Optimum design of B-series marine propellers

Transcription

1 Alexandria Engineering Journal (2011) 50, Alexandria Universiy Alexandria Engineering Journal ORIGINAL ARTICLE Opimum design of B-series marine propellers M.M. Gaafary *, H.S. El-Kilani, M.M. Mousafa Naval Arciecure and Marine Eng. Dep., Faculy of Engineering, Por Said Universiy, Egyp Received 19 December 2009; acceped 24 July 2010 Available online 5 Marc 2011 KEYWORDS Opimizaion; Single objecive; B-Series propeller Absrac Te coice of an opimum marine propeller is one of e mos imporan problems in naval arciecure. Tis problem can be andled using e propeller series diagrams or regression polynomials. Tis paper inroduces a procedure o find ou e opimum caracerisics of B-series marine propellers. Te propeller design process is performed as a single objecive funcion subjeced o consrains imposed by caviaion, maerial sreng and required propeller rus. Aloug opimizaion sofware of commercial ype can be adoped o solve e problem, e compuer program a as been specially developed for is ask may be more useful for is flexibiliy and possibiliy o be incorporaed, as a subrouine, wi e complex sip design process. ª 2011 Faculy of Engineering, Alexandria Universiy. Producion and osing by Elsevier B.V. All rigs reserved. 1. Inroducion Naval arciecs can easily design an opimized propeller wi e eoreical propeller design meods (lifing-line/surface eories) using a compuer wiou e geomery consrains seen in series propellers [1]. However, series propellers are sill valuable; ey are sill widely used in e preliminary design of lig or moderaely loaded propellers [2]. Moreover, for ose * Corresponding auor. address: ebaelkilani@gmail.com (M.M. Gaafary) ª 2011 Faculy of Engineering, Alexandria Universiy. Producion and osing by Elsevier B.V. All rigs reserved. Peer review under responsibiliy of Faculy of Engineering, Alexandria Universiy. doi: /j.aej Producion and osing by Elsevier wo canno afford lifing surface sofware, radiional series propellers are good coices. Among e propeller series, e B-series is one of e commonly used series. In is sudy, a compuer program as been specially developed o find e opimum caracerisics of any B-series propeller. Te propeller design process is andled as a single objecive funcion subjeced o several consrains suc as caviaion, maerial sreng and e required propeller rus. 2. Opimizaion problem Te main difficuly in mos opimizaion problems does no lie in e maemaics or meods involved. I lies in formulaing e objecive all e consrains. Te propeller design problem as been andled as a muliobjecive consrained opimizaion problem [1 4]. Tere are wo principal ways o andle muli-objecive problems, bo leading o single objecive opimizaion problems [5]: one objecive is seleced and e oer objecives are formulaed as consrains.

2 14 M.M. Gaafary e al. Nomenclaure A E propeller expanded area, m 2 A O propeller disk area, m 2 C 0.75R blade cord leng a 0.75R, m C mi regression coefficien of orque coefficien C si regression coefficien of rus coefficien D propeller diameer, m H PAP eig of propeller aperure, m J advance coefficien K Q orque coefficien K T rus coefficien N propeller roaing speed, rpm N = 60\ n N P number of propellers n propeller roaing speed, rps M number of e design variables P propeller blade pic, m P CL propeller immersion, m P D developed power, kw P S saf power per blade, kw P O saic pressure a e cenerline of e propeller saf, Pa P V vapor pressure, Pa P oal number of consrains q number of inequaliy consrains Rn Reynolds number R T sip oal resisance, N S C maximum allowable sress of e propeller maerial, MPa si si 0 T T Cal T R T d i i 0 min ui ui 0 V A V S Vi Vi 0 w Z DK Q DK T g g max exponen of (J) ink T equaion exponen of (J) ink Q equaion propeller rus, N calculaed propeller rus, N required propeller rus, N propeller blade ickness, m rus deducion exponen of (P/D) ink T equaion exponen of (P/D)in K Q equaion minimum ickness of propeller blade, m exponen of A E /A O in K T equaion exponen of A E /A O in K Q equaion speed of advance, m/sec sip speed, m/sec exponen of Z in K T equaion exponen of Z in K Q equaion wake fracion number of propeller blades correcion of orque coefficien correcion of rus coefficien propeller efficiency maximum propeller efficiency q waer densiy, kg m /m 3 m waer kinemaic viscosiy, m 2 /s a weiged sum of all objecives forms e opimizaion objecive funcion. Te raer arbirary coice of weig facors makes e opimizaion model obscure and e firs opion is mosly preferred [5]. In is sudy, e propeller design problem is andled as a consrained opimizaion problem according o e firs opion. Any consrained opimizaion problem can be andled according o e opimizaion model sown in Fig. 1. Tis opimizaion problem can be formulaed as follows [6]: 8 9 x 1 >< x 2 >= Find X ¼. >: >; x m wic maximizes an objecive funcion called f(x) subjeced o e following consrains: 9 g j ðxþ 6 0; j ¼ 1; 2;...; q >= and ð2þ >; j ðxþ ¼0; j ¼ q þ 1; q þ 2;...; p were, g j (X) and j (X) are e inequaliy and e equaliy consrains, respecively. 3. B-series propeller ð1þ No Figure 1 Sar New Design Feasibiliy Ceck of e Design Objecive Funcion Evaluaion Analysis of e resuls Opimum Design Reaced Yes End Opimizaion model for consrained problems. B-series propellers were developed in e Neerlands Sip Model Basin, and e secion of e blade was improved laer. For any B-series propeller, e rus and orque coefficiens can be expressed as funcions of e blade number (Z), blade

3 Opimum design of B-series marine propellers 15 area raio (A E /A O ), pic raio (P/D), and advance coefficien (J) as follows: K T ¼ X39 i ui C si J si P A E Z vi ð3þ D A i¼1 O and, K Q ¼ X47 C mi J P i 0 ui 0 AE si0 Z vi0 ð4þ D A i¼1 O were, C si and C mi are e regression coefficiens of e rus and orque coefficiens, respecively. Te values of e coefficiens and exponens involved in Eqs. (3) and (4) are given in [7]. If Reynolds number of a propeller a 0.75R is greaer an 2 \ 10 6, correcions o e rus and orque coefficiens mus be aken ino consideraion [1]. qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðrnþ 0:75R ¼ C 0:75R ½V 2 A þð0:75p ndþ2 Š ð5þ m were, C 0.75R is e blade cord leng a 0.75R and V A is e advance velociy (m/s) Objecive funcion Te mos common objecive funcion, for e opimum marine propellers, is e propeller maximum efficiency (g max ) [8]. Te efficiency of marine propellers can be compued as follows: g ¼ J 2p K T ð6þ K Q 3.2. Consrains Caviaion consrain Caviaion could affec a propeller s performance and needs o be considered in e propeller design process. A simple way o avoid caviaion is o increase blade area raio. Te minimum blade area raio required o avoid caviaion was suggesed by Keller [1] as follows: A E A O min ¼ ð1:3 þ 0:3ZÞT ðp O P V ÞD 2 þ K were, (A E /A O ) min is e minimum expanded area raio. Te coefficien K equals 0.1 for win-screws sips, and 0.2 for single-screw sips Sreng consrain To acieve adequae blade ickness and us ensure maerial sreng, e following formula can be used o deermine e minimum raio of blade ickness a 0.7R o e diameer [4]: i min D 0:7R sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3 ½3183: :15ðP=DÞŠP S ¼ 0:0028 þ 0: :04nD 3 ðs C þ 20:9D 2 n 2 Þ were, ( min /D) 0.7R is e minimum propeller blade ickness raio a 0.7R. According o e B-series propeller geomery [7], e raio of maximum ickness of propeller blade a eac secion o e propeller diameer is given as sown in Table 1. ð7þ ð8þ Table 1 Maximum blade ickness % of D for B-series propellers. r/r Max. blade ickness (% of D) Z =3 Z =4 Z = As sown in able 1, e raio of maximum ickness of propeller blade a 0.7R o e diameer is given as follows: 8 i >< 0:0171 for Z ¼ 3 ¼ 0:0156 for Z ¼ 4 ð9þ D 0:7R >: 0:0141 for Z ¼ 5 Using Eq. (8) and e geomery of B-series propellers, e required propeller blade ickness can be obained as follows: i P i min ð10þ D 0:7R D 0:7R Trus consrain Te calculaed propeller rus (T Cal ) mus be equal o e required rus (T R ). Te propeller rus can be calculaed as follows: T CAL ¼ K T q n 2 D 4 ð11þ R T T R ¼ ð12þ N P ð1 d Þ were, R T is e sip oal resisance, N P is e number of propellers and d is e rus deducion. 4. Developed compuer program A compuer program as been specially developed o find e opimum caracerisics of any B-series propellers. Tis program sars wi an iniial feasible poin and proceeds owards e opimum poin according o e seps sown in Fig. 1. Te deailed procedure of is program is summarized in e flow car sown in Fig Case sudy Te problem is o find e opimum caracerisics (D, A E /A O, P/D, J) of four bladed B-ype propeller of a win screw sip. Te propeller diameer (D) is resriced a specified value o sui e sip lines and o provide accepable ull clearances o avoid ull vibraion. Sip speed (V S ) and corresponding resisance (R T ) are given for is design condiion. Wake fracion (w) and rus deducion ( d ) are given for is design condiion. Tis problem can be formulaed as follows:

4 16 M.M. Gaafary e al. Figure 2 Flow car for B-series marine propellers design process.

5 Opimum design of B-series marine propellers 17 Table 2 Inpu daa. Sip oal resisance (R T ) Sip speed (V S ) Number of propellers (N P ) 2 Number of propeller blades (Z) 4 Heig of propeller aperure (H PAP ) 1.25 Propeller immersion (P CL ) 2.0 Wake fracion (w) 0.20 Trus deducion ( d ) 0.15 Max. allowable sress of propeller maerial (S C ) 260 Boundary consrain Design variable Lower Limi Upper Limi Propeller diameer (D) \ H PAP Area raio (A E /A O ) Pic raio (P/D) Advance coeff. (J) Figure 4 Open waer diagram. Figure 3 Oupu resuls of e developed compuer program. Use e inpu daa wic are sown in Table 2 o find (D, A E /A O, P/D, J) wic maximize e propeller efficiency (g) wen subjeced o e following consrains: Caviaion consrain: e expanded area raio sould be larger an a minimum value in order o avoid caviaion, (A E /A O ) P (A E /A O ) min were (A E /A O ) min can be calculaed using Eq. (7). Trus consrain: e calculaed propeller rus (T cal ) as o mac e design requiremen, as follows: T cal = T required. Sreng consrain: o ensure adequae maerial sreng, a minimum propeller blade ickness is required, as sown in Eq. (8). D i 0:7R > min D i 0:7R Boundary consrain: o obain an opimum design poin, e design variables mus lie in an accepable domain. Figure 5 Oupu resuls of Lingo sofware. In is par, e problem under consideraion is carried ou in deails by using e following ecniques: Using e developed compuer program. Using a commercial sofware (Lingo).

6 18 M.M. Gaafary e al. Table 3 Oupu resuls. Iems Developed program Lingo Objecive: g max Design variables D A E /A O P/D J [n] RPS Coefficiens K T K Q Power in kwa P D P D (Blade) Conclusions Te design process of a series propeller by radiional calculaion or car meods is a edious job due o e muliple parameers and consrains involved. To andle e muli-objecive consrained problem of e propeller design, one objecive is seleced and e oer objecives are formulaed as consrains. Te developed compuer program represens a ailored and simple ool o find e opimum caracerisics of any B-series marine propeller. I is more flexible o use i as a subrouine in global sip design problems an o use commercial opimizaion sofware. References 5.1. Using e developed compuer program Te daa wic are sown in Table 2 are used o perform e problem under consideraion by using e developed compuer program. Te oupu resuls of is program are sown in Fig. 3. Te obained opimum poin is ploed on e open waer diagram for is design condiion as sown in Fig Using a commercial sofware (Lingo) Lingo [9] is a compreensive ool designed o make building and solving linear, nonlinear and ineger opimizaion models. Te problem under consideraion is recalculaed using LINGO sofware [9] and e oupu resuls are sown in Fig Analysis of e resuls Te oupu resuls obained by e presened soluions are sown in Table 3. I is clear a, e oupu resuls of Lingo sofware are found in a good agreemen wi ose obained by e developed program. [1] C. Jeng-Horng, S. Yu-San, Basic design of a series propeller wi vibraion consideraion by geneic algorim, J. Mar. Sci. Tecnol. 12 (2007) [2] E. Benini, Muli-objecive design opimizaion of B-screw series propellers using evoluionary algorim design, J. Mar. Sci. Tecnol. 40 (2003) [3] J.-B. Suen, J.-S. Kouj, Geneic algorim for opimal series propeller design, in: Proceeding of e Tird Inernaional Conference on Marine Tecnology, ORDA 99, Poland, [4] M.M. Karim, M. Ikeaa, A geneic algorim (GA)-based opimizaion ecnique for e design of marine propeller, in: Proceeding of e Propeller/Safing Symposium, Virginia Beac, USA, [5] H. Scneeklu, V. Berram, Sip Design for Efficiency and Economy, Buerwor, Heinemann, [6] S.S. Rao, Opimizaion Teory and Applicaions, second ed., Wiley Esern Limied, India, [7] M.W.C. Ooserveld, V. Oossanen, Furer Compuer Analyzed Daa of Te Wageningen B-series, I.S.P., vol. 23, July [8] M.M. Gaafary, Compuerized meod for propeller design of opimum diameer and rpm, in: 13 Congress of IMAM, Isanbul, Turkey, Ocober [9] Lindo Sysems Inc., Lingo Sofware, Version (9), <p:// April 2006.