Water Hammer in Pipes

Transcription

1 Waer Haer Hydraulcs and Hydraulc Machnes Waer Haer n Pes H Pressure wave A B If waer s flowng along a long e and s suddenly brough o res by he closng of a valve, or by any slar cause, here wll be a sudden rse n ressure due o he oenu of he waer beng desroyed. Ths wll cause a wave of hgh ressure o be ransed along he e wh a velocy equal o he sound wave, whch ay seu noses known as nockng. The agnude of hs ressure wll deend on () The ean e flow velocy () The lengh of he e () The e aken o close he valve and (v) The elasc roeres of he e aeral and ha of waer. Ths sudden rse n ressure n he e due o he soage of he flow generang a hgh ressure wave, whch wll have a haerng effec on he walls of he e, s known as Waer Haer. The cases ha can be suded under hs are:. Gradual closure of valve. Sudden closure of valve and a. Pe s rgd b. Pe s elasc rcal T e: I s defned as he e requred for he ressure wave generaed due o closure of valve o ravel once fro he on of orgn o reservor over he lengh of e and back o he on of orgnaon. If T s he e requred by he ressure wave o ravel once u and down he e and s he velocy of he ressure wave equal o he velocy of sound wave n waer also called as elery, hen fro Newon s law, we have sance raveled = Average velocy x e.e. T r. M.N. Shesha Prakash

2 Waer Haer Hence T If s he acual e of closure, T s he crcal e, hen. f >. f < T T Hydraulcs and Hydraulc Machnes, hen s referred o as Gradual closure and, hen s referred o as Sudden closure Insananeous rse n ressure n a e runnng full due o Gradual closure of valve onsder a e AB of lengh conneced o a ank a A and a valve a B wh waer flowng n as shown n Fg. e be he ean flow velocy and a s he flow cross-seconal area, he nsananeous rse n ressure due o gradual closure of valve and be he acual e of closure of valve. Fro Newon s second law of oon, he reardng force generaed agans he flow drecon s gven by he rae of change oenu of he lqud along he drecon of he force. 0 Reardaon of waer = hange n velocy / Te = Reardng force = Mass of waer x Reardaon = a... (0) The force generaed due o ressure wave = Pressure nensy x area = a... (0) Fro qs. and, we ge a a Hence Insananeous rse n Pressure head = H g g g... (03) The above equaon s vald only for rgd es wh ncoressble fluds flowng hrough. Insananeous rse n ressure n a e runnng full due o Sudden closure of valve when he e s rgd When he valve rovded a he downsrea end s closed suddenly and he e s rgd, hen he convered ressure energy fro he knec energy due o closure s o be absorbed by he flud due o s coressbly only. r. M.N. Shesha Prakash

3 Waer Haer Hydraulcs and Hydraulc Machnes Pressure nergy Pressure energy convered fro = absorbed by waer due nec energy o s coressbly.e. k = w (0) onsder he e AB of lengh and cross-seconal area a n whch waer of ass densy, wegh densy and bulk densy s flowng wh a ean velocy be suddenly soed due o closure of valve rovded a B. The knec energy of flowng waer before closure of valve wll be convered o sran energy, when he effec of frcon and elascy of e aeral are gnored. oss of knec energy k = ½ x ass of waer x As ass = x volue = x a oss of knec energy k = ½ x a x... (0) Gan n sran energy = a olue Fro qs., and 3, we ge... (03) a a or or Bu elery. Subsung for he value of n he above equaon for ressure rse, we ge = Insananeous rse n ressure n a e runnng full due o Sudden closure of valve when he e s elasc When he valve rovded a he downsrea end s closed suddenly and he e s elasc, hen he convered ressure energy fro he knec energy due o he valve closure s o be absorbed by boh he flud due o s coressbly and he elascy of he e. Pressure nergy convered fro nec energy = Pressure energy absorbed by waer due o s Pressure.e. k = w + (0) k and w can be coued as n he revous dervaon. energy absorbed by he lasc e due o s exanson r. M.N. Shesha Prakash 3 +

4 Waer Haer ouaon of can be done by sulang he suaon o he hck cylnder subjeced o nernal flud ressure. e be he hckness of he elasc e wall and assue ha s sall coared o s daeer. e f be he hoo or crcuferenal and sress f be he longudnal sress as shown n fgure. Hydraulcs and Hydraulc Machnes e he Young s odulous of he e aeral be and osons raon / e he nsananeous flud ressure be. Fro he knowledge oaf Srengh of aerals, we can wre ha f and f Hoo Sress=f aeer = l ongudnal Sress=f Hence f = f Furher, he sran energy sored n e er un volue s gven by f f f f Subsung f = f, we ge l f f f l f Subsung for f, and = l we ge 6 l Fro qs. 0 and 03, we have... (0) k = ½ x a x... (0) k a r. M.N. Shesha Prakash

5 Waer Haer Hydraulcs and Hydraulc Machnes r. M.N. Shesha Prakash a w olue... (03) Subsung n q. 0, we have a a 6 Slfyng, we ge a a Bu a and hence a/ ges canceled on boh sdes... (0) The above exresson gves he nsananeous rse n ressure n an elasc e due o sudden closure of alve. If he Posons raon s no gven, can be assued as ¼. Then q. 0 reduces o (06) Nuercal Proble s July/Ag 00. A hydraulc elne 3 k long, 00 daeer s used s used o convey waer wh a velocy of. /s. eerne he ressure growh of he valve rovded a he ouflow end s closed n () 0 s () 3. s. onsder e o be rgd and ake bulk odulous of elascy of waer as waer = 0 x 0 8 N/ Soluon: (0) = 3000 ; d = 0. ; =. /s; = 0 s; = 3. s; = 0 x 0 8 N/ ; = 000 kg/ 3 (Assued) rcal e T, where elery /s

6 Waer Haer Hence ase () 3000 T.. s > T, Hence he valve closure s gradual. Insananeous rse n ressure s gven by ase () kpa (Ans) 0 < T, Hence he valve closure s Sudden wh e rgd. Insananeous rse n ressure s gven by MPa (Ans) July 006 Hydraulcs and Hydraulc Machnes. Waer flowng wh a velocy of. /s n a rgd e of daeer 00 s suddenly brough o res. Fnd he nsananeous rse n ressure f bulk odulous of waer s.6 GPa. (0) Soluon: =. /s; =.6 GPa; = 000 kg/ 3 (Assued) elery /s Insananeous rse n ressure s gven by MPa (Ans) ec/jan A seel ensock of 000 daeer has a hckness of 0. Waer s flowng nally wh a velocy of.0 /s. Flow velocy s brough o res by closng a valve a he end of he elne. Bulk odulous of waer s x 0 N/ and elasc odulous of e aeral s x 0 N/. If he lengh of he e s 000, fnd he ressure rse n ers of head of waer when: () Waer s coressble and e s elasc (08) () Waer s coressble and e s rgd (0) Soluon: =.0 /s; d = ; =0 x 0-3 ; =.0 GPa; = 00 GPa; ase () and = 000 kg/ 3 (Assued) The valve closure s Sudden wh e lasc. r. M.N. Shesha Prakash 6

7 Waer Haer Hydraulcs and Hydraulc Machnes Insananeous rse n Pressure head =.30 MPa (Ans) g of waer (Ans) ase () The valve closure s Sudden wh e rgd. elery /s Insananeous rse n ressure s gven by MPa Insananeous rse n Pressure head = g of waer (Ans) Jan/Feb 00. Wha s he axu ressure rse due o sudden closure of a valve n a e of 300 daeer conveyng waer wh a velocy of.8 /s? The e wall s 8 hck. The e = 0 GPa and waer =. GPa. Also fnd he hoo sress develoed. (06) Soluon: =.8 /s; d = 0.3 ; =8 x 0-3 ; =. GPa; = 0 GPa; and = 000 kg/ 3 (Assued) The valve closure s Sudden wh e lasc..8 Hoo sress develoed s gven by d f 7. MPa (Ans) 3 80 Hoework Jan/Feb MPa (Ans) r. M.N. Shesha Prakash 7

8 Waer Haer Hydraulcs and Hydraulc Machnes The waer s flowng wh a velocy of. /s n a e of lengh. k. A he end of he e, a valve s rovded. Fnd he ressure rse f he valve s closed n 0 seconds. Take he ressure wave velocy as 60 /s. If he valve s closed n seconds fnd he ressure rse assung he e o be rgd and bulk odulous as 000 MPa. (08) Ans: T = 3. s, ase () Gradual closure, = 87. kpa, ase () Sudden closure =. Ma Old Schee Aug 000 The waer s flowng wh a velocy of /s n a e of lengh 000 and daeer 60 c. A he end of he e a valve s rovded. Fnd he r se n ressure f he valve s closed n 0 s. Take he value of = 0 /s. (06) Ans: T =.87 s, Gradual closure, = 00 kpa Aug 00 A seel ensock 60 c daeer and wall hckness. c carres waer wh a ean velocy of. /s. Fnd he ressure rse due o a sudden valve closure by () neglecng e elascy () consderng e elascy =. x 0 N/, =. x 0 3 N/ Also fnd he sresses n he e aeral and crcal e, f lengh of e s. k for case () (0) Ans: =. /s, ase () = 3.03 MPa, ase () =.8 MPa T =.66 S, Hoo sress = 6. MPa and ongudnal sress = 3.06 MPa, Jan 003 A waer an of concree e 3. k long and 300 n daeer dscharges no a reservor a he rae of M. If hs lne s gradually closed by a valve a reservor end n 6 seconds, show ha here s rsk of e burs. Assue es ressure of concree e s (06) Ans: Q = /s, =.7 /s, = kpa, h =.7 > r. M.N. Shesha Prakash 8

9 Waer Haer Hydraulcs and Hydraulc Machnes Jan 00 The waer s flowng wh a velocy of. /s n a e of lengh 000 and of daeer 0. A he end of he e, a valve s rovded. Fnd he rse n ressure f he valve s closed n 7. s. Take = 60 /s (06) Ans: T =.7 s, Gradual closure, = 0. kpa Jan 00 A 300 concree e carres waer o a dsance of 3 k no a reservor a a rae of 0,000 3 /day. If he suly lne s gradually closed by oerang a valve near he reservor n 6 seconds, check wheher here s a rsk of e burs. The concree es are esed for a safe ressure of kn/. (06) Ans: Q = /s, =.637 /s, = 307 kpa > kn/ r. M.N. Shesha Prakash