal predominate, it is possible by pivotting to cause the matrix to become ill-conditioned. It has been found that one way to overcome this problem is to arrange the matrix rows in the order of decreasing product of area and conductivity: Another way around the problem is to omit the pivotting routine from the solution to avoid upsetting the inherent physical order of the matrix.

03--

F

Exact Theory

x l O ":

0

ComputeO Values

0.2-

O.t-

Possible extension to magnetic materials

The current ring model has much to commend it but as described above it is unable to allow for a core of magnetic material. Two important modifications to the solution as described are apparent if it is to be suitable for this. It will be necessary to model the core and its boundary conditions using current rings and it will then be necessary to manipulate the resulting, and more complex, matrix. Other modifications may appear but we suggest here an approach to these two. Modelling the magnetic core

s [oo

1 looo

I l 2000 3000

ExcJlalion Frequency, Hz

Fig. 25. Current ring model compared with exact theory for immersed flowmeter (non-magnetic core).

We shall consider only a nonconducting core in which ~ can be taken as infinite. As a first step this core is removed and in its place current rings are set where the core surface was. These are imagined to be of finite surface length defined by their spacip.g

Electromagnetic flowmete~ for fast reactors (axially or radially) and of negligible thickness. The problematic step is now to obtain the right boundary condition. We require that the magnetic field entering the core shall be perpendicular to the core surface. In other words we require that the combined tangential field of all the current rings which simulate the core shall at every point of the core surface exactly balance the tangential field from all other sources.

Modified equations Equations (39) and (40), (48) and (49) will now be modified to include the core terms. However we now have to consider the equations for IQ separately for currents in real conducting material and for currents in the core model. We will distinguish these by giving the latter the symbol tu. We now write for the equations

ffi Z(co, + CQ .)I p

+

(cQ. + cQ.,)1 °

+

+ cQ ,)x ° -

+

(55)

(Ew + Eup,)l 12

~(EuR + EuRo)I~/2

+ ~(EUT 4- EuT.)i~/2

(56)

and similar equations for Io° and i °. In these equations E is obtained from an equation like equation (34) which provides the axial field component for the cylindrical surface and from equation (34) for the end surfaces of the core. The value of E also includes the boundary condition discussed briefly in the previous section. A further similar set of equations will also replace equations (48) and (49).

59

The various regions of this may be identified. c, is the effect of core-model currents on the field in the fluid and other conducting material, ca is that part of the core boundary condition which relates the tangential field at the core ~u-face due to currents in the fluid and conducton to the core currem ring strength at that point, c, is the other part of the core boundary condition which relates the tangential field at the core surface due to core current rings to the core current ring strength at that point. It is reasonably straightforward to reduce equation (57) to the form of equation (44). It is first necessary to find the reciprocal matrix of c3 and with this it is possible to eliminate/o and i "/2. Once the form of equation (44) is recovered, the solution may proceed as before. CONCLUSIONS In this paper we have reviewed the state of the art of electromagnetic induced-voltage and flux-distortion finwmeters. We have also looked in some detail at methods of analysing the eddy-current fiowmeter (one type of flux-distortion flowmeter) using numerical techniques. The design of fast reactors for the future is still developing. The fiowmeters which have been discussed have certainly not arrived at their ultimate design and ff the fast reactor programme proceeds there is room for some useful further development of several types of flowmeter. The displaced electrode type of large finwmeter for secondary circuits has been mentioned. Other types such as the pulsed field type may also be useful. Recent experiments on correlation techniques look particularly promising for the future of in-core calibration and accurate flow measurement. The methods of analysis discussed in Sections 3 and 4 provide tools which could prove useful in this future development and the current ring model, in particular, which should be extended to magnetic cores, may have wider applications in pumps and other devices with axial symmetry.

Modified matrix We can now construct the general form which the resulting matrix will take 0

c2

C3 --OI-cl

0

0

o

CI

1

Ro

_0

---1¢2

Acknowledgements Sections 3 and 4 of this paper were undertaken as part of an agreement (5R52107B) with the United Kingdom Atomic Energy Authority, Reactor

-7-

C3 J

i=/

--

--

R~/2

r./2

(57)

60

Rotter C. BAr~R

Group, Risley, Warrington, Cheshire, England. I am grateful to UKAEA for permission to publish, to Mr. E. Duncombe for his interest and support and to Mr. G. Thatcher for many useful discussions. Sections 1 and 2 could not have been written without the co-operation and patience of Miss Edna Archer and her colleagues in the Mechanical Engineering Library at Imperial College and I acknowledge my indebtedness to them. I am also grateful to Mr. R. Puddy for producing the excellent diagrams. REFERENCES Akiyama S. and Yada H., (1969) Electromagnetic pump and electromagnetic flowmeter for a sodium cooled fast reactor. FAPIG (Tokyo), .~5, 193(43). Baker R. C. (1968) Solutions of the electromagnetic flowmeter equation for cylindrical geometrics. Br. J. appl Phys. (J. Phys. D) SER 2 1, 895. Baker R. C. (1969) Flow measurement with motion induced magnetic fields at low magnetic Reynolds numbers. Mngnetohydrodynamics 3, 69. Baker R. C. (1970a) Effects of non-uniform conductivity fluids in electromagnetic flowmeters. J. Phys. D: appl. Phys. 3, 637. Baker R. C. (1970b) Linearity of motion-inducedmagnetic-field flowrneter. Proc. IEE 117, 629. Bevir, M. K. (1970) The theory of induced voltage electromagnetic flowrneters. J. Fluid Mech. 43, 577. Bevir M. K. (1971a) The predicted effects of red blood cells on electromagnetic flowmeter sensitivity. J. Phys. D: appL Phys. 4, 387. Bevir M. K. (1971b) Sensitivity of electromagnetic velocity probes. Phys. med. Biol. 16, 229. Bevh" M. K. (1971c) Long induced voltages electromagnetic flowmeters and the effects of velocity profile. Q. JI. Mech. appl. Math. 24, 347. Brewer J., Jaross R. A. and Brown R. L. (1971) Eddycurrent probe-type sodium flowsensor for FFTF reactor fuel channel monitoring. IEEE Trans. nucl. Sci. NS-18(1) 372. Costello T. J., Laubham R. L., Miller W. R. and Smith C. R. (1972) FFTF probe-type eddy-current flowmeter wet vs dry performance evaluation in sodium. Am. nucl. Soc. Int. Meeting, Washington, 12th November. Cowley M. D. (1961) The distortion of a magnetic field by flow in a shock tube. J. FluidMech. 11, 567. Cowley M. D. (1965) Flowmetering by a motion-induced magnetic field. J. sol Instrum. 42, 406. Dean S. A., Harrison E. and Stead A. (1970) Sodium flow monitoring. Nuclear Engineerino International, 15, No. 174. Duncombe E. and Thomasson R. K. (1970) Sodium process instrumentation for the Dounreay PFR. Nuclear Engineering International 15, No. 172, 714. Dwight H. B. (1961) Tables of Integrals and other Mathematical Data. Mactm'llan, 4th Ed. Elrod R. G. and Fouse R. R. (1952) An investigation of Electromagnetic fiowmeters. Trans. ASME 74, 589. Evans P. B. F., Burton E. J., Duncombe E., Harrison D., Jackson G. O. and McAffer N. T. C. (1966) Control and instrumentation of the prototype fast reactor. British Nuclear Energy Society "Fast Breeder Reactors" Proc. of the London Congress BNES May. Feog C. C., Deeds W. E. and Dodd C. V., (1975) Analysis of eddy-current flowrneters, d. appl. Phys. 46 No. 7.

Forster G. A. (1971) Performance of permanent magnet flow-through-type sodium fiowmeters in EBR-II instrumented subassemblies. IEEE Trans. NS 18(1) 363. Forster G. A. (1973) Long-term stability of Alnico 5 and 8 Magnets at 700 to 1200°F. Argonne National Lab. Ill (USA) Report No. ANI.,-CF-73-16, November, 29p. Fuhs A. E. (1964a) Techniques for obtaining the electrical conductivity/velocity profile. Electromagnetic Aspects of Hypersonic Flight, Proceedings o f Second Plasma Sheath Symposium, Ed. Rotman W. Moore H. and Papa R. Paper 19, 337. Fuixs A. E. (1964b) Instrumentation for re-entry plasma sheath. Physico-cheraical diagnostics o f p/asmas, Ed. Anderson T. P., Springer R. W. and Warder R. C. Jr. Proceedings o f the fifth biennial gas dynamics symposium AIAA and Gas Dynamics Lab. North-western University, 383, paper 19. Fuhs A. E. (1964c) An instrument to measure velocity/ electrical conductivity of arc plasmajets. AIAA Journal 2, No. 4, 667. Fuhs A. E. and Oibb O. L. (1964) Plasma Sheath transducer for axisymmetric re-entry vehicles. AIAA Journal 2, No. 4, 773. Fuhs A. E. and Kelly J. A. (1964) Zero an#e-of-attack sensor. AIAA Journal 2, 1492. Griffin C. W. (1972) Response time of simulated large permanent magnet flowrneters. Liquid Metal En#neering Center (LMEC) semi-annual technical progress report. Jan.-June 1971 LMEC-71-7 134, Atomics International, Ca. 130. Kainin R. K., Kisis A. Yu. and Sermons G. Ya. (1966) On one contactless method of measuring the velocity in electrically conducting liquids. Magnetohydrodynamics 3, 150. Kisis A. Ya. (Ed). (1968) Electromagnetic methods of measuring the parameters of MHD processes. Riga. Komori Y., Someyarna T., lwamoto S., Yamada K. and Kobayashi H., (1974) Shimadzu electromagnetic flowmeter for liquid sodium. Shimadzu Hyoran 31, 163. Lehde H. and Lang W. T. (1948) Device for measuring rate of fluid flow. U.S. Patent 2 435 043. Libby H. L. and Jensen J. D. (1969) Feasibility study of probe-type liquid metal flowmeters for FTR subassemblies. Battelle Memorial-Institute, Washington 99352 Rep. No. BNWL-911 UC-37 Instruments, February. Lin S. C., Resler E. L. and Kantrowitz A. (1955) Electrical conductivity of highly ionized argon produced by shock waves. J. appl. Phys. 26, 95. Mayer F. (1964) Induction controlled flowmeters for conductive liquids. U.S. Patent No. 3 138 022, June 23; French Patent No. 1 288 806. Meyer R. X. (1961) Some remarks concerning magnetohydrodynamic applications to re-entry problems. Second Symposium on the Engineering Aspects of M. H. D. (Columbia University Press.) Meshii T. and Ford J. A. (1969) Calibration of electromagnetic flowmeters in the Enrico Fermi Atomic Power Plant Nuclear Applications and Technology 7, 76. Ohgushi K., Hayakawa S. and Hanamiya I. (1972) Research and development of in-core instruments for sodium-cooled fast reactors. Fuji Electric Journal 45, 257(71). Pain H. J. and Stay P. R. (1960) The electrical conductivity of shock-ionized argon. J. Fluid Mech. 2, 390.

Electromagnetic flowmeters for fast reactors Ptister C. G. and Dunham R. J. (1957) D-C Magnetic Flowmeter for Liquid Sodium Loops. Nucleonics 15, 122. Popper G. F. and Glass M. C. (1967) The design and performance of a 1200°F magnetic flowmeter for in-core applications in sodium cooled reactors. IEEE Trans. nucl. Sci. NS-14(I), 342. Popper G. F., Wiegand D. E. and Glass M. C. (1967) Summary review of flowmeters suitable for measuring sodium flow at tempemtur~ up to 1200°F in the fast flux test facility. Argonne National Lab., II1. ANL 7340 Report. Raptis A. C. and Forster G. A. (1975) A signal analysis method using the cross-correlation of turbulence flow signals to determine calibration of permanent magnet sodium flowmeters. Argonne National Laboratory, A~-76-8 July. SheffJ. R. and Lessor D. L. (1972) Optimization of eddy current flow meter performance. Battelle Memorial Institute, Pacific North-west Laboratory Rep. No. BNWL-SA-4236 to be presented at the 1972 Annual Meeting of the Am. Nucl. Soc. June 18-22 Las Vegas, Nevada. Sherciiff J. A. (1954) Relation between the velocity profile and the sensitivity of electromagnetic flowmeters. J. appl. Phys. 25, 817. Shercliff J. A. (1962) The Theory of Electromagnetic Flow-Measurement. Cambridge University Press. Smythe W. R. (1968) Static and Dynamic Electricity. McGraw-Hill, 3rd Ed. Thatcher G. (1971) Electromagnetic flowmeters for liquid metals. International Conference on Modern Developments in Flow Measurement, Harwell pp. 359-380.

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Thatcher G., Bentley P. G. and McGonigal G. (1970) Sodium flow measurement in PFR. Nuclear Engineering International 15, 822. Tsivkunov V. E., Zheigur B. D., Sermons G. Y. and Kalnin R. K. (1973) The contactless measurement of liquid metal fow. Riga. Turner G. E. (1960) The non-linear behaviour of large permanent-magnet flowrneters. Atomics International Report, NAA-SR-4544. Ulmanis L. Ya. (1962) A non-contact flowmeter for liquid metals. Byulleten lzohretenii (19). Verber F., Jaross R. A. and Mulcahey T. P. (1971) Development o f an in-core permanent-magnet probetype sodium flowsensor for operation in the fast flux test facility (FFTF). IEEE Trans. NS-18(I), 366. Wiegand D. E. (1967) Summary of an analysis of the eddy-current fowraeter. 14th Nuclear Science Syrup., IEEE, Los Angeles, October. Wiegand D. E. (1969) The eddy-current flowrneter. An analysis giving performance characteristics and preforced operating conditions. Argonne National Laboratores Engineering and Equipment, Rep. No. ANL7554, August. Wiegand D. E. and Michels C. W. (1969) Performance tests on an eddy-current flowmeter. IEEE Trans. NS-I 6(1-2), 192. Wiegand D. E. (1972) Magnetometer flowsensor. Argonne National Laboratories, Rep. No. ANL-7874, March. Yada H. (1970) Fuji electromagnetic flowmeters for liquid sodium. Fuji Electric Journal 43, 194(24). Zepgir B. D. and Sermons G. Ya. (1965) Impulse method of measuring the velocity in electrically conducting liquids. Magnetohydrodynamics 1, 131.