JP2011158362A - Thermal fatigue evaluation method - Google Patents

Abstract

<P>PROBLEM TO BE SOLVED: To provide a thermal fatigue evaluation method capable of estimating accurately in a short time, a temperature change on the inner surface from a measured temperature on the outer surface of a tubular structure. <P>SOLUTION: This method includes: the first process for expressing a change with time of the inner surface temperature and a change with time of the outer surface temperature of a structure in the form of Fourier series expansion so as to satisfy a heat conduction equation, and comparing each Fourier series expansion term of the change with time of the inner surface temperature and the change with time of the outer surface temperature, to thereby determine a coefficient ratio M<SB>j</SB>and a phase delay Δθ<SB>j</SB>of each Fourier series expansion term; the second process for measuring the outer surface temperature of the structure by a temperature sensor; and the third process for developing outer surface measured temperature measured in the second process by the Fourier series, multiplying the coefficient of each Fourier series expansion term of the outer surface measured temperature by the coefficient ratio M<SB>j</SB>determined in the first process, and shifting the phase of each Fourier series expansion term as much as the phase delay Δθ<SB>j</SB>determined in the first process, to thereby estimate the change with time of the inner surface temperature of the structure. The thermal fatigue damage degree of the structure is determined based on the change with time of the inner surface temperature estimated in the third process. <P>COPYRIGHT: (C)2011,JPO&INPIT

Description

  The present invention relates to a method for evaluating thermal fatigue of tubular structures such as pipes that undergo temperature fluctuations of fluid.

  Plants such as power plants that handle high-temperature fluids are provided with many branch pipes that branch from the main pipe. At the joint between the main pipe and the branch, temperature fluctuations occur in the part where high temperature water and low temperature water merge, and in the part where thermal stratification occurs due to the temperature difference between the two fluids when the flow from the main pipe flows into the branch pipe. May occur. This temperature fluctuation (vibration) causes a stress fluctuation in the branch pipe, which causes thermal fatigue of the branch pipe.

  In order to evaluate such thermal fatigue, it is necessary to grasp the temperature change of the inner surface of the pipe. As an example of a method for grasping the temperature change of the inner surface of the pipe, it is conceivable to attach a temperature sensor to the inner surface of the pipe and measure the temperature change of the inner surface directly by this temperature sensor. In this case, it is necessary to attach temperature sensors to all places where such temperature fluctuations are expected in advance, resulting in high costs.

  Thus, various methods for estimating the temperature change of the inner surface of the pipe from the measured temperature of the outer surface of the pipe and evaluating thermal fatigue based on the estimated temperature change of the inner surface of the pipe have been studied. For example, in the method disclosed in Patent Document 1, the temperature change of the pipe outer surface is measured by a temperature sensor attached to the pipe outer surface, and the inverse analysis using the heat conduction equation is performed based on the measured temperature change of the pipe outer surface. Thus, the temperature change (temperature time distribution) on the inner surface of the pipe is obtained. Then, using the temperature change of the inner surface of the pipe obtained by the above reverse analysis, the thermal stress change of the pipe is calculated by the Green function, and the fatigue damage degree change is calculated by the thermal stress change and the internal pressure stress change of the pipe.

JP-A-10-96704

  In the method using the Green function shown in Patent Document 1, the time change of the temperature appearing on the outer surface of the pipe when the temperature change that changes with time in the delta function is uniformly applied to the inner surface of the pipe using the structural analysis. A stress change with time is prepared as a database. At this time, in the reverse analysis step, the temperature of the pipe outer surface at a certain time t is obtained by superimposing the time variation of the temperature of the pipe inner surface generated in the past (0 ≦ τ <t) and the above database. The time change of the temperature of the inner surface is calculated. For this reason, the relationship between the time change of the temperature of the pipe inner surface and the time change of the temperature of the pipe outer surface becomes a large matrix corresponding to the time step as t increases. Therefore, it is possible to estimate the one-dimensional plate thickness direction, but in the case of elbow-shaped piping that takes into account the spatial spread, a large database using many structural analyzes is prepared, and the inverse analysis process is very large. It is difficult to apply because it requires the matrix to be solved.

  The present invention has been made in view of the above, and provides a thermal fatigue evaluation method capable of accurately estimating the temperature change of the inner surface from the measured temperature of the outer surface in a tubular structure such as a pipe in a short time. The purpose is to do.

In order to solve the above-described problems and achieve the object, the thermal fatigue evaluation method of the present invention is a method for evaluating thermal fatigue of a tubular structure having a hollow portion through which a fluid flows, and the structure The time variation of the inner surface temperature and the time variation of the outer surface temperature are expressed in a form expanded by Fourier series so as to satisfy the heat conduction equation, and each Fourier series expansion term of the time variation of the inner surface temperature and the time variation of the outer surface temperature is compared. Thus, the first step of obtaining the coefficient ratio M _j and the phase delay Δθ _j (j = 1, 2,...) Of each Fourier series expansion term, and the first step of measuring the outer surface temperature of the structure with a temperature sensor 2 steps, the outer surface measurement temperature measured in the second step is expanded by Fourier series, the coefficient of each Fourier series expansion term of the outer surface measurement temperature is multiplied by the coefficient ratio M _j obtained in the first step, and Fourier series expansion terms And by shifting only the phase lag [Delta] [theta] _j obtained in the first step, anda third step of estimating the time variation of the inner surface temperature of the structure, the inner surface temperature estimated in the third step The thermal fatigue damage degree of the structure is determined based on a change with time.

In the thermal fatigue evaluation method according to the present invention, a plurality of temperature measurement points B _p (p = 1, 2,... N) for measuring the outer surface measurement temperature in a cross section perpendicular to the axis of the structure. ) Is provided on the outer surface of the structure, and a plurality of temperature estimation points A _q (q = 1, 2,... M) for estimating the time change of the inner surface temperature are provided on the inner surface of the structure. In the first step, a coefficient ratio M _jpq and a phase delay Δθ _jpq are obtained between the arbitrary temperature measurement point B _p and the arbitrary temperature estimation point A _q, and in the second step, the plurality of temperature measurements The temperature at the point B _p is measured, and the time change of the temperature at the plurality of temperature estimation points A _q is estimated using the coefficient ratio M _jpq and the phase delay Δθ _jpq in the third step. To do.

  The thermal fatigue evaluation method according to the present invention includes a fourth step for obtaining a temporal change in the temperature inside the structure using the temporal change in the inner surface temperature estimated in the third step, and a temperature inside the structure. And a sixth step of obtaining a fatigue cumulative damage coefficient based on the temporal change of the thermal stress. The fifth step of obtaining a temporal change of the thermal stress generated in the structure using the temporal change of .

In the thermal fatigue evaluation method of the present invention, the time change of the internal surface temperature of the structure and the time change of the external surface temperature are expressed in a form developed by Fourier series so as to satisfy the heat conduction equation. By comparing each Fourier series expansion term of time change, a coefficient ratio M _j and a phase delay Δθ _{j of} each Fourier series expansion term are obtained. Then, the outer surface temperature of the pipe is measured with a temperature sensor, the outer surface measured temperature is expanded by a Fourier series, the coefficient of each Fourier series expansion term of the outer surface measured temperature is multiplied by the coefficient ratio M _j , and each Fourier series expansion is performed. By shifting the phase of the term by the phase delay Δθ _j , the time change of the temperature of the pipe inner surface is estimated. By performing the above procedure, it is possible to accurately estimate the temperature change of the inner surface of the pipe in a short time without requiring a huge calculation as in the prior art. As a result, the thermal fatigue evaluation of piping can be performed more efficiently than in the past.

  Further, according to the thermal fatigue evaluation method of the present invention, even in the case where the temperature distribution is in a cross section perpendicular to the axis of the structure, the temperature distribution on the inner surface of the structure is not required, as in the conventional case. In addition, the temperature change can be calculated accurately in a short time.

  Moreover, according to the thermal fatigue evaluation method of the present invention, it is possible to accurately evaluate the load applied to the structure, and accurately predict when to replace or repair the structure.

FIG. 1 is a cross-sectional view of a pipe to be evaluated, and is a diagram conceptually showing how temperature is transmitted from the inner surface of the pipe to the outer surface of the pipe. FIG. 2 is a flowchart showing the procedure of the thermal fatigue evaluation method of Examples 1 and 2. FIG. 3 is a block diagram of an analysis apparatus that executes the thermal fatigue evaluation method according to the first and second embodiments. FIG. 4 is a cross-sectional view of a pipe to be evaluated, and is a diagram conceptually showing how temperature is transmitted from the inner surface of the pipe to the outer surface of the pipe.

  Hereinafter, the present invention will be described in detail with reference to the drawings. It should be noted that the present invention is not limited by the mode for carrying out the invention. In addition, constituent elements in the following embodiments include those that can be easily assumed by those skilled in the art or those that are substantially the same.

  The thermal fatigue evaluation method according to the present invention evaluates thermal fatigue damage of piping provided in a nuclear power plant such as a pressurized water nuclear plant (PWR: Pressurized Water Reactor) or a boiling water nuclear plant (BWR: Boiling Water Reactor). It is used suitably for. In addition, although the case where thermal fatigue of piping is evaluated below is described, the structure to be evaluated is not limited to piping, and fluids such as reactor vessels and melting furnaces (including gas-liquid two-phase flows) General tubular structures having a hollow portion through which the gas is distributed are to be evaluated.

[Example 1]
FIG. 1 is a cross-sectional view of a circular pipe 10 to be evaluated cut along a plane orthogonal to the axis. Although illustration is omitted, it is assumed that a fluid flows in the hollow portion 11 of the pipe 10. Further, FIG. 1 conceptually shows a state in which the temperature is transmitted from the pipe inner surface (inner surface) 12 to the pipe outer surface (outer surface) 14 through the pipe interior 13. In the figure, reference numeral 15 denotes a temperature fluctuation (vibration) of the pipe inner surface 12, and reference numeral 16 denotes a temperature fluctuation (vibration) of the pipe outer surface 14. A temperature sensor 17 such as a thermocouple for detecting the temperature of the pipe outer surface 14 is installed at a predetermined portion on the outer surface of the pipe 10, and the detection signal of the temperature sensor 17 is an analysis device (see FIG. 3) that performs thermal fatigue evaluation. Reference) is configured.

FIG. 2 is a flowchart illustrating the procedure of the thermal fatigue evaluation method of the first embodiment. In the thermal fatigue evaluation method of the first embodiment, first, the time change of the pipe inner surface 12 and the time change of the temperature of the pipe outer surface 14 are expressed in a form developed by Fourier series so as to satisfy the heat conduction equation. and by comparing the respective Fourier series expansion terms of the time variation of the external surface temperature, determined a coefficient ratio M _j and the phase delay [Delta] [theta] _j of the Fourier series expansion terms, coefficient ratio M _j and phase for each Fourier series expansion terms A database for the delay Δθ _j is created in advance (first step: step S01). Next, the temperature of the pipe outer surface 14 is measured by the temperature sensor 17 (second step: step S02). Next, the measured temperature of the pipe outer surface 14 is expanded by Fourier series, and the time change of the temperature of the pipe inner surface 12 is calculated by the inverse analysis method using the coefficient ratio M _j and the phase delay Δθ _j obtained in step S01. (Third step: Step S03). Next, the temperature change of the pipe interior 13 is calculated by solving the heat conduction equation using the temperature of the pipe inner surface 12 obtained in step S03 as a boundary condition (fourth step: step S04). Next, thermal stress analysis is performed using the temperature change in the pipe interior 13 to calculate thermal stress acting on the evaluation target part (fifth step: step S05). And based on the thermal stress calculated | required by step S05, the thermal fatigue damage calculation of the evaluation object site | part is performed (6th process: step S06).

  The first to sixth steps (steps S01 to S06) can be realized by a computer and a program executed by the computer. The program includes a computer-readable hard disk, a flexible disk (FD), It can be stored in a recording medium such as a CD-ROM, MO, or DVD. This program can be distributed via a network such as the Internet.

  Below, first, the concept of the thermal fatigue evaluation method of Example 1 and the contents of data calculated in each step will be described.

(First step: calculation of coefficient ratio and phase delay)
In the first embodiment, the pipe 10 is regarded as an infinite flat plate, the entire pipe inner surface 12 is at a uniform temperature, and the entire pipe outer surface 14 is also at a uniform temperature. Is calculated.

It is assumed that the temperature T inside the pipe can be expressed in the form of e ^{at + bx} . Here, x is a coordinate taken in a direction perpendicular to the wall surface, and x = 0 on the pipe inner surface 12 and x = h (h: plate thickness) on the pipe outer surface 14. By substituting this into the heat conduction equation, the following equation (1) is obtained. By substituting x = 0 and x = h into this (Equation 1), the temperature of the inner and outer surfaces of the pipe is expressed as in the following (Equation 2) and (Equation 3).

Next, the ratio of the coefficients is obtained by comparing j-ths of the terms in the Fourier series expansion of the inner and outer surfaces in the formulas (2) and (3). As a result, the ratio of the obtained j-th Fourier series term coefficient (coefficient ratio) is M _j (j = 1, 2,...). Here, as shown in FIG. 1, the coefficient ratio M _j is such that when the temperature vibration 15 on the pipe inner surface 12 is transmitted to the pipe outer surface 14 through the pipe inner 13 and becomes the temperature vibration 16 of the pipe outer surface 14, the amplitude increases. 1 / M _j , and the reciprocal of the coefficient ratio M _j corresponds to the attenuation rate.

Further, the phase delay Δθ _j is obtained by comparing the j-ths of the terms in the Fourier series expansion of the inner and outer surfaces in the equations (2) and (3). Here, Δθ _j indicates that the temperature vibration 15 on the pipe inner surface 12 is transmitted to the pipe outer surface 14 with a delay of the phase Δθ _j and becomes the temperature vibration 16 of the pipe outer surface 14. The coefficient ratio M _j and the phase delay Δθ _j are expressed by the following equation (Equation 4).

A database of the coefficient ratio M _j and the phase delay Δθ _j obtained by the above method is constructed. Here, the database is constructed on an information recording medium such as a hard disk in a computer executing the calculation of the first step, or an information recording medium provided outside the computer.

(2nd step: Measurement of pipe outer surface temperature)
As shown in FIG. 1, a temperature sensor 17 is used to measure a temporal change in temperature at one place on the pipe outer surface 14. The detection signal of the temperature sensor 17 is input to the analysis device 20 (see FIG. 3).

(Third step: Calculation of pipe inner surface temperature)
Next, an inverse analysis method for estimating the time change of the temperature of the pipe inner surface 12 from the measured temperature of the pipe outer surface 14 measured in the second step will be described. The result of developing the measured temperature Tout of the pipe outer surface 14 measured in the second step by Fourier series is expressed by the following equation (Equation 5).

Then, using the coefficient ratio M _j and the phase delay Δθ _j stored in the above-described database, an operation of multiplying the coefficients a _j and b _j of each Fourier series expansion term of the above equation (5) by the coefficient ratio M _j. , And the operation of shifting the phase of each Fourier series expansion term in the above equation (5) by Δθ _j . Then, the time change Tin of the temperature of the pipe inner surface 12 is expressed as the following (Equation 6).

Thus, the time change of the temperature of the pipe outer surface 14 and the pipe inner surface 12 is expressed in a form developed by Fourier series, and a database about the coefficient ratio M _j and the phase delay Δθ _j is independently provided for each term of the Fourier series. Since it was prepared and the inverse analysis was performed to estimate the temperature of the pipe inner surface 12 from the measured temperature of the pipe outer surface 14 using the value, it does not require enormous calculation as in the past, and in a short time And the time change of the temperature of the pipe inner surface 12 can be estimated with high accuracy.

(4th step: Calculation of pipe internal temperature)
Using the time change of the temperature of the pipe inner surface 12 calculated in the third step as a boundary condition, the heat conduction equation is solved by numerical calculation, and the time change of the temperature at each point inside the pipe 13 is obtained.

(Fifth step: thermal stress calculation step)
The distribution and temporal change of the thermal stress generated in the pipe 10 are calculated using the temporal change in temperature at each point in the pipe interior 13 obtained in the fourth step. The thermal stress distribution and time change can be obtained from the following equation (7). A change in stress with respect to a stepwise temperature change of the inner surface is calculated by numerical calculation using a finite element method, a finite difference method, or the like, and the stress change is obtained by superimposing according to the temperature change amount of the inner surface.

(6th process: Thermal fatigue damage calculation process)
Based on the temporal change of the thermal stress obtained in the fifth step, the thermal fatigue damage calculation of the evaluation target part is performed. In this embodiment, as an example of thermal fatigue damage calculation, a fatigue cumulative damage coefficient is obtained as described below.

First, a relationship between the thermal stress σ generated in the pipe 10 and the fatigue limit number N of how many times the thermal stress is generated in the pipe 10 is considered to have reached the fatigue limit is obtained in advance. For example, the fatigue limit number Nmax ₁ when a thermal stress of 10 Mpa to 20 MPa is applied to the pipe 10 and the fatigue limit number Nmax ₂ when a thermal stress of 20 Mpa to 30 Mpa is applied to the pipe 10 are obtained. Similarly, the fatigue limit number is obtained for all stresses of 30 MPa or more.

Then, the number of times N ₁ when a stress of ₁₀ Mpa to 20 MPa is applied and the number of times N ₂ where a thermal stress of 20 Mpa to 30 Mpa is applied are obtained from the temporal change of the thermal stress obtained in the fifth step. Similarly, the number of times is obtained for a thermal stress of 30 Mpa or more. Next, a fatigue cumulative damage coefficient U _f at a predetermined portion of the pipe 10 is obtained from the following equation (8).

If U _f <1, it is determined that the thermal fatigue level of the predetermined portion of the pipe 10 has not reached the limit, and it is not necessary to replace or repair the pipe 10. On the other hand, if U _f > 1, it is determined that the thermal fatigue level of the predetermined part of the pipe 10 has reached the limit, and the pipe 10 needs to be replaced or repaired.

Next, an analysis apparatus that executes the above-described thermal fatigue evaluation method will be described. FIG. 3 is a block diagram showing an example of the configuration of the analysis apparatus 20 that executes the above-described thermal fatigue evaluation method. The analysis device 20 illustrated in FIG. 3 is realized by causing a calculation device such as a personal computer to read a program. The coefficient ratio and phase lag calculation unit 21, the pipe outer surface temperature measurement unit 22, and the pipe An internal surface temperature calculation unit 23, a pipe internal temperature calculation unit 24, a thermal stress calculation unit 25, and a fatigue cumulative damage coefficient calculation unit 26 are provided. The coefficient ratio and phase delay calculation unit 21 includes a database 27 that stores the data of the coefficient ratio M _j and the phase delay Δθ _j described above.

The coefficient ratio and phase lag calculation unit 21 performs the arithmetic processing of the first step described above. Specifically, the coefficient ratio and phase delay calculation unit 21 calculates the coefficient ratio M _j and the phase delay Δθ _j from the above equation (4), and stores the calculated coefficient ratio M _j and the phase delay Δθ _j in the database 27. Store.

  The pipe outer surface temperature measurement unit 22 performs the second step described above, and measures the temperature of the pipe outer surface 14 based on a detection signal from the temperature sensor 17.

The pipe inner surface temperature calculation unit 23 performs the arithmetic process of the third step described above. Specifically, the pipe inner surface temperature calculation unit 23 receives the measurement temperature data of the pipe outer surface 14 from the pipe outer surface temperature measurement unit 22, expands the measurement temperature data of the pipe outer surface 14 in Fourier series, and the above (Formula 5) Create an expression. Then, the pipe inner surface temperature calculation unit 23 uses the coefficient ratio M _j and the phase delay Δθ _j stored in the database 27 to calculate the coefficients a _j and b _j of the Fourier series expansion terms in the equation (5). By performing the process of multiplying by the ratio M _j and the process of shifting the phase of each Fourier series expansion term in Expression (5) by Δθ _j , the above-mentioned Expression (6), that is, the temperature change Tin of the pipe inner surface 12 is obtained. Is calculated.

  The pipe internal temperature calculation unit 24 performs the arithmetic process of the fourth step described above. Specifically, the pipe internal temperature calculation unit 24 receives data of time variation of the pipe inner surface temperature from the pipe inner surface temperature calculation unit 23, and uses this data as a boundary condition to calculate the heat conduction equation by numerical calculation such as a finite element method. By solving, the time change of the temperature at each point inside the heat pipe 13 is calculated.

  The thermal stress calculation unit 25 performs the above-described calculation process of the fifth step. The thermal stress calculation unit 25 receives data of time variation of the pipe internal temperature from the pipe internal temperature calculation unit 24, and uses this data to calculate the above (Equation 7). The time change of the thermal stress generated in the pipe 10 is calculated from the equation.

The fatigue cumulative damage coefficient calculation unit 26 performs the calculation process of the sixth step described above, receives the thermal stress data from the thermal stress calculation unit 25, and uses the thermal stress data, the above equation (8) A fatigue cumulative damage coefficient U _f at a predetermined portion of the pipe 10 is calculated.

  The values calculated in each part of the analysis device 20 can be output through output means (not shown) such as a display or a printer.

As described above, in the thermal fatigue evaluation method of the first embodiment, first, the time change of the temperature of the pipe inner surface 12 and the time change of the temperature of the pipe outer surface 14 are expressed in a form developed by Fourier series, and each development of the Fourier series is expressed. A database for the coefficient ratio M _j and the phase delay Δθ _j is created in advance for each term. Then, the outer surface temperature of the pipe 10 is measured by a temperature sensor, the outer surface measured temperature is expanded by a Fourier series, the coefficient of each Fourier series expansion term of the outer surface measured temperature is multiplied by the coefficient ratio M _j , and each Fourier series expansion is performed. By shifting the phase of the term by Δθ _j , the time change of the temperature of the pipe inner surface 12 is estimated. By performing such a procedure, it is possible to accurately estimate the temperature change of the pipe inner surface 12 in a short time without requiring a huge calculation as in the prior art. As a result, the thermal fatigue evaluation of the pipe 10 can be performed more efficiently than before.

[Example 2]
Next, the thermal fatigue evaluation method of Example 2 will be described. In addition, since the procedure of the thermal fatigue evaluation method of Example 2 is the same as that of Example 1 mentioned above, it demonstrates using the flowchart of FIG. Moreover, since the analyzer which performs the thermal fatigue evaluation method of Example 2 is the same structure as the analyzer 20 of Example 1 shown in FIG. 3, it demonstrates using the analyzer 20 of FIG.

  The nuclear power plant described above is provided with a number of branch pipes that branch from the main pipe. For example, thermal stratification (boundary layer formed between the high-temperature fluid and the low-temperature fluid) is formed at the portion where the high-temperature water and low-temperature water merge, such as the joint between the main pipe and the branch. In addition, the position of the interface of this thermal stratification varies with time in the branch pipe. For this reason, the temperature of the pipe inner surface is not uniform, and a temperature distribution is generated in the pipe circumferential direction in a cross section perpendicular to the central axis of the pipe. In such a case, the following method is applied to estimate the temperature distribution in the circumferential direction on the inner surface of the pipe.

(First step: calculation of coefficient ratio and phase delay)
FIG. 4 is a cross-sectional view of the pipe 10 cut along a plane orthogonal to the central axis. Although illustration is omitted, it is assumed that a fluid flows in the hollow portion 11 of the pipe 10. In addition, the plurality of arrows in FIG. 4 conceptually show that the temperature at an arbitrary point on the pipe inner surface 12 is transmitted to a plurality of points on the pipe outer surface 14 through the pipe interior 13.

  In the second embodiment, in the cross section perpendicular to the central axis of the pipe 10, the pipe inner surface 12 has a temperature distribution in the circumferential direction, and the temperature distribution in the length direction (direction parallel to the axis) of the pipe 10 is uniform. Assume that there is. Similarly, the pipe outer surface 14 has a temperature distribution in the circumferential direction on the pipe outer surface 14 in a cross section perpendicular to the central axis of the pipe 10, and the temperature distribution in the length direction of the pipe 10 (direction parallel to the axis) is one. Assuming that In this case, the temperature at each point on the pipe outer surface 14 is determined by the temperatures of all points on the pipe inner surface 12.

As shown in FIG. 4, a plurality of points are provided in each circumferential direction of the pipe outer surface 14 and the pipe inner surface 12 in a cross section perpendicular to the central axis of the pipe 10. The n points (temperature measurement points) on the pipe outer surface 14 are B _p (p = 1, 2,... N), and the temperature at each point is T out _p (p = 1, 2,... N). To do. Further, m points (temperature estimation points) on the pipe inner surface 12 are A _q (q = 1, 2,... M), and the temperature at each point is Tin _q (q = 1, 2,. ).

Next, the method applied in the first step (coefficient ratio and phase lag calculation step) in Example 1 described above is expanded, and the amount of increase from the initial value of the temperature at the point A _q of the pipe inner surface 12 is the following ( It is assumed that it is changed in a form expanded by a Fourier series as expressed by equation (9).

Then, the equation of equation (9) is used as a boundary condition to solve the heat conduction equation by numerical calculation, and the time change in temperature at the point B _{p on the} pipe outer surface 14 is obtained. Then, as a result of developing the time change of the temperature at the pipe outer surface 14 by Fourier series, it can be expressed by the following equation (10).

Next, the ratio of the coefficients is obtained by comparing the j-th term obtained by Fourier series expansion of the point A _q in the equation (9) and the j-th term obtained by Fourier series expansion of the point B _p in the equation (10). Ask for. As a result, the ratio of the obtained j-th Fourier series term coefficient (coefficient ratio) is M _jpq (j = 1, 2,...). Here, the coefficient ratio M _jpq has an amplitude when the vibration of the temperature at the point A _q on the pipe inner surface 12 is transmitted to the pipe outer surface 14 through the pipe inner 13 and becomes the temperature vibration at the point B _p on the pipe outer surface 14. Is attenuated to 1 / M _jpq , and the reciprocal of the coefficient ratio M _jpq corresponds to the attenuation rate R _jpq .

The attenuation rate R _jpq and the phase delay Δθ _jpq are obtained using the formula (9) and the formula (10). Further, using the attenuation rate R _jpq and the phase delay Δθ _jpq , the complex coefficient H _jpq is obtained by the following equation (11).

Then, the complex coefficient H _jpq obtained by the above method is stored in the storage area of the analysis apparatus 20 shown in FIG.

(2nd step: Measurement of pipe outer surface temperature)
As shown in FIG. 4, a temperature sensor 17 is provided at each point B _p (p = 1, 2,... N) on the pipe outer surface 14. And each temperature sensor 17 measures the time change of the temperature of the point B _p (p = 1, 2,... N).

(Third step: Calculation of pipe inner surface temperature)
The temperature measured at the point B _p of an outer surface of a pipe 14 obtained in the above second step to expand in Fourier series, expressed by equation (12) below.

On the other hand, it is assumed that the temperature change at the point A _{q on} the pipe inner surface 12 is expanded by a Fourier series and expressed by the following formula (13).

Since the time change in temperature at all points (q = 1 to m) of the point A _{q on} the pipe inner surface 12 appears in the temperature change at the point B _{p on} the pipe inner surface 14, equation (12) is expressed in the database 27. Using the stored complex coefficient H _jpq , it is expressed by the following equation (14).

This is a simultaneous linear equation of the coefficients of the Fourier series at all points A _{q on} the pipe inner surface 12 and the coefficients of the Fourier series at all points B _{p on} the pipe outer surface 14. Here, since the relationship between the coefficients of the Fourier series in the equations (12) and (14) can be calculated independently in the frequency domain, the simultaneous linear equations including only the coefficients of the j-th Fourier series are obtained. Become.

Therefore, if ΔTout _jp is the j-th Fourier series as a complex number (a _jp + i · b _jp ) and ΔTin _jq is the j-th Fourier series as a complex number (c _jq + i · d _jq ), Equation 14 Can be expressed in the form of the following (Equation 15).

Here, the matrix component H _jpq (p = 1, 2,... N, q = 1, 2,... _M ) is a complex number and has information corresponding to R _jpq and Δθ _jpq . The step of creating a database consisting of M _jpq and Δθ _jpq in advance in the first step (step S01) corresponds to the step of creating a database consisting of H _jpq in advance.

Here, the temperature at the point B _p on the pipe outer surface 14 is determined by the temperatures at all points on the pipe inner surface 12, but H _jpq is used to determine the temperature at the point B _{p on the} pipe outer surface 14. This is a quantity representing the ratio (cross term) to which the temperature at the point A _q contributes.

Furthermore, the temperature of the pipe inner surface 12 is obtained as follows using the above equation (15). First, the matrix H of Equation (15) is divided into a matrix having only a diagonal component and a matrix having only a non-diagonal component.

The above equation (16) is transformed into the following equation (17).

_First , an appropriate initial value is assumed as ΔTin _jq . Then, the initial value and ΔTout _jp obtained by measurement are substituted into the right side of Equation (17) to obtain ΔTin _jq on the left side. The obtained ΔTin _jq and ΔTout _jp obtained by the measurement are substituted again in the right side of Equation (17) to obtain ΔTin _jq on the left side. If it is confirmed that ΔTin _jq has converged by repeating this, the value at that time is set to ΔTin _jq . Normally, the above operation is repeated 4 or 5 times to achieve sufficient convergence.

  By such a method, even if there is a temperature distribution in the circumferential direction in a cross section perpendicular to the axis of the pipe 10, the temperature distribution and temperature change of the pipe inner surface 12 can be estimated in a short time with high accuracy. It becomes possible.

If the pipe inner surface 12 has a temperature distribution in the circumferential direction in a cross section perpendicular to the axis of the pipe 10 and further has a temperature distribution in the length direction (direction parallel to the axis) of the pipe 10, (Equation 15) may be expanded to expand ΔTout _jp and ΔTin _jq to a two-dimensional matrix and H _jpq to a three-dimensional tensor.

(4th step: Calculation of pipe internal temperature)
After obtaining the temperature distribution of the pipe inner surface 12 and the time change of the temperature in the third step, the thermal conduction equation is solved by numerical calculation using the obtained temperature distribution and time change of the pipe inner surface 12 as boundary conditions in the pipe inner surface 12. The time change at each point in the pipe interior 13 is obtained.

(5th step: thermal stress calculation)
The distribution and temporal change of the thermal stress generated in the pipe 10 are calculated using the temporal change in temperature at each point in the pipe interior 13 obtained in the fourth step. For the thermal stress distribution and the change with time, the above equation (6) is used.

(6th step: Thermal fatigue damage calculation)
Based on the temporal change of the thermal stress obtained in the fifth step, the thermal fatigue damage calculation of the evaluation target part is performed. The thermal fatigue damage calculation is performed in the same manner as in Example 1 to determine whether or not the pipe 10 has reached the fatigue limit. The cumulative fatigue damage coefficient U _f is calculated using the equation (Equation 7) described in the first embodiment.

  The first to sixth steps (steps S01 to S06) in the second embodiment can be realized by a computer and a program executed by the computer. The program can be a computer-readable hard disk or flexible disk. (FD), CD-ROM, MO, DVD and the like can be stored. This program can be distributed via a network such as the Internet.

Next, an analysis apparatus that executes the thermal fatigue evaluation method of Example 2 will be described. In addition, since the structure of the analyzer which performs the thermal fatigue evaluation method of Example 2 is the same structure as the analyzer 20 used in Example 1 mentioned above, it demonstrates below using FIG. Similar to the first embodiment, the analysis apparatus 20 that executes the thermal fatigue evaluation method of the second embodiment includes a coefficient ratio and phase lag calculating unit 21, a pipe outer surface temperature measuring unit 22, a pipe inner surface temperature calculating unit 23, and a pipe. An internal temperature calculation unit 24, a thermal stress calculation unit 25, and a fatigue cumulative damage coefficient calculation unit 26 are provided. The coefficient ratio and phase lag calculation unit 21 includes a database 27 that is a storage area for storing each data of the complex coefficient H _jpq composed of the coefficient ratio M _jpq and the phase lag Δθ _jpq described above.

The coefficient ratio and phase lag calculation unit 21 performs the arithmetic processing of the first step described above. Specifically, the coefficient ratio and phase delay calculation unit 21 calculates a complex coefficient H _jpq composed of the coefficient ratio M _jpq and the phase delay Δθ _jpq from the above equation (11), and calculates the calculated coefficient ratio M _jpq and the phase delay. A complex coefficient H _jpq composed of Δθ _jpq is stored in the database 27.

The pipe outer surface temperature measuring unit 22 performs the second step described above, and measures the temperature of each point B _{p on} the pipe outer surface 14 based on the detection signal from the temperature sensor 17.

The pipe inner surface temperature calculation unit 23 performs the arithmetic process of the third step described above. Specifically, the pipe inner surface temperature calculation unit 23 receives the measurement temperature data of each point B _{p on} the pipe outer surface 14 from the pipe outer surface temperature measurement unit 22, expands the measurement temperature data in the Fourier series, ) Create an expression. Then, the pipe inner surface temperature calculation unit 23 uses the coefficient ratio M _jpq and the phase delay Δθ _jpq stored in the database 27 to calculate the coefficients a _jp and b _jp of the respective Fourier series expansion terms in the equation (12). By multiplying the ratio M _jpq and processing for shifting the phase of each Fourier series expansion term in Equation (12) by Δθ _jpq , the above Equation (17), that is, the point A _q on the pipe inner surface 12 is obtained. Calculate the change in temperature over time.

The pipe internal temperature calculation unit 24 performs the arithmetic process of the fourth step described above. Specifically, the pipe internal temperature calculation unit 24 receives the data of the time-varying temperature distribution at the point A _q on the pipe inner surface 12, and uses this data as a boundary condition for the finite element method or the like to which the heat conduction equation is applied. By performing the numerical calculation, the time change of the temperature at each point in the pipe interior 13 is calculated.

  The thermal stress calculation unit 25 performs the above-described calculation process of the fifth step. The thermal stress calculation unit 25 receives data of time variation of the pipe internal temperature from the pipe internal temperature calculation unit 24, and uses this data to calculate the above (Equation 7). The time change of the thermal stress generated in the pipe 10 is calculated from the equation.

The fatigue cumulative damage coefficient calculation unit 26 performs the calculation process of the sixth step described above, receives the thermal stress data from the thermal stress calculation unit 25, and uses the thermal stress data, the above equation (8) A fatigue cumulative damage coefficient U _f at a predetermined portion of the pipe 10 is calculated.

As described above, in the thermal fatigue evaluation method of the second embodiment, a plurality of temperature measurement points B _p (p = 1, 2) for measuring the temperature of the pipe outer surface 14 in the cross section perpendicular to the axis of the pipe 10. ,... N) are provided on the pipe outer surface 14, and a plurality of temperature estimation points A _q (q = 1, 2,... M) are provided on the pipe inner surface 12 for estimating the time variation of the pipe inner surface temperature. In the first step, the coefficient ratios M _jpq and Δθ _jpq are obtained between the temperature measurement point B _p and the temperature estimation point A _q by the procedure described above, and in the second step, a plurality of temperature measurement points on the pipe outer surface 14 are obtained. The temperature at B _p is measured, and in the third step, the temperature at a plurality of temperature measurement points B _p is expanded by Fourier series, the coefficient of each Fourier series expansion term is multiplied by the coefficient ratio M _jpq , and each Fourier series expansion term _Is shifted by Δθ _jpq to estimate the time variation of the temperature at a plurality of temperature estimation points A _{q on} the pipe inner surface 12. By performing such a procedure, even in the case where the temperature distribution is in the circumferential direction of the pipe 10 in a cross section perpendicular to the axis of the pipe 10, the temperature distribution of the pipe inner surface 12 is not required, as in the conventional case. In addition, it is possible to accurately estimate the temperature change in a short time.

  As described above, the thermal fatigue evaluation method according to the present invention is usefully used for the thermal fatigue evaluation of piping such as a power plant.

10 Piping (structure)
DESCRIPTION OF SYMBOLS 11 Hollow part 12 Piping inner surface 13 Piping inside 14 Piping outer surface 17 Temperature sensor 20 Analyzing device 21 Coefficient ratio and phase delay calculation part 22 Piping outer surface temperature measurement part 23 Piping inner surface temperature calculation part 24 Piping internal temperature calculation part 25 Thermal stress calculation part 26 Fatigue damage factor calculator

Claims (3)

A method for evaluating thermal fatigue of a tubular structure having a hollow portion through which a fluid flows,
The time variation of the inner surface temperature of the structure and the time variation of the outer surface temperature are expressed in a form expanded by Fourier series so as to satisfy the heat conduction equation, and each Fourier series expansion of the time variation of the inner surface temperature and the time variation of the outer surface temperature. A first step of determining the coefficient ratio M _j and phase delay Δθ _j (j = 1, 2,...) Of each Fourier series expansion term by comparing the terms;
A second step of measuring the outer surface temperature of the structure with a temperature sensor;
The outer surface measurement temperature measured in the second step is expanded by a Fourier series, the coefficient of each Fourier series expansion term of the outer surface measurement temperature is multiplied by the coefficient ratio M _j obtained in the first step, and each Fourier series expansion term. And a third step of estimating a time change of the inner surface temperature of the structure by shifting the phase of the phase by the phase delay Δθ _j obtained in the first step,
The thermal fatigue evaluation method characterized by determining the thermal fatigue damage degree of the said structure based on the time change of the said internal surface temperature estimated at the said 3rd process. In a cross section perpendicular to the axis of the structure, a plurality of temperature measurement points B _p (p = 1, 2,... N) for measuring the outer surface measurement temperature are provided on the outer surface of the structure. A plurality of temperature estimation points A _q (q = 1, 2,... M) for estimating temporal changes in the inner surface temperature are provided on the inner surface of the structure,
In the first step, a coefficient ratio M _jpq and a phase delay Δθ _jpq are obtained between the arbitrary temperature measurement point B _p and the arbitrary temperature estimation point A _q, and in the second step, the plurality of temperatures The temperature at the measurement point B _p is measured, and the time change of the temperature at the plurality of temperature estimation points A _q is estimated using the coefficient ratio M _jpq and the phase delay Δθ _jpq in the third step. The thermal fatigue evaluation method according to claim 1.   A fourth step of obtaining a time change of the temperature inside the structure using the time change of the inner surface temperature estimated in the third step, and generated inside the structure using a time change of the temperature inside the structure. The thermal fatigue evaluation method according to claim 1 or 2, further comprising: a fifth step for obtaining a temporal change in thermal stress to be performed and a sixth step for obtaining a fatigue cumulative damage coefficient based on the temporal change in the thermal stress. .

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