E-Journal of Advanced Maintenance Vol.7-1 (2015) 108-116 Japan Society of Maintenology
Reliability Based Thermal Fatigue Evaluation Method against Random Fluid Temperature Fluctuations Masaaki SUZUKI1,* and Naoto KASAHARA1 1 The University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-8656, Japan ABSTRACT Japanese regulations for thermal fatigue issues are mainly based on the guideline of the Japan Society of Mechanical Engineers (JSME). However, the methods of the guideline include unclear safety margins, owing to existing uncertainties on loading and strength parameters. As a first step toward improving the JSME guideline to quantify safety margins with probabilistic approaches, a reliability evaluation procedure for thermal fatigue of mixing tees suitable for codes and standards has been discussed in this paper. While Monte Carlo simulations would be unsuitable for engineering use due to its cumbersome process and requirement for professional knowledge of probabilistic theory, simplified reliability analysis methods, such as the advanced first-order second-moment method, are more applicable in code designs. In addition, partial safety factors are easier to create a simpler measure of safety margins for code users who are unfamiliar with probabilistic reliability analysis methods. Thus, this paper has proposed a practical reliability evaluation procedure, which is based on the load and resistance factor design concept, for the thermal fatigue induced by random fluid temperature fluctuations in a mixing tee.
Thermal Fatigue, Mixing Tee, Best Estimate, Uncertainty, Reliability, Frequency Response, Power Spectral Density, Equivalent Stress, Load and Resistance Factor Design, Partial Safety Factor
ARTICLE INFORMATION
Article history: Received 21 November 2014 Accepted 13 April 2015
1. Introduction
Incomplete turbulent mixing of hot and cold fluids in a mixing tee induces random temperature fluctuations at the pipe wall. Due to heat transfer, local temperature gradients appear in the wall structure and cause transient thermal stress. Such a phenomenon is referred to as thermal striping. Thermal stress, which repeatedly occurs over the operational lifetime of a plant, can lead to thermal fatigue failure (e.g., [1], [2]). Since thermal fatigue is a coupled thermal, hydraulic and mechanical phenomenon, thermal fatigue evaluation contains many uncertainty factors in both load prediction and strength evaluation processes. A treatment of the existing uncertainties and their effect on fatigue damage calculation is crucial to assessing the actual safety of a design.
Current Japanese regulations for thermal fatigue issues are mainly based on the guideline of the Japan Society of Mechanical Engineers (JSME) published in 2003 [3]. The method behind the guideline is deterministic, conservative, and therefore, practical. However, this method includes unclear safety margins, owing to existing uncertainties on loading and strength parameters. Several researchers have examined the effect of loading and strength scattering on the prediction of fatigue damage by applying probabilistic reliability analysis that incorporates loading and strength uncertainties (e.g., [4], [5]). While a reliability analysis can quantitatively evaluate the safety margin of a design, the analysis would be impractical due to the cumbersome process (such as Monte Carlo sampling) and requirement for professional knowledge of probabilistic theory; consequently it would be unsuitable for codes and standards.
This paper aims at developing the reliability evaluation procedure for thermal fatigue of a mixing tee suitable for codes and standards. The evaluation procedure that is developed should simultaneously satisfy the following requirements: best estimation, probabilistic evaluation, and practicality. Figure 1 presents an approach for fulfilling each requirement. To rationalize and simplify
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the reliability evaluation procedure that is proposed, authors employ a thermal stress evaluation method based on power spectral density (PSD) and introduce equivalent stress (denoted by the red area in Fig. 1). We then develop a practical reliability evaluation procedure based on the load and resistance factor design (LRFD) concept by using a new reliability verification equation that contains equivalent stress and by deriving partial safety factors (PSFs) (denoted by the blue area in Fig. 1).
Fig. 1. Approach to develop the reliability evaluation procedure for thermal fatigue of a mixing tee
suitable for codes and standards.
2. Rational Evaluation of Thermal Stress Induced by Fluid Temperature Fluctuation
2.1. Rational evaluation of thermal stress based on the frequency response function and
PSD
Fluctuations in fluid temperature in a mixing tee have been measured experimentally and classified into three flow patterns: wall jet, deflecting jet, and impinging jet [6]. A non-dimensional PSD of the fluid temperature fluctuation was obtained for each type of flow pattern.
When a fluid temperature fluctuation is transferred to a pipe wall and converted to stress, the frequency of the fluctuation has a significant influence on the actual stress. In order to quantify the frequency-dependent attenuation characteristics of thermal stress, Kasahara et al. derived a frequency response function of thermal stress to fluid temperature fluctuations G(Bi, jf*, Rm, Rb), which is dependent on the Biot number Bi ∝ h (h: heat transfer coefficient), the non-dimensional frequency f* = fL2/α (f: frequency of the fluid temperature fluctuation, L: wall thickness of structure, α: thermal diffusivity of structural material), and the constraint efficiency factor of membrane stress (Rm) and bending stress (Rb) [7]. The gain of the frequency response function G represents the stress conversion ratio from the fluid temperature range. By using the frequency response function, the PSD of fluid temperature can be transferred to the PSD of thermal stress as follows [8]:
While the PSD-based thermal stress evaluation method can consider the frequency-dependent stress conversion ratio and therefore calculate thermal stress with high accuracy, current guideline of JSME employ the constant stress conversion ratio which corresponds to the maximum value of that used in PSD-based method (Fig. 2). In this study, the PSD-based method is employed to evaluate the thermal load in a best estimate manner. 2.2. Simplified evaluation method of the equivalent stress based on the PSD of stress
The PSD of thermal stress is generally transformed into a thermal stress time series using inverse Fourier transform by adding a random phase. The pairs of stress amplitude and number of cycles are then calculated by using a cycle counting algorithm, such as the Rain flow counting method. However, this option would also be impractical due to its cumbersome process.
To simplify the thermal stress evaluation process from the PSD of thermal stress, an equivalent stress amplitude method with a constant frequency has been proposed [9]. The equivalent stress amplitude Se and the constant frequency ν0, which approximate the original random stress time series (Fig. 3), can be derived directly from the spectral moment λ of the PSD of thermal stress based on
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statistical theory. The i th-order spectral moment λi of PSD P(ω) is calculated as follows:
𝜆𝑖 = ∫ |𝜔|𝑖 ∙ 𝑃(𝜔) 𝑑𝜔+∞−∞ (2)
By assuming that the thermal stress time series induced by thermal striping is a stationary random process, we can derive the equivalent stress amplitude Se as the expected value of amplitude of the stress time series, the constant frequency ν0 as the expected rate of occurrence of up-crossing of zero, and the equivalent number of cycles Ne from the spectral moments of the PSD of thermal stress as follows:
𝑃𝑠 = �𝜆0 (3)
𝑁𝑠 = 𝜈0 ∙ 𝑇𝑜𝑡, 𝜈0 = 12𝜋�
𝜆2𝜆0
(4)
where Top is the operational period. In Ref. [9], results of thermal fatigue evaluation obtained via three evaluation methods,
namely—finite element analysis (for reference), the JSME guideline, and equivalent stress amplitude method,—were compared under various conditions to validate the equivalent stress amplitude method. While the JSME guideline results were too conservative (a maximum of 104 times more conservative than the FEM results) and varied greatly depending on the evaluation condition, the equivalent stress amplitude method results showed moderate and almost homogeneous conservativeness (around 10 times more conservative than the FEM results). In this study, therefore, the equivalent stress amplitude method is employed to concisely evaluate the thermal load.
3. Development of a Practical Reliability Evaluation Procedure Based on LRFD 3.1. Load and Resistance Factor Design
There are multiple levels of structural reliability analysis, depending on their accuracy and simplicity. The simplest reliability analysis methods do not directly use probabilistic methods, such as Monte Carlo simulations, but rather use suitable safety factors in order to perform reliability evaluations. One such approach is the LRFD safety-checking format as expressed below [10]:
𝑅𝑃𝑃𝑃𝑅
> ∑ 𝑃𝑃𝑃𝐿,𝑖 ∙𝑖 𝐿𝑖 (5)
where R denotes the resistance, Li represents the load effects, and PSFR and PSFL,i indicate the partial safety factors for resistance and load effects, respectively. If code developers derive suitable partial safety factors depending on a target reliability from the probabilistic characteristics of random
Fig. 2. Comparison of the stress conversion ratio used in the JSME guideline and PSD-based method (Bi=2, Rm=0, Rb=1).
Fig. 3. Definition of equivalent stress amplitude Se and constant frequency ν0. (Si: i th stress level; nSi: number of cycles associated with each stress level Si; and N: total number of cycles included in the original stress time series)
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variables R and Li—mean value of R and Li (𝜇𝑅 and 𝜇𝐿𝑖), and standard deviation of R and Li (𝜎𝑅 and 𝜎𝐿𝑖)—by using accurate probabilistic methods, code users can utilize Eq. (5) to achieve the desired reliability without any professional knowledge of probability theory in a manner similar to conventional allowable stress design method.
Figure 4 illustrates a general flow chart of the reliability evaluation using LRFD. First, in the preparation phase of PSFs (denoted by the red area in Fig. 4), a code developer defines the limit state function Z (a generalized form is given by Z = R - L) that corresponds to the limit state for the failure mode selected. Random variables are then identified and their uncertainties are modeled/evaluated under some representative conditions. Finally, the PSFs of each random variable for each condition are derived from the probabilistic characteristics of each random variable and target reliability via a reliability analysis method, such as an Advanced First-Order Second-Moment (AFOSM) method [11]. In the actual evaluation phase (denoted by the blue area in Fig. 4), code users set the evaluation condition and the allowable probability of failure and then select the PSFs corresponding to those conditions. Finally, the reliability of the design is verified by using Eq. (5) (i.e., the requirement for the factored strength to be larger than a summation of the factored loadings).
Fig. 4. General flow chart of the reliability evaluation using LRFD. 3.2. Formulation of a new reliability verification equation using equivalent stress
A new reliability verification equation that contains the equivalent stress amplitude Se and number of cycles Ne has been developed [12]. The limit state for crack initiation can be expressed by using the Palmgren-Miner linear rule as follows:
1 = ∑𝑛𝑆𝑖
𝑁𝑓(𝑃𝑖)𝑖 (6)
Assuming a Stromeyer-type material fatigue curve for crack initiation with the stress factor FS:
𝑃𝑃 ∙ 𝑃 = 𝑎 ∙ 𝑁𝑓−1 𝑚⁄ + 𝑏 (7)
we can derive the limit state function for crack initiation with the equivalent stress amplitude Se and number of cycles Ne as follows:
1 = 𝑁𝑠 ∙1𝑡𝑚
(𝑃𝑃 ∙ 𝑃𝑠 − 𝑏)𝑚 (8)
where the stress factor FS is an adjustment factor intended to consider influences from factors such as fatigue data scatter, size effect, and surface roughness. While the best-fit fatigue curve corresponds to FS = 1, the fatigue strength is reduced when FS > 1 or magnified when FS < 1. For example, in the Rules on Design & Construction for Nuclear Power Plants of the JSME [13], the stress factor was set at 2 in order to obtain the design fatigue curve from the best-fit fatigue curve. Finally, substituting the partial safety factors yields the following new reliability verification equation:
1 > 𝑃𝑃𝑃𝑁𝑒∙𝑁𝑒𝑡𝑚
�𝑃𝑃𝑃𝑃𝑆 ∙ 𝑃𝑃 ∙ 𝑃𝑃𝑃𝑃𝑒 ∙ 𝑃𝑠 − 𝑏�𝑚 (9)
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where the loading variables are Se, the equivalent stress amplitude; and Ne, the equivalent number of cycles; and the strength variable is FS, the stress factor applied on the fatigue curve. Meanwhile, a, b, and m denote constant fatigue curve parameters. 3.3. Modeling uncertainties
Dominant uncertainty parameters are required to be extracted from the many uncertainty factors. In this study, two parameters, one from the calculation of the loadings, and the other from the strength, were selected as they convey much uncertainty due to their inherent randomness or scattering. These parameters were namely the heat transfer coefficient and the stress of the material fatigue curve. 3.3.1. Uncertainty of the heat transfer coefficient
The heat transfer coefficient is involved in the calculation of thermal stress through the Biot number, which appears in the frequency response function. In unsteady turbulent flow such as in a mixing tee, the heat transfer coefficient can become several times larger than its value in steady turbulent flows. Although the heat transfer mechanism is well understood in steady turbulent flows, a shortage of data and studies in unsteady turbulent flows has obliged the JSME guideline to adopt sometimes over conservative heat transfer coefficient values. The JSME guideline proposed a correction factor Fp to calculate the unsteady heat transfer coefficient hu from its steady value hs, which is determined by the Dittus-Boelter equation.
ℎ𝑡 = 𝑃𝑡 ∙ ℎ𝑠 (10)
The values of Fp can attain 4.3 or 5.4 in the case of a wall jet or an impingement jet. While the international benchmark analysis of thermal striping organized by the International
Atomic Energy Agency [14] indicates that the most dominant factor for crack initiation is the attenuation of fluid temperature fluctuation, which can be quantified by the frequency response function, the unsteady heat transfer coefficient has significant amounts of both aleatory and epistemic uncertainties. Therefore, the heat transfer coefficient is chosen as a dominant uncertainty parameter in this study. To model the uncertainty of the heat transfer coefficient of the unsteady turbulent mixing hu in a physical yet practical manner, we proposed a normal distribution on the correction factor Fp (Fig. 5). The maximum value Fp,max was selected from the JSME guideline. The minimum value Fp,min was defined as Fp,min = 1 in order for the heat transfer coefficient to be bounded from below by its fully-developed turbulent value (derived from the Dittus-Boelter equation). The standard deviation σ was defined so that the difference between Fp,min and Fp,max equated to 6σ. The probability distribution on the heat transfer coefficient of the unsteady turbulent mixing hu is obtained from Eq. (10). As the frequency response function depends on the Biot number, which is proportional to the heat transfer coefficient, the definition of uncertainty of the heat transfer coefficient has a direct effect on the gain of the frequency response function (Fig. 6), and therefore, on the equivalent stress amplitude Se and number of cycles Ne.
Fig. 5. Definition of uncertainty on the correction factor Fp (and therefore the heat transfer coefficient): Wall jet case.
Fig. 6. Effect of uncertainty on the heat transfer coefficient on the gain of the frequency response function (1,000 samples).
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3.3.2. Uncertainty on the material fatigue curve
Multiple studies aimed at determining the fatigue curves of materials have faced large scatterings in the experimental data obtained. Such scatterings introduce uncertainty on both the stress and the number of cycles (life) of the material fatigue curve. To model the uncertainty on the material fatigue curve, we select the stress factor FS as one of the dominant uncertainty parameters, since the uncertainty on the stress appeared to be dominant over the uncertainty related to life in the sensitivity analysis conducted in Ref. [5]. Assuming that the stress factor is regarded as the product of sub-factors on data scatter, size effect, and surface roughness, and those sub-factors are random variables, the probability distribution of the stress factor can be determined via Monte Carlo simulation [15, 16]. For the purposes of simplicity, in this paper, only data scatter is considered as the sub-factor in the evaluation example, which is presented in the next section (Section 3.4), to demonstrate the feasibility of our reliability evaluation procedure. We assumed that the probability distribution for data scatter follows the log-normal distribution by referring to Ref. [16] (Fig. 7). The mean material fatigue curve is considered as the mean of stress factor distributions. The maximum boundary was selected from the JSME criteria as being a factor of 2, and the minimum boundary as 1/2, its symmetrical value across the mean curve on a log scale. We set the standard deviation to be ±3σ for the maximum and minimum boundaries. Figure 8 shows the material fatigue curves with uncertainty on the stress factor FS.
3.4. Calculation of partial safety factors
Partial safety factors can be derived from the probabilistic characteristics (mean value and
standard deviation) of each random variables (Se, Ne, and FS) and target reliability by using a reliability analysis method. In this paper, the mean value and standard deviation of the loading variables Se and Ne have to be evaluated under representative conditions, while those of the strength variable FS can be obtained from its uncertainty model defined in Section 3.3.2. Applying the PSD-based method, equivalent stress amplitude method, and Monte Carlo method, which consider the uncertainty of the heat transfer coefficient, we calculate the mean values and standard deviations of the loading variables. The partial safety factors of each random variable for each condition are then derived by using the AFOSM method.
As an evaluation example, partial safety factors were derived for the conditions described in Table 1. Table 2 lists the partial safety factors obtained. In this evaluation example, each partial safety factor has a small magnitude as the allowable probability of failure Pf was set to 0.1, which is relatively high. Figure 9 shows the relationship between the partial safety factor of the equivalent number of cycles Ne and allowable probability of failure Pf. Notice that the partial safety factor PSFNe increases as the allowable probability of failure Pf decreases.
Fig. 8. Effect of uncertainty on the stress factor on the material fatigue curve (1,000 samples).
Fig. 7. Definition of uncertainty on the stress factor FS .
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Once the tables of partial safety factors are prepared, code users can evaluate the reliability of a
design using Eq. (9) without any detailed knowledge of probability theory. The proposed reliability evaluation procedure for the thermal fatigue of a mixing tee is summarized in Fig. 10. First, in the loading evaluation process, code users classify the flow pattern as a wall jet, a deflecting jet, or an impinging jet according to the momentum ratio between both the main and branch pipes, which can be calculated from design specifications (flow rates and pipe diameters). The non-dimensional PSD chart of fluid temperature [6], which corresponds to the flow pattern identified, can then be selected. By using the frequency response function G, which can be theoretically derived from design specifications (material properties, wall thickness, constraint condition, and unsteady heat transfer coefficient which is calculated from flow rates, fluid properties and mean value of the correction factor Fp), the PSD of fluid temperature is transferred to the PSD of thermal stress (Eq. (1)). The equivalent stress Se and the equivalent number of cycles Ne are then derived directly from the spectral moments λi of the PSD of thermal stress by employing Eq. (2)-(3). After calculating the loadings, code users select a material fatigue curve (which in this study is the Stromeyer type curve) and set its parameters (constant parameters a, b, m, and mean value of the stress factor FS=1). Next, the PSFs corresponding to the evaluation condition/design specification (such as the temperature difference between hot and cold fluids, the flow pattern, Biot number, and material) and the required reliability are selected. Finally, the reliability of the design is verified via Eq. (9).
To set the appropriate required (or target) reliability, the correspondence relationship between each evaluation result, which is obtained from the proposed procedure and the JSME guideline for the same evaluation conditions, should be clarified quantitatively.
Fig. 10. Proposed reliability evaluation procedure for thermal fatigue in a mixing tee.
Table 1. Evaluation conditions.
Fig. 9. Partial safety factor of equivalent number of cycles Ne for various values of allowable probability of failure Pf.
Table 2. Evaluation example of partial safety factors.
[17]
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4. Conclusions
A practical reliability evaluation procedure that incorporates loading and strength uncertainties
for the thermal fatigue induced by random fluid temperature fluctuations has been proposed. To evaluate thermal stress in a best estimate manner without any cumbersome process, a thermal stress evaluation method based on PSD has been employed. An equivalent stress method, which can be derived directly from the PSD of thermal stress based on statistical theory, has also been employed to simplify the loading evaluation process.
In order for code users to perform reliability evaluations without any professional knowledge of probability theory, a new reliability verification equation that explicitly contains probabilistic variables on loading and strength (Se, Ne, and FS), and a reliability evaluation procedure based on LRFD have been developed. Partial safety factors provided by the LRFD can simply quantify safety margins with considering failure probabilities.
Improving the uncertainty distribution to better describe the actual scattering is an area of future research. Additional data from the numerical analysis of thermal striping may allow the definition of more precise uncertainty models. It is also crucial to ensure consistency of concept/data of uncertainties with related codes and standards.
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