Pipe Flows (ecures 5 o 7). Choose he crec answer (i) While deriving an expression f loss of head due o a sudden expansion in a pipe, in addiion o he coninuiy and impulse-momenum equaions, one of he following assumpions is made: (a) head loss due o fricion is equal o he head loss is eddying moion (b) he mean pressure in eddying fluid is equal o he downsream pressure (c) he mean pressure in eddying fluids is equal o he upsream pressure (d) head loss in eddies is negleced [Ans.(c)] (ii) Which of he following saemens is crec? (a) Energy grade line lies above he hydraulic grade line and is always parallel o i (b)energy grade line lies above he hydraulic grade line and hey are separaed from each oher by a verical disance equal o he velociy head (c)the hydraulic grade line slopes upwards meeing he energy grade line only a he exi of flow (d) Hydraulic grade line and energy grade line are he same in fluid flow problems. [Ans.(b)] (iii) F pipes arranged in series (a) he flow may be differen in differen pipes (b) he head loss per uni lengh mus be me in a smaller pipe (c) he head loss mus be he same in all pipes (d) he flow rae mus be he same in all pipes [Ans.(d)]. A ank of area A 0 is draining in laminar flow hrough a pipe of diameer and lengh, as shown in he figure. (i)neglecing he exi-je kineic energy and assuming he pipe flow is driven by he hydrosaic pressure a is enrance, derive a fmula f he ank level h() if is iniial Area A 0 level is h 0, assuming he pipe flow o be ρ,µ laminar. h () (ii)repea he above derivaion wihou neglecing he exi-je kineic energy, and assuming he pipe flow o be highly urbulen. Soluion Applying energy equaion beween secions and, we obain pam pam h () 0 hl ρg + g + ρg + g + +
(i)neglecing and enry loss ( given) and as is negligible as compared o, g he above equaion becomes h () h l Considering laminar flow, we have Area A 3µ 0 hl ρ,µ ρ g h () Thus, one can wrie 3µ h (), ρ g ρg h () 3µ Again, from coninuiy equaion, we ge Apipe A0 d ρg h () A0 3µ d Inegraing he above equaion, we obain ρg h ( ) h0 exp 8 µ A0 (ii) Given ha he flow is highly urbulen, herefe fricion fac f consan. From energy equaion wih he consideraion of exi kineic energy, we have h () + f g g gh() + f Again, from coninuiy equaion, we ge Apipe A0 d gh() A0 + f d Inegraing he above equaion, we obain h () h0 8 A 0 g + f
3. A single unifm pipe joins wo reservoirs. Calculae he percenage increase of flow rae obainable if, from he midpoin of his pipe, anoher of he same diameer is added in parallel o i. Neglec all losses excep pipe fricion and assume a consan and equal f f boh pipes. Soluion e he diameer of he pipe be. Case : When he single pipe joins wo reservoirs, as shown in he figure below, he loss of head is f g where is he average velociy of fluid in he pipe. F his case, he discharge is given by A H H Case : When anoher pipe is added in parallel o he main pipe from he midpoin as shown in he figure below, he loss of head is f + f () g g H 3 H From coninuiy equaion, we have + 3 Since he diameer and lengh of boh he parallel pipes are same, we have 3 3
3 Subsiuing he value of in Eq.(), we ge 5 f + f f g g 8 g Equaing he head losses, we have h h 3 f f 5 f f g 8 g.6 F his case, he discharge is given by A.6A Therefe, he percenage increase in he flow rae is given by.6a A 0.6 6% A. There is a sudden increase in he diameer of a pipe from o. If he min loss is independen of he direcion of flow, wha would be he value of? Assume coefficien of conracion C 0.6. c Soluion The schemaic diagram of he pipe is shown in he figure below. e he velociies cresponding o diameers and be and respecively. From coninuiy equaion, we have A A A A
oss of head due o sudden enlargemen is given by g g g ( ) he If he direcion of flow is reversed, here will be a sudden conracion from o. Then he loss of head due o sudden conracion is given by hc 0.66 g Cc g 0.6 g When he min loss is independen of he direcion of flow, he loss of head due o sudden enlargemen should be equal o he loss of head due o sudden conracion. Therefe, we have g g 0.66 0.66 0.66 0.787 0.787 0.73 0.73 0.66 5