Jtam-A4.dvi JOURNAL OF THEORETICAL AND APPLIED MECHANICS 53, 3, pp. 723-730, Warsaw 2015 DOI: 10.15632/jtam-pl.53.3.723 GAS TURBINE RELIABILITY MODEL BASED ON TANGENT HYPERBOLIC RELIABILITY FUNCTION Ahmed Zohair Djeddi, Ahmed Hafaifa University of Djelfa, Faculty of Science and Technology, Djelfa, Algeria e-mail: a z djeddi@univ-djelfa.dz; hafaifa.ahmed.dz@ieee.org Abudura Salam University of Médéa, Faculty of Science and Technology, Médéa, Algeria e-mail: abudura.salam@univ-medea.dz The present work deals with the exploration of a new model proposed for the reliability analysis of industrial production systems. This proposedmodel is mainly based on the tan- gent hyperbolic function, where the survival function is determined and used in the lifetime distributionmodeling taking into accountof estimation theparameters of the proposed func- tion. On the other side, tests validation is performed using the real data of a gas turbine installation. The obtained results allow themodeling of damage effects, hence the prediction of the performance of the examined gas turbine using the proposedmodel gives good results in terms of validity. Keywords: reliability estimation, reliability algorithms, lifetime distribution,Weibull distri- bution, availability 1. Introduction Actually, the complexity of industrial plants and their equipment behavior led practically to very complex maintenance strategies that are containing a number of different tasks.Moreover, the random nature of degradation and failure that may occur makes the determination of the required strategy to fulfill thebestdecisions regarding theirmaintenance averydifficultpractical task (Costa et al., 2014; Lai, 1994; Guemana et al., 2011; Rao et al., 2005; Hasumi et al., 2009). The main aim of the proposed model is to analyze and measure industrial system reliability parameters to achieve the best time determination for making decision on the maintenance actions. Whereas, in the present work, this study is based on amodel of a gas turbine which is considered as the main important equipment of the compressor stations, gas pumps, oil pumps and petrol production. The reliability model proposed in this paper is based essentially on the elementary function of tangent hyperbolic which is proved in this paper to be theoretically and practically similar to the well know Weibull distribution which was introduced by the Swedish mathematician in 1951 (Weibull, 1951). The Weibull distribution is considered the first distribution used in reliability analysis due to its very big capacity to adapt a very large number of data sets. However, many lifetime distributions that have a bathtub-shaped hazard rate function have been introduced after Weibull distribution, in which the main aim was to improve the systems reliability. A survey presenting the state-of-the-art on the class of such distributions was presented byMoeini et al. (2013), Scott (1979), Lai et al. (1998, 2001, 2003). This work proposes an exploration of a newmodel whichwill contribute to the enhancement of the industrial systems reliability. It is obvious that the hyperbolic tangent function has a curve similar to the cumulative function of the probability distribution. Hence, a new model based on a tangent hyperbolic function defined on ℜ+ can be proposed. It is very clear that 724 A.Z. Djeddi et al. its form is similar to the form of Weibull’s model. In this model, the survival function will be studied based on it. For validation of the proposedmodel, a practical application has beenmade with real data. These data have been collected from a gas turbine operating in a natural gas transportation system which is used in the south of Algeria. The reliability approach developed in this paper allows the modeling of the effects of damage to predict the performance of the examined gas turbineoperation andgive good results in termsof validity comparedwith theWeibull approach. 2. Reliability model based on the tangent hyperbolic function The availability of the industrial equipment control allows the industries to act positively on the conformity of production, the exploitation costs and the production successful competitiveness. Indeed, the lifemanaging of the equipment that is being used in oil and gas facilities is based on taking into account theaspects of aging, the economic and regulatory factors in order to optimize the operation, maintenance and lifespan of the system structures and components. The main goal is tomaintain the level of safety and reliability aswell as tomaximize the investment return on the overall lifespan. The reliability of industrial systems is always essential and attractsmuch attention among scientific researches. It is especially important for componentswhose failure can cause major problems in terms of maintainability, availability and security (Halimi et al., 2014; Hafaifa et al., 2013a,b; Sturges, 1926; Ruji, 1990; Trofimov et al., 1978). As it is aforementioned, anewmodel basedon the tangent hyperbolic function is proposed to beused for the analysis and the study of the survival function in the reliability modeling. The tangent hyperbolic function considered in this paper is defined as a function onℜ+ (Fig. 1). It is expressed as follows F(x)= { 0 for x∈ℜ+ tanh(x) for x­ 0 (2.1) Fig. 1. Tangent hyperbolic function To assess the impact of the aging effects on the equipment by annual frequency and to identify the sensitive equipment and the requested strategies for service priority to manage the risks that are associatedwith the age of the facilities, a cumulative distribution function is used. In this case it is the limit of F(x) equal to 1 when x tends to +infinity and is equal to 0 when x tends to 0. The proposed function is a non-decreasing and a right-continuous function, so the proposed tangent hyperbolic function is fulfils these requirements and it presents a good choice as a candidate cumulative distribution function (Costa et al., 2014; Guaily and Epstein, 2013) lim x→0 F(x)= 0 lim x→+∞ F(x)= 1 (2.2) Gas turbine reliability model based on... 725 The distribution lifespan in the proposed tangent hyperbolic distribution form is deduced from theWeibull distribution. The survival function is represented as follows R(t)= 1− tanh(λt)β (2.3) where β is the shape parameter and λ is the scale parameter. The cumulative distribution function is expressed as follows F(t)= 1−R(t)= tanh(λt)β (2.4) Based on the derivative of the last function, the probability density function is obtained as follows:. f(t)= dF(t) dt = (λt)β−1λβ cosh2(λt)β (2.5) The hazard function is determined in the flowing equations Pr(t t)= f(t) R(t) ∆t=h(t)∆t (2.6) where h(t) = f(t) R(t) = (λt)β−1 λβ cosh2(λt)β 1− tanh(λt)β Using a variable converting the reliabilityR(t) replaced by the hyperbolic function, h(t) can be rewritten using exponential form as follows h(t) = 2λβ(λt)β−1 1+e−2(λt) β (2.7) 2.1. Hazard and density functions The hazard function in the proposed tangent hyperbolic model is expressed by equation (2.7) and is shown in Fig. 2. It is obvious that the curve of this function depends only on the parameter β. Therefore, the curve of this function can be discussed based on the value of β: • In the case when β=1, the hazard function tends quickly to 2λ lim t→∞ h(t) = 2λ (2.8) • In the case when β=2, the hazard function becomes h(t)= 4tλ2 1+e−2(λt) 2 → 4tλ2 (2.9) The hazard function h(t) is a straight line with slope equal to 4λ2. • In the case when 0<β< 1, the hazard function becomes h(t)→ 2λβ(λt)β−1 (2.10) The hazard function curve follows tp formwith−1 2, the hazard function curve follows tp with p> 1 (convex form). • In the case when 1<β< 2, the hazard function curve follows tp with 0>p> 1 (concave form). It can be concluded that the curves of the hazard function have theirWeibull equivalents cu- rves.On the other side, the density function of the proposed tangent hyperbolicmodel presented in equation (2.5) is presented in Fig. 3 for different values of β. 726 A.Z. Djeddi et al. Fig. 2. Hazard function determined using the tangent hyperbolic algorithm Fig. 3. Density function found using the hyperbolic model 3. Application results The data used in this paper are taken from the data history of a gas turbine (PGT10) used for natural gas transportation and installed in a gas plant located in the south of Algeria. Those data are presented in Table 1. Table 1.Examined gas turbine data N 1 2 3 4 5 6 7 8 9 10 TBF∗ 240 264 384 552 624 648 696 720 720 768 N 11 12 13 14 15 16 17 18 19 20 TBF 768 792 960 1272 1344 1464 1632 1680 1776 1944 N 21 22 23 24 25 26 27 28 29 30 TBF 1968 1992 2064 2136 2208 2376 2448 2448 2472 2664 N 31 32 33 34 35 36 37 38 39 40 TBF 2832 2880 3000 3600 3672 3720 4272 4656 5592 5856 ∗TBF is the Time Between Failures given in hours 3.1. Esytimation of the model parameters Using equation (2.4), the inverse of the tangenthyperbolic function canbeobtainedas follows tanh−1y=(λt)β (3.1) Gas turbine reliability model based on... 727 The tangent hyperbolic can be eliminated from this equation based on the following well known expression tanh−1y= 1 2 log 1+y 1−y (3.2) Furthermore, if the log is used to the twomembers of equation (3.1), the obtained expression is as follows log (1 2 log 1+y 1−y ) =β logλ+β log t (3.3) which can be transformed into a linear expression Y =A+BX (3.4) where Y = log (1 2 log 1+y 1−y ) X = log t A=β logλ B=β To estimate A and B, any graphical or analytical methods can be used, a possible approach is the simple regression analysis using (3.2). The estimation of the parameters based on the least- -squares fit is shown in Fig. 4. The parameters of the examined gas turbine using the proposed approach are presented in Table 2. The obtained estimated parameters are λ = 0.00034 and β=1.24568. Table 2.The obtained parameters for the examined gas turbine Estimate Std. Error t value Pr(> |t|) β 1.24569 0.03435 36.27 < 2E-16 β logλ −7.98420 0.02801 −285.07 < 2E-16 Fig. 4. The distribution function of the examined gas turbine determined using the hyperbolic algorithm The plots of Y versusX which are related to the cumulative distribution function and time for the examined gas turbine determined using the proposed tangent hyperbolic algorithm are shown in Fig. 4, where the plotted data are scattered close to the fitted straight line. It can be said that this model is acceptable (Murthy et al., 2004; Yang and Scott, 2013). 3.1.1. Cumulative distribution function The cumulative distribution function of the examined gas turbine found using the tangent hyperbolic model is shown in Fig. 5. The plotted data are scattered close to the fitted cu- rve. Following the cumulative distribution function expressed in equation (2.4), the parameters λ=0.00034 and β=1.24568. 728 A.Z. Djeddi et al. Fig. 5. The cumulative failure rate of the examined gas turbine using the hyperbolic model 3.1.2. Density and hazard functions The density function and the hazard function of the examined gas turbine determined by using the tangent hyperbolic function model are shown respectively in Figs. 6 and 7. These functions are obtained by using equation (2.5) and taking into account the estimated parameter values λ=0.00034 and β=1.24568. Fig. 6. The distribution function of the model of the examined gas turbine using the hyperbolic model Fig. 7. The hazard function of the examined gas turbine using the hyperbolic model 3.1.3. Mean time between failures The Mean Time Between Failures (MTBF) is the predicted elapsed time between inherent failures of the system during exploitation (Yang and Scott, 2013) which is expressed as follows MTBF= +∞ ∫ 0 tf(t) dt= +∞ ∫ 0 t (λt)β−1λβ cosh2(λt)β dt (3.5) Gas turbine reliability model based on... 729 It is worth noting that there is an alternative way for computing the expected value (Murthy et al., 2004) MTBF= +∞ ∫ 0 S(t) dt= +∞ ∫ 0 1− tanh(λt)β dt (3.6) Table 3 Model Weibull Hyperbolic function MTBF 2088.623h 2075.40h 4. Conclusion The goal achieved in this paper is concerned with industrial reliability, which requires the spe- cialists to bemore competitive andmore responsive to variable market conditions. It is obvious that to achieve better management of the facilities performance, it is necessary to improve the reliability. It has been shown that with the application of the proposed method based on the tangent hyperbolic function good results can be obtained. Themain objective is to analyze and tomeasure the reliability parameters of the examined system in order to determine the best time to take maintenance action based on reliability analysis. The developed in this paper reliability approach allows themodeling of the damage effects to predict the performance of the examined gas turbine operation and to lead to good results in terms of validity comparedwith theWeibull approach. This model can be improved to have a bathtub-shaped hazard form. Refernces 1. Costa F.M.P., Rocha A.M.A.C., Fernandes E.M.G.P., 2014, An artificial fish swarm al- gorithm based hyperbolic augmented Lagrangianmethod, Journal of Computational and Applied Mathematics, 259, Part B, 868-876 2. Guaily A.G., Epstein M., 2013, Boundary conditions for hyperbolic systems of partial differen- tials equations, Journal of Advanced Research, 4, 4, 321-329 3. 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Yang J., Scott D.W., 2013, Robust fitting of aWeibull model with optional censoring,Compu- tational Statistics and Data Analysis, 67, 1, 149-161 Manuscript received October 3, 2014; accepted for print March 11, 2015