E-Journal of Advanced Maintenance Vol.2 (2010) 101-109
Japan Society of Maintenology

Remote Detection and Quantitative Evaluation of Wall Thinning Volumes
in a Metal Pipe
Linsheng LIU*
Nagoya University, Furo-cho, Chikusa-ku, Nagoya, 464-8603, Japan
ABSTRACTWe report a nondestructive method to measure the pipe wall-thinning (PWT) volumes remotely using microwaves.
A microwave vector network analyzer (VNA) and a self-designed transmitting and receiving (T&R) coaxial-line
sensor are employed in the experiment to generate microwave signals propagating in the metal pipe where thefrequency was swept from 14.00 to 14.21 GHz. A brass pipe with inner diameter of 17.03 mm, 1.0 mm wall
thickness, 2.0 m length, and connected respectively with 9 joints having the lengths of 17.0 mm and PWT volumes
from 0 to 550 mm3 were measured. By taking the pipe as a circular waveguide of microwave, after building up aresonance condition and then solving the resonance equations, the remote detection method is achieved. By comparing
the experimental results with the evaluated ones using our method, it is found that the evaluated results agree
well with the experimental ones, it indicates that a high precision evaluation method is established.
KEYWORDS

ARTICLE INFORMATION

Remote detection, Wall thinning, Microwave, Nondestructive measurement,
Metal pipe

Article history:
Received 5 September 2010
Accepted 17 November 2010

1.

Introduction

Metal pipes are used widely in industry. From 20 years ago, accidents due to pipe wall thinning(PWT) were reported frequently all over the world. PWT is one of the most serious defects in pipes
used in industry [1,2]. Efficient detection and quantitative evaluation of wall thinning in pipes arevery important issues for prediction of lifetime of the pipes in order to avoid severe accidents.
Recently, many nondestructive testing techniques, such as x-ray [3], electrical potential drop [4],ultrasonic [5,6], magnetic flux leakage [7], eddy current testing [8] and so on, have been used for themeasurement of PWT. However, all of them can only inspect a pipe locally, and are difficult tomeasure pipes buried under ground, in walls of some structures, or other buried conditions. Because a
metal pipe can be taken as a circular waveguide of microwave [9], and based on the fact that microwave
can propagate to a very long distance with quite little attenuation in media as air, petroleum, gasoline,
or any other low-loss dielectric materials, and what is more, because all the energies are confined
inside the metal pipe and the propagation and attenuation of microwaves in the pipe are independentof the pipe’s surrounding conditions, microwaves are adopted here.
PWT problem generally has two aspects as the PWT locations and degrees. The time of flight ofmicrowave has been used to detect the locations of cracks in the pipe [10,11], however, the PWT
degree is generally more important than the location for predicting the life time of the pipe. In ourprevious work two years ago [9], the PWT in a metal pipe was firstly detected using the resonancephenomenon of microwave at a short-end condition with microwave frequencies higher than 47 GHz,and the possibility to quantitatively evaluate the PWT degrees using microwaves was firstly proved bycomparing the experimental and theoretical results. Recently, we made some progress and establisheda systematic nondestructive evaluation method using microwaves to inspect a pipe in a large scale at
an open-end condition and to measure the PWT degrees remotely with the dominant mode frequencies[12]. However, both of our previous studies [9,12] deal with PWT degrees under the condition that the
PWT lengths are known and to be the same, they are not available for more practical problems that the
PWT lengths and depths are generally both unknown. In this research, we focus on deriving a more
useful method that can quantitatively evaluate the PWT degrees without knowing the PWT lengths.
*

Corresponding author, E-mail: liu@mech.nagoya-u.ac.jp

ISSN-1883-9894/10 © 2010 – JSM and the authors. All rights reserved.

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L.S. Liu / Remote Detection and Quantitative Evaluation ofWall Thinning Volumes in a Metal Pipe

As a result, through analyzing the relationship between the PWT length and degree, and by introducing
a geometric approximation, the PWT volumes containing information of both the lengths and degrees
are quantitatively evaluated. The basic microwave detection theory is the same as what has been
described in our previous works [9,12], it can be described as that the wavelength of microwave in thepipe is a function of the frequency and the inner diameter of the pipe after the working mode of
microwave for certain frequencies has been known. Therefore, after tracing the route that microwaves
propagate in the pipe and building up the resonance condition of microwaves, and then by solving theresonance equations, a method to evaluate the PWT volumes remotely and quantitatively is established.

2.

Experimental Approach

The experimental instrument is composed of a set of pipe specimens, a microwave network
analyzer, and a T&R coaxial-line sensor (the distance between the two ports of the sensor is d 0  6.0mm). The photograph of the instrument is shown in Fig. 1. The T&R coaxial-line sensor has a simple
structure and was designed using standard coaxial-line cables (K118) and connectors (K101F), and its
schematic diagram is shown in Fig. 2. The schematic diagram of the inspected pipe is shown in Fig. 3.

Fig. 1. Overall photograph of the microwave network analyzer, pipe, sensor, and joints
Connector
K101F

Cable K118

d0

d1 + 2t

Fig. 2. Schematic diagram of the self-designed T&R coaxial-line sensor
l2

l1
d0
T
R

l22

t
l21

d1

gd

l0

1

Microwave
Instrument

ld d2

gd

2

Fig. 3. Schematic diagram of resonance structure in the pipe connected with PWT joint102

L.S. Liu / Remote Detection and Quantitative Evaluation ofWall Thinning Volumes in a Metal Pipe

The pipe specimens are composed of a brass pipe with inner diameter of d1  17.03 mm, wall
thickness of t  1.0 mm and length of l1  2.0 m, and three groups of PWT joints with a lengthl2  17.0 mm and different PWT volumes. The first group of joints is composed of three joints, whichhave the character that each of them has a homogeneous 17.0 mm long PWT part. The second groupof joints is also composed of three joints and their PWT part having a length of 12.0 mm. The thirdgroup is composed of two joints and their PWT part having length of 9.842 and 14.932 mm,
respectively. The three groups of joints are numbered as No. 1 to No. 8 successively, and whose
detailed geometric parameters are shown in Table 1. Also, to form the same total length, another joint
with the same length of l2  17.0 mm but without PWT (i.e., the inner diameter is the same as that ofthe pipe) is used in the experiment and numbered as No. 0, its geometric parameter is also shown in
Table 1. The schematic graph of the joint connecting with the pipe is shown in Fig. 3. It should be
noted that all the introduced PWTs here are axisymmetric and full-circumferential.
Table 1 Detailed geometric parameters of the joints
Joint No.
Diameter, d2 (mm)

0

1

2

3

4

17.03 17.4

17.8

18.2 17.57 18.13 18.70 18.36 18.36

17.0

17.0

12.0

5

12.0

6

7

8

Length of PWT part, l22 (mm)

0

17.0

12.0 9.842 14.93

PWT vol. (mm3)0170.9 358.9 551.2 175.7 366.1 562.9 364.3 552.7

In Fig. 3, the “Microwave Instrument” refers to the network analyzer; the T and R represent
transmitting and receiving port of the sensor, respectively. Symbol d 0 is the distance between thetwo ports; l0 is the path along which microwaves propagate directly to the receiving port; d1 is theinner diameter of pipe without PWT and having the length l1 ; l2 is the total length of the joint; l21
is length at the part in the joint without PWT; d 2 is the inner diameter of the PWT part with aconstant PWT value and having the length l22  l2  l21 , and t is the wall thickness of the pipe. gd1
and gd2 are the wavelengths of the microwaves propagating in the pipe across the areas without
and with PWT respectively. Symbol ld is an introduced fictitious length corresponding to the phaseshift occurred due to the discontinuity at the abrupt PWT interface.

3.

Theoretical Analysis

3.1. Resonance condition and equations
To evaluate the PWT quantitatively using microwave, the crucial hint for analyzing microwave
signals is the resonance condition at the receiving port, which are built up by the microwave signals
reflected from the terminal of the pipe (after propagating along the pipe and reflected from the terminal)
and that going directly to the receiving port along a route having length larger than d 0 , the direct
distance of the two ports, with length l0 , without propagating in the pipe.
When taking lTotal  l1  l 2 and expressing the propagation route in wavelengths of microwave,
the equation for the difference of distance that microwaves propagate along the two routes can be
written as follows [12],with

2lTotal  l0 ( f q )  2ld ( f q )  (m  x)gd1  (n  y)gd2(1)m, n  N and 0  x, y  1(2)

N is the set of natural number. f q is the qth resonance frequency of the pipe connected with a PWT

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L.S. Liu / Remote Detection and Quantitative Evaluation ofWall Thinning Volumes in a Metal Pipe

joint. The integral number m means the times of full wavelength in the pipe at the part without PWT,
the 0  x  1 means the time of wavelength less than a full one, and (m  x) is the total times ofwavelengths. Similarly, the integer n means the times of full wavelength in the round trip along which
the microwave propagating in the joint at the part with PWT, the 0  y  1 and (n  y) have thesimilar meanings as 0  x  1 and (m  x) . Therefore, (m  x)gd1 corresponds to the differenceof distance for the two different route along which microwave propagates in the pipe at the partwithout PWT, and (n  y)gd2 is the length of roundabout trip along which microwave propagatesin the pipe at the part having PWT.
Eq. (1) can be written in the separated form as2(l1  l21)  l0 ( f q )  (m  x)gd1 ( f q )
2l22  2ld ( f q )  (n  y )gd2 ( f q )(3)

The resonance is under condition that the difference of phase change is integral times of 2 , so
the resonance condition of Eq. (1) or (3) can be expressed asq  (m  n  x  y )  N(4)

From Eqs. (4) and (2), the resonance condition can also be written in a simpler form asx  y 1

(5)

Eqs. (4) and (5) mean that the resonance condition is formed only when the difference of the
distance that microwave propagates along the two routes in the pipe is natural times (q times) ofwavelength, i.e., the two routes of microwave signals can form the resonance only when they having
the phase difference of 2q .
Therefore, when the wall thinning degree is expressed as s  (d2  d1 ) / 2 , based on Eq. (3), Eq.
(1) can be written in function of PWT length and degree as

where

2lTotal  l0 ( f q )  2ld ( f q )  q  gd1  P( f q , l22 , s)

(6)

P( f q , l22 , s)  (n  y)(gd1  gd2 )(7)

Symbol P( f q , l22 , s) is an undetermined composite function of the resonance frequency f q and
the PWT length l22 and degree s . While using V to express the PWT volume, it can be expressed asV   [(s  d1 / 2)2  (d1 / 2)2 ]l22   (d1s  s 2 )l22

(8)

When s / d1  1/ 10 , which is the general condition of the usual PWT, the PWT volume can be
written in brief as

V  Vappr  d1sl22(9)with error of approximation (V  Vappr ) / V  9% .
In Eq. (7), (n  y) at the right part has a proportional relation with the PWT length l22 , i.e.,(n  y)  l22 . While (gd1  gd2 ) is function of the applied frequencies f q and the pipe’s
diameters d1 and d 2 . To expand the right part of Eq. (7), the detailed expression of wavelength
should be introduced. In general, the wavelength of a circular waveguide having a relation with theworking mode of microwave and being a function of applied frequencies can be expressed as [12]104

L.S. Liu / Remote Detection and Quantitative Evaluation ofWall Thinning Volumes in a Metal Pipe f 2   pnm /(d )2g  1(10)Here,  and  are the permeability and the permittivity of the media in the pipe (air is used as themedium here, so that   0 and    0 ), f is the applied frequency, d is the inner diameter of thepipe, p nm is the mth root of the first kind Bessel function for TM modes [12].
When the microwave signal is introduced directly into the pipe through a coaxial line, theelectromagnetic field at the terminal of the coaxial line sensor determines that the working modes
exist in the circular waveguide are all TM modes, and among which the dominant mode is TM01-mode.
When sweeping frequency is between the cut-off frequency of the dominant mode and that of the firsthigher order mode, the mode for applied frequencies is the single TM01-mode. In this paper, only thefrequency range including the dominant TM01-mode is used, for which we have pnm  p01  2.4048 .
From Eq. (10), it can be derived thatgd  gd  (U 2  U1 ) (U1U 2 )
1

2

(11)

where U i   f q2   p01 /(di ) , ( i  1 or 2 ).
2While the numerator of the right part of Eq. (11) can be written as

U 2  U1  (U 22  U12 ) /(U 2  U1 )  (d22  d12 )[ p01 /(d1d2 )]2 /(U 2  U1 )  sD /(U 2  U1 )(12)

where D  2(d1  d 2 )[ p01 /(d1d2 )]2 .
Therefore, the difference of wavelengths can be written from Eqs. (11) and (12) asgd  gd  sD [(U1U 2 )(U1  U 2 )](13)

P( f q , l22 , s)  sl22 D [(U1U 2 )(U1  U 2 )]

(14)

1

2

Eq. (7) can be written as

At the condition that satisfies the approximate PWT volume expressed in Eq. (9), Eq. (14) can be
expressed asP( f q , l22 , s)  P( f q ,V )  V [(U1U 2 )(U1  U 2 )](15)

where (U1U 2 )(U1  U 2 ) has an approximately proportional relation with f q3 . Therefore, it isapproximately

P( f q , l22 , s)  P( f q ,V )  V f q3

(16)

In this meaning, Eq. (16) can be written in the separate form as

P( f q , l22 , s)  p( f q )  V(17)

where p( f q ) is an undetermined function of f q3 , and it will be calibrated using a PWT joint whose
PWT volume is known in the next section.

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L.S. Liu / Remote Detection and Quantitative Evaluation ofWall Thinning Volumes in a Metal Pipe

When using joint No. 0, the joint without PWT, only omitting the parameter l d taking accountfor the fictitious path at the interface of discontinuity in Eq. (1), the equation describing the differenceof distance that microwave propagates along the two routes in the pipe connected with the joint thatwithout PWT can be written as follows,

2lTotal  l0 ( f q)  q  gd1 ( f q)(18)

where f q is the resonance frequency of the pipe connected with the joint without PWT, and q is thesame as that of the pipe connected with a PWT joint.
3.2. Solving the resonance equations
Now, there are two groups of resonance equations, one is Eqs. (6) and (10) for pipe with PWT,
the another is Eq. (18) for pipe without PWT, the resonance conditions are both expressed in Eq. (4).
The method to solve the resonance equations and to evaluate the PWT volume can be separated
to the following two steps. The first step is using the comparatively simpler Eq. (18) to construct and solve
l0 and to solve q simultaneously, and the second step is to using l0 and q to solve Eqs. (6) and (17).
The inhomogeneity of the inner diameter of the pipe will aggravate the evaluation precision.
Therefore, before carrying out the calculation and evaluation, a high-precision method to detect the
average of the inner diameter of a pipe is derived from the cut-off frequency of TM01-mode measured
in the experiment and the expression is shown as follows [13],d E  p01  f cTM 01 



(19)

d E is the average inner diameter of the pipe, and f cTM 01 is the cutoff frequency of TM01-mode.
In our previous research [12], a method to determine l0 and q has been established anddemonstrated in detail.
In Eq. (6), ld ( f q ) is generated by the discontinuity at the PWT interface, i.e., for the pipe
without PWT ld ( f q )  0 . So ld ( f q ) should be a function of PWT degree of the pipe and of theapplied frequencies. For simplification, l d can be expressed as ld  a0 ( f q )V / 2 .
When considering Eq. (17) and the simplified expression of l d shown above, Eq. (6) describing
the PWT condition can be written as follows,

2lTotal  l0 ( f q )  qgd1  p( f q )V(20)

where p( f q )  p( f q )  a0 ( f q ) .
Eq. (20) also satisfies the limit condition that when the pipe is without PWT, then V equals 0
and f q becomes f q , and then Eq. (20) degenerates to Eq. (18).
Based on Eqs. (16) and (17), p( f q ) can be written as a1 f q3  a0 ( f q ) . Because the less joints
are needed for calibration, the easier use of the method, for simplification and easy to calibrate, takingp( f q )  a2 f q3(21)

when using one joint whose PWT volume is known (i.e., V is known) for calibration, it is easy tosolve the a 2 from Eq. (20), and then from Eq. (21), the p( f q ) can be easily calculated.
In this paper, joint No. 2 (with known PWT volume of 358.9 mm3, see Table 1) is used forcalibration.
Finally, from Eq. (20), the PWT volume is evaluated to be as follows,

Veval  [qgd1  l0 ( f q )  2lTotal ] / p( f q )
106

(22)

L.S. Liu / Remote Detection and Quantitative Evaluation ofWall Thinning Volumes in a Metal Pipe

4.

Results Analysis and Conclusion

When calibrating l0 using the method described in detail in previous work [12], the microwave
signals at the frequency range from 14.00 to 14.21 GHz which contains ten NRFs was measured, andall these ten NRFs were used for curve-fitting. In the evaluation of PWT degrees, the same frequency
range was measured and analyzed. It is found that for all the resonance frequencies, the proposed
method gives almost the same evaluation result. For conciseness, only the experimental results at thefrequency range from 14.08 to 14.12 GHz are presented here.
-48-50

Amplitude, A / dB

-52
-54
-56
No. 0
No. 1
No. 2
No. 3
No. 4
No. 5
No. 6
No. 7
No. 8

-58
-60
-62
-64
14.08

14.09

14.10

14.11

14.12x109Sweeping frequency, f / Hz
Fig. 4. Experimental results of amplitudes versus sweeping frequencies of microwaves when
the pipe is connected with different PWT joints

Fig. 4 shows the measured amplitudes of microwave signal versus the sweeping frequencies, inthe case that the pipe connected with the joints from No. 0 to No. 8. It can be found that the resonancefrequencies (peaks of waveforms) are changed due to the wall thinning, with the increase of the PWTvolume, the resonance frequencies decrease step by step. It is in accordance with the fact that the
wavelength of guided wave is correlative with the inner diameter of the waveguide.
From the waveforms at frequencies 14.08 ~ 14.10 GHz, it is found that for the PWT joints No. 0
to No. 8, with the increase of 562.9 mm3 PWT volume, the resonance frequencies decrease from
14.0828 to 14.0923 GHz, i.e., 9.5 MHz frequency change is found, considering the resolution of the
microwave instrument, this method is quite useful for detecting the PWT values.
To determine the high precision inner diameter of the pipe, the cut-off frequency for TM01-mode
is found to be between 13.47670 and 13.476775 GHz from 4 times of repetitive measurement(disassemble and reassemble the pipe and sensor for each time), and then from Eq. (19), the average
inner diameter d E is calculated to be 17.02823 mm with evaluation error less than 0.05 μm.
To determine the parameter l0 , ten neighboring resonance frequencies were measured in theexperiment by sweeping frequency at 14.00 ~ 14.21 GHz, and from Eq. (10), ten corresponding
wavelengths are calculated. Then using the calibration method described in Ref[11], the q and theten different l0 corresponding to the ten wavelengths are solved. It is solved as q0  51 . Then theten calculated l0 and the corresponding wavelengths are shown together in Fig. 5 in form of blue
triangle markers. In Fig. 5, using Least Square method, linear curve-fitting is used, and the linear107

L.S. Liu / Remote Detection and Quantitative Evaluation ofWall Thinning Volumes in a Metal Pipe

curve-fitting results matches well with the experimental ones.
With two undetermined coefficients, linear expression of l0 can be written asl0  a1gd1  b1

(23)

By using the linear curve-fitting in Fig. 5, the a1 , b1 are solved to be a1  3.612239210-3
and b1  7.5125426  10-5 , and then the expression of l 0 is achieved.

0

Propagation route parameter, l / mm

35

Experimental results
Linear curve fitting

34

33
32
31
30
29

28
27
66

68

70

72

74

76

78

Wavelength in the main pipe, λgd1 / mm
Fig. 5. Experimental method to determine the path length l0 using neighboring resonance frequencies

As described in Section 3.2, to solve the a2 in Eq. (21), joint No. 2 (with known PWT volumeof 358.9 mm3, see Table 1) is used for calibration. It is solved as a2  1.57942 1026 (m2  s3 ) .
q are q0  4 and q0  5 for the two groups of resonance frequencies shown in Fig. 4. To thecomparatively lower frequency results at range 14.08 ~ 14.10 GHz shown in Fig. 4, the PWT volumes
are evaluated by using the resonance frequencies extracted from Fig. 4 and Eqs. (21), (22) and theparameters a1 , b1 , a2 , q solved above. Fig. 6 shows the evaluated PWT volumes in comparison with
the nominal ones of the 9 joints shown in Table 1.

Resonance frequency, f / GHz14.092

Experimental results

14.091

Evaluated results

14.09014.08914.08814.08714.08614.08514.08414.083
0

100

200

300

400

Wall thinning volume, V / mm

500

600

3

Fig. 6. The evaluated results comparing with the experimental ones of the comparatively lower
frequency results in Fig. 4

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L.S. Liu / Remote Detection and Quantitative Evaluation ofWall Thinning Volumes in a Metal Pipe

The triangle markers in Fig. 6 show the relationship of the resonance frequencies and the PWTvolumes. It can be found that with the increase of the PWT volume (i.e., with the aggravation of thePWT degree), the resonance frequency decreases step by step. The PWT volumes of the experimental
results shown in Fig. 6 are in correspondence with the ones shown in Table 1, the corresponding
numbers of the data for the joints can be confirmed by comparing the volume values.
The circle markers in Fig. 6 show the evaluated results when using the same resonance frequencies
obtained from the experiment, i.e., the resonance frequencies used for obtaining the evaluated results
are completely the same to that of the experimental ones.
From Fig. 6, for pipe having 17.03 mm inner diameter, it is found that most of the evaluatedresults agree well with the experimental ones, and the maximum evaluation error is less than 13.0%
except for only one joint (joint No. 1). It indicates that this method can be used for remote detection
and quantitative evaluation of PWT volumes. It is estimated that the evaluation errors mainly comefrom the approximations approached in solving the resonance equations derived in this method.

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