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This 33 page Class Notes was uploaded by Jada Daniel on Wednesday September 23, 2015. The Class Notes belongs to MEM220 at Drexel University taught by YoungCho in Fall. Since its upload, it has received 19 views. For similar materials see /class/212396/mem220-drexel-university in Mechanical Engineering at Drexel University.
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Date Created: 09/23/15
Chapter 3 Bernoulli Equation We neglect friction Why For mathematical simplicity For quick approximation Energy equation without frictional term 31 Newton s Second Law Dividing streamline Do you see streamlines V0 0 gt 6 Po Stagnation point a Do you see velocity At any point velocity is to streamline 91 V710 Streamlines Fig 31 a b What is streamline u axial component of velocity vector V vertical component of velocity vector Slope of streamline in terms of u and v Derivation of uv dydx Velocity vector V ui vj and ds dxi dyj Since velocity vector is tangent to streamline V and ds must be parallel Thus des 0 or Vds0 32 Derivation of Bernoulli equation V T StL Q ream Ines 1 V 2 lFuid particle U D We apply ZF ma along a streamline S Figure 31 p 95 a acceleration along S coordinate dVdt 1 C1 32 Derivation of Bernoulli equation We apply 2F ma along a streamline S Figure 33 p 96 39 Freebody diagram of a fluid particle 2F sum of all the forces along the streamline g g p 57quot is 5y T 6S 5y 0 Particle thickness 6y 6 p 66115y saw 139 55 639 O p 5pv6n 6y 39 5n 5z 53 57 p 5pquot63 5 Normal to streamline Along streamline 32 Bernoulli equation dPpgdZdeVO ntegrating this we have Pressure unit Pa Energy per unit mass Jkg head unit Show why gZ has Jkg Show why v2 has Jkg 34 Physical interpretation of Bernoulli equation Il l i I I X R I i I L 1 n 1 Ir 31 7 v I erg P Which one is greater V1 or V2 V1 3 mS V2 39 Which one is greater P1 or P2 P1 P2 Example V13ms A12cm2 A21cm2 P1 75 kPa Determine V2 and P2 34 Physical interpretation of Bernoulli equation Il P l i I I I R I In IL 1 n 1 Ir 31 7 v I erg Example Given V13ms d101 m d2005m P1 125 kPa Density of water 1000 kgm3 Objective to determine V2 and P2 r v Conservation of mass quotii1 151 5 Find v2 Find P2 Progression of atherosclerosis vascular disease MONOCYTE 516 What happens at stenosis Velocity increases or decreases Pressure increases or decreases RUPTURED ENDoi39HELiUM What can happen to coronary artery TISSUE DAMA D IN A HEART ATTACK MYOCARDIAL lNFARCnON Scientific American Quarteriy 2000 Summer Voi11 NoZ Example Bernoulli equation l J i v f g 7 V r i 77 1 arm irtara w 4 it wanMm Objective to determine the maximum height h Solution Let us consider three locations indicated by p 85 kPa What is V3 01 m Figure P344 5th ed What is P2 Why is the pressure at the tip of a nozzle zero If it is not zero what happens 3 0 Apply Bernoulli eq between 1 and 2 Answer V2 1283 ms Vl 39 A as J aux 11 1 Ifampil litg1 inwfvr f I 39l 4 Elymil If i p85 kPa Figure P344 5th ed Apply Bernoulli eq between 2 and 3 Answer h 84 m i I 394 i 25 i 39 i i v 1 I i ligpy I D r quot 35 Static Pressure and Stagnation Pressure Stagnation point Figure 35 p 106 F39gure E32 39039 101 Stagnation points on bodies in flowing fluids Stagnation pOlnt Stagnation streamline Stagnation point a b How to measure Static Pressure and Stagnation Pressure Open 4 if H What IS the velouty at pomt 2 l I1 1743 V l l V 39 z mealin 4L 7 l 2 1le 120 Figure 34 p 105 Measurement of static and stagnation pressures Note that we made two holes on the pipe wall to measure these pressures Can we measure both pressures with only one hole How to measure Static Pressure and Stagnation Pressure 3 Using only one hole on the I i e wall American Blower company 1 National Physical laboratory England V I 0 q I l 2 American Society of Heating amp Ventilating Engineers I o o o Figure 36 p 107 The Pitotstatic tube I F I G U R E 37 Typical Pitotstatic tube designs Typical Pitotstatic tube designs How to measure a speed of an aircraft Figure E36a p 110 V1 100 mihr 27 1 Pitotstatic tube Does is matter where we locate a Pitot tube Spotlighf Air France Flight 447 vanished mm min m mm wmnn Lm mmmum could huw mm the m hmk w Mimian lmpa l yingauhzwmngspee 711mm lheory is lluloneofirsmree Pun nftlmsz u messngzsmdicamd um m Momentum behind the Pilot theory is gmwing Airbus after all recom mend ed nearly two years lgu that air incs replace Pitot tubes like thnse aboar 447 with an improved model less prone to idngr While aviation authorities in Europe and the 15 never made the change mandalury Air France said it had begun replacing he in es In Mayiand d the crash a the demand uf pilol unions Meanwhile such an US cxpcrls continue to listen for the elusive ping ing nfl lighl 44739 black boxes sitting somewhere on the oor of the Atlantic Ifihcy are never found a theory may est we ever get to the truth in runs THOMPSON TIME June 222009 PRISM Static air sensors Drain J Dulclng heater Hawa Flint tube works Alrspeed Is measured by onmparlng the pressure difference hetwaan a flowing rhmugn the Flint mine and nonmmlng air in a stark Ghamlmr In a similar manner where do you position a torpedo launcher 1 1 361 Free Jet FL we 193 5 Wm 1993 mm Objective to determine the exit velocity at point 2 as a function of h Solution Assume A1 gtgt A2 Apply Bernoulli eq between 1 and 2 Figure 311 p 110 Vertical flow from a tank 361 Free Jet 55 r twillEly r Jewis Question Doe the diameter of free jet remain constant or decrease axially Objective to determine the diameter of free jet as a function of H Figure 311 p 110 Vertical flow from a tank The column of uid descending from a spigot comrach as it acrelcmlcs as it must according to the principle of coudnuity 4 le n n pg 3quot 39 39i IE 4 Iquot Flows from atank ph lmzl 93 Example m p113 quot Objective to determine the inlet flow rate Qto maintain water level h 20 m Solution Apply Bernoulli eq between 1 and 2 Find relationship between V1 and V2 using conservation of mass 11 mt Figure E37 p 113 I1 020 m b Flows from a tank 1 I 1 PHquot Em hwza fi EW p93 mmt Can we explain what we see Ill 1 V FIOWS from a tank Fl E 41593quot 13193 g nt Example m p113 39 Objective When we stop the inlet flow rate Q how long will it take to drain ie h becomes zero Treat h ht Solution Apply Bernoulli eq between 1 and 2 Figure E37 p 113 I1 020 m b Ill 1 CaVItatIon FL Lag33 F H 4 REE 1 3939 i39l39 Objective to estimate p2 as a fuinction of Q or V1 Solution Apply Bernoulli eq between 1 and 2 V1 ltlt V2 gt p 2 gtlt If P2 decreases to the pressure 12 Absolute Water Will b0il or evaporate pressure What is the pressure at the room temperature Large Q Incipient cavitation Figure 316 p 117 Pressure variation and cavitation in a variable area pipe J 1 CaVItatIon F 39 agEL 3quot ll ll 159375 cmt rm Figure 317 p 117 Tip cavitation from a propeller V2 circumferential velocity R on 0 angular velocity 2nn60 where n is rpm Given Vapor pressure at room temperature 1770 Pa absolute Radius of propeller R 1 m Objective to estimate the maximum rpm to avoid cavitation Example 310 Siphon and Cavitation Objective to determine the maximum value of H Question Why How does siphon fail What is the failure mode of siphon Do you see that P2 decreases as we raise the hose ie increasing H Answer At the moment of failure P2 becomes s P1 zero s P3 zero What is P2 Parabsolute Figure E310b p 118 l a i Luv ll H V i Ersrgzlzr j algavitation air writ E f b I dte were quotHarv e til3235 wetter Wm Solution Apply Bernoulli eq between point 1 and point 3 Find V3 359 fts Apply Bernoulli eq between point 2 and point 3 note P2 is absolute pressure Figure E310b p 118 in Figure 318 p 119 Typical devices for measuring flowrate in pipes 39r quota a r T 39v 311 i M 39i L sji si quotquot49 I 39g 39i i v h l quotWarquot rquotInquot Mei 39J39MM Orifice m x w W Nozzle DREXEL UNIVERSITY m w College of Engineering name Bernoulli prinCIple L Mk m Wm an For unsteady ow WWMZ Th DSdlld iUK 4 a minimian lir39ci in M Li Me Exampie 316 p129 Jim W M 7 Q be m OSCIIIatlng flows MM u 13 w 52 askMmish l 7 A In U tube m immgim Open We fulukmn 0 ma a 1M manger Bannlli 9 3 MW D and 4 Fusszu i v5 h rf z iwiz quot mikymsufiq wa M mime a3 Fa f o 0 jFVz v M0 mus Me mc ei mnbx 0 is L o 351 3 Dltig 0 o 32 o Equilibrium z gt Q position i1 3 39 392 3932 V vmies nulde Hm u V is inhfewiw f 5 Tins UM wvik n4 I ma Wlw 5 6 2L0 5 EEC am I av av 1 NM War 1 Egg alumni U Me 0 em Figure E316 p 129 L asazu s m i we Clinical disposable viscometer Cho and Kensey LED Array CCD Computer data acquisition system Blood from vacutainer Capillary tube A new scanning capillary tube viscometer Rheolog with disposable capillary tube LiBlood enters Pressure drop and flow rate from single measurement of ht AP pgm 0 142 0 9 Calculate Shear stress Fluid falls Velocity is a derivative of height mm L212 9 Calculate Shear rate CapillaIy tube 71d4AP 128 L Q Conceptually u Disposable tube of Rheolog Att0 Att 00 Scanning capillary tube viscometer Cuntact image Sensuv Cuntact image Sensuv HW Figure P348 p 137 Wind tunnel Used for measurement of Drag of a car V10 V260 mph Objective to calculate h P2 is vacuum Yes No
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