Evaluation of general applicability of microwave NDT to pipes with a bend focusing on mode conversion
公開日:
カテゴリ: 第15回
モード変換を考慮したマイクロ波探傷法の曲がり部を持つ配管への一般適用性評価
Evaluation of general applicability of microwave NDT to pipes with a bend focusing on mode conversion
東北大学大学院
陳
冠任
Guanren CHEN
Non-Member
東北大学大学院
片桐
拓也
Takuya KATAGIRI
Non-Member
東北大学大学院
遊佐
訓孝
Noritaka YUSA
Member
東北大学大学院
橋爪
秀利
Hidetoshi HASHIZUME
Member
Abstract
This study evaluates the key factors impacting on multi-mode conversion of microwave when applying microwave NDT to bent pipe inspection. Mathematical derivation demonstrates the dependence of multi- mode conversion due to a bend on three factors: the ratio of bending radius to pipe’s diameter, the bend angle and the frequency normalized with the cut-off frequency of an arbitrary mode. To verify the deducting results, numerical simulations were conducted to evaluate multi-mode conversion of bent pipe adopting TE01 microwave mode as excitation. Simulation results showed that in spite of different inner diameters, transmission characteristics of microwaves propagating in bent pipes manifest excellent consistency if the pipes are characterized by the same three aforementioned factors.
Keywords: Microwave NDT, mode conversion, bend, pipe
1 Introduct on
Metallic pipes are widely utilized in many industrial facilities, where failures of pipes [1] may cause immense economic losses, severe environmental pollutions and even personnel casualties. Therefore, developing efficient non-destructive pipe inspecting methods becomes increasingly essential and imperative.
Microwave NDT [2] has been proposed as an advanced technology which can implement high-speed and long-range pipe inspection. Microwaves are excited at one end of the pipe and propagate inside it with very low attenuation. Areflection will occur if a flaw is situated on the inner surface of the pipe, and the location of the flaw, relative to the excitation probe, can be determined by analyzing the TOF (time of flight) of reflection signal caused by the flaw.
The applicability of microwave NDT to bent pipe inspection
has been verified experimentally in a former study [3]. Reflection signals from circumferential pipe wall thinning located behind a bend are discernable for detection, whereas they are not as clear as the results obtained using straight pipes. It is because the mode conversion of microwave at bends causes the presence of several modes, which gives rise to multiple peaks and obscure reflection in time domain. Factors affecting mode conversion due to bend were investigated preliminarily [4]. The results showed that mode conversion due to a bend was determined by three factors: the ratio of pipe’s bending radius to diameter, the normalized frequency and the bend angle. However, mathematical derivation in the former study only demonstrated the dependence of single- mode conversion, while multi-mode conversion usually occurs with the width of sweeping frequency span increasing. Moreover, in the numerical simulation of the former study, only TM01 mode was used as excitation, and only one scenario of simulation condition was discussed.
In order to evaluate the general applicability of microwave NDT to bent pipe inspection, the generality of the dependence of multi-mode conversion due to a bend should be studied deeply
and comprehensively. In this study, the dependence of multi- mode conversion due to one bend on aforementioned three factors was demonstrated theoretically. Corresponding numerical
proportional relationship between wavenumber k and frequency
f, r?m'n' is calculated as follow:
2u r
simulations for different scenarios were conducted with TE01 microwave mode employed as excitation.
2 Mathemat caI der vat on
Fig. 1 illustrates the mode conversion at a bend. r, D and α
r?m'n' ? r
? 2u01r
D
?01
D
? f (fr
0 f,)D
cM01
(3),
denote bending radius, pipe’s diameter and bend angle. z is the length of the curve along which microwaves propagate. Mode conversion of microwave occurs at the bend region.
where Xm’n’ is the nth zero of Jm (X) (for TM modes) or Jm’(X) (for TE modes), Jm (X) is m-rank Bessel function. kcM01 and fcM01 denote the cut-off wavenumber and the cut-off frequency of
TM01 mode, and
u01
is given by u01 = 2kcM01/D. From (4),
it is evident that rβm’n’ is dependent on r/D as well as f/fcM01. Virtually, f can be normalized with the cut-off frequency of any mode.
Besides, taking the coupling coefficient from TE to TM mode
(m'n')(mn)
as an instance,
rC士can be expressed as follows:
Fig. 1. Illustration of mode conversion at a bend
According to the previous study on mode conversion due to bend [5], mathematical expressions of multi-mode coupling can be interpreted in (1).
士
(m 'n ')(mn)
rC
??
f
f fcM01
?dA???
こ 士士
? f1 ( f)
m'n '
?
? dz
j?m'n ' Am 'n ' ? j
mn
C(m 'n ')(mn) Amn
cM01
?
dA-
?m'n ' ? j?
A?? jこC士
(1),
A
where umn and vm’n’ are constants determining the cut-off
frequencies of TMmn and TEm’n’ modes. The result indicates that
ヽ dz
m'n ' m 'n '(m 'n ')(mn) mn mn
rC士
only hinges on f/fcM01. Similarly, by substituting other
where A, C and β denote the amplitude of the converted mode, coupling coefficient and phase constant, respectively. m, n, m’ and n’ are mode numbers, and | m ? m’ | = 1. ‘+’ and ‘?’ refer to the forward direction and backward direction of propagation.
Since it is obvious to see that dz = r ? da, from Fig. 1, (1) can be rewritten as:
(m'n')(mn)
coupling coefficients (TM to TE, TM to TM, TE to TE) into rC士 , the same consequences are able to be acquired accordingly.
(m'n')(mn)
Based on the above deductions, it can be easily concluded that the coefficients of (2) depend on r/D and f/fcM01, while the solution to (2) is a function of α Technically, how modes are converted at
?dA???
こ士士
a bend can be determined if r/D, f/fcM01 and α are designated.
m'n '
?
? d?
jr?m'n ' Am'n ' ? j
mn
rC(m'n ')(mn) Amn
3 Numer caI s muIat on
?
dA-
?m'n ' ? jr?
ヽ d?
(2)
士
A? jこrCA
m'n ' m'n '(m 'n ')(mn) mn
mn
3 1 S muIat on modeI and parameters
Three-dimensional finite element simulations were performed
According to the definition of phase constant and theusing commercial COMSOL Multiphysics v5.0 with its RF
module. The governing equation of the simulation is:transmission characteristics of bent pipes with the same r/D, the
?? ??1(?? E) ? k 2[? ? j? / (?? )]E ? 0 (5),
r0r0
where k0 = ?E0?0 is the propagation constant in a vacuum, ε0 and μ0 are permittivity and permeability in a vacuum, ω is the
angular velocity. E denotes the vector of electric field intensity, whilst σ is the electrical conductivity. εr and μr are relative permittivity and relative permeability.
The geometric model of simulation is illustrated in Fig. 2. TE01 microwave mode was excited at one end of the pipe (I), while transmission characteristics of converted modes as well as TE01 mode were evaluated at the other end (II). An additional PML (perfectly matched layer) was deployed at the end II to eliminate the reflections at the surface. Second-order tetrahedral were used for discretization. The simulation parameters are listed in Table 1.
Fig. 2. Geometric model of numerical simulation
Table 1 Simulation parameters
Parameter (unit)
value
D (mm)
12.7, 57.5, 74
r/D (-)
1, 2, 3, 4, 5
α (o)
90, 180
f (GHz)
30.5- 38.5 (D = 12.7 mm)
10 ? 12.4 (D = 39 mm)
5.2 - 6.6 (D = 74 mm)
f/fcM01 (-)
1.68 - 2.12 (approx.)
f/fcE01 (-) *
1.05 - 1.33 (approx.)
(*: fcE01 is the cut-off frequency of TE01 mode)
3 2 S muIat on resuIts
Simulation results of two scenarios (a = 180°, r /D = 1 and
a = 90°, r /D = 3) are displayed in Fig. 3 and Fig. 4. In both two scenarios, when microwaves propagate through three bent pipes of different D, the transmission characteristics are different if f were not normalized with fcM01 or fcE01. After normalization, the
same α and the same range of f/fcM01 reveal good consistency, despite different D (Some modes were not plotted in Fig. 4 because their values are too small). Also, from (b) and (c) in Fig. 3 and 4, it is easy to see that the cut-off frequency used for normalization can be arbitrary and has no influence on results.
Without normalization
With normalization (f normalized with fcM01)
With normalization (f normalized with fcE01) Fig. 3. Transmission characteristics (ex = 180°, r /D = 1)
Without normalization
With normalization (f normalized with fcM01)
With normalization (f normalized with fcE01) Fig. 4. Transmission characteristics (ex = 90°, r /D = 3)
4 ConcIus on
This study theoretically demonstrated the dependence of
multi-mode conversion due to a bend on the ratio of bending radius to diameter, the normalized frequency and the bend angle. The results of numerical simulations using TE01 mode as excitation comply with those of mathematical derivation, and the generality with respect to the dependence of multi-mode conversion due to bend has been verified.
References
http://www.sozogaku.com/fkd/en/
K. Sugawara et al., "Development of NDT method using electromagnetic waves", JSAEM Stud. Appl. Electromagn. Mech., Vol. 10, 2001, pp. 313-316.
S. Uoshita et al., “Long-range inspection of a pipe with a bend using microwaves”, the 18th international symposium on applied electromagnetics and mechanics, Chamonix- Mont-Blanc, France, September 2017, pp. 3-6.
陳冠任ら、“マイクロ波探傷による配管検査を考慮し た曲がり部におけるモード変換の影響因子の評価”、日本非破壊検査協会東北支部第6 回支部会・講演会、
、2018, pp. 3
H. Li and M. Thumm, “Mode conversion due to curvature in corrugated waveguides”, International Journal of Electronics, Vol. 71, No. 2, 1991, pp. 333-347.