Pipe flow friction, small vs. big pipes

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Everett Fowler
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1 Friction actor (t/0 t o pip) Friction small vs larg pips J. Chaurtt May 016 It is an intrsting act that riction is highr in small pips than largr pips or th sam vlocity o low and th sam lngth. Friction is dtrmind by th Darcy-Wisbach quation which in turn rlis on th riction paramtr givn rom th Colbrook quation. H F L t luid ( v( t / s)) 0 0 t o pip D ( in) g ( t / s ) Darcy-Wisbach log 7. D R Colbrook In th lattr quation th rlativ roughnss /D is a major componnt o riction. This numbr is larg or small tubs which maks riction highr. Th ollowing graph shows th variation in riction actor or typical pump discharg vlocitis btwn 8-1 t/s and pips ranging in siz rom 0.5 to 1 inchs. For ths calculations I hav usd th absolut roughnss or stl ( t) which is a vry common matrial. Small tubs could b in polythyln or som clar sot plastic matrial both o which would hav a similar roughnss or lowr than stl Pip low riction, small vs. big pips Absolut roughnss t, viscosity 1 cst, watr Vlocity (t/s) Dia (in)

2 Powr (hp) As w can s, th riction actor is low or larg pips and high or small ons. It is as much as 50 tims highr or th vlocity rang and pip sizs considrd. Thankully, sinc most projcts that us small tubs hav gnrally short distancs to covr, th highr riction dos not bcom a problm. Th othr actor is that most small projcts us pumps that produc rlativly low had that will gnrat low vlocitis thror lowr riction. What about powr? I w compar th riction powr or powr rquird to ovrcom riction w ind that th largr pips will rquir mor powr bcaus o th highr low rats as xpctd..0 Friction powr, small vs. big pips 0 t long pip, SG 1, watr Dia (in) Vlocity (t/s) Calculations Th riction paramtr was dtrmind by using th Moody diagram (shown blow). It is possibl to calculat th riction paramtr dirctly by th Colbrook quation by using th Nwton-Raphson itration tchniqu which is in my book and I hav includd hr or convninc. Th rst o th calculations ar straight orward and hr is an xampl: Vlocity is givn

3 Th low rat or powr calculation is: ( ) Th rlativ roughnss is: Th riction actor rom Darcy-Wisbach H F L t 0 t luid o pip 0 D ( v( t / s)) ( in) g ( t / s ) 0,5.17 Friction or L = 0 t Friction powr (

4 4

5 Th Nwton-Raphson Itration Tchniqu Sinc th valu or in th Colbrook quation cannot b xplicitly xtractd rom th quation, a numrical mthod is rquird to ind th solution. Lik all numrical mthods, w irst assum a valu or, and thn, in succssiv calculations, bring th original assumption closr to th tru valu. Dpnding on th tchniqu usd, this can b a long or slow procss. Th Nwton-Raphson mthod has th advantag o convrging vry rapidly to a prcis solution. Normally only two or thr itrations ar rquird. Th Colbrook quation is: log 7. D R Th tchniqu can b summarizd as ollows: 1. R-writ th Colbrook quation as: F log 7. D R 0. Tak th drivativ o th unction F with rspct to : df 1 /.51 1 d.51 log R.7D R. Giv a trial valu to. Th unction F will hav a rsidu (a non-zro valu). This rsidu (RES) will tnd towards zro vry rapidly i w us th drivativ o F in th calculation o th rsidu. F n n1 RES with RES df d For n = 0 assum a valu or 0, calculat RES and thn 1, rpat th procss until RES is suicintly small (or xampl RES < 1 x l0-6 ).

6 Th Nwton-Raphson itration tchniqu is a mthod that convrgs vry rapidly to a solution. You nd to provid a sd valu or n-1 to start th itration and an accptabl rror RES which you can mak vry small. You can ind an xampl o th solution at this link which maks us o an Excl spradsht and th Goal/Sk unction.

Section 11.6: Directional Derivatives and the Gradient Vector

Section 11.6: Directional Derivatives and the Gradient Vector Sction.6: Dirctional Drivativs and th Gradint Vctor Practic HW rom Stwart Ttbook not to hand in p. 778 # -4 p. 799 # 4-5 7 9 9 35 37 odd Th Dirctional Drivativ Rcall that a b Slop o th tangnt lin to th