Introduction to Fluid Mechanic
Introduction to Fluid Mechanic CE 321
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This 32 page Class Notes was uploaded by Jolie Shields on Saturday September 19, 2015. The Class Notes belongs to CE 321 at Michigan State University taught by Roger Wallace in Fall. Since its upload, it has received 46 views. For similar materials see /class/207369/ce-321-michigan-state-university in Civil Engineering at Michigan State University.
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Date Created: 09/19/15
Lecture 3 Pressure in Liquids Manometers Announcements Quiz 1 next week Monday Topics Sections 11 to 29 in your text HW 1 opens TODAY Office Hours today and Friday 13 in 3550 EB Course Learning Objective 1 Calculate pressure in a static liquid oday derive manometer equations today calculate the magnitude and location or TlUlCl rorces on a plane surface next class Pressure at a point Pressure 5 e wrmal ow pe ut urea acting at a point on a given plane You can draw an infinite number of I lanes woinw through a given point Does the pressure at a point depend on the orientation of the plane Answer Answer To answer his consider M Ma c a some angle to the horizontal Let us consider the forces acting w t c rla c Forces The weight of the fluid element acting vertically down and the pressure acting on the plane See figure l3 p 51 ii 539 i gt W R I I I 39 gll v V W 65 ff 1 f i t E 5r 1 I 111311 Forces on an arbitrary wedgeshaped element Pressure at a point No Shear ZFy py x z ps x s sin6 p 5X5y5Z ay ZFZ pz5x5y ps5x5s cos6 y 5X52y Z pWaZ 5y 2 5s cos 6 52 5s sin6 From geometry py ps 2 pay Equations of motion pz ps paz 7 5y gtu5z gtu PASCAL S LAW Pressure at a point in a fluid at rest or in motion pz ps without shear is independent of direction Py Ps Q LI IUL tl IVIe HIV IIV ShVUI Inu stresses in the fluid how does the pressure var fw Wint w pout To answer this question we need to develop an equation for the pressure menu by applying Newton s second law to a small fluid element First some background Body Forces and Surface Forces Body forces as the name implies as w a volume of fluid from a distance Examples Gravitational forces weight Surface forces Act on the surface Taylor s Series See Thomas and Finney Calculus for details 1x 5 l 639 1 I if 1 3 39I 4 3 5 39l Arreh39 11 TI Ju 7 M 1 ellll39 quot it lii39l39ilj High a l J T mm r kl 6 gt m 39 a 39 391 p 1 l Vp y 2021 Equation for the Pressure Field 6F 2 p a p 6x62 pa p 6x62 y 6y 2 6y 2 gt 6Fy a p6x6y62 5y 6F x a p6x6y62 6x 6FZ a p6x6y62 62 Resultant surface force on the element FS FXE ij6512 or A 6F A A 6Fsz aii yjaik 6x6y62 6x 6y 62 A 8F A A 6F i4jkj x y zzVp x y z z 6x 8y V a ia ja 1 0x Q 62 FS H x y z W12 y x y5212 Z F ma 6mp6x y z 25F FS W12 ma gt Vp x y z 7 x y z I 0 x y za gt Vp 71Apa Special Case Fluid at Rest Vp y pa a0 Vp y 0 gtVp 1Alt 0r apzo p0ap ax a 82 5p 82 lncompressible Fluids Since 70g changes in specific weight are caused by changes in density primarily A fluid whose densit does not var is called an incompressible fluid For this case the h drostatic e uation can be integrated Incompressible Fluids Compressible Fluids Specific Weights for 7pg gases are small p compared to liquids PZPRTZWZE Pressure changes dp gp due to elevation 5 E changes can be IZdp1namp 2f ignored for gases for A 19 p1 R21 T small distances but m not for large p2 91 exp RT distances 0 Isothermal 7 The Standard Atmosphere Idealized representation of mdatitude year round mean conditions of the earth s atmosphere US Standard Atmosphere T 28815 k 15 C 51867 R 59 F P 10133 kPa abs 147 bin2 abs Diniiil 122 kg m3 0002377 slugsft3 Sp Weight 12014 Nm3 00764739b 3 Viscosity 1789e05 Nsm2 3737907 IbSftZ Pressure Gage and Absolute Abs Pressure is always positive Gage pressure can be positive or negative Negative gage pressure is called vacuum pressure Atmospheric 95 Pressure m mlllimet rs 15139 mercury Barometer Mercury xx patm thpvap0r my Manometers are devices for measuring pressure and use liquid columns in vertical or inclined tubes they use columns of fluid at rest so the fundamental equation is p7hp0 Examples the piezometer tube the Utube manometer and the inclinedtube manometer In a fluid at FGSt pressure ucurcasc Upvval us and increases downwards why Piezometer Tube Open at the top PO can be set equal to zero gage pressure so that PA 71h1 What are some potential problems with this device Suitable only if pressure in the container is gt patm Heiht of the column for lare pressures is an issue No gases D pt 1 UTube Manometer Gage fluid is different from fluid in the container Start at the end A and work towards the open end Fluid in the container can be a gas or a liquid PA 71h172hz 0 PA 72h2 71h1 pAzyZh2 ifl is agas Differential UTube Manometer a tube manometer used to measure the difference in pressure between two containers Start at one end and work to the other end PA71h1 72h2 73ha PB pAps 72hz73h3 wit Inclined Tube Manometer is used to measure small pressure changes For small angles the reading along the inclined tube pA 714 772125111677ng pB can be made large 7 even for small pA pB 72125m673h3 71h1 pressure differences 9 Xv A g m v V v V V a m 9 lt gt2 60 80 0 20 40 Angle Deg MATLAB code 1 If gases are used pB 72 81119 I 2 pA pB 72 sin 9 Mechanical amp Electronic Devices Manometers are not wellsuited for very high pressures Measuring th eight ol b time consuming The Bourdon pressure gage work principle that pressure results in a deformation however small Pressure transducer electrical output Bourdon Gae Contains a hollow elastic tube Bourdon tube As P in the tube increases it tends to straighten Can measure both positive and negative gage pressure Needs calibration 21 What p1 essm39ecxpresscd in pascals will a skin diver be subjected to at u dsplh 01 50 m in seawater 1 am39 fK4fi 41 m gar e a 3 me 3 3N 12 112 73 50m rano Z1 2 5054 9 gym TabLa I5 1er 22 In a certain chemical plant a closed tank contains ethyl alcohol to a depth of 50 ft Air at a pressure of 25 psi fills the gap at the top of the tank Determine the pressure at a closed valve attached to the tank l0 1quot above its bottom NV 73 3 r 41 77 surface 7 vzjlpsg and le Hum deli11 Song Mpg 6 0 1457 P w 9 0 4 4 31 lb H4quot 5 570 It 38765 23 For an atmospheric pressure of 101 kPa abs def termine the heights of the fluid columns in barometcrs con taining one of the following liquids a mercury 1 water and c ethyl alcohol Calculate the heights including the effect of vapor pressure and compare the results with those obtained neglecting vapor pressure Do these results suppon the widespread use of mercury for barometcrs Why gt
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