International Journal of Mechanical Engineering and Applications

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1 Intnatonal Jounal of Mchancal Engnng and Applcatons 018; 6(): do: /j.jma ISSN: 0-0X (Pnt); ISSN: (Onln) Th Effct of th Flud Flm Vaal Vscosty on th Hydostatc Thust Sphcal Bang Pfomanc n th Psnc of Cntptal Inta and Sufac Roughnss Pat, Rcssd Fttd Bang Ahmad Waguh Yacout Elscandaany Mchancal Dpatmnt, Faculty of Engnng, Alxanda Unsty, Alx, Egypt Emal addss: To ct ths atcl: Ahmad Waguh Yacout Elscandaany. Th Effct of th Flud Flm Vaal Vscosty on th Hydostatc Thust Sphcal Bang Pfomanc n th Psnc of Cntptal Inta and Sufac Roughnss Pat, Rcssd Fttd Bang. Intnatonal Jounal of Mchancal Engnng and Applcatons. Vol. 6, No., 018, pp do: /j.jma Rcd: July1, 018; Accptd: August14, 018; Pulshd: Sptm11, 018 Astact: Th study dals wth th flud scosty aaton, th cntptal nta and th ang sufac oughnss affctng th xtnally pssuzd thust sphcal ang pfomanc wh th patal dffntal quaton of th tmpatu gadnt pously dd fom th flud gonng quatons, s ntgatd and appld to ths typ of angs to calculat and pdct tmpatu dstuton along th flud flm. In ths pat of th sach, th cssd fttd typ of angs has n studd dng mathmatcal xpssons that not only co ths confguaton ut also co th fttd typ of ths ang wth ts dffnt confguatons and showng also th css ffct on th ang pfomanc. Th sults showd th ffct of th scosty aaton on th pssu, th load cang capacty, th flud flow at, th fctonal toqu, th fcton facto, th pow facto, th stffnss facto and th cntal pssu ato as wll as th ffct of th spd paamt and th ccntcty on th tmpatu s. Applyng ths dd mathmatcal xpssons, whch could consdd as a gnal soluton fo th fttd typ, an optmum dsgn fo th fttd typ wth ts dffnt confguatons (wth and wthout css; hmsphcal and patal hmsphcal sats) has n pfomd. Usng th sam ang dmnsons, th applcaton of ths quatons pod th xcllnc of th afomntond optmum dsgn of ths ang n ou pous paps wh th tmpatu of th outlt flow was lss than 14 dgs cntgad o ts nlt tmpatu. Kywods: Hydostatc Bangs, Sphcal Bangs, Sufac Roughnss, Inta Effct, Vscosty Effct 1. Intoducton Followng th 1 st pat of ths sach, studyng th uncssd fttd typ, th cssd typ of th ang has n handld. Rynolds quaton modfd y Dowson and stochastcally dlopd y th autho to applcal to th hydosph s thmally -dlopd to study and xamn such typ of angs [1-4]. Essam Salm and Fad Khall [5] studd th hydosph and found that th tmpatu s ducs th load and th fctonal toqu whl ncass th flow at. Kth Bockwll t al [6] studd th tmpatu chaactstcs of a (PSJ) potd sho jounal ang wh t s statd that th tmpatu ncass wth spd and load. SB Glaatskh and S D Camllo [7] studd th scosty ffct on thust pad ang and concludd that th ang tmpatu s sgnfcantly affctd y th ol scosty gad; ncass wth th man sldng spd and th thck ol ncass th pow losss. Mnhu H t al [8] xamnd th flud flm jounal ang and concludd that th scosty shang gnats hat n th flud flm whch lads to duc th scosty and to ncas th tmpatu and th pow losss. Ian Flpoć and DžadBć [9] studd th ffct of th ol scosty on th jounal ang n ntnal comuston

2 Intnatonal Jounal of Mchancal Engnng and Applcatons 018; 6(): ngns and found that ducng th ol scosty dctly nfluncs thmnmum thcknss of th ol flm and th sstanc of jounal n th ang wh t sults n ncasng th fcton du to th dcas n th mn. flm thcknss and ough sufacs on th ang whch also contut to sgnfcant ncas of mchancal losss. Snasan V. [10, 11] studd, thotcally and xpmntally, th annula cssd ang consdng th nta and th scosty aaton concludng that th spd ncas lads to as th tmpatu and pssu. N. B. Nadunaman and A. K. Kadad [1] studd th ffct of th scosty aaton on a shot jounal ang hao statng that th ffct of th aaton n th scosty lads to dcas th load cayng capacty. Shgang Wang t al [1] studd th tmpatu ffct on th hydostatc ang pfomanc n oth cass of suffcnt and dfcnt flud flm usng th Flunt 6.5 Softwa to smulat th tmpatu fld. Accodng to th otand sults, t s concludd that th ol scosty wll affct th tmpatu dstuton and th tmpatu s. B. Bouchht1 t al [14] studd a fol lucatd ang and concludd that th tmpatu has a notcal ffct on th ang pfomanc. NOMENCLATURE A = (6 µ q π p ) R a =Bang pojctd aa ( π ). C= (6 µ π ). q c = Lucant spcfc hat E (f) =Expctd alu of. = Eccntcty & = + f = Dmnsonlss fcton facto. F= Fcton facto. H = Dmnsonlss flm thcknss. h= Flm thcknss. h o = Dtmnstc (man) pat of th flm thcknss ( cos). h = Random stochastc pat of th flm thcknss. st h = Pow facto f K = Constant of scosty aaton K = ( R)& K = ( R). m= Fctonal toqu. M = Dmnsonlss fctonal toqu ( m πµω R ) m = Dtmnstc pat of th fctonal toqu. o m st= Random stochastc pat of th fctonal toqu. N= Shaft spd (pm). P = Dmnsonlss pssu ( p p ) p= pssu along th flud flm p = Inlt pssu. 5 p s=supply pssu ( 5x10 N m ). Q = Dmnsonlss olum flow at (Q = A). q= Flow olum flow at o 4 R = Bang adus (50 mm). S = Spd paamt ( ρ Ω R 40p ). SF = Stffnss facto T = Tmpatu W = Dmnsonlss load cayng capacty( w π R p ) w= Load cayng capacty. z = ( )& α = ( ) z β=( p p s ). = Angl co-odnat. ϕ = Sat out m angl. ϕ =Rcss angl =Inlt flow angl. = Outlt flow angl. = Rcss dpth 4 ρ = Lucant dnsty(867 N. s m ) σ= Dmnsonlss sufac oughnss paamt. σ o= Vaanc of th flm thcknss. λ= Bang stffnss. µ =Lucant scosty(0.068 N. s m ) Ω=Rotatonal spd. Thotcal Analyss Th modfd fom of Rynolds quaton dd y Dowson and Taylo [1-] and th suggstd quaton fo th lucant scosty aaton applcal to th hydosph Fgu (1) ha n adoptd n ths study. Fgu 1. Bang confguaton. Basd on Dawson and Taylo [1-]: dp C h ( ρω R sn ) = (1) d 0 sn µ = µ (1 K sn ) () dp C(1 K sn ) [ sn ] h ρω R = () d 0 sn And fom Elscandaany [-4], th pssu dffntal quaton and ts ntgaton a:

3 75 Ahmad Waguh Yacout Elscandaany: Th Effct of th Flud Flm Vaal Vscosty on th Hydostatc Thust Sphcal Bang Pfomanc n th Psnc of Cntptal Inta and Sufac Roughnss Pat, Rcssd Fttd Bang Wh: dp C(1 k sn ) = + ρω R d ( h σ h )sn 0 o + o o 1 sn A P = dh 4S HdH ( H + H)(1 H ) + Ak ( H + H)(1 H ) dh A 1 P = { ln(1 + sc ) + ln(tan( )} k A 1+ sn sn {ln( ) + ln( )} 1 sn sn Scos + B P = p p ; H = ho = cos; S = (40) ρ Ω R, A C P =, σ o = σ σ =, C = 6 d = (1sn ) dh. Thotcal Solutons.1. Pssu Dstuton µ Q π.1.1. Rcssd Zon Followng Yacout [4] n tatng th cssd zon as a fttd typ ang of adus (R) and ccntcty ( ): h( ) = ( ) h& A Equaton (5) and ts ntgaton com: 1 = αa α A P( ) = dh 4S HdH ( H + H)(1 H ) + P αak ( H + H)(1 H ) ( ) dh αa 1 = { ln(1 + sc ) + ln(tan( )} (4) (5) (6) (7) α Ak 1+ sn sn {ln( ) + ln( )} (8) 1 sn sn Scos + B Applyng th ounday condtons: At = P ( ) = B = 1+ Scos αa[ { ln(1 + sc ) ln(tan( )} sn sn 1 sn k {ln( ) + ln( )}] 1 1 sn 1 1 (1 + sc ) (tan( ) P = 1 + αa[ { ln + ln } + 1 (1 + sc ) (tan( ) k 1+ sn 1sn {ln( * ) + 1 sn 1+ sn sn sn ln( * )}] sn + 1 sn + S(cos cos ).1.. Sat Zon Applyng th ounday condtons: At = P = P At = P = 0 Thn: 1 1 B = Scos A[ { ln(1 + sc ) ln(tan( )} sn sn 1 sn k {ln( ) + ln( )}] 1 1 sn 1 1 (1 + sc ) (tan( ) P = A[ { ln + ln } + 1 (1 + sc ) (tan( ) k 1+ sn 1sn {ln( * ) + 1 sn 1+ sn sn sn ln( * )}] sn + 1 sn + S(cos cos ) Equatng th two quatons (9) and (10) gs: Wh: 1+ S(cos cos ) A = L L (1 + sc ) (tan( ) L1 = [ { ln + ln } + 1 (1 + sc ) (tan( ) k 1+ sn 1sn {ln( * ) + 1 sn 1+ sn sn sn ln( * )}] sn + 1 sn (9) (10) At = P ( ) = P Thn:

4 Intnatonal Jounal of Mchancal Engnng and Applcatons 018; 6(): L α 1 (1 + sc ) (tan( ) = [ { ln + ln } + 1 (1 + sc ) (tan( ) α k 1+ sn 1sn {ln( * ) + 1 sn 1+ sn Gnally: sn sn ln( * )}] sn + 1 sn αa 1 P = { ln(1 + sc ) + ln(tan( )} α Ak 1+ sn sn {ln( ) + ln( )} (11) 1 sn sn Scos + D D = B at( α < 1) Wh: D = B at( α = 1) Hnt: Ths pssu fomula cos th un-cssd ang also whn ( α = 1).. Load Cayng Capacty Fom th Appndx (A 1 ): W = sn + [( a + ) ( a )] (1).. Tmpatu Dstuton Fom Elscandaany [4]: 1 1 dp dp ( Ssn ) ( ) 4 S( )sn( ) + dt p = ( ) d d 0.1 d ρc dp (sn ) d 4 k ( Const)sn sc (1 sn ) + = X dp (sn ) d.4. Fctonal Toqu T = X T = T + X n T = T 1 n1 µ 4 160S Const = ( ) ρ p R K Followng Yacout and Dowson [1-4]: 4 sn 4 sn cos cos m R d R = πµ Ω + πµ Ω d (1) π µ ΩR + m µ = µ (1 k sn ) 4 = z (1 ksn )sn d cos M = z + (1 ksn )sn d cos (1 ksn )sn d h (1 ksn )sn d h Th Intgaton could found n th Appndx (A) as: Wh: M s M = z M + M (14) σ σ cos = [ + ( 1)ln(cos ) + ] cos k [( σ 1){sn ln(tan + sc )} σ sn {ln(sc + tan ) sctan } ] M k s σ σ [( σ 1){sn ln(tan + sc )} cos = [ + ( 1)ln(cos ) + ] cos σ sn {ln(sc + tan ) sctan } ].5. Volum Flow Rat Fom [1-4].6. Fcton Facto Fom [1-4]:.7. Pow Facto Fom [1-4]:.8. Pump Pow Fom [- 5]: h f Q = A (15) F = M W (16) π Q = (17) W cos s π β p Pt = Q (18) 6 µ

5 77 Ahmad Waguh Yacout Elscandaany: Th Effct of th Flud Flm Vaal Vscosty on th Hydostatc Thust Sphcal Bang Pfomanc n th Psnc of Cntptal Inta and Sufac Roughnss Pat, Rcssd Fttd Bang.9. Fctonal Pow Fom [- 5]: P.10. Total Losss.11. Stffnss Facto f 4 π R Ω = M (19) Pt = Pp + Pt (0) dstuton, tmpatu dstuton, load cayng capacty, flow at, fctonal toqu, fcton facto and stffnss facto. Th ang chaactstcs ha n xamnd fo th ang wth hmsphcal and patally hmsphcal sats. Th css ffct on th ang hao has n also xamnd and an optmal dsgn asd on th mnmum losss s pfomd. Ths mathmatcal xpssons could consdd as gnal xpssons fo th fttd typ of angs wh thy could also appld to xamn and dsgn th un-cssd ang. Fom [1-4]: SF = ( β W + βw) (1) 5. Dscusson Studyng th thmal hao of th cssd fttd ang n th psnc of nta and sufac oughnss, th sons of fncs [1-5] ha n followd. 4. Rsults As pously mntond, th man tagt of th sach s to thmally xamn th dsgn of ths typ of angs. Mathmatcal xpssons ha n dd to xamn th cssd ang chaactstcs such as pssu 5.1. Th Pssu Dstuton Th pssu dstuton und th ffct of dffnt paamts s psnt y fgus (, )fo a ang wth hmsphcal and patal hmsphcal sats. Hmsphcal Sat Fgu. Effct of dffnt paamts.

6 Intnatonal Jounal of Mchancal Engnng and Applcatons 018; 6(): Patal Hmsphcal Sat Fgu. Effct of dffnt paamts Th Effct of th Vscosty Fom sufgus (a, a) t could osd that th scosty has nsgnfcant ffct on th pssu n cas of th hmsphcal sat du to th hgh nta whl ts ffct could appcal n th cas of th patal hmsphcal sat du to lss nta Th Effct of th Inta Sufgus (, ) show th pssu und th ffct of th nta whch has a domnant ol n ncasng th pssu spcally wth th hmsphcal sat dspt ts lss ffct wth th patal hmsphcal on. Th nta s psntd y th spdpaamt [1-4], Th Effct of th Eccntcty Th ccntcty ffct on th pssu s psntd y Sufgus (c, c) wh t s cla that th pssu dcass wth th ncas n th ccntcty. Ths nsgnfcant dcas nth pssu dspt th psnc of th costy aaton poosth hghly stffnd ang confguaton dsgnd n Yacout [] Th Effct of th Sufac Roughnss Fom sufgus (d, d) t s cla that th two ang confguatons hm and patal hmsphcal sats han t n sgnfcantly affctd y th sufac oughnss. Compang ths sults wth thos n Yacout [-4] no makal dffnc could osd dspt th scosty aalty whch lads to th pdcton that th ang stffnss wll not sously affctd y th scosty aaton [4]. 5.. Th Vscosty Effct on th Pfomanc Th ffct of th scosty on th pfomanc of a ang wth hm and patal hmsphcal sats s psntd y fgus (4-7). Sufgus (4a, 6a) show th load dcas wth th ncas n oth of th scosty aaton costant and th sufac oughnss wh t could asly pdctd fom th pssu fgus. It could sn fom th sufgus (4, 6) that th olum flow at ncass wth th ncas n th scosty aaton constant and th sufac oughnss paamt n tun th pow factosufgus (5d, 7d)dpndntly ncass wh t could sctly pdctd fom th mathmatcal quaton numd (17). Sufgus (4c, 6c) and sufgus (4d, dc) show dcasng nth fctonal touqu and th fcon facto spctly. Th stffnss facto, sufgus (5a, 7a), shows nsgnfcant ffct spcally n cas of th patal hmsphcal sat at th hgh alus of thsufac oughnss paamt and whn nomalzd, sufgus (5, 7), t showsconsstancy. Th cntal pssu ato, psntd y sufgus (5c, 7c), shows nsgnfcant ffct n cas of th patal hmsphcal sat.

7 79 Ahmad Waguh Yacout Elscandaany: Th Effct of th Flud Flm Vaal Vscosty on th Hydostatc Thust Sphcal Bang Pfomanc n th Psnc of Cntptal Inta and Sufac Roughnss Pat, Rcssd Fttd Bang Fgu 4. Hmsphcal sat. Fgu 5. Hmsphcal sat.

8 Intnatonal Jounal of Mchancal Engnng and Applcatons 018; 6(): Fgu 6. Patal hmsphcal sat. Fgu 7. Patal hmsphcal sat. 5.. Th Tmpatu Dstuton Fgus (8, 9) psnt th ffcts of th scosty aaton, th nta, th ccntcty and sufac oughnss on th tmpatu dstuton Th Effct of th Vscosty Sufgus (8a, 9a) clalt show that th tmpatu dcass wth th ncas n th scosty aaton constant.

9 81 Ahmad Waguh Yacout Elscandaany: Th Effct of th Flud Flm Vaal Vscosty on th Hydostatc Thust Sphcal Bang Pfomanc n th Psnc of Cntptal Inta and Sufac Roughnss Pat, Rcssd Fttd Bang 5... Th Effct of th Inta Sufgus (8, 9) shows that th nta plays an mpotant ol n asng th tmpatu wh t s mo ffcnt n cas of th patal hm sphcal sat [4] Th Effct of th Eccntcty Sufgus (8c, 9c) shows that th tmpatu s postly affctd y th dcas n th ccntcty n cas of th hmsphcal sat whl ts ffct could consdd nsgnfcat n cas of th patal hmsphcal sa Th Effct of th Sufac Roughnss Sufgus (8d, 9d) shows that th sufac oughnss has th last ffct on th tmpatu wh t could pactclly gnod [4]. Fgu 8. Hmsphcal sat. Fgu 9. Patal hmsphcal sat.

10 Intnatonal Jounal of Mchancal Engnng and Applcatons 018; 6(): Th Effct of th Rcss Fgus (10-1) psnt th ffct of th css on th ang pfomanc Th Load Cayng Capacty Sufgus (10 a, 1 a) show notal ncas n th load achs to aout % n cas of th hmsphcal sat whl t achs to aout % 0 n cas of th patal hmsphcal on. Dpnng th css gts nsgnfcant ncas n th load Th Volum Flow Rat Sufgus (10, 1 ) show ncas n th olum flow at wh t s compatly mo n cas of th patal hmsphcal sat. Incasng th css dpth has nsgnfcant ffct Th Fctonal Toqu and th Fcton Facto Sufgus (10 c, d & 1 c, d) show dcas n th fctonal toqu and th fcton facto and th dcas s latly mo n cas of th patal hmsphcal sat. Dpnng th css has nsgnfcant ffct Th Pow Facto Sufgus (11 a, 1 a) show dcas n th pow facto ut t could y small n cas of th hmsphcal sat compaatly to that n cas of th patal hmsphcal on. Hgh alus of th css dpth ha nsgnfcant ffct Th Cntal Pssu Rato Sufgus (11, 1 ) show appoxmatly no ffct n cas of th hmsphcal sat and notal dcas n cas of th patal hm-sphcal on spcally at hgh alus of th sufac oughnss paamt. Dpnng th css has nsgnfcant ffct Th Stffnss Facto Sufgus (11 c, 1 c) shownotal dcas n cas of th patal hm-sphcal sat spcally at hgh alus of th sufac oughnss paamt and nsgnfcant ffct n cas of th hmsphcal sat. Nomalzng th Stffnss, facto, sufgus (11 d, 1 d) show no ffct whch als th ang consstancy. Dpnng th css has nsgnfcant ffct. Fgu 10. Hmsphcal sat.

11 8 Ahmad Waguh Yacout Elscandaany: Th Effct of th Flud Flm Vaal Vscosty on th Hydostatc Thust Sphcal Bang Pfomanc n th Psnc of Cntptal Inta and Sufac Roughnss Pat, Rcssd Fttd Bang Fgu 11. Hmsphcal sat. Fgu 1. Patal hmsphcal sat.

12 Intnatonal Jounal of Mchancal Engnng and Applcatons 018; 6(): Fgu 1. Patal hmsphcal sat Th Bang Dsgn As n Yacout [], aang wth th sam dmnsons s dsgnd und th sam condtons n th psnc of th flud aal scosty. R = 50mm, φ = 5 o, φ = 4 o, η = (1& 0.85), ξ = 0.05, K = (110), N = 100 ps, S = ps 5 = 5x1o N m, β =, ρ = 867 N. s m, µ = N. s m, C = 1880 m s. c, K = (0&0.5) o Dtmnng th Eccntcty Fgus (14, 15) show losss at dffnt ccntcts fo th fttd typ cong ts ght possl cass. Sufgus (14 a, & 15 a, ) fo th ang (wth and wthout) css, (hm and patal hm) sats at constant scosty. Sufgus (14 c, d & 15 c, d) fo th ang (wth and wthout) css, (hm and patal hm) sats at aal scosty. Th optmum ccntcty fo ach s dtmnd at th mnmu total Pow losss fom th gaphs [14, 15]. Fgu 14. Hmsphcal sat.

13 85 Ahmad Waguh Yacout Elscandaany: Th Effct of th Flud Flm Vaal Vscosty on th Hydostatc Thust Sphcal Bang Pfomanc n th Psnc of Cntptal Inta and Sufac Roughnss Pat, Rcssd Fttd Bang Fgu 15. Patal hmsphcal sat Th Bang Chaactstcs and Bhao Th concnd ang n ths pat of th sach s th cssd on at aal scosty wth hn snd patal hmsphcal sats. Fgus (16-1) psnt th dsgnd ang pfomanc and ts hao f opatd und condtons dff fom th thos of th dsgn Chckng th Bang Dsgn To chck th ang dsgnd n Yacout [], Sufgus (14 a, & 15 a, ) show th ccntcty atos (K = 1/17.606) footh th un-cssd and cssd hmsphcal sats whch concd wth thos n Yacout [] and (K = 1/41.81) founcssdpatal hmsphcal sat and (K = 1/4.67) fo cssdpatal hmsphcal sat spctly. In Elscandaany [4] th study dosn t handl th dsgn. Fgu 16. Hmsphcal sat.

14 Intnatonal Jounal of Mchancal Engnng and Applcatons 018; 6(): Fgu 17. Hmsphcal sat. Fgu 18. Hmsphcal sat.

15 87 Ahmad Waguh Yacout Elscandaany: Th Effct of th Flud Flm Vaal Vscosty on th Hydostatc Thust Sphcal Bang Pfomanc n th Psnc of Cntptal Inta and Sufac Roughnss Pat, Rcssd Fttd Bang Fgu 19. Patal hmsphcal sat. Fgu 0. Patal hmsphcal sat.

16 Intnatonal Jounal of Mchancal Engnng and Applcatons 018; 6(): Fgu 1. Patal hmsphcal sat. 6. Concluson Ths pat of th sach concns wth th cssd fttd typ of angs whn th flud flm scosty s aal n th psnc of nta and sufac oughnss. Th man tms could astactd fom th afomntond dscusson a: a) Dng mathmatcal xpssons co ths typ of angs wth ts dffnt confguatons and whth th flud flm scosty s constant o aal. ) Optmal dsgn asd on th mnmum losss. c) Chckng ang dsgn wth constant scosty. Futu Wok Th d pat of th sach handls th dsgn of th fttd typ wth ts dffnt confguatons compang twn th pfomancs alng th adantags and dawacks of ach. Appndx A 1 Th Load Cayng Capacty Daton Followng Dowson and Yacout [1-4]: w = πr sn p + π R p sn( )cos( ) d + o ( ) πr psn( )cos( ) d W = sn + P sn( )cos( ) d + Psn( )cos( ) d Wh: ( ) W = sn + ( F + F ) 1 F1 = P( ) sn( )cos( ) d F = Psn( )cos( ) d Th ntgaton of (F) could dctly found n Yacout [, 4] as: F = a 1 +a

17 89 Ahmad Waguh Yacout Elscandaany: Th Effct of th Flud Flm Vaal Vscosty on th Hydostatc Thust Sphcal Bang Pfomanc n th Psnc of Cntptal Inta and Sufac Roughnss Pat, Rcssd Fttd Bang a + cos sn 1 a = A[ ln( + cos ) ln(sn ) 4 ( + 1) ( + 1) cos S B ln(cos )] [cos ] [cos ] 4 + KA = (cos ) (1 sn ) + cos ( + 1 sn ) [ ln ( )ln ] 4 (1 + sn ) + 1 ( sn ) And th ntgaton of (F 1 ) has xactly th sam fom as: F 1 = cos sn 1 = A [ ln( + cos ) ln(sn ) 4 ( + 1) ( + 1) cos S B ln(cos )] [cos ] [cos ] 4 + K A (cos ) (1 sn ) = [ ln 4 (1 + sn ) + cos ( + 1 sn ) ( )ln ] + 1 ( sn ) M s + )(1 ksn )sn (cos σ = d cos Th ntgaton could found n Yacout [, 4] as: M σ σ cos = [ + ( 1)ln(cos ) + ] cos K [( σ 1){sn ln(tan + sc )} σ sn {ln(sc + tan ) sctan } ] M σ σ cos = [ + ( 1)ln(cos ) + ] cos K [( σ 1){sn ln(tan + sc )} s σ sn {ln(sc + tan ) sctan } ] KA (cos ) (1 sn ) = [ ln 4 (1 + sn ) + cos ( + 1 sn ) ( )ln ] + 1 ( sn ) Hnc: A = α A W = sn + [( a + ) ( a )] 1 1 Rfncs [1] Dowson D. and Taylo M. 1967, Flud nta ffct n sphcal hydostatc thust angs, ASLE Tans. 10, A Th Intgaton of th Fctonal Toqu k (1 ksn )sn (1 sn )sn M = z d + d h h Followng Yacout [, 4]: 1 h + σ cos + σ E( ) = = (Fttd) h h cos Takng th xpctaton of oth sds: M = z + M (cos (cos + )(1 ksn )sn + )(1 ksn )sn σ d cos σ d cos M = z M + M s + )(1 ksn )sn (cos σ = d cos [] Dowson D. and Taylo M. 1967, A R- Examnaton of hydosph pfomanc, ASLE Tans. 10, 5-. [] Ahmad W. Yacout, Ashaf S. Ismal, Sadk Z. Kassa, 007, Th comnd ffcts of th cntptal nta and th sufac oughnss on th hydostatc thust sphcal ang pfomanc, Tolgy Intnatonal Jounal Vol. 40, No., 5-5. [4] Ahmad W. Y. Elscandaany, 018, Th Effct of th Flud Flm Vaal Vscosty on th Hydostatc Thust Sphcal Bang Pfomanc n th Psnc of Cntptal Inta and Sufac Roughnss (Pat 1 Un-cssd fttd ang), Th Intnatonal Jounal of Mchancal Engnng and Applcatons Vol. 6, No. 1, pp [5] Essam Salm and Fad Khall, Vaal scosty ffcts n Extnally Pssuzd sphcal Ol Bangs, Jounal of Wa 1978, 50, 1-5. [6] Kth Bockwll, Scan Dcamllo and Waldma Dmochowsk, 001, Masud tmpatu chaactstcs of 15 mm damt potd sho jounal angs wth floodd lucaton, Tology Tansacton ol. 44, No. 4, [7] B Glaatskh and S D Camllo, 004, Influnc of ol scosty gad on thust pad ang opaton, Poc. Inst. Mch. Eng. Vol. 18, pat j: J. Engnng tology. [8] Mnhu H, Cloud C. Hunt and Jams M. Byn, 005, Fundamntals of Flud Flm, Jounal Bang Opaton and Modlng, Pocdngs of th thty fouth Tuo-machny Symposum.

18 Intnatonal Jounal of Mchancal Engnng and Applcatons 018; 6(): [9] Ian Flpoć and DžadBć, 010, Impact of ol scosty on functonal paamts of jounal angs n ntnal comuston ngns, Goamaza, 49, 4, 4-51 [10] Snasan V. 01, Analyss of Statc and Dynamc Load on th Hydostatc Bang wth Vaal Vscosty Affctd y th Enonmntal Tmpatu, Jounal of Enonmntal Rsach and Dlopmnt Vol. 7, No. 1A, [11] Snasan V. 01, Analyss of Statc and Dynamc Load on Hydostatc Bang wth Vaal Vscosty and Pssu, Jounal of Enonmntal Rsach and Dlopmnt Vol. 6 (6s), [1] N. B. Nadunaman and Achana K. Kadad, 01, Th ffct of scosty aaton on th mco-pola flud squz flm lucaton of a shot jounal ang, Adancs of Tology Jounal, Vol. 01, Atcl ID [1] Shgang Wang, Xanfng Du, Mngzhu L, Zhonglang Cao, Janja Wang, 01, Analyss of tmpatu ffct on th lucatng stat of hydostatc ang, Jounal of Thotcal and Appld Infomaton Tchnology Vol. 48, No., [14] B. Bouchht, B. Bou-Saïd and M Gaca, 016, Statc and dynamc pfomancs of fgant- lucatd fol angs, 7 th ntnatonal confnc on adancd concpts n mchancal ngnng. [15] Camon A. 1981, Basc lucaton thoy, Longman.