Introduction to Fluid Mechanic
Introduction to Fluid Mechanic CE 321
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This 29 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 42 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 8 Fluids in Motion Bernoulli Equation Course Learning Objective CLO 2 Determine changes in fluid pressure with the Bernoulli Equation Today Q What equation that you already know describes the flow of fluids Newton s Second Law of Motion F ma Describing Fluid Flows Two Approaches Follow individual particles in the fluid The Lagrangian Approach Difficult to do du 8L1 dt 8t x The lazy person s approach Sit at one point and observe The Eulerian Approach Usually followed in Fluid Mechanics The two can be related remember chain rule in calculus Inviscid Fluid Viscosity 0 If viscosity is zero what forces are absent Shear What forces are present Pressure and gravity forces Fma 39 F ma 9 One force 39ZFi ma 9 Multiple forces Net pressure forcenet gravity force particle mass x particle acceleration Acceleration amp Velocity dV a d VVu2v2 V S E v V Steady Flow Flow doesn t change with time at a given location 20 dt This means 9 Streamlines iV 2 L um pamde u m Streamlines Lines that are tangent to the velocity vectors throughout the flow 5 5t field are called d V streamlines a gt a a dt 8 TI d8 gtvv dt 8 TI dV av as av 8 dt 88 8t 88 v2 a R F ma along a streamline vx p 5p 5 5 Particle Wickrtm iv Lu 5px 151i 5 6n Ln pjbnm r A J H 5 6 i Z 53 a p pnj s 5y Nurr r39ual to Streamline Along SUEBH IHI IE F ma along a streamline 26E 6mas 6mVZ V2 pWVa V 3 83 6W3 6W sin 6 75V Sin 9 6 39 9T S39 p pg 1 as 2 ayor erles N 5 pg 83 2 p5 5F p 6ps6n6y p 5P56n5y 26P56n5y 2p 6357159 2 p 39 V 3 8 Z5F6W6F ysin9 6V 3 5 P5 88 8 8V 9 V ysm as p as 3 8 ysin9 a OV 8V 83 7 951 VaV Bernotilll WD 537 as Equation Em 9 d1 Aiunqsmdmm d3 2 V E 1 CW J d8 2 d3 83 d 1 2 7 gt dp 5mm J dz 7 0 For constant density this can be integrated p lsz v2 onstant I 2 along a streamline Physical Interpretation 1 p l Epl2 l 72 constant 2 p V l l 2 Constant 2 Hl 7 9 Elevation W 39 Head Pressure Veloc1ty Head Head The sum of the pressure head the velocity head and the elevation head is constant along a streamline Limitations of Bernoulli s Equation Steady Flow Frictionless Flow No Shaft Work Incompressible Flow No Heat Transfer Flow along a Streamline Qwewwe Do Not Use Bernoulli s Law Here Sudden Long and narrow expansion tubes 6 D gt gt Flow through a valve A heating section Static Stagnation Dynamic amp Total Pressure Stagnation Pressure Pressure at the Stagnation Point 39 W Newman smmw D C 425 UhltS p pvz 72 constant Each term has dimensions of force per unit area p is the actual thermodynamic pressure called static pressure Open Static relative to moving fluid pV22 is called the dynamic pressure gt yz is the hydrostatic pressure PTstatic dynamic hydrostatic gt Why is pV22 called the dynamic pressure Open Consider pressure at the end of a small tube inserted into the flow pointing upstream Liquid fills tube to a height H after initial transients have died out Velocity of fluid in tube and at the tip is zero Applying Bernoulli formula 9 P at stagnation point is greater than the static pressure p1 by an amount pV22 the dynamic pressure Pitot Tube Pitot Tube If we know the static and stagnation ressures we can calculate the fluid speed 7 Two concentric tubes Inner tube measure 1 2 stagnation pressure 193 p Epl Outer tube measures 194 p1 1 static pressure 2ltp3 p4 Problem 31 31 Water ows steadily through the vari able area horizontal pipe shown in Fig P31 The veiocitv is given by V 101 xi fts where x is in feet Viscous effects are neglected a De termine the pressure gradient Sip6x as a func tion of x needed to produce this ow b If the pressure at section 1 is 50 psi determine the pressure at 2 by i integration of the pressure gradient obtained in a ii application of39 the Bernoulli equation i 1 V 2 Problem 31 Solution a b sxh9 JV3 m 90 and Vox W sill ev or 15 9V pox0 65 777w 5 ir ig 0 ilX 1 MM XI feet 99 X7g f1 Xz3 b 39 47 HX fin 7 HmX 0 so a 12m it or 02 fopsi ma 50 o349 E ii plf39MzJquotZ z 379quot W172 or WWI 2 22 2LMv v were 140000 i V1 o3 W 75 f1 sapsz 77 13 02 W 3 5 g 377 m Problem 35 3 5 What pressure gradient along the streamline dpds is required to accelerate water upward in a vertical pipe at a rate of 30 flsz What is the answer if the ow is downward 3 Aquotsn QV 3Z lube 5 6 70 for upfau I e 470 For down fow and VL 45 30 T us for o fau lb 33 6z 4 fgg gog 1206 77741 unease and for plow fow if M 52y 1f 44 59 7420 l Hooz92 If 313 Carbon tetrachloride lows in a pipe of variable diam eter with negligible viscous effects AI point A in the pipe he pressure and velocity are 20 psi and 30 his respectively Al lo cation 8 the pressure and velocity are 23 psi and 14 fls Which point is al the higher elevation and by how much vquot w L u 421Tgz 7zi4 zs WI 339 7257 392 z or f r 5 W V 20 23739 4 lym 39 30 4 28 Z x 2 57 23zzg or ZR 392 659 7 B i abave 9 Problem 325 p95 325 Air is drawn into a wind tunnel used for testing auto mobiles as shown in Fig P7325 a Determine the manometer reading h when the velocity in the test section is 60 mph Note that there is a 1 in column of oil on the water in the manome ter b Determine the difference between the stagnation pres sure on the front of the automobile and the pressure in the test section lmrl Umilcl 60 m m Vf WJIEV INK Wind tunnel 325 m 60 mph 1 solution gt i 3 Open Fan 1quot 11 in Wa er Oil 50 09 a Hz g 3 E zz Wier e 24 mm W 140 ms wn 1z sawA My 41quot g 0quot P2 fl F V12 1 a00238if Hg1 222 73141 wuf