Series Sytem of Pipes Parallel Sytem of Pipes

Transcription

1 ANALYSIS OF VISCOUS FLOW IN PIPES Te vicou flow in pipe ytem may be analyzed by claifiying tem in te following manner: i) Serie ytem of pipe, ii) Parallel ytem of pipe, iii) Pipe network, iv) Interconnected reervoir ytem. In deigning or analyi of pipe flow ytem, tere are ix primary parameter involved, wic are 1) te total ead lo of te ytem, f, ) te volumetric flow arte of te fluid, Q 3) te diameter of te pipe, d 4) te lengt of te pipe, L 5) te rougne of te pipe, e f 6) te fluid denity r and vicoity, m Uually, one of te firt tree parameter i to be determined, wile te reamaining one are eiter known or can be pecified by te deigner. Te metod of performing te deign or te analyi i different depending on wat i unknown. Serie Sytem of Pipe Parallel Sytem of Pipe Pipe network Interconnected reervoir ytem 7 Head Lo 1

2 i) Serie Pipe Sytem If te pipe ytem i arranged uc tat te fluid flow troug a ingle pat witout brancing, it i referred a a erie pipe ytem. In a erie pipe ytem, te total ead lo i te um of te ead loe in eac erially connected pipe, and te volumetric flow rate in pipe ould be te ame. l la lb lc Q Qa Qb Qc Serie pipe ytem analyi and deign problem can be claified into tree clae a follow: Cla I ytem: Te total ead lo, l, of te ytem i to be determined. Cla II ytem: Te volumetric flow rate, Q, of te ytem i to be determined. Cla III ytem: Te diameter of te pipe, d, of te ytem i to be determined. 7 Head Lo

3 CLASS I SYSTEMS In cla I ytem, te volumetric flow rate Q, te diameter of te pipe d, te lengt of te pipe L, te rougne of te pipe e, and fluid propertie denity r and vicoity m are known and te total ead lo l i to be determined. Total and lo between two point of te ytem i um of te major and all local (minor) loe, i.e. L V V l l, major l f l,min or K D g g To complete tee calculation, friction factor for eac pipe and lo coefficient for all fitting need to be determined. Te following procedure may be followed: Te procedure for olving Cla I problem 1) Determine te relative rougne, e /d, for eac pipe. ) Calculate te average velocity in eac pipe, V=4Q/pd 3) Calculate Reynold number, Re=rVd/m 4) Determine te friction factor f 5) Calculate te major ead lo, l =f(l/d)(v /g) 6) Determine minor ead lo coefficient K and ten calculate te minor ead loe for eac variable are portion, l =K(V /g). 7) Evaluate te total ead lo of te ytem by adding te major and minor loe calculated in tepe 5 and 6. 7 Head Lo 3

4 Example: In te ytem own, water at 0 o C i flowing at te volumetric flow rate of m 3 /. Te uction line i made from commercial teel pipe of diameter 4 inc and lengt i 15 m. Te dicarge line i alo commercial teel pipe and it diameter i inc and lengt i 00 m. Te entrance to te uction line from te reervoir i troug a quare edged inlet. Te elbow i 90 o tandad elbow and te globe valve i fully open. Calculate te power upplied to te pump if it efficiency i 76%. Solution: Te required pump ead for tranporting water between te two reervoir may be determined by applying te extended Bernoulli equation between point and 1 along te treamline, wic i own in Figure. P1 V1 P V z1 z f rg g rg g t t1 f 1 or olving for te pump ead, we obtain, t t 1 f 1 Since te area of te two reervoir are very large, wen compared to te croectional area of te uction and te dicarge pipe, te velocitie at te free urface of te reervoir are negligible, tat i V 1 =0, and V =0. Alo, te free urface of te reervoir are expoed to te atmopere, o tat P 1 =P =P atm. 1 P V Patm rg g rg 1 1 t1 z1 P V P z z rg g rg atm t z Terefore, te required pump ead may be expreed a, f1 7 Head Lo 4

5 Te frictional ead lo, f1-, may now be evaluated by following te procedure wic i preented in ti ection. i) Te relative rougne for te 4" commercial teel uction pipe and te " commercial teel dicarge pipe may be obtained from Figure a follow: (ϵ /d) = (ϵ /d) d = ii) Te average velocity in te uction and te dicarge pipe may ten be evaluated a follow: V 3 4 Q (4)(0.015m /) 1.85 m/ pd p(0.1016m) V d 3 4 Q (4)(0.015m /) 7.4 m/ pd p(0.0508m) d iii) Te kinematic vicoity of water at 0 0 C i 1.003x10-6 m /. Terefore, te Reynold number for te uction and te dicarge pipe are a follow: Vd (1.85)(0.1016) (1.003)(10 ) 5 Re (1.87)(10 ) 6 Vd (7.4)(0.0508) (1.003)(10 ) d d 5 Re d (3.75)(10 ) 6 iv) Now, te friction factor for te uction pipe correponding to a relative rougne of (ϵ /d) = and a Reynold number of Re = 1.87x10 5 from te Moody diagram i, f =0.019 Similarly, te friction factor for te dicarge pipe correponding to relative rougne of (ϵ /d) d = and a Reynold number of Re d = 3.75x10 5 i, f d =0.00 v) Te major ead loe in te uction and te dicarge pipe may be evaluated by uing te Darcy-Weibac equation a f L V (15m) (1.85m/) f (0.019) 0.49m d g (0.1016m) ()(9.81m/ ) fd Ld Vd (00m) (7.4m/) fd (0.00) 19.77m d g (0.0508m) ()(9.81m/ ) d 7 Head Lo 5

6 vi) For te quare edged inlet, te ead lo coefficient, k i, may be obtained from te Figure a 0.5. Ten te minor ead lo troug te inlet i fi V (1.85m/) ki (0.5) 0.09 m g ()(9.81m/ ) Te ead lo coefficient, k o, for te outlet from te dicarge pipe into te reervoir may be obtained from figure a 1.0, ten te minor ead lo i fo Vd (7.4m/) ko (1.0).79m g ()(9.81m/ ) For full open globe valve lo coefficient k v = 10 fv Vd (7.4) kv m g ()(9.81) For 90 o tandard elbow lo coefficient k=0.3, fe Vd (7.4) ke m g ()(9.81) vii) Te total ead lo i te um of te major and minor loe, ten f f fd fi fo fv fe f 1 f m Te power required by te pump i p rgq p 3 3 (998.kg/m )(9.81m/ )(0.015m /)(51.89m) (0.76) 50.6kW 7 Head Lo 6

7 CLASS II SYSTEMS In cla II ytem, te total ead lo of te ytem l, te diameter of te pipe d, te lengt of te pipe L, te rougne of te pipe e, and fluid propertie denity r and vicoity m are known and te volumetric flow rate of flow Q i to be determined. Since te volumetric flow rate i unknown, te fluid velocity and ence Reynold number can not be calculated directly. Tu, te friction coefficient can not be determine directly. Hence an iterative olution procedure i applied to determine te volumetric flow rate of te fluid. Te main tep of ti iterative olution procedure i given below: Te procedure for olving Cla II problem 1) Determine te relative rougne, e /d, for eac pipe. ) Aume a friction factor f for eac pipe from te Moody diagram uing te relative rougne e /d in fully roug region. Te reaon for uing te fully roug region in Moody diagram i tat in ti region friction factor i independent from te Reynold number. If ti aumption i true no iteration i neceary. 3) Calculate te major ead lo, l =f(l/d)(v /g) in term of te unknown velocity. 4) Determine minor ead lo coefficient K and ten calculate te minor ead loe for eac variable are portion, l =K(V /g) in term of te unknown velocity. 5) Evaluate te total ead lo of te ytem by adding te major and minor loe calculated in tep 4 and 5. 6) Relate te unknown velocitie in eac pipe by uing te continuity equation. 7) Expre te total ead lo l of te ytem in term of one of te unknown velocitie, and ten olve te unknown velocitie. 8) Calculate Reynold number, Re=rVd/m for eac pipe. 9) Determine te improved friction factor f for eac pipe from te Moody diagram uing Reynold number and e /d. 10) Compare te aumed and te improved value of te friction factor and repeat tep 3 troug 9 a many time a needed in order to obtain deired accuracy in te friction factor. 11) Evaluate volumetric flow rate of te fluid, Q=pd V/4 7 Head Lo 7

8 Example Te piping ytem, wic i own in te figure i ued to tranfer water at 5 o C from one torage tank to te oter. Te larger pipe i a 6 inc commercial teel pipe aving a total lengt of 30 m. Te maller pipe i a inc commercial teel pipe aving a total lengt of 15 m. Te entrance to te larger pipe from te torage tank i troug an inward projecting pipe. 90 o tandard elbow are ued, and te gate valve i alf open. Determine te volumetric flow rate troug te ytem. Solution: Te total ead lo during te tranfer of water from one torage tank to te oter may be determined by applying te extended Bernoulli equation between point and 1 along te treamline wic i own in Figure. Ten t t1 f 1 or olving for te total ead lo f 1 t 1 t Since te area of te two reervoir are very large wen compared to te croectional area of te larger and te maller pipe, ten te velocitie at te free urface of te torage tank are negligible tat i, V 1 =0 and V =0. Alo te free urface of bot torage tank are expoed to te atmopere, o tat P 1 =P = P atm. Terefore, P V P rg g rg 1 1 atm t1 z 1 10 P V Patm rg g rg t z Patm Patm Terefore, te total ead lo of te ytem i f m rg rg 7 Head Lo 8

9 Te volumetric flow rate of water may now be evaluated by following te procedure decribed above, i) Te relative rougne of 6 and inc commercial teel pipe are, (ϵ /d) l = (ϵ /d) = ii) Te friction factor correponding to te relative rougnee can be obtained from Moody Cart a, for (ϵ /d) l = f l = and for (ϵ /d) = f = iii) Te major ead loe in te larger and te maller pipe may be evaluated by uing te Darcy-Weibac equation a L V f d g (30) ( V ) l l l f (0.015) 0.151V l l l l (0.154m) ()(9.81m/ ) L V (15m) ( V ) S f f (0.019) 0.86V d g (0.154m) ()(9.81m/ ) iv) For an inlet projecting pipe inlet, te ead lo coefficient k i = 1.0 and for te reducer, k r =0.4. Uing tee coefficient ead loe for te inlet and te reducer can be calculated a follow, For te inlet, fi Vl ( Vl) ki (1) 0.051V g ()(9.81) l For reducer, fr V ( V) kr (0.4) 0.01V g ()(9.81) Te ead lo coefficient for te outlet k o =1.0, ten te minor ead lo for te outlet i fo V ( V) ko (1.0) 0.051V g ()(9.81) 7 Head Lo 9

10 Lo coefficient for alf open gate valve i k v =.1, fv V ( V) kv (.1) 0.11V g ()(9.81) Lo coefficient for 90 tandard elbow i 0.3 V ( V) g ()(9.81) l l fe ke (0.3) 0.035Vl Total ead lo can be calculated from te um of te major and minor loe a, f f fl f fi fr fo fv fe V 0.86V 0.051V 0.01V 0.051V 0.11V 0.035V f l l l V 0.468V l v) Te velocitie in te larger and te maller pipe may now be related by uing continuity equation a, pd V / 4 pd V / 4 l l V ( d / d ) V l l V V (0.154 / ) 9V l V l Subtituting into total ead lo equation, V 0.468(9 V ) Solving for V l and V, V l = 0.51 m/ V = 4.60 m/ l l ix) Te kinematic vicoity of water at 5 o C i 0.893x10 6 m / 7 Head Lo 10

11 Reynold number for te larger and maller pipe, Vd (0.49m/)(0.154m) l l 5 Re l (8.36)(10 ) 6 water 0.893(10 )m / Vd (4.60m/)(0.0508m) 5 Re (.54)(10 ) 6 water 0.893(10 )m / x) Now for bot large and mall pipe, te improved value of friction factor can be read from Moody cart uing for relative rougne value and te Reynold number calculated. For larger pipe, (ϵ /d) = and f l = 0.0 For maller pipe, (ϵ /d) = and f = xi) Tee tep given above ould be repeated until te friction factor converge. Te iteration reult are given in te table below. Te volumetric flow rate can be obtained uing te velocity value obtained in te current tep a, Q V pd / 4 (0.473m/)( p)(0.154) / (10 )m / l 3 3 l Iteration 1 f l (Aumed) f (Aumed) fl (m) V l 0.01 V l f (m) 0.86 V V fi (m) V l V l fr (m) 0.01 V 0.00 V fo (m) V V fv (m) 0.11 V 0.11 V fe (m) V l V l f1- (m) 0.37 V l V V l V V l (m/) V (m/) Re l 8.36x x10 4 Re.51x10 5.4x10 5 f l (Improved) f (Improved) Head Lo

12 CLASS III SYSTEMS Cla III ytem are true deign problem. In cla III ytem, te total ead lo of te ytem l, te volumetric flow rate Q, te lengt of te pipe L, te rougne of te pipe e, and fluid propertie denity r and vicoity m are known and te pipe diameter d i to be determined. Since te diameter i unknown, te fluid velocity and ence te Reynold number can not be calculated directly. Tu, te friction coefficient can not be determine directly. Hence, an iterative olution procedure i applied to determine te pipe diameter. Te main tep of ti iterative olution procedure i given below: Te procedure for olving Cla III problem 1) Aume te diameter d of eac pipe ) Determine te relative rougne, e /d, for eac pipe. 3) Calculate te average velocity V=4Q/pd for eac pipe. 4) Calculate Reynold number, Re=rVd/m for eac pipe. 5) Determine te friction factor f for eac pipe from te Moody diagram uing Reynold number and e /d. 6) Calculate te major ead lo for eac pipe uing l =f(l/d)(v /g). 7) Determine minor ead lo coefficient K and ten calculate te minor ead lo for eac variable are portion uing l =K(V /g). 8) Evaluate te total ead lo of te ytem by adding te major and minor loe calculated in tepe 6 and 7. 9) Compare te calculated total ead lo wit te given ead lo. If te calculated ead lo i le tan te given ead lo, ten decreae pipe diameter. If te calculated ead lo i more tan te given ead lo, ten increae te pipe diameter. Repeat tep () troug (8) until te calculated and te given ead loe are cloe enoug. 7 Head Lo 1

13 Example: In a cemical proceing ytem, benzene at 0 o C i taken from te bottom of a larger tank and tranferred by gravity to anoter part of te ytem, a own in te figure. Te lengt of te pipe between two tank i 7 m and it i commercial teel pipe. A filter i intalled in te line, and i known to ave a lo coefficient of 8.5. Te gate valve i fully open. Determine te diameter of te pipe wic would allow a volumetric flow rate of m 3 / troug ti ytem. Solution: Te total ead lo during te tranportation may be determined by applying extended Bernoulli Eq. between point and 1 along te treamline own. t t1 f 1 Solving for te total ead lo, f 1 t1 t Since te urface area of te reervoir are muc larger tan te cro-ection area of te piping, velocitie at te free urface are negligible, tat i V 1 =0 and V =0. Alo, P 1 =P =P atm. P V P rg g rg 1 1 atm t1 z 1 8 P V Patm rg g rg t z Te total ead lo of te ytem i Patm Patm f 1 8 8m rg rg 7 Head Lo 13

14 Step 1 In order to tart iteration procedure, a pipe diameter of may be aumed. Step Te relative rougne for commercial teel pipe i (ϵ /d) = Step 3 Te average velocity in te pipeline may be calculated a V 3 4 Q (4)(0.005m /) 1.3m/ pd p(0.0508m) Step 4 Calculation of te Reynold number r 895kg/m m (10 )Pa 3 rvd (895kg/m )(1.3m/)(0.0508m) 4 Re 8.6(10 ) 4 m 6.5(10 )Pa Step 5 Now te friction factor correponding to a relative rougne of (ϵ /d) = and a Reynold number of 8.6x10 4 can be read from Moody diagram a f=0.0 Step 6 Te major ead lo in pipeline may now be evaluated uing te Darcy-Weibac equation a LV (7) (1.3) fp f (0.0) 0.34m d g (0.0508m) ()(9.81m/ ) Step 7 For an inlet troug an inward projecting pipe, te ead loo coefficient k i i 1.0. fi V (1.3) k (1.0) 0.077m i g ()(9.81m/ ) 7 Head Lo 14

15 And for te filter, lo coefficient k f for te filter wa given a 8.5. fi V (1.3) k (8.5) 0.655m f g ()(9.81m/ ) And for te outlet, lo coefficient k o for te dicarge pipe i 1.0. fo V (1.3) k (1.0) 0.077m o g ()(9.81m/ ) Lo coefficient for te fully open gate-valve i fv V (1.3) k (0.15) 0.011m v g ()(9.81m/ ) Step 8 Total ead lo i te um of te major and minor loe. f fp fi ff fo fv m Step 9 Calculation in Step to Step 8 mut be repeated until te total ead lo converge. Te ummary of te calculation from te iteration are given in te table below. Iteration 1 t nd 3 rd 4 t d (Aumed) 1 ½ 1 1 ¼ ϵ /d V (m/) Re 8.16x x x x10 4 f fp (m) fi (m) ff (m) fo (m) fv (m) f1- (m) Cange in d Decreae Decreae Increae O.K. 7 Head Lo 15

16 Parallel Sytem of Pipe If te pipe ytem caue te flow to branc into two or more line, ten it i referred a a parallel ytem of pipe. A typical parallel ytem of pipe i own in Figure. Te flow in te main line plit into tree brance at ection 1, and ten rejoin at ection. Te ead lo in eac branc between ection 1 and mut be equal, tat i f1 = fa = fb = fc Alo, one ould oberve tat te volumetric flow rate in te main line i equal to te um of te volumetric flow rate troug eac branc. Hence Q = Q a + Q b + Q c Uually, two type of problem occur in parallel ytem of pipe. Tee are: i) Cla I Sytem for wic te ead lo in eac branc i known, and te volumetric flow rate in eac branc and in te main line are to be determined. ii) Cla II Sytem for wic te total volumetric flow rate i known, and te volumetric flow rate and te ead lo in eac branc are to be determined. Parallel ytem of pipe 7 Head Lo 16

17 Cla I Sytem In Cla I problem, te preure drop or te ead lo acro te parallel brance i known, and it i deirable to determine te volumetric flow rate in eac branc and in te main line. Te procedure for olving Cla I problem may be given a follow: 1) Determine te relative rougne, e /d, for eac branc by uing te pipe diameter, d, and te pipe material. ) Aume a friction factor, f, for eac branc from te Moody diagram by uing te relative rougne, ɛ /d, in te fully roug region. Te reaon for te aumption of, te flow in te fully roug region i tat no iteration i neceary, if ti aumption i true. 3) Calculate te major ead loe for eac branc by uing te Darcy-Weibac equation, f = f (L/d)(V /g) in term of te unknown branc velocitie. 4) Determine te minor ead lo coefficient K and ten calculate te minor ead loe for te variable area portion in eac branc by uing f = kv /(g) in term of te unknown branc velocitie. 5) Evaluate te total ead lo, f, of eac branc by adding up te major and te minor ead loe from tep (3) and (4). 6) Equate ti ead lo to te given ead lo acro te parallel brance and olve for te unknown branc velocitie. 7) Calculate te Reynold number, Re = ρvd/μ for eac branc, 8) Determine te improved value of te friction factor, f, for eac branc from te Moody diagram by uing te Reynold number, Re, and te relative rougne, ɛ /d. 9) Compare te aumed and te improved value of te friction factor for eac branc and repeat tep (3) troug (8) a many time a needed in order to obtain te deired accuracy in te friction factor. 10) Evaluate te volumetric flow rate of te fluid, Q = πd V/4, in eac branc and in te main line. 7 Head Lo 17

18 Example 10.4 Te arrangement, wic i own in Figure, i ued to upply lubricating oil to te bearing of a large macine. Te bearing act a retriction to te flow. Te ead lo coefficient for te journal bearing in brance x and y are 11 and 4, repectively. Te lengt of 1" commercial teel branc x i 10 m, wile te lengt of 1 ½ inc commercial teel branc y i 5 m. Eac of te four bend in te tubing a a radiu of 100 mm. Te preure before and after te brancing are 75 kpa and 195 kpa repectively. Te denity and te abolute vicoity of te lubricating oil are 881 kg/m 3 and.x10-3 Pa.. Determine: a) te volumetric flow rate troug eac bearing, and b) te total volumetric flow rate. Solution Te ead lo acro te parallel brance may be determined by applying te extended Bernoulli Equation between ection and 1 in Figure a P ρg + V g + z = P 1 ρg + V 1 g + z 1 f1 Note tat V 1 =V and z 1 = z, o tat te ead lo between ection 1 and may be evaluated a f1 = P 1 P ρg N = m N m (881 kg/m 3 )(9.81 m/ ) = 9.6 m a) Te volumetric flow rate troug eac branc may now be determined by applying te procedure wic i preented above. i) Te relative rougne for te 1 and 1/ commercial teel pipe may be obtained from a follow: (ε /d) x = (ε /d) y = Head Lo 18

19 ii) Te friction factor in brance x and y correponding to relative rougne value of (ε /d) x = and (ε /d) y = in te fully flow region may be obtained from te Moody diagram a f x = 0.05 f y = 0.0 iii) Te major ead lo in brance x and y may be evaluated by uing te Darcy- Weibac equation a L x V x fpx = f x d x g = (0.05) (10 m) V x (0.054 m) ()(9.81 m/ ) fpx = 0.45 V x L y V y fpy = f y d y g = (0.05) (10 m) V y (0.054 m) ()(9.81 m/ ) fpy = V y iv) Te minor ead loe in te journal bearing are V x fjx = k jx g = 11 V x 9.81 m = V x fjy = k jy V y g = 4 V y 9.81 m = 0.04 V y For tandard tee wit flow troug branc, lo coefficient from Table i obtained a k=1.0 ftx = k V x g = (1) V x 9.81 m = 0.10 V x fty = k V y g = (1) Te minor ead lo coefficient for bend k b = 1. 3 V y 9.81 m = 0.10 V y V x fbx = k b g = (1.3) V y fby = k b g = (1.3) V x 9.81 m = 0.13 V x V y 9.81 m = 0.13 V y 7 Head Lo 19

20 v) Ten te total ead loe in brance x and y may be evaluated a fx = fpx + fjx + ftx + fbx fx = 0.45 V x V x V x V x = 1.45 V x fy = fpy + fjy + fty + fby fy = V y V y V y V y = 0.57 V y vi) Te velocity in branc x i V x = ( fx 9.6 m 1.45 )1/ = ( 1.45 )1/ =.7 m/ and te velocity in branc y i V y = ( fy 9.6 m 0.57 )1/ = ( 0.57 )1/ = 4.04 m/ vii) Now, te Reynold number in brance x and y may be calculated a Re x = ρv xd x (881 kg = m 3)(.80 m )(0.054 m) μ (.x10 3 N. /m =.85x10 4 ) Re y = ρv 881 yd kg y m = m m μ.x10 3 = 6.67x10 4 N. m viii) Now, te improved value of te friction factor for branc x correponding to a relative rougne of (ε /d) x = and a Reynold number of Re x =.81x10 4 from te Moody diagram i f x = 0.08 Similarly, te improved value of te friction factor for branc y correponding to a relative rougne of (ε /d) y = and a Reynold of Re y = 6.67x10 4 i f y = 0.04 ix) A long a te aumed and te improved value of te friction factor are not te ame, ten te calculation in tep (iii) troug (vii) mut be repeated. Te ummary of tee iteration i preented in te Table. 7 Head Lo 0

21 x) Te volumetric flow rate troug brance x and y may now be evaluated a Q x = πd x V x = π m (.64 m/) = 1.34x10 3 m 3 / 4 4 Q y = πd y V y 4 = π m (4.14 m/) 4 = 4.7x10 3 m 3 / b) Te volumetric flow rate troug te main line i te um of te volumetric flow rate in brance x and y. tat i m3 m3 3 Q = Q x + Q y = 1.34x x10 3 m3 3 Q = 6.06x10 Iteration 1 f x (aumed) f y (aumed) fpx 0.45 V x 0.56 V x fpy V y V y fjx V x V x fjy 0.04 V y 0.04 V y ftx 0.10 V x V x fty 0.10 V y V y fbx 0.13 V x V x fby 0.13 V y 0.08 V y fx V x V x fy V y V y V x V y Re x.65x x10 4 Re y 6.67x x10 4 f x (improved) f y (improved) Head Lo 1

22 Cla II Sytem In Cla II problem, te volumetric flow rate in te main line i known, and it i deirable to determine te volumetric flow rate in eac branc and te ead lo acro te parallel brance. Te procedure for olving Cla II problem may be given a follow 1) Determine te relative rougne, ɛ /d, for eac branc by uing te pipe diameter, d, and te pipe material. ) Aume a friction factor, f, for eac branc from te Moody diagram by uing te relative rougne, ɛ /d in te fully roug region. Te reaon for te aumption of te flow in te fully roug region i tat no iteration i neceary, if ti aumption i true. 3) Calculate te major ead loe for eac branc by uing te Darcy-Weibac equation, f = f(l/d)(v /g) in term of te unknown branc velocitie. 4) Determine minor lo coefficient k and ten calculate te minor ead loe for te variable area portion in eac branc by uing f = kv /g in term of te unknown branc velocitie. 5) Evaluate te total ead lo, f, of eac branc by adding up te major and te minor ead loe from tep (3) and (4). 6) Equate te te total ead lo of eac branc and expre all unknown branc velocitie in term of one unknown branc velocity. 7) A long a te volumetric flow rate in te main line i te um of te volumetric flow rate of eac branc, ten expre it in term of te unknown branc velocitie. 8) Solve for te unknown branc velocitie. 9) Calculate te Reynold number, Re = ρvd/μ, for eac branc. 10) Determine te improved value of te friction factor, f, for eac branc from te Moody diagram by uing te Reynold number, Re, and te relative rougne, ɛ /d. 11) Compare te aumed and te improved value of te friction factor for eac branc and repeat tep () troug (10) a many time a needed in order to obtain te deired accuracy in te friction factor. 1) Evaluate te volumetric flow rate of te fluid, Q = πd V/4 in eac branc. 7 Head Lo

23 Example m 3 / of water at 15 C i flowing in a " commercial teel pipe at ection 1, a own in Figure. Te eat excanger in branc x a a ead lo coefficient of 1.5. Branc y i a bypa line, wic i compoed of 1 1/4" commercial teel pipe wit a lengt of 6 m. All tree valve are wide open, and te elbow are tandard. Determine; a) te volumetric flow rate of water in eac branc and b) te preure drop between point 1 and. Solution a) To determine te volumetric flow rate of water in eac branc one may apply te procedure wic i preented in ti ection i) Te relative rougne for te and 1 1/4 commercial teel pipe may be obtained from Figure 10.4 a follow: (ε /d) x = (ε /d) y = ii) Te friction factor in brance x and y correponding to relative rougne value of (ε /d) x = and (ε /d) y = in te fully flow region may be obtained from te Moody diagram in Figure 10.5 a f x = f y = 0.01 iii) Te major ead lo in brance y may be evaluated by uing te Darcy- Weibac equation a L y V y fpy = f y d y g = (0.01) (6 m) V y ( m) ()(9.81 m/ ) fpy = 0.0 V y 7 Head Lo 3

24 iv) Te minor ead loe in te journal bearing are V x fx = k g = 1.5 V x 9.81 m = V x For fully open gate valve; lo coefficient k v = Ten, te minor ead lo in brance x and y are; V x fvx = k v g = 0.15 (13) V x ()(9.81 m/ ) fvx = V x fvy = k V y V y g = (0.15) ()(9.81 m/ ) fvy = V y For tee wit flow troug lo coefficient k t = 0.; V x ftx = k t g = 0. V x ()(9.81 m/ ) ftx = 0.00 V x For tee wit flow troug branc, te lo coefficient k t = 1.0 V y fty = k t g = () 1 V y 9.81 m = 0.10 V y Lo coefficient for tandard elbow i k e = 0.3 V y fey = k e g = ()(0.3) V y ()(9.81 m/ ) fvy = V y v) Now, te total ead loe in brance x and y are fx = fx + fvx + ftx fx = V x V x V x = 0.68 V x fy = fpy + fvy + fty + fey fy = 0.0 V y V y V y V y = V y 7 Head Lo 4

25 vi) A long a te total ead lo in eac branc mut be equal ten fx = fy = 0.68 V x = V y or V x = V y vii) Te volumetric flow rate in te main line may now be expreed a Q = πd x V x 4 = πd y V y 4 Now, te numerical valve may now be ubtituted to obtain or Q = π( m) V x 4 + π( m) V y 4.07 V x V y = 6.3 = m3 viii) Te velocity in brance x and y may now be evaluated a V x =.05 m/ V y =.69 m/ ix) Te kinematic vicoity of water at 15 C i 1.139x10-6 m / from Table A.. Ten te Reynold number in brance x and y may now be evaluated a Re x = V xd x ν = (.05 m )( m) (1.139x10 6 m /) = 9.14x104 Re y = V yd y ν =.69 m m 1.139x10 6 m / = 7.50x10 4 x) Now, te improved value of te friction factor for branc x correponding to a relative rougne of (ε /d) x = and a Reynold number of Re x = 9.19x10 4 from Moody diagram in Figure 10.5 i f x = 0.0 Similarly, te improved value of te friction factor for branc y correponding to a relative rougne of (ε /d) y = and a Reynold of Re y = 7.50x10 4 i f y = Head Lo 5

26 xi) A long a te aumed and te improved value of te friction factor are not te ame, ten te calculation in tep (iii) troug (xii) mut be repeated. Te ummary of tee iteration i preented in Table 10.9 xii) Te volumetric flow rate troug brance x and y may now be evaluated a Q x = πd x V x = π m (.10m/) = 4.6x10 3 m 3 / 4 4 Q y = πd y V y 4 = π m (.59 m/) 4 =.05x10 3 m 3 / b) Te preure drop between ection 1 and may be evaluated by uing te extended Bernoulli Equation a P ρg + V g + z = P 1 ρg + V 1 g + z 1 f1 At ti point, one ould note tat V 1 =V and z 1 =z. Alo te frictional ead lo between ection 1 and i f1 = fx = fy = V x = (.10 m ) = 3.14 m Te denity of water i kg/m 3 from Table, terefore p p 1 = ρ g f1 = (999.1 kg/m 3 )(9.81 m/ )(3.14 m) p p 1 = kpa Iteration 1 f x (aumed) f y (aumed) fpy 0.0 V y 0.31 V y fx V x V x fvx V x 0.09 V x fvy V y V y ftx 0.00 V x V x fty 0.10 V y V y fey V y V y fx 0.68 V x V x fy V y V y V x V y Re x 9.14x x10 4 Re y 7.50x x10 4 f x (improved) f y (improved) Head Lo 6

27 Pipe Network In municipal water ditribution ytem, te pipeline are frequently brancing and looping in a complex manner wic i often referred a a pipe network. A imple pipe network i own in Figure. Te mot important factor wic make te analyi of pipe network difficult, i te uncertainty about te direction of te flow in different pipe one ould note tat, it i almot impoible to tell te direction of te flow, uc a in pipe c of Figure. Te fluid flow in a pipe network mut atify te baic principle of te conervation of ma and te conervation of energy. Tee principle can be ummarized a follow i) At eac junction of te pipe network, te um of te volumetric flow rate into te junction mut be equal to te um of te volumetric flow rate out of te junction. ii) Te algebraic um of te ead loe around any cloed loop mut be zero. Te pipe network problem are generally olved by a trial-and-error procedure. Te mot practical and widely ued metod of flow analyi in pipe network i tat of ucceive approximation, wic i developed by Hardy Cro. In te olution procedure, wic i outlined below, te pipe network ould be divided into everal cloed loop. 1) Determine te relative rougne, ɛ /d, for eac pipe from Figure 10.4 by uing te pipe diameter, d, and te pipe material. ) Aume a friction factor, f, for eac pipe from te Moody diagram by uing te relative rougne, ɛ /d in te fully roug flow region. 3) Expre te total ead lo, f, for eac pipe in te form of f = KQ were K i te equivalent reitance to te flow in te pipe, and Q i te unknown volumetric flow rate troug te pipe. 7 Head Lo 7

28 4) Aume a value for te volumetric flow rate in eac pipe uc tat te um of te volumetric flow rate into eac junction i equal to te um of te volumetric flow rate out of te junction. Note tat te fluid tend to follow te pat of leat reitance troug te pipe network, o tat te pipe aving ig value of K ave relatively low volumetric flow rate. 5) Determine te algebraic um of total ead loe, σ f, for eac cloed loop by uing te following ign convention. If te flow i clockwie, ten te total ead lo and te volumetric flow rate are poitive. However, if te flow i counterclockwie, ten te total ead lo and te volumetric flow rate are negative. 6) Determine te aritmetic um of KQ, tat i σ KQ for eac cloed loop. During ti ummation, conider all KQ value a poitive. 7) Calculate te correction for te volumetric flow rate, ΔQ, for eac loop from Q = f σ(qk) 8) Calculate te improved value of te volumetric flow rate, Q', by uing Q = Q Q 9) Repeat calculation in tep (5) troug (8) until ΔQ from tep (7) become negligibly mall. Te Q' value i ued for te next cycle of iteration. 10) Calculate te average velocity, V = 4Q/(πd ) for eac pipe. 11) Calculate te Reynold number, Re = ρvd/μ for eac pipe. 1) Determine te improved value of te friction factor, f, for eac pipe from te Moody diagram in Figure 10.5 by uing te Reynold number, Re, and te relative rougne, ɛ /d. 13) Compare te aumed and te improved value of te friction factor for eac pipe and repeat tep (lii) troug (xii) a many time a needed in order to obtain te deired accuracy in te friction factor. 7 Head Lo 8

29 Example Water at 0 C i flowing troug te pipe network, a own in Figure. All pipe are 1" cat iron, and te elbow are tandard. Determine te volumetric flow rate of water in pipe a, b, c, d and e. Solution: Te volumetric flow rate in eac pipe of te network may be determined by applying te procedure, wic i preented in ti ection: i) Relative rougne for te 1 cat iron pipe may be obtained from Figure a e D 0.01 ii) Te friction factor d correponding to ti relative rougne in te fully roug flow region may be obtained from te Moody diagram a f=0.038 iii) For tandard elbow, te tandard tee wit flow troug run and te tandard tee wit flow troug branc lo coefficient can be obtained from te table. Tee troug branc- K t1 =1 Tee wit line flow- K t =0. Elbow- K e =0.3 L 1 4Qa f a f Kt1 Ke Kt D g p D 6 1 4Qa (0.038) m ()(9.81 m / ) p (0.054 m) Q 6 a 7 Head Lo 9

30 Lb 1 4Qb fb f Kt Ke Kt1 D g p D 5 1 4Qb (0.038) m ()(9.81 m / ) p (0.054 m).310 Q 6 b Lc 1 4Qc fc f Kt D g p D 4 1 4Qc (0.038) (1) 6 c m ()(9.81 m / ) p (0.054 m).0910 Q Ld 1 4Qd fd f Kt Ke Kt1 D g p D Qd (0.038) m ()(9.81 m / ) p (0.054 m).1710 Q 6 d Le 1 4Qe fe f Kt Ke Kt1 D g p D Qe (0.038) m ()(9.81 m / ) p (0.054 m) Q 6 e iv) A long a te K value for pipe i a larger tan te K value for pipe B, ten te volumetric flow rate in pipe will be larger. Te continuity equation at junction J 1 yield 3 ( a b) 0.01 m / Q Q Terefore it i poible to aume Q a and Q b a Q a =0.004 m 3 / and Q b =0.006 m 3 / Now te continuity equation may be applied to junction J a (Q c +Q d ) = Q a m 3 /=0.003 m 3 / 7 Head Lo 30

31 Wen te flow in pipe C i aumed to proceed from junction J1 to junction J3. Since te K value for pipe C i maller tan te K value in pipe d, ten it i probable tat te volumetric flow rate in pipe C will be larger. Hence Qc and Qd may be aumed Qc=0.00 m3/ and Qd=0.001 m3/ Finally, te continuity equation for junction J3 i Qe=(Qc+Qb) m3/=0.004 m3/ v) Te um of te total eat loe in loop 1 may be now be calculated a K Q K Q K Q m f 1 a c b a a c c b b (.6110 )(0.004) ( )(0.00) (.3 10 )(0.006) 33.4 Similarly, te um of te ead loe in loop i K Q K Q K Q m f d e c d d e e c c ( )(0.001) ( )(0.004) ( )(0.00) vi) Te um of te value KQ for loop 1 i ( KQ) K Q K Q K Q (.6110 )(0.004) ( )(0.00) (.310 )(0.006) / m 1 a a c c b b Similarly, te um of te value of KQ for loop i ( KQ) KdQd KeQe KcQ ( )(0.001) ( )(0.004) (.0910 )(0.00) 7640 / m c vii) Te etimate of error in te aumed value of te volumetric flow rate for loop 1 and are: p Q m / ( KQ) / m 1 Q m / f ( KQ) 7640 / m viii) Te volumetric flow rate in pipe a, b and c of loop 1 may now be corrected a Q Q Q 410 m / ( m / ) m / ' a a 1 Q Q Q 610 m / ( m / ) m / ' b b 1 Q Q Q 10 m / ( m / ).5910 m / ' c c 1 Similarly, te correction may be applied to te volumetric flow rate of pipe c, d and e a Q Q Q.5910 m / ( m / ) m / '' ' c c Q Q Q 110 m / ( m / ).3110 m / '' d d Q Q Q m / ( m / ).6910 m / '' e e 7 Head Lo 31

32 ix) In order to make Q 1 and Q negligibility mall, it i neceary to repeat te calculation. Tee iteration are preented in Table. x) Te velocity in eac pipe may now be evaluated a V V V V V a b c d e Qa (4)( m / ) 9.45 m / pd p(0.054 m) Qb (4)(5.110 m / ) 10.8 m / pd p(0.054 m) Qc (4)(1.310 m / ) 1.43 m / pd p(0.054 m) Qd (4)(.5610 m / ) 5.05 m / pd p(0.054 m) 4Q p D e 3 3 (4)(.4410 m / ) 4.8 m/ p (0.054 m) xi) Te kinematic vicoity of water at 0 C i 1.003x10-6 m / from Table A., o tat te Reynold number for eac pipe may be calculated a ad (9.45 m / )(0.054 m) Rea m ( m / ) ad (10.8 m / )(0.054 m) Reb m ( m / ) ad (.43 m / )(0.054 m) Rec m ( m / ) Re d ad (5.05 m / )(0.054 m) m ( m / ) ad (4.8 m / )(0.054 m) 5 Ree m ( m / ) xii) Te friction factor correponding to a relative rougne value of e/d = 0.01 and Reynold number in te range 6.15x10 4 to.6x10 5 may be determined from te Moody diagram a f = 0.08 xiii) A long a te aumed and te improved value of te friction factor are te ame, ten tere i no need to iterate. Conequently, te volumetric flow rate in eac pipe may be given a follow: 3 3 Q m / a 3 3 Qb m / 3 3 Qc m / 3 3 Qd.5610 m / 3 3 Qe m / Head Lo 3

33 Iteration Q a (m 3 /) Q b (m 3 /) Q c (m 3 /) Q c (m 3 /) Q d (m 3 /) Q e (m 3 /) S S S(kQ) 1 (/m ) S(kQ) (/m ) DQ 1 (m 3 /) DQ (m 3 /) Q a (m 3 /) Q b (m 3 /) Q c (m 3 /) Q c (m 3 /) Q d (m 3 /) Q e (m 3 /) Head Lo 33

34 Interconnected Reervoir Sytem Anoter example of pipe ytem, wic a practical importance in water upply ytem i wen tree or more reervoir at variou elevation are interconnected at a joint. Figure ow tree interconnected reervoir. Reervoir A i at te iget elevation, reervoir B i at te intermediate elevation and reervoir C i at te lowet elevation. Tree pipe 1, and 3 meet at a common junction J, and Q 1. Q and Q 3 are te volumetric flow rate in te repective pipe. It i obviou tat te flow take place from te iget reervoir, tat i reervoir A, toward te junction, and ten from te junction to te lowermot reervoir, tat i reervoir C. However, te direction of te flow in te econd pipe i till uncertain. If te total ead at te free urface of reervoir B i greater tan te total ead at junction J, ten te flow take place from reervoir B toward junction J. In ti cae, te continuity equation at junction J become Q 1 + Q = Q 3 wen tb > tj However, wen te total ead at te free urface of te reervoir B i maller tan te total ead at junction J, ten te flow take place from junction J toward reervoir B. Hence, te continuity equation at junction J i Q 1 = Q + Q 3 wen tb < tj 7 Head Lo 34

35 Te procedure for determining te volumetric flow rate for eac pipe may be given a below: 1) Calculate te total ead, t, at te free urface of eac reervoir. ) Determine te relative rougne, ɛ /d, for eac pipe by uing te pipe diameter, d, and te pipe material. 3) Aume a friction factor, f, for eac pipe from te Moody diagram by uing te relative rougne, ɛ /d, in te fully roug flow region. Te reaon for te aumption of te flow in te fully roug region i tat no iteration i neceary if ti aumption i true. 4) Expre te total ead lo, f, for eac pipe in te form of f = KQ, were K i te equivalent reitance to te flow and Q i te unknown volumetric flow rate troug te pipe. 5) Aume te total ead for te junction, and determine te direction of flow in pipe wic are connected to intermediate reervoir. Write te continuity equation for te junction. 6) Write te extended Bernoulli equation along te treamline between te free urface of eac reervoir and te junction. 7) Determine te volumetric flow rate, Q, for eac pipe, 8) Determine weter te continuity equation i atified at te junction or not. If te continuity equation i not atified, ten repeat te calculation in tep (5) troug (7) a many time a neceary. 9) Calculate te average velocity, V = 4Q/(πd ) for eac pipe, 10) Calculate te Reynold number, Re = ρvd/μ for eac pipe, 11) Determine te improved value of te friction factor, f, for eac pipe from te Moody diagram in Figure 10.5 by uing te Reynold number, Re, and te relative rougne, ɛ /d. 1) Compare te aumed and te improved value of te friction factor for eac pipe and repeat tep (4) troug (9) a many time a needed in order to obtain te deired accuracy in te friction factor. 7 Head Lo 35

36 Example: Water at 0 C i flowing troug te tree reervoir ytem, wic i own in Figure. All pipe are cat iron. Neglecting all minor ead loe, determine te volumetric flow rate for eac pipe. Solution Te volumetric flow rate in eac pipe may be determined by applying te procedure, wic i preented in ti ection: 1) Since te cro-ectional area of te reervoir are very large compared to te croectional area of te pipe, ten te velocitie at te free urface of te reervoir are negligible tat i V A = V B = V C = 0 Terefore, te total ead at te free urface of te reervoir are PA VA N / m ta za 700 m m 3 rg g (998. kg / m )(9.81 m / ) PB VB N / m tb zb 450 m m 3 rg g (998. kg / m )(9.81 m / ) PC VC N / m tc zc 100 m m 3 rg g (998. kg / m )(9.81 m / ) 7 Head Lo 36

37 ) Te relative rougne of cat iron pipe are obtained from te diagram: For d 1 = 0.3 m (e / d) 1 = For d = 0.35 m (e / d) = For d 3 = 0.4 m (e / d) 3 = ) Te friction factor for correponding relative rougne value are obtained from te fully roug region a f 1 = f = f 3 = ) Since te minor ead loe are neglected, te total ead lo for eac pipe may be evaluated from te Darcy-Weibac equation a follow: L 1 4Q 00 m 1 4Q f Q f d1 g pd1 0.3 m 9.81 m / p(0.3 m) L 1 4Q 300 m 1 4Q f Q f d g pd 0.35 m 9.81 m / p(0.35 m) f 3 3 f 3 3 d3 g p d3 L 1 4Q 400 m 1 4Q Q 0.4 m 9.81 m / p (0.4 m) 3 5) Te total ead lo at junction J may me aumed a tj = 400 m For ti cae ta > tb > tj > tc Hence water i flowing from reervoir A and B into reervoir C. A a reult, te continuity equation at junction J become Q 1 + Q = Q 3 6) Te extended Bernoulli equation may now be applied along te treamline between te free urface of te reervoir and te junction J a ta tj f 1 tb tj f tj tc f 3 7 Head Lo 37

38 7) Te volumetric flow rate in eac pipe may now be determined a 1/ 1/ ta tj Q m / / 1/ tb tj Q 0.84 m / / 1/ tj tc Q3.7 m / ) Continuity equation at junction J Q 1 + Q = Q =.7.41 >.7 A een te continuity i not atified at junction J. Fluid flowing into te junction i greater tan te fluid flowing out from te junction. To atify te continuity, flow rate from reervoir A and B to junction J ould decreae and flow rate from junction J to reervoir C ould increae. For ti to appen, te total ead at junction J ould be increaed and te calculation in tep 7 ould be repeated. Te ummary of te iteration i preented in te table below. A a reult of 4 iteration, te flow rate in pipe are obtained a, Q 1 = 1.54 m 3 / Q = 0.76 m 3 / Q 3 =.30 m 3 / 9) Now te velocity in eac pipe may be calculated a below: V V V 4Q m / 1 1 pd1 p(0.3 m) 4Q m / pd p(0.35 m) 4Q m / 3 3 pd3 p(0.4 m) 7 Head Lo 38

39 10) Te kinematic vicoity of water at 0 o C i m /, ten te Reynold number for eac pipe may be evaluated a Vd Re Vd Re Vd Re ) Friction factor correponding to te relative rougne and Reynold number for eac pipe are obtained from Moody diagram a follow: f 1 = f = f 3 = ) A een te aumed and te improved value of friction factor are te ame for eac pipe. Hence tere i no need to iterate. Volumetric flow rate for eac pipe will be a Q 1 = 1.54 m 3 / Q = 0.76 m 3 / Q 3 =.30 m 3 / 6 6 Iteration tj (m) Q 1 (m 3 /) Q (m 3 /) Q 3 (m 3 /) Q 1 +Q +Q 3 (m 3 /) Cange in tj (m) increae Decreae Increae O.K. 7 Head Lo 39