SANDIA REPORTSAND97-0170 . UC-506Unlimited ReleasePrinted January 1997

A Tool to Detect External Cracks fromWithin a Metal Tube

Thurlow W. H. Caffey

Prepared bySandia National Laboratories

Albuquerque, New Mexico 87185 and Liverrnore, California 94S50for the United States Depamnent of Energy

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under Contract DE-AC04-94AL85000 ,. ‘,

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Acknowledgment

I wish to thank James R. Wait, Professor Emeritus of the Universityof Arizona, and David A. Hill, of the National Institute of Scienceand Technology at Boulder, Colorado, for discussions of theirpaper which was the basis for the computation of theelectromagnetic fields. Herman A. Watts, Department 1273,provided a derivation which significantly decreased the computationtime.

On page 7, replace the paragraph beginning Distance with:

Distance. Increasing the antenna separation distance is dubious fortwo reasons: The allowance for the scattering loss would have to beincreased because of the greater scattering angle, and the increasedpath length in the metal would further decrease the signal becauseof the exponential attenuation.

SAND97-0170Unlimited Release

Printed January 1997

DistributionCategory UC-506

A Tool to Detect External Cracksfrom within a Metal Tube

Thurlow W. H. CaffeyGeophysical Technology Department

Sandia National LaboratoriesAlbuquerque, NM 87185-0705

A tool using a continuous electromagnetic wave from a transversemagnetic-dipole source with a coaxial electric-dipole receiver is outlinedfor the detection of external sidewall cracks in boiler tubes. A numericalstudy of the distribution of the fields shows that the direct transmissionfrom the source to the receiver is reduced from that in free space.Further, if the diameter of the receiver dipole is made sufficiently small, itshould be possible to detect cracks with a scattering loss of up to -40dBin thin-walled boiler tubes.

Contents

Preface ....................................................................................................

Introduction ..............................................................................................

The Primary Field .....................................................................................

The Total Field within the Tube ................................................................

An Example ...............................................................................................

Conclusion ................................................................................................

References ...............................................................................................

Appendix: Electromagnetic Fields in a Cylindrical Void

within a Complex Media .................................................................

1

1

2

4

4

11

11

12

1

2

3

4

5

Al

A2

A3

Relative Ez-Field from an X-directed magnetic dipoleat any spherical radius ...............................................................................

Relative Ez-Field within a2.2cmmetal tubeat Z= O...................................

Relative Ez-Field within a2.2cmmetal tubeat Z= l.lcm ............................

Relative Ez-Field titina2.2cm metd~be at Z=2.2cm .............................

Relative Ez-Field within a2.2cm air tube at Z=2.2cm ................................

Cross.setiion geomet~for acylindricd void ..............................................

Magnitude of the relative Ez-Field in airwithin a radius of l.lcminthe XY-plane .....................................................

5

6

8

9

10

13

16

Difference between computations with azimuth as a parameter ..................... 17

A Tool to Detect External Cracksfrom within a Metal Tube

Preface

Historically, and even to the present day, the major cause of boiler outages is the degrada-tion of the steam tubes by which heat is extracted from the boiler. Although there is depo-sition and corrosion upon the internal surfaces of the tubes, the principal cause of fdure isby the development of exterior cracks. Of course, when a boiler is otherwise down forrepair, the exteriors of all apparently satisfactory tubes are inspected for cracking. Un-fortumtely, some cracks may not be discovered because the exterior surfaces are more orless covered by combustion or corrosion products. A inspection tool that could be passedthrough a boiler tube, and detect an unseen sidewall crack from within, would increase theintervals between boiler outage and provide substantial cost savings.

This work was done as a six-week effort under Laboratory Directed Research& Devel-opment Project #3536.

Introduction

The basic idea is to transmit an electric field which is parallel to the axis of the tube, andwhich will reflect off the exterior wall and return to an internal receiver. It is assumed thatthe return from a crack-anomaly will be distinguishable from that of an acceptable part ofthe tube wall,-and that the interior of the tube is filled with air. A pulse-type system can-not be used because the small dimensions of boiler tubes, whose inner diameters are gen-erally 10cm or less, would require such a high-frequency system that the thickness of themetal wall would completely absorb the electromagnetic field before the field could reachthe exterior.

A single-frequency, continuous-wave syste~ in which the transmitting and receiving an-tennas are designed to minimize the crosstalk between them, is presented here. The cross-talk constitutes a ‘self-clutter’ which will set the lower bound to the signal-to-noise ratioof the system. The choice of operating frequency must iidfill these opposing criteria:

1. The frequency must be low enough that the round-trip attenuation along thetwo-way path in the metal will be small enough to allow a detectable returned sig-nal level at the receiver. This attenuation constraint upon the frequency will ensurethat the operating frequency is below the cutoff frequency of the cylindricalwaveguide formed by the tube. -In other words, the receiver will not be excited bya waveguide mode per se. As a practical matter, to ensure detectability, I requirethat the rms returned signal be at least twice as great as therms level of the clutter.

1

2. The frequency must be high enough so that cracks of some specified minimumsize can be detected.

These two criteria are not separable, and, to make matters worse, the fractional amount ofthe field incident upon an anomaly that will be returned toward the intenor of the tube isnot yet known. I will assume that this scattering loss is fixed at 1/100 (-40dB) for presentpurposes.

One way to examine the effect of both frequency and clutter is to compare the magnitudeof the field transmitted into the tube wall, attenuated by the both path loss and scatteringloss, with the magnitude of the self-clutter field. This signal-to-clutter ratio, or SCR, canbe written as follows:

[1

-2%E slgnal,ms = EP_/it,m~ >

100 ‘(1)

where d = wall-thickness of the tube, and C$= the skindepth (later defined). The SCR isdefined as

goE~igml,rm,SCR(dB) = 20L0 E

clutter~s

> (2)

which becomes:

&z5-17”:d-40SCR(dB) = 20L0 (3)

In this formula the 2ndterm is the two-way path loss, the 3d term is the scattering loss, andE

transmit and Eclufler are computed according to Wait and Hill, (1977).

The target will be considered detectable only if the SCR is 6dB or more. A discussion ofthe computation of the fields with application of the SCR to a metal boiler tube is givenbelow.

The Primary Field

Because the use of an electric field parallel to the tube axis is considered, it is desirable tohave a transmitter which not only provides such a field at the intenor wall of the tube, butwhich also has an absence of such a field elsewhere in the tube where a receiving elementcould be placed.

2

Consider a Cartesian coordinate system in which the Z-axis will later be placed along theaxis of an air-filled tube. In the absence of the tube, the Z-component of the electric fieldprovided by an infinitesimal magnetic dipole whose moment is along the X-axis is givenby:

()- jptiidA sin ~sin jkR -1 ~JmEz = An R2 Y V/m (4)

where:j= J-l;

v = permeabtity of free space, Henries/meteqo = 2nf = radian iiequency, seconds-l;k = a + j ~ = the propagation fictor, meters-l;

~=L= the loss tangent;o&r&o

SO=the dielectric constant of free space, Farads/m;

S,= the relative dielectric constant;

c= conductivity, Siemens/m;c = speed of ele.gtromagnetic propagation in free space, m/s;

I = dipole current, peak Amperes;dA = dipole arm mete~;

and (~ e p) are the usual spherical coordinates. This formula for Ez is not defined at theorigi~ R = O, where the magnetic dipole is located. The magnitude of Ez is a maximum inthe plane O= 90° and along the direction q = MO”, and is zero whenever either t?or p areeither 0° or 180°. In the Cartesian coordinates, Ez is a maximum along the i Y-axis andis zero in the XZ-plane aside from the origin. An infinitesimal electric dipole, aligned par-allel to the Z-axis and placed anywhere in the XZ-plane, aside from the origi~ would becompletely uncoupled from the transmitter. However, a physical dipole will always have anon-zero cylindrical radius, and its effect on self-clutter will be examined later.

If the magnitude of Ez is written-out, the terms in R may be collected into one factor, andthe angular distribution of IEz I is seen to be the same for any spherical radius greater thanzero:

~ = /fd-iA

[

J-Z 47r R*

‘1

exp – ~R) sin ~sin$ (5)

This distribution is shown in Fig. 1 in which the half-power bearnwidth of each lobe is 90°.The hi-lobed distribution describes the primary field even when the magnetic dipole is en-closed by a cylindrical metal tube of radius ‘p’ as long as R <p.

The Total Field within the Tube

The theory of the total field within a metal tube is described in Append~ A. The Z-component of the internal electric field is the sum of the primary field fi-omthe magneticdipole, EPMMmY, and the field reflected horn the wall, EmFMmw:

Ez=EPMMARY+EmFEGED (6)

The field transmitted into the tube wall, EwsMm, or ET, is equal to Ez evaluated at theinterior air/tube bounday because of the continuity of the tangential components of the E-field. In a metal tube, at frequencies below the itiared, the parts of the propagation con-

stant are equal to each other, namely cz= B = ~=, and attenuation through the

wall thickness ‘d ‘ occurs as exp(– d / @ where 6 is the ‘skindepth’ given by d= l/a.

An Example

An Inconel tube with an inner diameter of 2.2c~ a wall-thickness of O.127cm, and a con-ductivity of 8.2E6 S/m is typical of the smallest tube commonly found in boilers. The fre-quency of 47.88KHZ will be used to make the two-way path through the wall equal to oneskindepth so that the second term in Eq.(3) becomes -8.7dB. In the following figures, thefields will be presented as normalized surfaces in cylindrical coordinates in which radialdistances are relative to the tube radius. The increments of normalized radius and polarangle will be 0.1 and 10° respectively.

Figure 2 illustrates the relative magnitude of Ez in the XY-plane. The figure is dominatedby the double-peaks near the Z-axis. These occur because the field is computed ever moreclosely to the magnetic dipole located at the origin (where the graph is set to zero for dis-play purposes). When the field along the tube wall at p = 1 is examined closely, the fieldis seen to vary as the cosine of the polar angle rather than as the sine-fbnction of the pri-mary field, Eq.(4). This effect is caused by the large loss tangent which occurs in themetal tube. A small, Z-oriented, electric dipole placed anywhere within Fig. 2 would beexcited by strong EZ -fields, and provide a large self-clutter.

hmdxezl.dcfhmdezf(ll .dat

Figure 1Relative IEz-field I from an X-directed magnetic dipole

at any spherical radius greater than zero

1.0

IIE Iz

1-

ctube6a.dcf- ctube6a.dat

ctube6a.wklctubeo.06a

Figure 2Relative Ez-field with a 2.2cm metal tube at Z = O

-1.

6

Figure 3 shows the relative magnitude of Ez in the plane where Z is one tube radius. Thecosine-nature of the field along the boundary is now very apparent. The surface variesrapidly, but the Z-axis could be considered as a location for the receiving antenna.

Figure 4 shows the relative magnitude of l?z in the plane where Z is one tube diameter.The surtlace appears as a folded disc, curved slightly downward at the outer edge, andwith an undulation along the Y-axiswhich falls to zero at the center and at Y= Al. Theprimary field alone is shown in Fig. 5, and, by comparison with Fig. 4, vividly demon-strates the rotation due to the reflecting boundary.

Suppose a small electric dipole, with a radius of one-tenth the tube radius, is placed at theorigin in Fig. 4. The self-clutter field at the receiver is estimated as the mean value of thefield along p = 0.1. Suppose fbrther that a crack is located along the exterior tube wall in

the ~-plane which is also the location of the source at Z = O. The ratio of l?~w~~~m tol?c~um~~, the first term in Eq.(3) is about 43.6dB, but the SCR is -5. ldB which is belowthe detectability criteria. Any other Z-location of a crack will fhrther reduce the SCR.For example, if the crack is located at Z = one tube radius, the l!7~field is reduced by 3dBbecause of the bearnwidth of the primary field, and the SCR decreases to -8. ldB.

The problem is that the reflected field does not sufficiently reduce the primary field at thelocation of the receiving antenna. For example, using Figures 4 and 5, the ratio of theself-clutter field with the tube present, to the field with the tube absent, is only -12.6 dB.There are three changes that can be made:

. Increase the antenna separation distance● Change the operating frequency. Reduce the dk.rneter of the electric dipole.

Distance. Increasing the antenna separation distance is dubious for two reasons: The al-lowance for the scattering loss would have to be increased because of the increased pathlength in the metal.

Freauency. If the frequency is increased by a factor of 4, the ET field increases by7.3dB, but the skin depth is halved and the path loss increases by 8.7dB. The self-clutterincreases by about 5.4dB, and the net result is that the SCR decreases from -5.1dB to -11.9dB. If the frequency is decreased by a factor of 4, the l?T field decreases by 8.4dB,but the skin depth is doubled and the path loss decreases by 4.3dB. The self-clutter de-creases by 5.5dB, and the net result is that the SCR decreases from -5.1 dB to -3.6dB. Byhow much can the Iiequency be decreased in an effort to raise the SCR to 6dB? Thisquestion cannot be answered without a model Iibrmy of cracks and their scattering loss,but the scattering loss will eventually increase with decreasing frequency as the maximumdimension of the crack becomes a smaller fraction of the wavelength in the metal.

DiRole Diameter. In the present example the dkuneter of the electric dipole is taken asone-tenth the inner dkuneter of the tube or 0.22cm. Reducing the dipole thickness by a

7

Figure 3Relative Ez-field within a 2.2cmmetal tube at Z = 1.Icm

ctube8adcfctuba8a.dat -ctube8awklctubeo.08a I

ctube7a.dcfctube7a.datctube7a.wklctubeo.07a

Figure 4Relative Ez-field with a 2.2cm metal tube at Z = 2.2cm

Figure 5Relative G-field within a 2.2cm air tube at Z = 2.2cm

cpril a.dcfcpril a.datcpril a.wklcprio.01a

factor of four with the use of printed circuit techniques, and coupling directly to an inte-grated preamplifier, would raise the SCR from -5. ldB to 6.9dB and make the crack de-tectable.

Conclusion

Numerical computations of the fields show that the use of a printed-circuit receiving di-pole and integrated amplifier could detect cracks whose scattering loss is not less than-40dB. It would be desirable to develop a catalog of practical cracks which are found in avariety of pipe sizes and conductivities. A three-dimensional model code (Newman andAlumbaufi 1996% 1996b) could be used as a means of estimating the scattering loss’ofeach catalog entry. The design and development of an instrument could be undertaken ifthe modeling showed that practical cracks have scattering losses greater than -40dB.

References

G. A. Newman and D. L. Alumbaug~ 1996%Electromagnetic Modeling of Subsurface3D Structures, Proceedings of the 1996 IEEE International Geoscience and RemoteSensing Symposium, Vol. IV, pp 1941-1944, Lincoln, Nebrask~ May 1996. IEEE #96CH35875.

G. A. Newman and D. L. Alumbaug~ 1996b, 7hree-Dimensions/ ElectromagneticMba?ding and Inversion on AhssiveIy ParalIeI Computers, Sandia National Labor-atories report SAND96-0582, March 1996. Available from the U.S. Department ofCommerce, Clearing-house for Federal Scientific and Technical Itiormation, Springfield, VA.

James R. Wait and David A. Hill, 1977, “Electromagnetic Fields of a Dipole Source in aCircular Tunnel Containing a Surface Wiwe Line”, International Journal of Electronits, 1977, Vol. 42, No. 4, pp 377-391.

11

APPENDIX

Electromagnetic Fields in a Cylindrical Void within a Complex Media

The solution for the electromagnetic fields within a borehole surrounded by a complexhomogeneous medium was included by J. R. Wait and David A. Hill as part of an articledevoted to another topic wait and Hill, 1977]. The theory is summarized below with theauthors’ notation in which time-harmonic fields are assumed in accordance with exp(jd).

GeometryFigure Al depicts the XY-plane intercepted by a circular cylinder of radius ‘ao’whoseaxis is coincident with the Z-axis. A magnetic dipole, with its moment parallel to the X-

axis, is located at cylindrical coordinates (pJ,PI,0) where pl e ao, and the field point is at

@ q,z) with P< a.. The cylinder is air-filled with permeability ~ and dielectric constant~, and the surrounding homogeneous media has conductivity O, with permeability ~ anddielectric constant Se..

EZ FieldThe Ez field is represented as a superposition of modes in terms of the Hertz potentialswhich supply the modal contributions:

where l.(vp) and K~(vp) are modified Bessel fbnctions, V*= A2+ y: and Y. = jo\c. The

fi,mctionsAm@ and B.(A) are the primary Hertz potentials determined by the strength andorientation of the magnetic dipole source, and the secondary potential ~~(1) is related to~~(1) and B~(A)by the boundary conditions at the cylinder/media interface.

Hertz PotentialsWhen the X-directed magnetic dipole is located at Z = O,the electric and magnetic Hertzpotentials are given by these expressions when pl <p

~vz2 [~~-,(vP,)exp(-j@,)+~~+,(vp,)exp(j4,)]Aom2Bin(A) = - ‘ad

(A2)

(A3)

12

Y

-

Figure AlCross-section geome~ for a cylindrical void

13

where:

Boundary ConditionsThe boundary conditions at the cylinder/media interface can be succinctly expressed interms of a radial wave impedance L and a radial wave admittance Y. at p = a. [Stratton,1941]:

Eh = a~Em + Z.Hz~ (A4)

H@ = –Y~Em + amHm (A5)

a. = mAJ(aou2) (A6)

Y: = jp.m(~. + joz.) (A7)

U== A=+y: (A8)

Ym= [jy;f(u~e~)]K;(uao) fKm(uao) (A9)

‘m= ‘(J~ealu)K~(ua.)lK.(ua.) (A1O)

The application of the boundary conditions establish the relation of the secondarypotential, Pm, to the primary potentials Amand B.:

Pm= -[Am(2)r. +Bm(@m]/~m

where

rm=[[%-aml+(%+~oy

Dm =

[) a Woomltm =

aov= m aOv2

(3-42+(2+4(F+?)]’:

Vemz (All)

lmKm (A12)

(A13)

(A14)

14

and qO= r PO% is assumed to be the characteristic imPedance of the air inside thecylinder. The argument of the modified Bessel iimctions is aOv, and the prime, ‘ ,denotes

differentiation with respect to v, the wavenumber within the cylinder.

Computational NotesIn the special case where the magnetic dipole is located at the origi~ the representationsof the Hertz potentials are simplified and only two modes are needed, namely m = +1.For example, Eq.(A2) for A~@) becomes:

(A15)

with AO = – poaib!4/(8v7r2 ). Using the properties of the Bessel finctions (Abramowitz

and Stegu~ 1972), A.l = A“, and A+l = -A”, but Am is zero for m = O,m <-2, and m >2.Similarly, B.l = B“, and B+I = -B” where B“ is the leading factor, and all other B~ are zero.

As a check on both the code and the theory, it is of interest to see what happens when

y.=yo, orwhenao+~. The secondary potential Pn becomes zero as required, and

the field reduces to:

*

Ez = ~ ~- V2{A~(l)K~(vp)}exp[- jm+] exp[- jJ.zldl (A16)m+1-~

The closed-form solution for the dipole source in air is given by:

– jpcoId4E, = ~z

()

jkR– 1 ~j@sinesin ~

R2 .(A17)

Eq.(A16) is undefined when p is zero, and Eq.(Al 7) is undefined when R is zero, becausethese respective radii would place the field-point at the location of the magnetic dipole.

As a check the magnitudes of these two formulas were compared for a frequency of47.88KHz and p = 1.lcm in the XI’-plane. The magnitude-plots overlap as shown inFigure A2 where the figure is dominated by double-peaks near the Z-axis. These occurbecause the field is computed ever more closely to the magnetic dipole located at theorigin (where the graph is set to zero for display purposes). Along the rim of the figure,

1(1where the normalized radius is one, the magnitude varies as sin $) with maximums at

90° and 180° as required by Eq.(Al 7). The maximum difference between the twoformulas is about 0.045% and is shown in Figure A3. This relative accuracy of less thanone-percent is sufficient for practical purposes.

15

cpriol b.dcfcpriol b.datCPriil b.wklcprio.Ola

Figure A2Magnitude of the relative Ez-field in air

within a radius of 1.1cm in the XY-plane

4

1

Figure A3 Difference between computationswith azimuth as a parameter.

1.0

0.8

Max dif “ = 1 /2075.

0.2 \ \ \/ \

0.0 I I I I I I

0.0 0.2 0.4 0.’6 0

\

10 deg

,150

4

/,//

\\\\l

\

) I I I I I I I iI) 1.

CDIF1BE.PLTCDIF1BE.GRFCDIF1BE.DATCPRIOIBE.WK1CPRIO.01A

)

Relative radial distance

17

References

M. Abramowitz and I. A. Stegu~ 1972, “Handbook of Mathematical Functions”,Dover Publications Inc. New York Sections 9.6.1-9.7.11.

J. A. Stratton, 1941, “Electromagnetic l%eory”, McGraw-Hill, New Yorlqpp 354-361, p532.

James R. Wait and David A. Hill, 1977, “Electromagnetic Fields of a Dipole Source in aCircular Tunnel Containing a Surface Wwe Line”, International Journal of Electronics,1977, Vol. 42, No. 4, pp 377-391.

18

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