Application of time-temperature superposition method in thermal aging life prediction of shipboard cables

Vol. 13 No. 4 December 2014 Journal of Chongqing University (English Edition) [ISSN 1671-8224] doi:10.11835/j.issn.1671-8224.2014.04.04 o cite this article: DENG Wen-dong, CHEN Yi-yuan. Application of time-temperature superposition method in thermal aging life prediction of shipboard cables [J]. J Chongqing Univ: Eng Ed [ISSN 1671-8224], 2014, 13(4): 142-150. Application of time-temperature superposition method in thermal aging life prediction of shipboard cables DENG Wen-dong 1,, CHEN Yi-yuan 2 1 State Grid Chongqing Jiangbei Power Supply Company, Chongqing 401147, P. R. China 2 Chongqing Electric Power College, Chongqing 400053, P. R. China Received 17 September 2014; received in revised form 28 September 2014 Abstract: he life of shipboard cables will decrease due to the complex aging processes. In terms of the safety perspective, remaining life prediction of the cable is essential to maintain a reliable operation. In this paper, firstly, based on Arrhenius equation, residual life of new styrene-butadiene cable is calculated; result indicates that the degradation rate which changes with time is proportional to thermal temperature. hen second order dynamic model is adopted into the residual life prediction, combined with the time-temperature superposition method (SP), and a new residual life model is proposed. According to the accelerated thermal aging experiment data and Arrhenius equation, SP method demonstrates to be an efficient way for life prediction, and life at normal temperature can be estimated by this model. In order to monitor the state of styrene-butadiene cable more accurately, an improved residual life model based on equivalent environment temperature of cable is proposed, and life of cable under real operation is analyzed. Result indicates that this model is credible and reliable, and it provides an important theoretical base for residual life of cables. Keywords: insulated cable; thermal aging; Arrhenius equation; SP; residual life CLC number: M247 Document code: A 1 Introduction a he safety of cable operation mainly depends on cable insulation [1], especially for shipboard cables which are subjected in service to many operating environmental conditions simultaneously, including oxidative atmosphere and elevated temperature which cause deterioration due to thermal aging [2]. In terms of economic and safety perspective, residual life of a DENG Wen-dong ( 邓文东 ): 12799118@qq.com. cable is essential to maintain an efficient operation, especially for a shipboard cable. hrough accelerated thermal aging experiment, appropriate data extrapolation method and a life prediction model are essential to predict the operation state of cable. Efforts to confirm material aging states completely have been made, for example, Oyegoke et al. [3] and Wang et al. [4] adopted isothermal relaxation current technique in cable life assessment, and the main method is based on Arrhenius equation [5-7] and first order reaction dynamic model which focus on the calculation of active energy; residual life of styrene-butadiene rubber cables has 142

been researched by Wang [9] and Wei [2] ; based on Arrhenius curve and extrapolation method, the aging condition of insulation material at other temperatures was researched by Dixon [8] and Wise et al. [9] and DSC [10] method is also used widely for life prediction. However, the degradation rate of insulation material is not a constant all the time; it changes with time after the value of aging characteristic parameter drops below a certain threshold [11-12]. Hence, in this study, the second order dynamic model was used in the residual life model, combined with the time-temperature superposition method (SP). A new residual life model based on SP and second order dynamic model was proposed. According to accelerated thermal aging experiment data and Arrhenius equation, SP method demonstrates to be an efficient way for life prediction, and residual life at normal temperature can be estimated by this model and can prove whether the aging characteristic of high temperature accords with low temperature or not. In order to monitor the state of styrene-butadiene cable accurately, an improved life prediction model based on equivalent environment temperature of cables was proposed, and the cable life under real operation was analyzed. his paper is organized as follows: in Section II, the accelerated experimental setup is described in detail; life prediction model based on Arrhenius equation and the active energy are analyzed in section III; in Section IV, the SP method for life prediction is discussed firstly, then, combined with second order dynamic model, a new life prediction model is proposed, finally, a residual model based on equivalent environment temperature is proposed and cable life under real operation is analyzed; and Section V concludes the study. 2 Experimental setup he styrene-butadiene cable was used for accelerated thermal aging experiment, and elongation at break of cable was selected as deterioration characteristic. According to standard GB/20028-200, the failure point was selected as 50% retention of the origin value. Dumb-bell samples shown in Fig. 1 were used for the experiment, made strictly according to GB/ 2951.1-1997; each group included 5 samples [13]. An constant air aging oven was used for thermal experiment, and the experiment temperature was selected as 125 C, 135 C and 145 C,the time interval was 24 h, and the aging period was chosen as arithmetic progression. Samples of different aging degrees must be cooled for 24 h at indoor temperature before the tension test. Accelerated aging experiment lasted till the desired extent of degradation was achieved. Fig. 1 Dimensions of a dumb-bell test piece, unit: mm 3 Life prediction based on Arrhenius equation A. Life prediction equation he method widely used for life prediction is based on Arrhenius equation, and the cable life under thermal aging can be described as[ 10] : b lg t a, (1) where t is the life of insulation material, h; b=0.401 E a /R, E a is the active energy, kj/mol, R is the gas constant which is 8.314 J/mol, is thermal temperature; a is the constant of material characteristic. J. Chongqing Univ. Eng. Ed. [ISSN 1671-8224], 2014, 13(4): 142-150 143

By processing the experiment data with regression analysis method [14], the elongation at break changing with aging time at different temperatures 125 C, 135 C, and 145 C is shown in the following equations, respectively. 2 3 Y 2.704 17.689 lg t 32.173( lg t) 13.395( lg t), (2) 2 3 Y 15.417 23.353lg t 16.118( lg t) 3.693( lg t), (3) 2 3 Y 47.527 65.811lg t 33.09( lg t) 5.527( lg t), (4) where Y is the elongation at break; t is the aging time, h. Curves of elongation at break at different thermal temperature varying with logarithm of aging time are shown in Figs. 2 to 4. he elongation at break at different temperature varying with aging time is shown in Fig. 5. Results indicate that elongation at break decreases with the increasing of aging time, and the degradation rate is not constant but changing with the aging time; in addition, the degradation is positively related to temperature. In conclusion, temperature is a critical factor for aging of shipboard cables. As the failure point is 50% retention of the origin value. According to Eqs. (2) to (4), the time-to-failure at different temperature are t 1 =89.80 h, t 2 =196.84 h, t 3 =475.87 h, respectively. Based on Eq. (1), 1/ and lgt can be assumed as x and y. Eq. (1) is simplified as: y a bx. (5) By processing the experiment data with the least square method, the coefficients a and b can be calculated: b a xy x y /3 6 269.1, 2 2 x ( x) / 3 y b x 13.073. 3 (6) (7) Fig. 2 Elongation at break of new cable samples at different aging times under 398 K Fig. 3 Elongation at break of new cable samples at different aging times under 408 K 144 J. Chongqing Univ. Eng. Ed. [ISSN 1671-8224], 2014, 13(4): 142-150

Fig. 4 Elongation at break of new cable samples at different aging times under 418 K Fig.5 Curves of elongation at break of new styrene-butadiene cable during different aging days Hence, the life prediction equation of styrene-butadiene cable can be described as: lgt 13.073 6 269.1/. (8) he active energy E a is calculated by E a =b R/0.401 which is 129.97 kj/mol. 4 Life prediction model based on SP theory 4.1 Second order dynamic model As research has indicated that degradation rate of insulation material is not a constant, but changes with time [15]. Hence, in this work, the second order dynamic model is adopted for life prediction, and it can be described as: Y k 2 2 0 (1 e kt t Y ), (9) k10 where Y 0 is the origin value of elongation at break, Y t is the elongation at break at the time t, k 2 and k 10 are coefficients of degradation rate he residual life model of shipboard cables derived from Eq. (9) is as follows. k t (10) 1 ln[1 2 (Y end Y0 )], k2 k10 where Y end is the elongation at break at the time to failure he coefficients of k 2 and k 10 can be achieved by fitting the data curve at different temperatures. However, because the degradation rate is not a constant in the second order dynamic model, the residual life at other temperature is hard to be derived [16-17]. hus, SP theory is adopted into the life prediction of styrene-butadiene cables, and an improved residual life model is proposed. 4.2 SP theory and procedure he main goal of accelerated thermal experiment is to obtain operation characteristics of the material under low temperature through principle data at a high temperature. SP is based on the theory that parameters changing in chemical structure under accelerated experiment accords with that at low temperature for a long time [1,18], and can be used for judging whether the characteristic of a cable at a high J. Chongqing Univ. Eng. Ed. [ISSN 1671-8224], 2014, 13(4): 142-150 145

temperature coincides with that at a low temperature. As a method of extrapolation, the main procedures include the following steps. Step 1. he construction of master curve and calculation of shift factor α Curves of elongation at break of new styrenebutadiene cable varying with aging time at different temperature must be given in the same coordinate, and reference temperature t ref is selected, other curves move along the time axis until they become one part of the reference curve and form a smooth master curve. he shift factor α for temperature can be defined as: /, tref t (11) same thermal characteristic, ln(e a ) varies with (1/ ref 1/) linearly, and the slope is E a /R. So the active energy E a can be easily calculated. 4.3 Application of SP By processing the experiment data under three different temperatures with SP method, 125 C is set as the reference temperature, the procedure and master curve are shown in Fig. 6 and 7. Based on the shift factor data α obtained by applying SP method, the curve about ln(e a ) and (1/ ref 1/) is shown in Fig. 8. where t is the aging time at temperature before moving, t ref is the aging time at the reference temperature, and the corresponding shift factor α ref =1. Step 2. he calculation of active energy E a of Arrhenius equation. he aging characteristic of insulation material obeys the Arrhenius equation at a low temperature: k E (12) R a () Aexp( ), where A is the pre-exponential factor. Because the chemical reaction rate k relatives with life and changing of aging characteristic, applying internal theorem on equation (12) yields: Fig.6 Curves of elongation at break of styrene-butadiene cable varying with exponential logarithmic of aging time t Y Y E 2 0 a exp( ). (13) A R Substituting Eq. (11) into (13), the shift factor for the same insulation material can be evaluated by E 1 1 a exp( ( )). R ref (14) More detailed information can be seen Ref. [18-19]. It is indicated that the shift factor α under different temperature can be described by Eq. (14) under the Fig.7 SP master curve using shift factors 146 J. Chongqing Univ. Eng. Ed. [ISSN 1671-8224], 2014, 13(4): 142-150

Lref L, 1 0.002 71 Lref ln[1 (Yend Y 0 )], 0.002 71 0.002 3 123.7610 1 1 exp( ( )). 8.314 273 125 (16) 4.4 Residual life model based on load Fig. 8 temperature Relationship between shift factor and thermal According to the straight line slope as shown in Fig. 8, the active energy magnitude is 123.76 kj/mol, which is almost equal with that calculated by the least square method in Section III. Hence, SP can be an efficient way for life prediction. hen, the life prediction model based on second order dynamic and SP is proposed. By fitting the master curve with the second order dynamic model, the coefficients k 2 and k 10 at reference temperature can be calculated. So the residual life L ref can be easily achieved and by dividing the relevant shift factor α, life of styrene-butadiene cable under other temperature can be obtained. Hence, the residual life model based on second order dynamic and SP is written as: Lref L, 1 k2 Lref ln[1 (Y end Y 0 )], k2 k10 Ea 1 1 exp( ( )). R ref (15) By fitting the master curve, the values of k 2 and k 10 are 0.002 71 and 0.002, respectively. herefore, the residual life model based on the second order dynamics and SP can be rewritten as: According to heat transfer calculation theory, heat spreads from the core of cable, and the temperature near the core is higher than other parts. his position will first probably break down, especially under overload condition [20]. Hence, the temperature in Eq. (16) should be core temperature. Based on different loads and environment temperature, the core temperature can be calculated as [20] : K (17) 2 c e R, where c is the core temperature; e is the equivalent environment temperature; K R is the coefficient of load, which relates with rated load and real load; Δ is the temperature difference between core and environment at rated load. Because the environment temperature changes with season and other factors, and the aging life of a cable varies with temperature nonlinearly. So, the mean temperature cannot be regarded as the equivalent value. In this work, only season factor is taken into account. According to Eq. (15), the exponential form of aging equation can be simplified as: Ea t K0 exp( ). (18) R o obtain the equivalent environment temperature, one year is divided into twelve months, and statistic temperature data of each month must be known. Applying the normalization method, the aging life for the ith month is t e /nt i, and for the whole aging process, J. Chongqing Univ. Eng. Ed. [ISSN 1671-8224], 2014, 13(4): 142-150 147

n te 1 1. (19) n t i1 i Substituting (18) into (19) results in n 1 Ea Ea e exp( ) exp( ) 1, (20) n R R e i1 where the active energy E a is 123.76 kj/mol. i Finally, the equivalent environment temperature can be rewritten as: e R E a n Ea [ln n ln exp( )] i1 R i. (21) For styrene-butadiene cable, the current capacity is 426 A, and the core temperature and environment temperature are 80 C and 45 C respectively. In real operation, the load is 380 A, and the monthly average temperature of cable laying environment is listed in able 1. able 1 Monthly average temperature of cable laying environment [21] Month emperature /C Month emperature /C 1 25.4 7 46.0 2 26.7 8 48.8 3 28.6 9 41.8 4 30.1 10 36.1 5 32.0 11 32.5 6 42.4 12 29.4 According to able 1, the equivalent temperature is 38.8 C, the load coefficient is: 380 KR 0.892, (22) 426 0 353 K 318 K 35 K. (23) he core temperature c is K (24) 2 c e R 339.6 K. he residual life of a styrene-butadiene cable is about 33 a, which coincides well with the real condition. Result indicates that residual life model based on second order dynamics and SP can be an efficient and reliable method, and it provides an important theoretical and practical basis for preventive replacement of cables. 5 Conclusions Residual life model based on second order dynamic model and SP method is proposed in this paper, and its application into styrene-butadiene cables is analyzed. he calculated results presented in this paper lead to the following conclusions. 1) Based on the accelerated thermal experiment data, the degradation rate, which changes with time, is proportional to thermal temperature. By combinedly use of the Arrhenius equation and the least square method, the life model is achieved and the calculated active energy is 129.97 kj/mol. 2) Considering the change of degradation rate of styrene-butadiene with time, a second order dynamic model is proposed, and SP method demonstrates to be an efficient method for life prediction, and the relative active energy is 123.76 kj/mol, which coincides well with that of Arrhenius equation. SP method can be a way to judge whether the characteristic of a cable at high temperature accords with that at low temperature. 3) A new residual life model based on SP and second order dynamic model is proposed. Life at normal temperature can be estimated. In order to monitor the state of styrene-butadiene cables more accurately, equivalent environment temperature of cable is taken into account. Based on this model, the residual life of a styrene-butadiene cable under the load 148 J. Chongqing Univ. Eng. Ed. [ISSN 1671-8224], 2014, 13(4): 142-150

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