Development and Applicability Evaluation of Frequency Response Function of Structures to Fluctuations of Thermal Stratification

Transcription

1 E-Journal of Advanced Maintenance Vol.7- (05) -7 Japan Society of Maintenology Development and Applicability Evaluation of Frequency Response Function of Structures to Fluctuations of Thermal Stratification Kohei SODA,, Masaaki SUZUKI, Naoto KASAHARA, Daniel CONSTANDA, and Hiroshi KURIBAYASHI The University of Tokyo, School of Engineering, 7-3-, Hongo, Bunkyo-ku, Tokyo , Japan ABSTRACT The oscillation of a thermal stratification layer can induce thermal fatigue damage on structures with nuclear components. To evaluate the thermal stress induced by thermal stratification oscillation, a frequency response function was developed in our previous research. However, this function does not consider the thickness of the stratified layer. Thus, it is difficult to evaluate the stress generated by actual thermal stratified layers having finite thicknesses with sufficient accuracy. To clarify the effects of layer thickness on induced thermal stress, finite element simulations were conducted under various fluid conditions. As a result, it was clarified that the non-dimensional layer thickness Ht, which is the ratio of layer thickness to layer oscillation length, can explain the thermal stress response mechanism with layer thickness. Based on the clarified mechanisms, the frequency response function was improved. Applicability of the proposed function to a closed branch pipe of a Light Water Reactor (LWR) and the upper plenum of a pressure vessel of a Fast Breeder Reactor (FBR) was validated through comparison with finite element simulations. KEYWORDS Thermal Stress, Thermal Fatigue, Thermal Stratification Oscillation, Frequency Response Function, Finite Element Simulation, Temperature Attenuation ARTICLE INFORMATION Article history: Received 0 November 04 Accepted 4 December 04. Introduction High cycle thermal fatigue induced by temperature fluctuations of a coolant has varies with the type of thermal load. Such typical cases are thermal stratification oscillation in a closed branch pipe of a Light Water Reactor (LWR) (Fig.) and thermal striping at a Tee-junction []. In a closed branch pipe of a LWR, fluid oscillation occurs with random frequencies at a stratified layer between a hot and cold fluid. This induces a temperature distribution in the structure, which leads to thermal fatigue damage on materials. In the current guidelines of the Japan Society of Mechanical Engineers (JSME) [], thermal stratification oscillation is evaluated by the location of a stratified layer. If the stratified layer exists at an elbow of the pipe, the design of the pipe must be modified. Thermal fatigue damage is not quantitatively evaluated in the JSME guidelines. To evaluate thermal fatigue quantitatively, two types of Frequency Response Function (FRF) of thermal stress (the non-layer-thickness FRF) was already developed [3, 4] One is the one-dimensional FRF, which evaluates the thermal stress induced by the thermal striping [3]. The other FRF is the non-layer-thickness FRF [4], which evaluates thermal stress induced by the oscillation of thermally stratified layer. The non-layer-thickness FRF was developed by adding the effects of the oscillation of the layer to the one-dimensional FRF. The frequency of the fluid temperature fluctuations has a significant influence on the actual stress. FRFs can consider the frequency-dependent attenuation characteristics of thermal stress [5]. However, the model used in developing the non-layer-thickness FRF function does not take the thickness of the stratified layer into account. Thus, it is impossible to evaluate the stress generated by the oscillation of the thermal stratified layer, which has finite thickness, with sufficient accuracy. Corresponding author, soda@n.t.u-tokyo.ac.jp ISSN /0 00 JSM and the authors. All rights reserved.

2 To evaluate the effect of layer thickness, finite element simulations were conducted with models that have finite thicknesses of a stratified layer in this study. Based on the Finite Element Method (FEM) simulation results, the thermal stress response mechanisms were clarified, which enabled development of a finite-layer-thickness FRF. Thermal stratification oscillation also occurs at an upper plenum of a pressure vessel of a Fast Breeder Reactor (FBR) [6]. Thus, applicability of the developed FRF to the upper plenum of a pressure vessel of a FBR was investigated. Fig.. Thermal stratification oscillation at closed branch pipe of LWR []. Frequency Response Function of Thermal Stress Thermal stress generated by fluid temperature fluctuations attenuates in accordance with the frequency of the fluid temperature fluctuations. Frequency response characteristics of the structure to the fluid temperature fluctuations are shown in Fig.. When the frequency of the fluid temperature fluctuations is low, the temperature in the structure becomes so homogenized that the temperature gradient in the wall-thickness direction, which leads to the thermal stress, is small. Thus, low thermal stress occurs at low frequencies. Meanwhile, when the frequency of the fluid temperature fluctuations is high, the fluid temperature changes so rapidly that the surface of the structure cannot respond to the fluid temperature change. Thus, low thermal stress occurs at high frequencies. Therefore, the maximum thermal stress occurs at intermediate frequencies [3]. The one-dimensional FRF was developed for considering frequency dependent thermal stress of one dimensional temperature gradient [3]. Furthermore, this model was extended to the non-layer-thickness FRF G(Bi,jf,Rm,Φ ) [4] for evaluation of thermal stress by thermal stratification attenuation effects. Thermal stress σ(bi,jf,rm,φ ) can be calculated by multiplying the fluid temperature fluctuations Tf (jf) and the non-layer-thickness FRF G(Bi,jf,Rm,Φ ) as shown in Eq.(). This function consists of H(Bi,jf) and S(jf,Rm,Φ ), where H(Bi,jf) represents the effective heat transfer function, which evaluates the heat transfer process from the fluid to the inner surface of the structure; and S(jf,Rm,Φ ) denotes the effective thermal stress function, which evaluates both the heat conduction process from the inner surface to the inside of the structure and the thermal stress generation process from the structural temperature distribution. Eα σ ( Bi, f ) = G( Bi, f ) Tf ( jf ) () ν G Bi, f ) = H ( Bi, jf ) S( f ) () ( Bi H ( Bi, jf ) = (3) Bi + jπf tanh( jπf ) S( f ) = f + ( f Rm)( Bm + jcm) + ( f Rb)( Bb jcb) (4) +

3 sin πf cos πf + sinh πf cosh πf Bm = (5) π f cos ( πf )cosh ( πf ) + sin ( πf )sinh ( πf ) sin πf cos πf sinh πf cosh πf (6) Cm = π f cos ( πf )cosh ( πf ) + sin ( πf )sinh ( πf ) sinh πf sin πf sin πf cos πf + sinh πf cosh πf 3 πf Bb = (7) π f cos ( πf )cosh ( πf ) + sin ( πf )sinh ( πf ) sinh πf cos πf sinh πf sin πf sin πf cos πf sinh πf cosh πf + 3 πf πf (8) Cb = π f cos ( πf )cosh ( πf ) + sin ( πf )sinh ( πf ) where ν denotes the Poisson's ratio, E represents the Young's modulus, and α indicates thermal diffusivity. Bi=hL/k denotes Biot number and f=fl /a represents the non-dimensional frequency of fluid temperature fluctuations (h is the heat transfer coefficient, L is the thickness of the structure, k is the thermal conductivity, f is the frequency of the fluid temperature fluctuations, and a denotes the thermal diffusivity). Furthermore, Rm is the membrane constraint factor, which indicates the conversion efficiency from the membrane component of the temperature distribution to thermal stress; and Rb denotes the bending constraint factor, which indicates the conversion efficiency from the bending component of the temperature distribution to thermal stress. These factors are calculated as follows: σ ( Tm), Rm = Eα Tm ν σ ( Tb) (9) Rb = Eα Tb ν where Δσ(ΔTm) shows the amplitude of the thermal stress, which is caused by the amplitude of membrane temperature ΔTm, Δσ(ΔTb) shows the amplitude of the thermal stress, which is caused by the amplitude of bending temperature ΔTb. Δσ(ΔTm) and Δσ(ΔTb) can be calculated by static FEM simulations. In addition, Φ is a waveform parameter [4]. Φ was introduced to develop the non-layer-thickness FRF based on the one-dimensional FRF. One major difference between the non-layer-thickness FRF and the one-dimensional FRF is the assumption of the waveform of the temperature history of the fluid that is in contact with one point of inner surface of the structure. In the non-layer-thickness FRF, the waveform is rectangular. On the other hand, the waveform is sinusoidal in the one-dimensional FRF. The amount of heat transferred from a fluid to the surface of the structure, which affects the thermal stress, varies with the waveform. The difference of the waveform can be covered by Φ. Φ is calculated using Eq.(0). represents the amount of heat transferred. depends on the waveform. Meanwhile, SinusoidalWave is the amount of heat transferred when the waveform is a sinusoidal wave, which one-dimensional FRF assumes. φ = (0) Sinu soidalwave and SineWave can be calculated theoretically using Eq.(). T represents the period of the oscillation of the layer, Tf(t) denotes the temperature of the fluid, and Ts(t) shows the temperature of the inner surface of the structure. Ts(t) can be theoretically obtained by approximating the structure as a semi-infinite solid. If the waveform is a sine wave, Φ =.0, and if the waveform is a rectangular wave, Φ =.6. In 3

4 the model of the non-layer-thickness FRF, the waveform is a rectangular wave. Thus, Φ =.6 [4]. T = h ( Tf ( t) Ts( t)) dt () 0 However, the non-layer-thickness FRF does not consider the thickness of a stratified layer. As the actual stratified layer has some thickness in thermal stratification oscillation as shown in Fig. 3, the non-layer-thickness FRF faces difficulty in evaluating the stress generated by oscillations of the thermal stratified layer having a finite thickness with a sufficient accuracy. Fig.. Frequency response characteristics of the structure to fluid temperature fluctuations Fig. 3. Fluid temperature distribution around the thermal stratified layer 3. The finite element model for thermal stress simulations FEM simulations were conducted to investigate the influences of layer thickness. Figure 4 shows the model for fluctuations of thermally stratified layer. The fluid temperature is divided into three sections, namely the hot section, cold section, and temperature transition layer. In the transition layer, the temperature changes linearly along the axial direction. The thickness of the transient section in the axial direction is defined as the layer thickness Ht. The layer of this model moves in the axial direction with a sinusoidal velocity at certain frequencies with a layer oscillation length Lm. The non-layer-thickness FRF assumes fluctuations of thermally stratified layer when Ht = 0. Figure 5 shows the structural model with boundary conditions. A straight pipe is made of 36FR, where the inner surface has a thermal boundary with a Biot number of.0. From the FEM results, gains of thermal stress, which show the conversion efficiency of thermal stress from fluid temperature amplitude, were evaluated as shown in Fig. 6. This figure shows thermal stress response to the frequency of layer oscillation. It was clarified that both layer thickness Ht and layer oscillation length Lm influences thermal stress response. Ht and Lm can vary independently by the design specification. Thus, there are enormous combinations of the values of Ht and Lm, which are used to check the influence on thermal stress response. Fig. 4. Fluctuations of thermally stratified layer Fig. 5. Mechanical and thermal boundary conditions of the pipe model Fig. 6. Thermal stress gain from FEM Results. 4. Relationship between layer thickness and layer oscillation length In order to clarify the influence of layer thickness Ht and layer oscillation length Lm on thermal 4

5 stress response, the relationship among Ht, Lm, and gain of thermal stress was investigated. Since the thermal stress mainly occurs by the temperature gradient in the wall-thickness direction as described in section, the temperature history of the fluid in contact with the inner surface of the structure mainly influences the thermal stress. Therefore, we focused on the temperature history of the fluid that was in contact with the point of interest. Figure 7 shows the temperature histories of the fluid at various Ht and Lm. When Ht = 0, the waveform of the fluid temperature was rectangular; when 0 < Ht <, the waveform was similar to a trapezoid; and when Ht, the waveform was sinusoidal. Figure 8 shows the temperature histories of the fluid when the ratio of Ht and Lm is identical. The waveform of fluid temperature history was identical when the ratio of Ht and Lm was identical. Fig. 7. Fluid temp. history at fixed Lm and various Ht Fig. 8. Fluid temp. history focusing on the ratio of Lm and Ht Therefore, it is conceivable that the thermal stress response is similar if the ratio of Ht and Lm is identical. The non-dimensional layer thickness Ht was defined by Eq.() and a description of thermal stress response by Ht was proposed. Ht Ht = () Lm Figure 9 shows the thermal stress response for cases with the same Ht. It was clarified that the influence of Ht and Lm on thermal stress could be represented by Ht. Thermal stress response was rearranged as shown in Fig. 0. The blue line in Fig. 0 represents the gain of thermal stress obtained by the non-layer-thickness FRF. The FRF overestimated thermal stress when Ht. Meanwhile, the FRF slightly underestimated thermal stress when Ht <. Fig. 9. Thermal stress gain for cases with the same Ht. Fig. 0. Rearrangement of thermal stress by Ht. 5

6 5. Thermal stress response mechanism and improvement of the frequency response function In order to clarify the mechanism of thermal stress response, focus was given to fluid temperature history. When the non-dimensional layer thickness Ht, the waveform of fluid temperature was a sinusoidal wave as shown in Fig.. On the other hand, the non-layer-thickness FRF assumed the model where Ht = 0 and the fluid temperature waveform was rectangular. Different waveforms caused differences in thermal stress response. Figure also shows that the amplitude of fluid temperature halved when Ht = and decreased by three times when Ht = 3 than when Ht <. Therefore, the amplitude of fluid temperature became Ht times smaller when Ht.The thermal stress is proportional to the fluid temperature as shown in Eq.(). Thus, thermal stress reduced as Ht increased. Based on the two cases of Ht area (i.e., when Ht and Ht = 0), the non-layer-thickness FRF was improved. First, in the model of the non-layer-thickness FRF, the waveform of fluid temperature was rectangular. Thus, the waveform parameter Φ, which was described in section, was defined as.6. However, when Ht, the waveform of the fluid temperature history was sinusoidal. Thus, the waveform parameter Φ should be changed from.6 to.0. When Ht, the fluid temperature amplitude became Ht times smaller than when Ht <. Thus, the induced thermal stress was inversely proportional to Ht compared with the prediction of the non-layer-thickness FRF. The improved finite-layer-thickness FRF proposed was given by the following formula: H ( Bi, jf ) S( f ) G'( Bi, f ) = (3) Ht φ =.0 (4) The FEM results and the finite-layer-thickness FRF are compared in Fig.. Note that the finite-layer-thickness FRF could evaluate thermal stress rationally when the non-dimensional layer thickness Ht. Fig.. Fluid temperature histories. Fig.. Prediction results of the proposed finite-layer-thickness FRF. 6. Application to the upper plenum of a reactor vessel of a FBR In addition to a closed branch pipe of a LWR, thermal stratification oscillation also occurs at the upper plenum of a reactor vessel of a FBR. There are two characteristics at the reactor vessel of a FBR. The first is a high Biot number, which is around 7.0. Since the coolant is liquid metal, the heat transfer coefficient is large. The second characteristic is a very thin wall. The inner radius is 0700 mm and the wall thickness is 50 mm. Thus, investigation on whether the finite-layer-thickness FRF could evaluate thermal stress induced by thermal stratification oscillation at the reactor vessel of a FBR was conducted. 6

7 Figure 3 shows the thermal stress response for cases with the same Ht. Note that the influence of Ht and Lm on thermal stress could be represented by Ht in the case of the reactor vessel of a FBR. The improved FRF was applied to evaluate thermal stress when Ht as shown in Fig. 4. The finite-layer-thickness FRF could evaluate the maximum thermal stress induced at intermediate frequencies precisely. However, this developed function slightly overestimated thermal stress at low frequencies and underestimated thermal stress at high frequencies. The reason for overestimation at low frequencies and underestimation at high frequencies was not clarified. For further precise evaluation at the upper plenum of a reactor vessel of a FBR, a study on this imprecise evaluation was proposed for future research. Fig. 3. Thermal stress gain for cases with the same Ht Fig. 4. Prediction results of finite-layer-thickness FRF for upper plenum of FBR 7. Conclusion The influence of stratified layer thickness on the thermal stress response to thermal stratification oscillation was studied in this research. The ratio of layer thickness to layer oscillation length, which was defined as the non-dimensional layer thickness Ht, was found to be an important factor to thermal stress response. The influence of layer thickness and layer oscillation length to thermal stress could be represented by Ht. Based on the clarified mechanisms, a finite-layer-thickness FRF was proposed. The proposed function could evaluate thermal stress induced by thermal stratification oscillation at the closed branch pipe of a LWR when Ht. Furthermore, applicability of the proposed finite-layer-thickness FRF to the reactor vessel of a FBR was studied. The proposed function could evaluate the maximum gain of thermal stress, which was generated at intermediate frequencies, induced by thermal stratification oscillation when Ht. However, the proposed function could not evaluate thermal stress at low frequencies and high frequencies precisely. Investigation of this imprecise evaluation should be conducted in the future. References [] N. Kasahara, T. Itoh, M. Okazaki, Y. Okuda, M. Kamaya, A. Nakamura, H. Nakamura, H. Machida, M. Matsumoto: ''Development of Thermal Fatigue Evaluation Methods of Piping Systems'', E-Journal of Advanced Maintenance, Vol. 6, No., pp. 4-3 (04) [] The Japan Society of Mechanical Engineers: ''Guideline for Evaluation of High-cycle Thermal Fatigue of a Pipe'', JSME S07 (003). [3] N. Kasahara, A. Yacumpai, H. Takasho: ''Structural Response Diagram Approach for Evaluation of Thermal Striping Phenomenon'', Proc. SMiRT5, F05/4 (999). [4] K. Soda, T. Mizutani, N. Kasahara: ''Thermal Stress Response to Boundary Oscillation between Hot and Cold Fluid Temperature'', Proc. ASME PVP04, PVP (04). [5] O. Gelineau, J. P. Simoneau, M. Sperandio, J. Guinovart: ''Review of Predictive Methods Applied to Thermal Striping Problems and Recommendations'', Proc. SMiRT5, F06/3 (999). [6] Y. Chikazawa, K. Aoto, H. Hayafune, Y. Ono, S. Kotake, M. Toda, T. Ito: ''Conceptual Design for a Large-Scale Japan Sodium-Cooled Fast Reactor () Feasibility of Key Technologies'', Proc. ICAPP ', Paper 78 (0). 7