Fluid Properties Examples

Transcription

1 CIVE400: Flui Mecanic Example: nwer Flui Propertie Example. Explain wy te vicoity of a liqui ecreae wile tat of a ga increae wit a temperature rie. Te following i a table of meaurement for a flui at contant temperature. Determine te ynamic vicoity of te flui. u/y (ra - ) (N m - ) CIVE400: Flui Mecanic Example: nwer. Te enity of an oil i 850 kg/m. Fin it relative enity an Kinematic vicoity if te ynamic vicoity i kg/m. oil = 850 kg/m water = 000 kg/m oil = 850 / 000 = 0.85 Dynamic vicoity = = 50 - kg/m Uing Newton' law of vicoity u y were i te vicoity. So vicoity i te graient of a grap of ear tre againt vellocity graient of te above ata, or u y Plot te ata a a grap: Sear tre u/y Calculate te graient for eac ection of te line u/y ( - ) (N m - ) Graient Kinematic vicoity = = / m. Te velocity itribution of a vicou liqui (ynamic vicoity = 0.9 N/m ) flowing over a fixe plate i given by u = 0.68y - y (u i velocity in m/ an y i te itance from te plate in m). Wat are te ear tree at te plate urface an at y=0.4m? t te plate face y = 0m, Calculate te ear tre at te plate face t y = 0.4m, u 0.68y y u 0.68 y y u 0.68 y u N / m y u y te velocity graient i zero at y=0.4 ten te ear tre mut alo be zero. Tu te mean graient = vicoity = 4.98 N / m Example: nwer CIVE400: Flui Mecanic Example: nwer CIVE400: Flui Mecanic

2 CIVE400: Flui Mecanic Example: nwer m of oil weig N. Fin it ma enity, an relative enity,. Weigt = mg Ma m = / 9.8 = kg Ma enity = Ma / volume = / 5.6 = 85 kg/m 85 Relative enity water 5. From table of flui propertie te vicoity of water i given a poie. Wat i ti value in N/m an Pa unit? = poie poie = 0. Pa = 0. N/m = Pa = N/m 6. In a flui te velocity meaure at a itance of 75mm from te bounary i.5m/. Te flui a abolute vicoity Pa an relative enity 0.9. Wat i te velocity graient an ear tre at te bounary auming a linear velocity itribution. = Pa = 0.9 u y u y Pa CIVE400: Flui Mecanic Example: nwer Preure an Manometer. Wat will be te (a) te gauge preure an (b) te abolute preure of water at ept m below te urface? water = 000 kg/m, an p atmopere = 0kN/m. [7.7 kn/m, 8.7 kn/m ] a) pgauge g N / m,( Pa) 7. 7 kn / m,( kpa) b) pabolute pgauge patmoperic ( 770 0) N / m,( Pa) 8. 7 kn / m,( kpa). t wat ept below te urface of oil, relative enity 0.8, will prouce a preure of 0 kn/m? Wat ept of water i ti equivalent to? [5.m,.m] a) water kg / m p g p 0 0 m g of oil b) 000 kg / m 0 0. mof water Wat woul te preure in kn/m be if te equivalent ea i meaure a 400mm of (a) mercury =.6 (b) water ( c) oil pecific weigt 7.9 kn/m () a liqui of enity 50 kg/m? [5.4 kn/m,.9 kn/m,.6 kn/m,.04 kn/m ] a) water kg / m p g N / m Example: nwer CIVE400: Flui Mecanic Example: nwer CIVE400: Flui Mecanic 4

3 CIVE400: Flui Mecanic Example: nwer b) p g N / m c) g p g N / m ) p g N / m.4 manometer connecte to a pipe inicate a negative gauge preure of 50mm of mercury. Wat i te abolute preure in te pipe in Newton per quare metre i te atmoperic preure i bar? [9. kn/m ] 5 patmopere bar 0 N / m pabolute pgauge patmoperic g patmoperic N / m,( Pa) 9. kn / m,( kpa).5 Wat eigt woul a water barometer nee to be to meaure atmoperic preure? [>0m] 5 patmopere bar 0 N / m 5 0 g 5 0 mof water m of mercury (. ). Example: nwer CIVE400: Flui Mecanic 5 CIVE400: Flui Mecanic Example: nwer.6 n incline manometer i require to meaure an air preure of mm of water to an accuracy of +/- %. Te incline arm i 8mm in iameter an te larger arm a a iameter of 4mm. Te manometric flui a enity 740 kg/m an te cale may be rea to +/- 0.5mm. Wat i te angle require to enure te eire accuracy may be acieve? [ 9 ] p p iameter z iameter D θ Scale Reaer Datum line p p mang man g z z z Volume move from left to rigt = z x in D z z x 4 in 4 4 z z x in D D p p man gxin D water g man gxin D water g 074. water gxin x (in 0. ) Te ea being meaure i % of mm = 0.00x0.0 = m Ti % repreent te mallet meaurement poible on te manometer, 0.5mm = m, giving (in 0. ) in [Ti i not te ame a te anwer given on te quetion eet] Example: nwer CIVE400: Flui Mecanic 6 x z

4 CIVE400: Flui Mecanic Example: nwer.7 Determine te reultant force ue to te water acting on te m by m rectangular area B own in te iagram below. [4 560 N,.7m from O] O P CIVE400: Flui Mecanic Example: nwer.8 Determine te reultant force ue to te water acting on te.5m by.0m triangular area CD own in te figure above. Te apex of te triangle i at C. [4.50 N,.8m from P] b B.m.0 m D.0m C.0 m Te magnitue of te reultant force on a ubmerge plane i: R = preure at centroi area of urface R gz N / m Ti act at rigt angle to te urface troug te centre of preure. IOO n moment of area about a line troug O Sc x t moment of area about a line troug O By te parallel axi teorem (wic will be given in an exam), Ioo IGG x, were I GG i te n moment of area about a line troug te centroi an can be foun in table. b I GG Sc x x 45 G G / For a triangle I 86 N / m GG b 6 Dept to centre of gravity i z 0. co m. R gz Ditance from P i x z / co m Ditance from P to centre of preure i I oo Sc x I I x oo GG I GG 5. Sc x. 748 x m G G For a rectangle I GG b te wall i vertical, Sc D an x z, Sc.. 7. m from O Example: nwer CIVE400: Flui Mecanic 7 Example: nwer CIVE400: Flui Mecanic 8

5 CIVE400: Flui Mecanic Example: nwer Force on ubmerge urface. Obtain an expreion for te ept of te centre of preure of a plane urface wolly ubmerge in a flui an incline at an angle to te free urface of te liqui. orizontal circular pipe,.5m iameter, i cloe by a butterfly ik wic rotate about a orizontal axi troug it centre. Determine te torque wic woul ave to be applie to te ik pinle to keep te ik cloe in a vertical poition wen tere i a m ea of fre water above te axi. [76 Nm] Te quetion ak wat i te moment you ave to apply to te pinle to keep te ic vertical i.e. to keep te valve ut? So you nee to know te reultant force exerte on te ic by te water an te itance x of ti force from te pinle. We know tat te water in te pipe i uner a preure of m ea of water (to te pinle).75 CIVE400: Flui Mecanic Example: nwer I GG ' 4 r 4( r ) r. 06m So te itance from te pinle to te line of action of te force i x ' m n te moment require to keep te gate ut i moment Fx kn m. ock gate i to be reinforce wit tree orizontal beam. If te water act on one ie only, to a ept of 6m, fin te poition of te beam meaure from te water urface o tat eac will carry an equal loa. Give te loa per meter. [ N/m,.m, 4.m, 5.47m] Firt of all raw te preure iagram, a below: F Diagram of te force on te ic valve, bae on an imaginary water urface. m, te ept to te centroi of te ic = ept to te centre of preure (or line of action of te force) Calculate te force: F g kn Calculate te line of action of te force,. n moment of area about water urface ' t moment of area about water urface Ioo By te parallel axi teorem n moment of area about O (in te urface) Ioo IGG were I GG i te n moment of area about a line troug te centroi of te ic an I GG = r 4 /4. x R / Te reultant force per unit lengt of gate i te area of te preure iagram. So te total reultant force i R g = N ( per m lengt) lternatively te reultant force i, R = Preure at centroi rea, (take wit of gate a m to give force per m) R g N ( per m lengt) Ti i te reultant force exerte by te gate on te water. Te tree beam oul carry an equal loa, o eac beam carrie te loa f, were R f N f f f Example: nwer CIVE400: Flui Mecanic 9 Example: nwer CIVE400: Flui Mecanic 0

6 CIVE400: Flui Mecanic Example: nwer If we take moment from te urface, DR f f f D f f Taking te firt beam, we can raw a preure iagram for ti, (ignoring wat i below), H/ H F=58860 CIVE400: Flui Mecanic Example: nwer. Te profile of a maonry am i an arc of a circle, te arc aving a raiu of 0m an ubtening an angle of 60 at te centre of curvature wic lie in te water urface. Determine (a) te loa on te am in N/m lengt, (b) te poition of te line of action to ti preure. [ N/m lengt at ept 9.0m] Draw te am to elp picture te geometry, R a 60 We know tat te reultant force, F F gh, o H g R F R y F H g m n te force act at H/, o ti i te poition of te t beam, poition of t beam H. m Taking te econ beam into conieration, we can raw te following preure iagram, F F v =. H H/ f F=58860 f Te reaction force i equal to te um of te force on eac beam, o a before F ( 58860) H 49. m g Te reaction force act at H/, o H=.7m. Taking moment from te urface, ( 58860) ept to econ beam 4. m For te tir beam, from before we ave, ept to tir beam m Example: nwer CIVE400: Flui Mecanic 0in m a 0co m Calculate F v = total weigt of flui above te curve urface (per m lengt) F v g( area of ector - area of triangle) = kn / m Calculate F = force on projection of curve urface onto a vertical plane F g kn / m Te reultant, F F F R v kn / m acting at te angle Fv tan 089. F 9. Example: nwer CIVE400: Flui Mecanic

7 CIVE400: Flui Mecanic Example: nwer ti force act normal to te urface, it mut act troug te centre of raiu of te am wall. So te ept to te point were te force act i, y = 0in 9.=9m.4 Te arc of a brige over a tream i in te form of a emi-circle of raiu m. te brige wit i 4m. Due to a floo te water level i now.5m above te cret of te arc. Calculate (a) te upwar force on te unerie of te arc, (b) te orizontal trut on one alf of te arc. [6.6 kn, 76.6 kn] Te brige an water level can be rawn a:.5m a) Te upwar force on te arc = weigt of (imaginary) water above te arc. Rv gvolume of water volume ( 5. ) m R kn v b) Te orizontal force on alf of te arc, i equal to te force on te projection of te curve urface onto a vertical plane..5.0 m F preure at centroi area g kn CIVE400: Flui Mecanic Example: nwer x = 7.0 m, o = = 9.5. x = 9.5/tan 60 = m. Vertical force = weigt of water above te urface, Fv g x 05. x kn / m Te orizontal force = force on te projection of te urface on to a vertical plane. F g kn / m Te reultant force i FR Fv F kn / m n act at te angle Fv tan F tank wit vertical ie i quare in plan wit m long ie. Te tank contain oil of relative enity 0.9 to a ept of.0m wic i floating on water a ept of.5m. Calculate te force on te wall an te eigt of te centre of preure from te bottom of te tank. [65.54 kn,.5m] Conier one wall of te tank. Draw te preure iagram: 60.5 Te face of a am i vertical to a ept of 7.5m below te water urface ten lope at 0 to te vertical. If te ept of water i 7m wat i te reultant force per metre acting on te wole face? [56.9 kn] Example: nwer CIVE400: Flui Mecanic Example: nwer CIVE400: Flui Mecanic 4

8 CIVE400: Flui Mecanic Example: nwer F f f f enity of oil oil = 0.9 water = 900 kg/m. Force per unit lengt, F = area uner te grap = um of te tree area = f + f + f ( ) f 5974 N f ( ) N ( ) 5. f 09 N F f f f N To fin te poition of te reultant force F, we take moment from any point. We will take moment about te urface. DF f f f D ( ) 09 ( 5. ) D. 47m( from urface) 5. m ( from bae of wall) CIVE400: Flui Mecanic Example: nwer pplication of te Bernoulli Equation. In a vertical pipe carrying water, preure gauge are inerte at point an B were te pipe iameter are 0.5m an 0.075m repectively. Te point B i.5m below an wen te flow rate own te pipe i 0.0 cumec, te preure at B i 475 N/m greater tan tat at. uming te loe in te pipe between an B can be expree a k v were v i te velocity at, g fin te value of k. If te gauge at an B are replace by tube fille wit water an connecte to a U-tube containing mercury of relative enity.6, give a ketc owing ow te level in te two limb of te U-tube iffer an calculate te value of ti ifference in metre. [k = 0.9, m] Part i) = 0.m B = 0.m B Rp 05. m B m Q 0. 0 m / p p 475 N / m B f kv g Taking te atum at B, te Bernoulli equation become: p u p u z z k u B B B g g g g g z 5. z 0 B By continuity: Q = u = u B B u u giving B 0. 0 / m/ 0. 0 / m/ Example: nwer CIVE400: Flui Mecanic 5 Example: nwer CIVE400: Flui Mecanic 6

9 CIVE400: Flui Mecanic Example: nwer Part ii) p p u u z k u B B 000g g g k k 0. 9 pxxl w gzb pb pxxr m grp w gz w grp p pxxl pxxr wgz B pb mgr p wgz wgr p p pb p wgz zbgrpm w R p R m p. Venturimeter wit an entrance iameter of 0.m an a troat iameter of 0.m i ue to meaure te volume of ga flowing troug a pipe. Te icarge coefficient of te meter i uming te pecific weigt of te ga to be contant at 9.6 N/m, calculate te volume flowing wen te preure ifference between te entrance an te troat i meaure a 0.06m on a water U-tube manometer. [0.86 m /] = 0.m CIVE400: Flui Mecanic Example: nwer Wat we know from te quetion: Calculate Q. For te manometer: For te Venturimeter Combining () an () g C g 9. 6 N / m m 0. m u Q/ u Q/ p gz p g z R gr g g p w p p p 9. 6 z z ( ) p u p u z z g g g g g g p p 9. 6 z z 0. 80u ( ) 0. 80u u ieal m/ 0. Qieal m / Q C Q m / iea = 0.m Z Z Rp Example: nwer CIVE400: Flui Mecanic 7 Example: nwer CIVE400: Flui Mecanic 8

10 CIVE400: Flui Mecanic Example: nwer. Venturimeter i ue for meauring flow of water along a pipe. Te iameter of te Venturi troat i two fift te iameter of te pipe. Te inlet an troat are connecte by water fille tube to a mercury U-tube manometer. Te velocity of flow along te pipe i foun to be 5. H m/, were H i te manometer reaing in metre of mercury. Determine te lo of ea between inlet an troat of te Venturi wen H i 0.49m. (Relative enity of mercury i.6). [0.m of water] CIVE400: Flui Mecanic Example: nwer Subtitute in () / Loe L m.4 Water i icarging from a tank troug a convergent-ivergent moutpiece. Te exit from te tank i roune o tat loe tere may be neglecte an te minimum iameter i 0.05m. If te ea in te tank above te centre-line of te moutpiece i.8m. a) Wat i te icarge? b) Wat mut be te iameter at te exit if te abolute preure at te minimum area i to be.44m of water? c) Wat woul te icarge be if te ivergent part of te mout piece were remove. (ume atmoperic preure i 0m of water). [0.075m, 0.066m /, 0.08m /] Z Z H For te manometer: For te Venturimeter p gz p g z H gh w w m p p gz gh gh gz () w w m w p u p u z z Loe g g g g w w wu wu p p w gz wgz Lwg ( ) Combining () an () p u p u z z Loe g g g g w w w Lwg Hgm w u u () but at. From te quetion u 5. H 75. m/ u u 75. u 4 0 u m/ Example: nwer CIVE400: Flui Mecanic 9 From te quetion: pply Bernoulli: If we take te atum troug te orifice: 005. m p minimum preure 44. m g p m p 0 g g p u p u p u z z z g g g g g g z 8. m z z 0 u negligible Between an u g Between an p p u 57. m/ Q u.. m / Example: nwer CIVE400: Flui Mecanic 0

11 CIVE400: Flui Mecanic Example: nwer u 8. g u 599. m/ Q u m If te mout piece a been remove, p p p p u z g g g u gz 599. m/ Q m / 4.5 cloe tank a an orifice 0.05m iameter in one of it vertical ie. Te tank contain oil to a ept of 0.6m above te centre of te orifice an te preure in te air pace above te oil i maintaine at 780 N/m above atmoperic. Determine te icarge from te orifice. (Coefficient of icarge of te orifice i 0.6, relative enity of oil i 0.9). [ m /] CIVE400: Flui Mecanic Example: nwer 780 u 06. o g g u 65. m/ 005 Q m.... /.6 Te icarge coefficient of a Venturimeter wa foun to be contant for rate of flow exceeing a certain value. Sow tat for ti conition te lo of ea ue to friction in te convergent part of te meter can be expree a KQ m were K i a contant an Q i te rate of flow in cumec. Obtain te value of K if te inlet an troat iameter of te Venturimeter are 0.0m an 0.05m repectively an te icarge coefficient i [K=060].7 Venturimeter i to fitte in a orizontal pipe of 0.5m iameter to meaure a flow of water wic may be anyting up to 40m /our. Te preure ea at te inlet for ti flow i 8m above atmoperic an te preure ea at te troat mut not be lower tan 7m below atmoperic. Between te inlet an te troat tere i an etimate frictional lo of 0% of te ifference in preure ea between tee point. Calculate te minimum allowable iameter for te troat. [0.06m] = 0.5m P = 780 kn/m 0.66m oil From te quetion: From te quetion o = 0.05m 05. m Q40m / r m / u Q/ 77. m/ p p 8m 7m g g pply Bernoulli, o o C 06. w Friction lo, from te quetion: pply Bernoulli: f p p 0. g p u p u z z g g g g Take atmoperic preure a 0, Example: nwer CIVE400: Flui Mecanic Example: nwer CIVE400: Flui Mecanic

12 CIVE400: Flui Mecanic Example: nwer p u p u f g g g g p p u u f g g g g u g g u. 46 m/ Q u m.8 Venturimeter of troat iameter 0.076m i fitte in a 0.5m iameter vertical pipe in wic liqui of relative enity 0.8 flow ownwar. Preure gauge are fitte to te inlet an to te troat ection. Te troat being 0.94m below te inlet. Taking te coefficient of te meter a 0.97 fin te icarge a) wen te preure gauge rea te ame b)wen te inlet gauge rea 570 N/m iger tan te troat gauge. [0.09m /, 0.04m /] = 0.076m = 0.5m CIVE400: Flui Mecanic Example: nwer a) p p By continuity: b) p u p u z z g g g g u u z z g g Qu u u u u u 6u g g u 094. m/ 5 Q C u Q m / 4 p p 570 p p u u g g 570 Q g g Q Q 0. 05m / From te quetion: pply Bernoulli: 05. m m m m 800 kg / m C 097. Example: nwer CIVE400: Flui Mecanic Example: nwer CIVE400: Flui Mecanic 4

13 CIVE400: Flui Mecanic Example: nwer Tank emptying 4. reervoir i circular in plan an te ie lope at an angle of tan - (/5) to te orizontal. Wen te reervoir i full te iameter of te water urface i 50m. Dicarge from te reervoir take place troug a pipe of iameter 0.65m, te outlet being 4m below top water level. Determine te time for te water level to fall m auming te icarge to be 075. a gh cumec were a i te cro ectional area of te pipe in m an H i te ea of water above te outlet in m. [5 econ] 50m From te quetion: H = 4m a = (0.65/) = 0.m Q 075. a g 096. In time t te level in te reervoir fall, o Qt t Q Integrating give te total time for level to fall from to. T Q te urface area cange wit eigt, we mut expre in term of. = r But r varie wit. It varie linearly from te urface at H = 4m, r = 5m, at a graient of tan - = /5. r = x = x + 5(4) x = 5 o = ( ) = ( ) Subtituting in te integral equation give x 5 Example: nwer CIVE400: Flui Mecanic 5 r H CIVE400: Flui Mecanic Example: nwer T / / / 764. / 5 / 4 / From te quetion, = 4m = m, o / T / 4 4 / / 5 5 / 4 / ec 4. rectangular wimming pool i m eep at one en an increae uniformly in ept to.6m at te oter en. Te pool i 8m wie an m long an i emptie troug an orifice of area 0.4m, at te lowet point in te ie of te eep en. Taking C for te orifice a 0.6, fin, from firt principle, a) te time for te ept to fall by m b) te time to empty te pool completely. [99 econ, 66 econ].0m.0m Te quetion tell u a o = 0.4m, C = 0.6 pply Bernoulli from te tank urface to te vena contracta at te orifice: p u p u z z g g g g p = p an u = 0. u g We nee Q in term of te eigt meaure above te orifice..6m Q Caou Cao g Example: nwer CIVE400: Flui Mecanic 6 L

14 CIVE400: Flui Mecanic Example: nwer n we can write an equation for te icarge in term of te urface eigt cange: Qt t Q Integrating give te total time for level to fall from to. T Q 68. ( ) a) For te firt m ept, = 8 x = 56, watever te. So, for te firt perio of time: T ec b) now we nee to fin out ow long it will take to empty te ret. We nee te area, in term of. 8L L So T ec Total time for emptying i, T = = 66 ec / / / / CIVE400: Flui Mecanic Example: nwer 4. vertical cylinrical tank m iameter a, at te bottom, a 0.05m iameter arp ege orifice for wic te icarge coefficient i 0.6. a) If water enter te tank at a contant rate of cumec fin te ept of water above te orifice wen te level in te tank become table. b) Fin te time for te level to fall from m to m above te orifice wen te inflow i turne off. c) If water now run into te tank at 0.0 cumec, te orifice remaining open, fin te rate of rie in water level wen te level a reace a ept of.7m above te orifice. [a).4m, b) 88 econ, c) 0.5m/min] Q = m / o = 0.005m From te quetion: Q in = m /, o =0.05m, C =0.6 pply Bernoulli from te water urface () to te orifice (), p u p u z z g g g g p = p an u = 0. u g. Wit te atum te bottom of te cyliner, z =, z = 0 We nee Q in term of te eigt meaure above te orifice. Qout C aou C ao g ( ) For te level in te tank to remain contant: inflow = out flow Q in = Q out m (b) Write te equation for te icarge in term of te urface eigt cange: Example: nwer CIVE400: Flui Mecanic 7 Example: nwer CIVE400: Flui Mecanic 8

15 CIVE400: Flui Mecanic Example: nwer Qt t Q Integrating between an, to give te time to cange urface level T Q / / / / 06 = an = o T = 88 ec c) Q in cange to Q in = 0.0 m / From () we ave Qout Te quetion ak for te rate of urface rie wen =.7m. i.e. Qout m / Te rate of increae in volume i: QQin Qout m / Q = rea x Velocity, te rate of rie in urface i Q u Q 0. 0 u m/ 0. 5m/min orizontal boiler ell (i.e. a orizontal cyliner) m iameter an 0m long i alf full of water. Fin te time of emptying te ell troug a ort vertical pipe, iameter 0.08m, attace to te bottom of te ell. Take te coefficient of icarge to be 0.8. [70 econ] = m m From te quetion W = 0m, D = 0m o = 0.08m C = 0.8 o = 0.08 m Example: nwer CIVE400: Flui Mecanic 9 CIVE400: Flui Mecanic Example: nwer pply Bernoulli from te water urface () to te orifice (), p u p u z z g g g g p = p an u = 0. u g. Wit te atum te bottom of te cyliner, z =, z = 0 We nee Q in term of te eigt meaure above te orifice. Qout C aou C ao g Write te equation for te icarge in term of te urface eigt cange: Qt t Q Integrating between an, to give te time to cange urface level T But we nee in term of Q Surface area = 0L, o nee L in term of L a a ( ) L ( ) L 0 Subtitute ti into te integral term, L.0m a Example: nwer CIVE400: Flui Mecanic 0.0m.

16 CIVE400: Flui Mecanic Example: nwer 0 T / ec 4.5 Two cyliner taning uprigt contain liqui an are connecte by a ubmerge orifice. Te iameter of te cyliner are.75m an.0m an of te orifice, 0.08m. Te ifference in level of te liqui i initially.5m. Fin ow long it will take for ti ifference to be reuce to 0.66m if te coefficient of icarge for te orifice i (Work from firt principle.) [0.7 econ] =.75m =.0m =.5m CIVE400: Flui Mecanic Example: nwer ( ) Qt ( ) From te Bernoulli equation we can erive ti expreion for icarge troug te ubmerge orifice: Q C a g So Integrating o Ca o gt t C a g o T C a g o C a g o ec 4.6 rectangular reervoir wit vertical wall a a plan area of 60000m. Dicarge from te reervoir take place over a rectangular weir. Te flow caracteritic of te weir i Q = H / cumec were H i te ept of water above te weir cret. Te ill of te weir i.4m above te bottom of te reervoir. Starting wit a ept of water of 4m in te reervoir an no inflow, wat will be te ept of water after one our? [.98m] o = 0.08m by continuity, efining, = m 0785m o 008m ao 00050m C ,.. Q t () From te quetion = m, Q = / Write te equation for te icarge in term of te urface eigt cange: Subtituting ti in () to eliminate Example: nwer CIVE400: Flui Mecanic Example: nwer CIVE400: Flui Mecanic

17 CIVE400: Flui Mecanic Example: nwer Qt t Q Integrating between an, to give te time to cange urface level T Q / / From te quetion T = 600 ec an = 0.6m m Total ept = =.98m / / CIVE400: Flui Mecanic Example: nwer Notce an weir 5. Deuce an expreion for te icarge of water over a rigt-angle arp ege V-notc, given tat te coefficient of icarge i 0.6. rectangular tank 6m by 6m a te ame notc in one of it ort vertical ie. Determine te time taken for te ea, meaure from te bottom of te notc, to fall from 5cm to 7.5cm. [99 econ] From your note you can erive: Q 8 C gh 5 5 / tan For ti weir te equation implifie to Q 44. H 5 / Write te equation for te icarge in term of te urface eigt cange: Qt t Q Integrating between an, to give te time to cange urface level T = 0.5m, = 0.075m Q 6 6 / 44. / T ec / / Example: nwer CIVE400: Flui Mecanic Example: nwer CIVE400: Flui Mecanic 4

18 CIVE400: Flui Mecanic Example: nwer 5. Derive an expreion for te icarge over a arp crete rectangular weir. arp ege weir i to be contructe acro a tream in wic te normal flow i 00 litre/ec. If te maximum flow likely to occur in te tream i 5 time te normal flow ten etermine te lengt of weir neceary to limit te rie in water level to 8.4cm above tat for normal flow. C =0.6. [.4m] From your note you can erive: Q Cb / g From te quetion: Q = 0. m /, = x Q =.0 m /, = x were x i te eigt above te weir at normal flow. So we ave two ituation: 0. Cb / / gx 80. bx ( ) / / 0. Cb g x bx ( ) From () we get an expreion for b in term of x b 0. x / Subtituting ti in () give, x x / x x x m So te weir breat i b m / / Example: nwer CIVE400: Flui Mecanic 5 CIVE400: Flui Mecanic Example: nwer 5. Sow tat te rate of flow acro a triangular notc i given by Q=C KH 5/ cumec, were C i an experimental coefficient, K epen on te angle of te notc, an H i te eigt of te uniturbe water level above te bottom of te notc in metre. State te reaon for te introuction of te coefficient. Water from a tank aving a urface area of 0m flow over a 90 notc. It i foun tat te time taken to lower te level from 8cm to 7cm above te bottom of te notc i 4.5econ. Determine te coefficient C auming tat it remain contant uring i perio. [0.65] 8 Te proof for Q C gh CKH 5 5 / 5 / tan i in te note. From te quetion: = 0m = 90 = 0.08m = 0.07m T = 4.5ec So Q =.6 C 5/ Write te equation for te icarge in term of te urface eigt cange: Qt t Q Integrating between an, to give te time to cange urface level T Q 0 C 5/ C / C C / / 5.4 reervoir wit vertical ie a a plan area of 56000m. Dicarge from te reervoir take place over a rectangular weir, te flow caracteritic of wic i Q=.77BH / m /. t time of maximum rainfall, water flow into te reervoir at te rate of 9m /. Fin a) te lengt of weir require to icarge ti quantity if ea mut not excee 0.6m; b) te time neceary for te ea to rop from 60cm to 0cm if te inflow uenly top. [0.94m, 09econ] From te quetion: = m Q =.77 B H / Q max = 9 m / a) Fin B for H = =.77 B 0.6 / B = 0.94m Example: nwer CIVE400: Flui Mecanic 6

19 CIVE400: Flui Mecanic Example: nwer b) Write te equation for te icarge in term of te urface eigt cange: Qt t Q Integrating between an, to give te time to cange urface level T Q B / / B / / T 09ec 5.5 Develop a formula for te icarge over a 90 V-notc weir in term of ea above te bottom of te V. cannel convey 00 litre/ec of water. t te outlet en tere i a 90 V-notc weir for wic te coefficient of icarge i t wat itance above te bottom of te cannel oul te weir be place in orer to make te ept in te cannel.0m? Wit te weir in ti poition wat i te ept of water in te cannel wen te flow i 00 litre/ec? [0.755m,.8m] Derive ti formula from te note: Q 8 C gh 5 5 / tan From te quetion: = 90 C 0.58 Q = 0. m /, ept of water, Z = 0.m giving te weir equation: Q 7. H 5 / a) H i te eigt above te bottom of te V, te ept of water = Z = D + H, were D i te eigt of te bottom of te V from te bae of te cannel. So 5/ Q7. ZD 5/ D D m b) Fin Z wen Q = 0. m / Z Z 8. m 5 / CIVE400: Flui Mecanic Example: nwer 5.6 Sow tat te quantity of water flowing acro a triangular V-notc of angle i 8 5 / Q C tan gh. Fin te flow if te meaure ea above te bottom of te V i 8cm, wen 5 =45 an C =0.6. If te flow i wante witin an accuracy of %, wat are te limiting value of te ea. [0.6m /, 0.77m, 0.8m] Proof of te v-notc weir equation i in te note. From te quetion: H = 0.8m = 45 C = 0.6 Te weir equation become: 5 / Q 47. H 5 / m / Q+% = 0.9 m / 5 / H H 0. 8m Q-% = 0.4 m / 5 / H H 077. m Example: nwer CIVE400: Flui Mecanic 7 Example: nwer CIVE400: Flui Mecanic 8

20 CIVE400: Flui Mecanic Example: nwer pplication of te Momentum Equation 6. Te figure below ow a moot curve vane attace to a rigi founation. Te jet of water, rectangular in ection, 75mm wie an 5mm tick, trike te vane wit a velocity of 5m/. Calculate te vertical an orizontal component of te force exerte on te vane an inicate in wic irection tee component act. [Horizontal.4 N acting from rigt to left. Vertical 4.6 N acting ownwar] CIVE400: Flui Mecanic Example: nwer mm iameter pipeline carrie water uner a ea of 0m wit a velocity of m/. Ti water main i fitte wit a orizontal ben wic turn te axi of te pipeline troug 75 (i.e. te internal angle at te ben i 05). Calculate te reultant force on te ben an it angle to te orizontal. [ kn, 5 9 ] y u 45 5 x From te quetion: a u m/ Q m / a a, o u u m Calculate te total force uing te momentum equation: F T Q u u x co5 co co5 5co N F Q u in 5 u in 45 T y in55in N Boy force an preure force are 0. So force on vane: R F. 44N x t x R F 4. 6N y t y Example: nwer CIVE400: Flui Mecanic 9 u θ From te quetion: a m m m u u m/ Q m / Calculate total force. F Q u u F F F Tx x x Rx Px Bx F Tx co kn F Q u u F F F Ty y y Ry Py By F Ty in kN Calculate te preure force p = p = p = g = = 94. kn/m F p a Tx co pa co co kn F p a in p a in Ty in kn Tere i no boy force in te x or y irection. F F F F Rx Tx Px Bx kn Example: nwer CIVE400: Flui Mecanic 40

21 CIVE400: Flui Mecanic Example: nwer FRy FTy FPy FBy kN Tee force act on te flui Te reultant force on te flui i FR FRx FRy kn FRy tan 5 9' FRx 6. orizontal jet of water 0 mm cro-ection an flowing at a velocity of 5 m/ it a flat plate at 60 to te axi (of te jet) an to te orizontal. Te jet i uc tat tere i no ie prea. If te plate i tationary, calculate a) te force exerte on te plate in te irection of te jet an b) te ratio between te quantity of flui tat i eflecte upwar an tat ownwar. (ume tat tere i no friction an terefore no ear force.) [8N, :] y u x u θ CIVE400: Flui Mecanic Example: nwer Q = a u = 0-5 = 0.0 Q = (a + a ) u Q = a u Q = (a - a )u Calculate total force. FTx Q ux u x FRx FPx FBx F Tx in N Component in irection of jet = 90 in 60 = 8 N tere i no force parallel to te plate F ty = 0 FTy ua ua ua co 0 a a aco 0 a a a a aco aa 4 4a a a a a Tu /4 of te jet goe up, /4 own mm iameter jet of water aving a velocity of 5m/ trike a flat plate, te normal of wic i incline at 0 to te jet. Fin te force normal to te urface of te plate. [.9kN] y u x u From te quetion a = a =x0 - m u = 5 m/ pply Bernoulli, p u p u p u z z z g g g g g g u Cange in eigt i negligible o z = z = z u = u = u =5 m/ By continuity Q = Q + Q an preure i alway atmoperic p = p = p =0. So u θ u a = u a + u a o a = a + a Put te axe normal to te plate, a we know tat te reultant force i normal to te plate. Example: nwer CIVE400: Flui Mecanic 4 From te quetion, jet = 0.075m u =5m/ Q = 5(0.075/) = 0. m / Force normal to plate i Example: nwer CIVE400: Flui Mecanic 4

22 CIVE400: Flui Mecanic Example: nwer F Tx = Q( 0 - u x ) F Tx = ( 0-5 co 0 ) =.9 kn 6.5 Te outlet pipe from a pump i a ben of 45 riing in te vertical plane (i.e. an internal angle of 5). Te ben i 50mm iameter at it inlet an 00mm iameter at it outlet. Te pipe axi at te inlet i orizontal an at te outlet it i m iger. By neglecting friction, calculate te force an it irection if te inlet preure i 00kN/m an te flow of water troug te pipe i 0.m /. Te volume of te pipe i 0.075m. [.94kN at to te orizontal] y p u x p u 45 m CIVE400: Flui Mecanic Example: nwer u u m/ 05. / m/ F T co x N an in te y-irection F Q u u T y y y Qu in in N 4 Calculate te preure force. F P preure force at - preure force at F p co0 p co p p co P x & Draw te control volume an te axi ytem p = 00 kn/m, Q = 0. m / = 45 = 0.5 m = 0. m = 0.77 m = m F p in0 p in p in P y We know preure at te inlet but not at te outlet. we can ue Bernoulli to calculate ti unknown preure. Calculate te total force in te x irection FT Q u u x x x Qu co u by continuity u u Q, o p u p u z z g g g g were f i te friction lo In te quetion it ay ti can be ignore, f =0 Te eigt of te pipe at te outlet i m above te inlet. Taking te inlet level a te atum: z = 0 z = m So te Bernoulli equation become: p p 564. N / m f Example: nwer CIVE400: Flui Mecanic 4 Example: nwer CIVE400: Flui Mecanic 44

23 CIVE400: Flui Mecanic Example: nwer F F P x P y co kn 564. in Calculate te boy force Te only boy force i te force ue to gravity. Tat i te weigt acting in te y irection. F g volume B y N Tere are no boy force in te x irection, F B x 0 6 Calculate te reultant force F F F F T x R x P x B x F F F F T y R y P y B y F F F F R x T x P x B x N CIVE400: Flui Mecanic Example: nwer F F F R R x R y kn n te irection of application i F R y tan tan F R x Te force on te ben i te ame magnitue but in te oppoite irection R F R 6.6 Te force exerte by a 5mm iameter jet againt a flat plate normal to te axi of te jet i 650N. Wat i te flow in m /? [0.08 m /] y u u x F F F F R y T y P y B y N u n te reultant force on te flui i given by F Ry F Reultant φ F Rx From te quetion, jet = 0.05m F Tx = 650 N Force normal to plate i F Tx = Q( 0 - u x ) 650 = 000Q ( 0 - u ) Q = au = ( /4)u 650 = -000au = -000Q /a 650 = -000Q /(0.05 /4) Q = 0.08m / Example: nwer CIVE400: Flui Mecanic 45 Example: nwer CIVE400: Flui Mecanic 46

24 CIVE400: Flui Mecanic Example: nwer 6.7 curve plate eflect a 75mm iameter jet troug an angle of 45. For a velocity in te jet of 40m/ to te rigt, compute te component of te force evelope againt te curve plate. (ume no friction). [R x =070N, R y =5000N own] y x u CIVE400: Flui Mecanic Example: nwer reucing ben, 0.6m iameter uptream, 0.m iameter owntream, a water flowing troug it at te rate of 0.45m / uner a preure of.45 bar. Neglecting any lo i ea for friction, calculate te force exerte by te water on te ben, an it irection of application. [R=4400N to te rigt an own, = 4] y x ρ u u θ ρ u θ From te quetion: a u / m/ m Q m / a a, o u u Calculate te total force uing te momentum equation: F T Q u u x co co45 40 F T y N Q u in in N Boy force an preure force are 0. So force on vane: R F 070N x t x R F 4998N y t y & Draw te control volume an te axi ytem p = N/m, Q = 0.45 m / = 45 = 0.6 m = 0. m = 0.8 m = m Calculate te total force in te x irection FT Q u u x x x Qu co u by continuity u u Q, o u u m/ 06. / m/ Example: nwer CIVE400: Flui Mecanic 47 Example: nwer CIVE400: Flui Mecanic 48

25 CIVE400: Flui Mecanic Example: nwer F T co x 0 N an in te y-irection F Q u u T y y y Qu in in N 4 Calculate te preure force. F P F p co0 p co p p co P x F p in0 p in p in P y preure force at - preure force at CIVE400: Flui Mecanic Example: nwer Te only boy force i te force ue to gravity. Tere are no boy force in te x or y irection, F B x F B y 0 6 Calculate te reultant force F F F F T x R x P x B x F F F F T y R y P y B y F F F F R x T x P x B x N F F F F R y T y P y B y N We know preure at te inlet but not at te outlet. we can ue Bernoulli to calculate ti unknown preure. p u p u z z g g g g were f i te friction lo In te quetion it ay ti can be ignore, f =0 ume te pipe to be orizontal z = z So te Bernoulli equation become: p p 6007 N / m F F P x P y N 600 N co in f n te reultant force on te flui i given by F Ry F Reultant φ F Rx F F F R R x R y kn n te irection of application i F R y 800 tan tan 6. F 45 R x Te force on te ben i te ame magnitue but in te oppoite irection R F R 5 Calculate te boy force Example: nwer CIVE400: Flui Mecanic 49 Example: nwer CIVE400: Flui Mecanic 50

26 CIVE400: Flui Mecanic Example: nwer Laminar pipe flow. 7. Te itribution of velocity, u, in metre/ec wit raiu r in metre in a moot bore tube of 0.05 m bore follow te law, u =.5 - kr. Were k i a contant. Te flow i laminar an te velocity at te pipe urface i zero. Te flui a a coefficient of vicoity of kg/m. Determine (a) te rate of flow in m / (b) te earing force between te flui an te pipe wall per metre lengt of pipe. [6.4x0-4 m /, 8.49x0 - N] Te velocity at itance r from te centre i given in te quetion: u =.5 - kr lo we know: = kg/m r = 0.05m CIVE400: Flui Mecanic Example: nwer 7. liqui woe coefficient of vicoity i m flow below te critical velocity for laminar flow in a circular pipe of iameter an wit mean velocity u. Sow tat te preure lo in a lengt of pipe i um/. Oil of vicoity 0.05 kg/m flow troug a pipe of iameter 0.m wit a velocity of 0.6m/. Calculate te lo of preure in a lengt of 0m. [ 50 N/m ] See te proof in te lecture note for Conier a cyliner of flui, lengt L, raiu r, flowing teaily in te centre of a pipe δr We can fin k from te bounary conition: wen r = 0.05, u = 0.0 (bounary of te pipe) 0.0 =.5 - k0.05 k = 6000 r r R u = r a) Following along imilar line to te erivation een in te lecture note, we can calculate te flow Q troug a mall annulu r: Q ur annulu annulu ( rr) r rr Q r rr b) Te ear force i given by F = (r) 005. Q. 5r6000r r 0 5. r m / r Te flui i in equilibrium, earing force equal te preure force. rl p pr p r L Newton law of vicoity u y, We are meauring from te pipe centre, o u r Giving: p r u L r u p r r L From Newton law of vicoity In an integral form ti give an expreion for velocity, u r u 6000r 000r r F ( 0. 05) N p u L rr Te value of velocity at a point itance r from te centre u r p r C L 4 t r = 0, (te centre of te pipe), u = u max, at r = R (te pipe wall) u = 0; p R C L 4 t a point r from te pipe centre wen te flow i laminar: Example: nwer CIVE400: Flui Mecanic 5 Example: nwer CIVE400: Flui Mecanic 5

27 CIVE400: Flui Mecanic Example: nwer u r p R L 4 r Te flow in an annulu of tickne r Q ur annulu annulu ( rr) r rr p Q R r r r L 4 R p Q Rrr r L p R p L 8 L8 So te icarge can be written p Q 4 L 8 To get preure lo in term of te velocity of te flow, ue te mean velocity: u Q/ p u L Lu p u p per unit lengt b) From te quetion = 0.05 kg/m = 0.m u = 0.6 m/ L = 0.0m p 50 N / m 0. CIVE400: Flui Mecanic Example: nwer 7. plunger of 0.08m iameter an lengt 0.m a four mall ole of iameter 5/600 m rille troug in te irection of it lengt. Te plunger i a cloe fit inie a cyliner, containing oil, uc tat no oil i aume to pa between te plunger an te cyliner. If te plunger i ubjecte to a vertical ownwar force of 45N (incluing it own weigt) an it i aume tat te upwar flow troug te four mall ole i laminar, etermine te pee of te fall of te plunger. Te coefficient of velocity of te oil i 0. kg/m. [ m/] F = 45N = 5/600 m Q plunger cyliner 0.8m Flow troug eac tube given by Hagen-Poieuille equation p Q 4 L 8 Tere are 4 of tee o total flow i 4 4 p 4 ( 5/ 600) Q 4 p p60. 0 L Force = preure area / 600 F 45 p 4 p N / m 0 0. m Q m / Flow up troug piton = flow iplace by moving piton Q = v piton Example: nwer CIVE400: Flui Mecanic = 0.04 v piton Example: nwer CIVE400: Flui Mecanic 54

28 CIVE400: Flui Mecanic Example: nwer v piton = m/ 7.4 vertical cyliner of metre iameter i mounte concentrically in a rum of 0.076metre internal iameter. Oil fill te pace between tem to a ept of 0.m. Te rotque require to rotate te cyliner in te rum i 4Nm wen te pee of rotation i 7.5 rev/ec. uming tat te en effect are negligible, calculate te coefficient of vicoity of te oil. [0.68 kg/m] From te quetion r - = 0.076/ r = 0.075/ Torque = 4Nm, L = 0.m Te velocity of te ege of te cyliner i: u cyl = 7.5 r = =.767 m/ u rum = 0.0 Torque neee to rotate cyliner T urface area 4r L 654. N / m Ditance between cyliner an rum = r - r = = 0.005m Uing Newton law of vicoity: u r u r kg / m ( N / m ) Example: nwer CIVE400: Flui Mecanic 55 CIVE400: Flui Mecanic Example: nwer Dimenional analyi 8. tationary pere in water moving at a velocity of.6m/ experience a rag of 4N. noter pere of twice te iameter i place in a win tunnel. Fin te velocity of te air an te rag wic will give ynamically imilar conition. Te ratio of kinematic vicoitie of air an water i, an te enity of air.8 kg/m. [0.4m/ 0.865N] Draw up te table of value you ave for eac variable: variable water air u.6m/ u air Drag 4N D air 000 kg/m.8 kg/m Kinematic vicoity i ynamic vicoity over enity = u u Te Reynol number = Re Cooe te tree recurring (governing) variable; u,, From Buckingam teorem we ave m-n = 5 - = non-imenional group. u,,, D, 0, 0 a b c u D a b c u eac group i imenionle ten coniering te imenion, for te firt group, : (note D i a force wit imenion MLT - ) a b c M L T LT L ML MLT M] 0 = c + c = - L] 0 = a + b - c + -4 = a + b T] 0 = -a - a = - b = - u D D u Example: nwer CIVE400: Flui Mecanic 56

29 CIVE400: Flui Mecanic Example: nwer n te econ group : a b c M L T LT L ML L T M] 0 = c L] 0 = a + b - c + - = a + b T] 0 = -a - a = - b = - 0 u u So te pyical ituation i ecribe by ti function of nonimenional number,,, 0 D u u For ynamic imilarity tee non-imenional number are te ame for te bot te pere in water an in te win tunnel i.e. air water For air water D D u u air Dair ( ) D N For u air u water uair 6. u 0. 4m/ air air water Example: nwer CIVE400: Flui Mecanic 57 CIVE400: Flui Mecanic Example: nwer 8. Explain briefly te ue of te Reynol number in te interpretation of tet on te flow of liqui in pipe. Water flow troug a cm iameter pipe at.6m/. Calculate te Reynol number an fin alo te velocity require to give te ame Reynol number wen te pipe i tranporting air. Obtain te ratio of preure rop in te ame lengt of pipe for bot cae. For te water te kinematic vicoity wa.0-6 m / an te enity wa 000 kg/m. For air toe quantitie were m / an.9kg/m. [447, 8.4m/, 0.57] Draw up te table of value you ave for eac variable: variable water air u.6m/ u air p p water p air 000 kg/m.9kg/m m m 000 kg/m.8 kg/m 0.0m 0.0m Kinematic vicoity i ynamic vicoity over enity = u u Te Reynol number = Re Reynol number wen carrying water: u Re water To calculate Re air we know, Re water Reair uair u 8. 44m/ air To obtain te ratio of preure rop we mut obtain an expreion for te preure rop in term of governing variable. Cooe te tree recurring (governing) variable; u,, From Buckingam teorem we ave m-n = 5 - = non-imenional group. u,,,, p 0, 0 a b c u a b c u p eac group i imenionle ten coniering te imenion, for te firt group, : a b c M L T LT L ML L T Example: nwer CIVE400: Flui Mecanic 58

30 CIVE400: Flui Mecanic Example: nwer M] 0 = c L] 0 = a + b - c + - = a + b T] 0 = -a - a = - b = - 0 u u n te econ group : (note p i a preure (force/area) wit imenion ML - T - ) a b c M L T LT L ML MT L M] 0 = c + c = - L] 0 = a + b - c - - = a + b T] 0 = -a - a = - b = 0 u p p u So te pyical ituation i ecribe by ti function of nonimenional number,,, p u u 0 For ynamic imilarity tee non-imenional number are te ame for te bot water an air in te pipe. air water air water We are interete in te relationip involving te preure i.e. CIVE400: Flui Mecanic Example: nwer p p u u air water pwater wateruwater pair airuair Sow tat Reynol number, u/, i non-imenional. If te icarge Q troug an orifice i a function of te iameter, te preure ifference p, te enity, an te vicoity, ow tat Q = Cp / / / were C i ome function of te non-imenional group ( / / /). Draw up te table of value you ave for eac variable: Te imenion of tee following variable are ML - u LT - L ML - T - Re = ML - LT - L(ML - T - ) - = ML - LT - L M - LT = i.e. Re i imenionle. We are tol from te quetion tat tere are 5 variable involve in te problem:, p,, an Q. Cooe te tree recurring (governing) variable; Q,, From Buckingam teorem we ave m-n = 5 - = non-imenional group. Q,,,, p 0, 0 a b c Q a b c Q p eac group i imenionle ten coniering te imenion, for te firt group, : a b c M L T L T L ML ML T M] 0 = c + c = - L] 0 = a + b - c - - = a + b T] 0 = -a - a = - b = Example: nwer CIVE400: Flui Mecanic 59 Example: nwer CIVE400: Flui Mecanic 60

31 CIVE400: Flui Mecanic Example: nwer Q Q n te econ group : (note p i a preure (force/area) wit imenion ML - T - ) a b c M L T L T L ML MT L M] 0 = c + c = - L] 0 = a + b - c - - = a + b T] 0 = -a - a = - b = 4 4 Q p 4 p Q So te pyical ituation i ecribe by ti function of non-imenional number, 4,, 0 p Q Q or 4 p Q Q p p Te quetion want u to ow : Q f / / / / Q Take te reciprocal of quare root of : / p Convert by multiplying by ti number / Q a Q p p a ten we can ay / / / a, CIVE400: Flui Mecanic Example: nwer / / / p p / a, a, / 0 or / / / p p Q / 8.4 cyliner 0.6m in iameter i to be mounte in a tream of water in orer to etimate te force on a tall cimney of m iameter wic i ubject to win of m/. Calculate () te pee of te tream neceary to give ynamic imilarity between te moel an cimney, (b) te ratio of force. Cimney: =.kg/m = 60-6 kg/m Moel: = 000kg/m = 80-4 kg/m [.55m/, 0.057] Draw up te table of value you ave for eac variable: variable water air u u water m/ F F water F air 000 kg/m.kg/m kgm kg/m 0.6m m Kinematic vicoity i ynamic vicoity over enity = u u Te Reynol number = Re For ynamic imilarity: Re water Reair 000uwater u 55. m/ water To obtain te ratio of force we mut obtain an expreion for te force in term of governing variable. Cooe te tree recurring (governing) variable; u,, F, From Buckingam teorem we ave m-n = 5 - = non-imenional group. u,,,, F 0, 0 a b c u a b c u F eac group i imenionle ten coniering te imenion, for te firt group, : Example: nwer CIVE400: Flui Mecanic 6 Example: nwer CIVE400: Flui Mecanic 6

32 CIVE400: Flui Mecanic Example: nwer a b c M L T LT L ML ML T M] 0 = c + c = - L] 0 = a + b - c - - = a + b T] 0 = -a - a = - b = - u u i.e. te (invere of) Reynol number n te econ group : a b c M L T LT L ML ML T M] 0 = c + c = - L] 0 = a + b - c - - = a + b T] 0 = -a - a = - b = - u F F u So te pyical ituation i ecribe by ti function of nonimenional number,,, F u u 0 For ynamic imilarity tee non-imenional number are te ame for te bot water an air in te pipe. air water air water To fin te ratio of force for te ifferent flui ue Example: nwer CIVE400: Flui Mecanic 6 CIVE400: Flui Mecanic Example: nwer air water F F u u air water F F u u air water Fair F water 8.5 If te reitance to motion, R, of a pere troug a flui i a function of te enity an vicoity of te flui, an te raiu r an velocity u of te pere, ow tat R i given by ur R f Hence ow tat if at very low velocitie te reitance R i proportional to te velocity u, ten R = kru were k i a imenionle contant. fine granular material of pecific gravity.5 i in uniform upenion in till water of ept.m. Regaring te particle a pere of iameter 0.00cm fin ow long it will take for te water to clear. Take k=6 an =0.00 kg/m. [8min 9.ec] Cooe te tree recurring (governing) variable; u, r, R, From Buckingam teorem we ave m-n = 5 - = non-imenional group. ur,,,, R 0, 0 a b c u r a b c u r R eac group i imenionle ten coniering te imenion, for te firt group, : a b c M L T LT L ML ML T M] 0 = c + c = - L] 0 = a + b - c - - = a + b T] 0 = -a - a = - b = - u r ur i.e. te (invere of) Reynol number Example: nwer CIVE400: Flui Mecanic 64

33 CIVE400: Flui Mecanic Example: nwer n te econ group : a b c M L T LT L ML ML T M] 0 = c + c = - L] 0 = a + b - c - - = a + b T] 0 = -a - a = - b = - u r R R ur So te pyical ituation i ecribe by ti function of nonimenional number,,, R ur ru 0 or R ru ur CIVE400: Flui Mecanic Example: nwer r = m ept =.m 4 mg kru u u5. 0 m/. t min 9. ec ur e quetion ak u to ow R f or R ur f Multiply te LHS by te quare of te RHS: (i.e. (/ ) ) R ru ur R So R ur f Te quetion tell u tat R i proportional to u o te function f mut be a contant, k R k ur R kru Te water will clear wen te particle moving from te water urface reace te bottom. t terminal velocity tere i no acceleration - te force R = mg - uptrut. From te quetion: =.5 o = 500kg/m = 0.00 kg/m k = 6 Example: nwer CIVE400: Flui Mecanic 65 Example: nwer CIVE400: Flui Mecanic 66