2 An Entropy Model of the Aircraft Gas Turbine Engine Blades Restoration Method Choice Andriy Goncharenko Department of Aircraft Airworthiness Retaining, National Aviation University, UKRAINE, Kyiv, 1 Kosmonavta Komarova ave., email: andygoncharenco@yahoo.com Abstract: The paper goal is to investigate the influence The developed herein concept seems promising to the of a few specified methods of the gas-turbine engine variety of adjacent scientific areas applications, for instance, blades repair upon a choice of a preferable blades like for those ones considered and discussed in publications geometry restoration technology. There is a scientific [12–19]. proof for the good blades shape restoration method choice that fits the customer needs, being taking into account the II. MATHEMATICAL MODELING AND DEVELOPED subjective preferences of the available technologies and METHODS extremizing the preferences uncertainty. Keywords: mathematical model, alternative, economic A gas-turbine engine (GTE) blades apparatus is designed activity, subjective preference, subjective entropy. on one hand to ensure the required compression of the air supplied with the necessary parameters to the combustion I. INTRODUCTION chamber by the compressor part of the engine and on the other hand the blades apparatus of the turbine part allows For an active system economic activity management it is getting enough work and gas parameters for the compressor used the theory of subjective analysis [1]. A system governed driving and aircraft propulsion. by a decision making person (or a group of such persons in a The important thermodynamic parameter here is the certain hierarchical structure) is considered as active system polytropic process index n which magnitude must lie within since the system is under the managerial influence of the the very narrow designed value diapason. In operation the individuals (they are deemed to be reckoned with as the blades wear out, their geometrical shape distorts, as a result active elements of their own active system). The foundation of this the engine cannot perform the designed work. stone of the subjective analysis theory (the theory of The restoration of the blades geometry form has to be individual subjective preferences optimal distribution upon a made. certain, taken by an individual into her/his consideration, set of some attainable alternatives) is the entropy paradigm, the A. Theoretical Problem Setting so-called Subjective Entropy Maximum Principle (SEMP), Polytropic process index n approximate mean value, in which is going to be used in the presented paper as a research real engines thermodynamic processes (such as, for instance, tool, although with the application into an objectively expansion of combustion gasses in the cylinder of an internal existing optimum sphere rather than just to the subjectively combustion engine, or in our case, the other examples are preferred matters and only. This creates a background for the compression and expansion of gasses in an AGTE) information and analytical support of the economic activity. calculations, can be found from polytropic process equation: In aviation industry, in particular, in aeronautical pV n = Const , (1) engineering design and its further operation modes, the individuals’ subjective preferences play a crucial role. The where p is pressure; V is volume; provided the values of same is to the aircraft as a whole, as well as to its power plant the pressure p1 , p2 and volume V1 , V2 are known at some specifically. Definitely, those individuals’ subjective points 1 and 2 of the process. preferences are intruding the fields of both aircraft and their Indeed powerplants maintenance and repair as that follows the p1V1n = p2V2n . (2) references [2, 3]. The other area of subjective preferences application here is ln p1 + n ln V1 = ln p2 + n ln V2 , alternatives in technologies. Concerning an aviation gas- n ln V1 − n ln V2 = ln p2 − ln p1 , turbine engine (AGTE) repair it deals with the techniques ln p2 − ln p1 proposed in multiple and developed, described, and discussed n= . (3) in publications of both directions of practitioners/engineers ln V1 − ln V2 and academicians [4–6]. This simplest method of (1) – (3) for polytropic process The objectives of the presented paper are to demonstrate index n determination can be found from practically any the multi-optional optimality doctrine newest developments reference, guidance or study book on either theoretical or applicability initiated in works [7–11] to the problems of engineering thermodynamics either heat engines. aeroengines technical operation. ACIT 2018, June 1-3, 2018, Ceske Budejovice, Czech Republic 3 B. Optional Functions Entropy Problem Setting III. ANALYSIS The other concept also proposed in references [7–11] and hereinafter is based upon an optimization principle close to A. Polytropic Equation Setting SEMP [1] application in the context close to the described in Thus, we have got a parallel result to the law of subjective papers [9, 10]. conservatism if the values of parameters n , V1 , and V2 are Let us consider the thermodynamic states 1 and 2 of a gas given. in polytropic process as some optional states in a certain In case as in work [9]: h1 (V1 ) = xp2 , h2 (V2 ) = xp1 , respect. Thus we come to a multi-optional problem. (10) Now, the other sub-problem of the polytropic process given description is to discover the options’ objective where x is unknown, uncertain multiplier in type of the effectiveness functions related to those two optional states. Lagrange one, we obtain with the help of the procedure Let us presuppose the objective effectiveness functions for considered through (4) – (10) the needed polytropic process the considered two-optional problem of the polytropic index (3). process considered description are lnV1 and lnV2 . This Indeed, substituting equations (10) for their values into expression (9) it yields might be reasonably natural with regards to apparent perception of the obvious quantitative characteristic of the xp ln 2 existing reality. xp1 n= , (11) With the use of the supposed multi-optional optimality, ln V1 − ln V2 likewise in subjective analysis [1, 10] conditional optimality finally, formula (3). of the individual’s subjective preferences distribution, with The sense of the uncertain multiplier x becomes obvious extremizing subjective entropy, that is applying the doctrine with the use of the normalizing condition of the initial analogous to SEMP concept, we have the right to write down functional (4). That is the postulated functional in the view of: 2 2 xp1 + xp2 = 1 . (12) Φh = − ∑ i =1 hi (Vi )ln hi (Vi ) + n ∑ h (V )lnV i =1 i i i Hence, 1 (4) x= . (13)  2  p1 + p2  i =1 ∑ + γ  hi (Vi ) − 1 ,  Remarkable here is that the multi-optional hybrid function has the view of where hi (Vi ) are specific hybrid-optional effectiveness 2 2 functions, similar to the preferences functions of [1, 10], ∑ i =1 pi ln Vi ∑ p ln Vi i however in this problem setting the assumed specific hybrid- n or n i =1 2 , (14) optional effectiveness functions hi (Vi ) are not relating with 2 anybody’s preferences or choice; γ is normalizing ∑p i =1 i ∑p i =1 i coefficient (function). where subscript i means: “pertaining not to the i th but to The first member of expression (4) is the hybrid-optional the other option of the two-optional situation”. effectiveness functions entropy (like subjective entropy of the Moreover, in case expressed with (4) – (14) the sought preferences). polytropic process index n has been got for the given values The necessary conditions for the functional (4) extremum of hybrid-optional effectiveness functions hi (Vi ) and at this existence: the found value of n can make the hybrid-optional ∂Φ h =0, (5) effectiveness functions of hi (Vi ) also be optimal for the ∂hi (Vi ) objective functional (4). yield ∂Φ h B. Aeroengine blades alternative restoration technique = − ln hi (Vi ) − 1 + n ln Vi + γ = 0 , ∀i = 1, 2 (6) ∂hi (Vi ) problem solution on the subjective entropy paradigm basis This inevitably means in turn ln h1 (V1 ) − n ln V1 = γ − 1 = ln h2 (V2 ) − n ln V2 . Now, in order to retain polytropic process index within the (7) required designed value interval, so that to ensure the desired From where engine performance, periodical restoration of the specified ln h1 (V1 ) − ln h2 (V2 ) = n(ln V1 − ln V2 ) . (8) GTE blades apparatus geometry has to be executed. And Supposedly, there are competing methods. These are, for example, plasma (Pl), laser (La), and electro-arc (El). Each of ln h1 (V1 ) − ln h2 (V2 ) which are the alternatives for a GTE repair plant to n= . (9) implement. Here we have a three-alternative problem. Then, ln V1 − ln V2 there are corresponding subjective effectiveness functions: Pl(·), La(·), and El(·) that have relations to each of the being considered alternatives. ACIT 2018, June 1-3, 2018, Ceske Budejovice, Czech Republic 4 Let us apply although simplified, rough, however possible In Fig. 2 it is depicted, with the Pr characters, the model for the objective effectiveness functions Pl(·), La(·), subjective preferences functions distributed optimally, and El(·). accordingly with the procedures of (4) – (6) [1], upon the set The results of the numerical modeling (in conditional of the alternative AGTE blades restoration technologies: units) are shown in Fig. 1–3. Pl(·), La(·), and El(·), correspondingly. In the considered example the objective effectiveness For the preferences, their four arguments out of the five functions Pl(·), La(·), and El(·) have five independent introduced are fixed exactly as for the corresponding variables. But in actual result presented in Fig. 1 only one: effectiveness functions (see designations in Fig.1 and 2). # 4 (productivity of the alternative technology: P) is being Subjective entropy of preferences is presented in Fig. 3. variated. The rest of the parameters are the corresponding constant values accepted for the: 1) thickness of the metal 1.2 layer welded onto the blades prepared surface; 2) thickness of 1.2 1.1 the metal layer removed out from the blades bodies down to 1.8 5.2 0.87747 H_Pr( P) 1 the nominal size; 3) number of blades undergoing the 0.9 treatment; and 5) is the number of the laborers supposed to be 0.6933 ln( 3) 0.8 involved into the alternative technological process. 0.7 ln( 2) 0.6 3 8000 7.548×10 0.5 7500 1.8 5.2 0.5 0.4 ( − 3 , 1.12⋅10− 3 , 45 , P , 1) El 3⋅10 7000 0.6 0.3 6500 La( 3⋅10 , 5⋅10 , 45 , P , 4) −3 −4 0.2 6000 5533.1 0.1 Pl( 3⋅10 , 1⋅10 , 45 , P , 1) −3 −3 5500 0.069 0 4794.9 0 1 2 3 4 5 6 7 8 9 10 5000 4500 1 P 9 3 4.702×10 4000 0 1 2 3 4 5 6 7 8 9 10 Fig.3. Subjective entropy of corresponding alternatives preferences 1 P 9 From Fig. 3 it is visible that there are areas with respect to the productivity P of the blades restoration methods from Fig.1. Objective effectiveness functions related to corresponding alternatives approximately 1.5 up to 6.5 kg/h where it is almost impossible to make a decision concerning which technology C. Results of the Numerical Experiment is better to apply (see and compare multi-alternativeness knots of the individual subjective preferences, noticeable in For the available three alternatives: plasma, laser, and Fig. 2 at P = 1.8 and 5.2 kg/h, with the subjective entropy electro-arc, all the described approach variables may also be climaxes appeared in Fig. 3). It might be suspected because functions of their arguments. Even with the simplified set of of the effectiveness functions positioning (also see Fig. 1 at the independent variables the situation depending upon the those values). arguments combination remains uncertain. The inter- The additional information is required to decrease the influence of the parameters can lead to variants when at some subjective entropy. In the framework of the presented model circumstances it is hard to give the preference to a specific and developed doctrine the mathematical expressions alternative. constructed and the major parameters being considered will The preferences obtained by the objective functional undoubtedly lead to factual realizations. This process of the treatment like (4) – (6) are visible in Fig. 2. modifications creation is a challenging task; and it is absolutely clear that the necessary additional information 1 0.987 needed to decrease the uncertainty of the alternative AGTE 0.9 1.8 5.2 blades restoration technologies (subjective entropy of the ( , 1.12⋅10 , 45 , P , 1) Pr_El 3⋅10 −3 −3 0.8 0.65901 available preferences) lies in the sphere of the technological 0.7 processes intrinsic values selected each time by the Pr_La ( 3⋅10 , 5⋅10 , 45 , P , 4) −3 −4 0.6 0.49507 researcher to compare the alternatives. Therefore, the 0.5 Pr_Pl ( 3⋅10 , 1⋅10 , 45 , P , 1) −3 −3 essential parameters of the processes models, as well as the 0.4 models’ own plausible mathematical constructions, for every 0.5 0.3 stated problem setting, embody the significant information 0.2 that finally decreases the entropy value. 0.1 Thus, if we suppose existence of the different thresholds of −8 1.916×10 0 0 1 2 3 4 5 6 7 8 9 10 the entropy: 0.5; 0.6; 0.7 (see Fig. 3) for the corresponding 1 P 9 choice, then the subjective entropy as the measure of uncertainty will be higher or lower than mentioned Fig.2. Subjective preferences of corresponding alternatives thresholds. More to the point, there are two extremums at ACIT 2018, June 1-3, 2018, Ceske Budejovice, Czech Republic 5 P = 1.8 and 5.2 kg/h; the latter says of practically two- [9] A. V. Goncharenko, “Aeronautical and aerospace alternative situation since the uncertainty almost coincides materials and structures damages to failures: theoretical with the two-alternative situation uncertainty maximal value concepts,” International Journal of Aerospace ln2 (see Fig. 3). Engineering, Article ID 4126085, 7 pages, 2018 IV. CONCLUSION https://doi.org/10.1155/2018/4126085. [10] A. V. Goncharenko, “Aircraft operation depending upon In view of the stated problem, the developed doctrine the uncertainty of maintenance alternatives,” Aviation, allows formulating the following new scientific results. vol. 21, no. 4, pp. 126-131, 2017. The proposed approach gives a possibility of the https://doi.org/10.3846/16487788.2017.1415227 alternative aviation gas-turbine engine blades restoration [11] A. V. Goncharenko, “Multi-Optional Hybrid technologies comparison based upon the application of the Effectiveness Functions Optimality Doctrine for entropy extremization principle. The doctrine implementation Maintenance Purposes,” in Proceedings of the 14th IEEE reflects the results of the goal achieving for both objectively International Conference on Advanced Trends in existing and subjectively preferable optimums, as well as for Radioelectronics, Telecommunications and Computer their combinations. Engineering (TCSET-2018), IEEE, Lviv-Slavske, Thus, for the absolutely objectively existing polytropic Ukraine, February 2018, pp. 771-775. process, the pressure in polytropic process is the optimal [12] M. Dyvak, V. Tymets, and V. Brych, "Improving the hybrid-optional function, measured with certain units, of the Effectiveness of Electrophysiological Monitoring of the “logarithmic measureless volume”. Furthermore the Recurence Laryngeal Nerve During Surgery on Neck polytropic process index is obtained on the basis of the multi- Organs," in Proceedings of the IEEE 14th International optional entropy conditional optimality doctrine rather than Conference on Advanced Trends in Radioelectronics, on the absolutely thermodynamic derivations. Telecommunications and Computer Engineering (TCSET- For the subjective component the preferences functions 2018), Lviv-Slavske, Ukraine, February 2018, Paper No. allow the alternatives assessing with the uncertainty measure. 153, Paper ID 387. In further research it should be considered some other [13] M. Dyvak, I. Oliynyk, Y. Maslyiak, and A. Pukas, "Static effectiveness functions and their variables, as well as found interval model of air pollution by motor vehicles and its more theoretical results and applicable areas of the hybrid- identification method," in Proceedings of the IEEE 14th optional optimality doctrine. 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