TRANS PROC IN PETE PROD
TRANS PROC IN PETE PROD PETE 314
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Fluid Statics Fluid Statics The word statics is derived from Greek word statikos motionless For a fluid at rest or moving in such a manner that there is no relative motion between particles there are no shearing forces present Rigid body approximation Pete 314 22 Pressure Force l Area In the hydraulic lift the total mass of car rack and piston is 1800 kg The piston has a cross sectional area of 02 m2 What is the gauge pressure in the hydraulic uid in the cylinder ifthe car is not moving Reserve I 13 mp Hydl39mlljc cylinder Pete 314 23 Pascal s Laws Pascals laws Pressure acts uniformly in all directions on a small volume point of a fluid In a uid confined by solid boundaries pressure acts perpendicularto the boundary it is a normal force PFA Pete 314 24 Direction of fluid pressure on boundaries lt TTHTT gt HHH Furnace duct Pipe or tube Heat exchanger Pressure is due to a Normal Force acting perpendicularto the surface It is also called a Surface Force Pete 314 25 Pressure distribution for a fluid at rest We will determine the pressure distribution in a uid at rest in which the only body force acting is due to gravity The sum of the forces acting on the fluid must equal zero Consider an in nitesimal rectangular uid element ofdimensions Ax Ay Az Pete 314 Pressure distribution for a fluid at rest Let P2 and PZAZ denote the pressures at the base and top ofthe cube where the elevations are 2 and zAz respectively Force at base of cube PZ A PZ Ax Ay Force at top of cube PZAZ A PZAZ Ax Ay Force due to gravity m g p V g p Ax Ay Az g A force balance in the z direction gives BAzAxAy Pzmv pgAxAyAz Z PzAz Pz AZ Pg For an infinitesimal element AZ gt0 39y dP pg 21 x Pete 314 27 Pressure in a Fluid Copyngm v 2004 Paavson Educalmn pumlshma as Addlson Weslsy H H ifnouu mm G Pete 314 28 Incompressible fluid Liquids are basically incompressible ie their density is assumed to be constant Pg B pglt22 zlgtl lt22gt When we have a liquid with a free surface the pressure P at any depth below the free surface is where Po is the pressure at the free surface 2 3 P P0 P0Patm and h Zfree surface 39 Z I By using gauge pressures we can simply write P pgh 24 gauge Pete 314 29 The product pg is also called specific weight Hydrostatic pressure gradient is another expression for the same thing Example 21 Calculate the speci c weight of water at a place where the w m 2 acceleration of maturity us 322 iii 5 quot41 523 W 623 b ft 5quot 332 Ihm it it ji 139 0 J Give the result also in psift and in Pam 0433 and 9810 Pete 314 210 Exercise Pressure in an Oil Storage Tank The figure below shows a schematic of a crude oil storage tank What is the absolute pressure at the bottom ofthe cylindrical tank if it is filled to a depth of H with crude oil with its free surface exposed to the atmosphere The speci c gravity of the crude oil is 085 Give the answers for a H50 m pressure in Pa b H150 ft pressure in lbf in2 What is the purpose of the surrounding dike Pete 314 211 Compressible fluid Gases are compressible ie their density varies with temperature and pressure p P M RT M is in kgmol For small elevation changes as in engineering applications tanks pipes etc we can neglect the effect of elevation on pressure For larger elevation difference the isothermal case start from Eq 21 dP PM 0rTTconstz 2 dz pg f 0 p RTO g RM dz RTO g 132 P1 eXp Isothermal ideal gas RTO For nonisothermal case read page 43 of the book Pete 314 212 Exercise Compute the atmospheric pressure at an altitude of 5000 m ifthe pressure at sea level is 1013 kPa by the following methods a assume air of constant density p124 kgm3 and b assuming that the density of air changes with altitude but temperature remains constant T 15 oC Hint change T into keVin and use for air molecular mass M 0029 kgmo Pete 314 213 Hydrostatic forces on plane surfaces Case 1 Horizontal surface exposed to a gas F Pconstant everywhere TTTT TTTT F Case 2 Horizontal surface exposed to a liquid P Patm FPA Pconstant along the horizontal surface F P A Pete 314 214 Exercises The crude oil storage tank shown on slide 211 has a at horizontal circular roof 150 it in diameter The atmospheric pressure is 147 psia What force does the atmosphere exert on the roof from outside A layer of rainwater 4 in deep collects on the roof ofthe tank What net pressure force does it exert on the roof ofthe tank Typical values of density of water can be found in the back cover of your textbook How about 623 lbmlt3 Pete 314 215 Hydrostatic forces on plane surfaces Case 3 Vertical surface exposed to air F ll Pressure varies little with height see also equation 24 For air and moderate heights one can totally neglect the change in pressure Then it is enough to use with a constant pressure PFA Pete 314 216 Hydrostatic forces on plane surfaces Case 4 Vertical surface exposed to liquid Example The lock gate ofa canal is rectangular 20 m wide and 10 m high One side is exposed to the atmosphere and the other side to the water What is the net force on the lock gate Here the pressure varies linearly with depth see also equation 24 Pgpgh Vertical rectangular wall horizontal wall width Vl0 Note the width cannot be shown on the picture Pete 314 217 Case 4 Vertical plane surface exposed to liquid For an infinitesimal area dA the normal force due to the pressure is dF p dA Find resultant force acting on a nite surface by integration 2 H FdeApghdApgthdhpgw h 0 For a vertical rectangular wall F 12 p g W H2 I E gmc Pete 314 2 18 Vertical surfaces General For surface of arbitrary shape we can write h dA F nghdAngA jhdA By de nition the centroid of the surface hC is hC A Therefore F p g hCA 25 The force exerted on a submerged plane surface is given by the product ofthe area and the pressure at the centroid The location ofthe centroid is known for several geometries imam ancnuim 6 Pete 314 219 Forces acting on the wall of an oil tank Cylindrical tan Tensile force Pressure force PAEFGH FDA Tensile force 2 cml a II b FIG DRE 2 lg A cylindrical atbottomed tank showing a horizontal slice used in part bl 1le force balance on half of the horizontal slice shaving the pliessure fame in the 139 direction and the tensile force in the two piooes of In tank g shell which resists this presstue force u mg 39 Pete 314 220 Needed wall thickness Example 29 Ifthe design tensile stress normally of the stress at rupture so safety factor is 4 of the tank wall is 20000 psi how thick must the shell ofthe tank be at the bottom ofthe tank The diameter ofthe tank is 120 ft the height is H 15 ft and SG ofoil is 0845 First calculate gauge pressure at the bottom P i D i A Ilatensile A it where rwmuc is the tensile stress in the shell P is the gauge pressure also assumed to be unlfotm and r is the thickness of the metal shell Now we make the thin walled assumption that rrmm c uniform over the wall thickness see below Solving for the required thickness of the shell we nd PD I V lCquotlll39lCl139lL Ell tlunwalled assumption L125 Ttensilc The tensile stress in Eq 235 is resisted by the external metal hoops in baITels and in wooden water tanks it is normally called the hoop Efrem Pete 314 221 Pressure vessels consider a constant working pressure Estimate the necessary wall thickness for a horizontal cylindrical pressure vessel with a diameter of 10 ft a working pressure of 250 psig and a design tensile stress of 20000 psi This is similar to Example 29 in which the pressure was due to gravity 250 inff in 10 ft 0625 ft 075 in mum 2 20000 lbe m Ham u ancmun c Pete 314 222 Buoyancy Laws of buoyancy discovered by Archimedes A body immersed in a uid experiences a vertical buoyant force equal to the weight ofthe uid it displaces A oating body displaces its own weight in the fluid in which it oats Free liquid surface 2 The upper surface of the body is subjected to a smallerforce than the lower surface A net force is acting 2 upwards Pete 314 223 Buoyancy The net force due to pressure in the vertical direction is FB F239 F1 Pbottom 39 Ptop AXAY 26 The pressure difference is Pbottom Ptop p g h239h1 P g H From 26 FB P g H AXAY Thus the buoyant force is where p the uid density Vo volume Pete 314 224 Example Consider a solid cube ofdimensions 1ft x1ft x1ft Its top surface is 10 it below the surface ofthe water The density of water is pf 623 Ibmft3 Considertwo cases a The cube is made of cork SG 016 b The cube is made of steel SG 785 In what direction does the body tend to move Pete 314 225 Pascal s principle The hydrostatic paradox From equation 23 the pressure at a point in a fluid depends only on density gravity and depth The pressure in a homogeneous incompressible uid at rest depends on the depth of the fluid relative to some reference plane and it is not in uenced by the size or shape ofthe tank or container Fluid level is the same in all containers l l Pressure is the same at the bottom ofall containers Pete 314 226 Measurement of Pressure Mechanical and electronic pressure measuring devices When a pressure acts on an elastic structure it will deform This deformation can be related to the magnitude of the pressure Bourdon pressure gage Pressure transducers convert pressure into an electrical output Straingage pressure transducers are suitable for rapid changes in pressure and cover big ranges of pressure values I E gmc Pete 314 2 27 Measurement of Pressure The atmospheric pressure can be measured with a barometer patm pvapor For mercury barometers atmospheric pressure 101325 kPa corresponds to h760 mmHg 292 in Hg B T Ghsslube ii we can neglect the vapor pressure for Hg Ifwater is used h 1033 m H20 34 ft FIGl39RIZ 21 Mercury barometer Pete 314 228 Measurement of Pressure Manometers are devices in which one or more columns of a liquid are used to determine the pressure difference between two points Utube manometer Inclinedtube manometer negligible pressure pl 7 px change in a gas since they are at px py the same height pl p2 Pa py pl pgh p1 prgh So a manometer measures gauge pressure Pete 314 229 Example 213 Manometer A I h u q The manometer is lled with colored water We want to know the gauge pressure at point D l V F l Water lbm ft lbf 52 ft2 lbf P P h 0 623 3227 3ft 129871298 s1 3 A pm g AB ft3 j 32 3221bmft 144in2 in2 p PD 2 PB pwalerghBC pgmghpc 2 1298 psi623 05ft01 lbm lbm ft3 t3 2 2 f 05ft322 M S ft 108psi 39322 lbmft 39 144 in2 Pete 314 230 Example Measurement of Pressure Differences Given 01 02 and h Find the pressure difference PAPB expressed with the given variables Pete 314 231 Bouran tuba 39 Emmet for N Flattened tube de ects dial gauge outwmd undel p1 assure k High pressure FIGURE 2 Reuniontube pressure gauge The whole assembly is in 1 shallow cylindrical containel The tube and linkage are at the back a sheet with numbers comes next the pointer is in from uf mm and a glass cover plate protects the whole assembly E mu s39 n a Pete 314 232 How does the coffee machine work Loose fitting cover prevents mixing but allowes pressure equilibration Coffee gm FICI RE ll Coffee percolator shaming uid ow driven by boiling lnt he riser tube the density becomes smaller when it starts to boil The level in the riser tube must increase to keep the pressure difference equal to ZEI39O unds When the level is high enough then a Ift mixture of steam and water will flow out the top of the tube he rate of generation of steam bubbles increases even more then the average density ofthe steamwater mixture in the riser tube will fall low enough Then the pressure force due to gravity will force water from the pot under the loosefitting cover and a continuous circulation will be established Pete 314 Pressure in accelerated rigidbody motions z y dP dzz EZ 8 Upordown X Example An open tank containing water 5 m deep is sitting on an elevator Calculate the gauge pressure at the bottom ofthe tank a when the elevator is standing still b when the elevator accelerates upward with 5 ms2 0 when the elevator accelerates downward with 5 ms2 Pete 314 234 A P For simple fluids at rest the pressuredepth relationship is given by the basic equation of fluid statics dPdz pg For constantdensity fluids the basic equation can be integrated to pgh This equation is an excellent approximation for liquids and a good approximation for gases when the change in elevation is small For changes in elevation measured in thousands of feet gases cannot be treated as constantdensity fluids Problems involving liquids with free surfaces are generally easiest to work in gauge pressure The force exerted by a static fluid on any infinitesimal surface is given by dFPdA The necessary wall thickness of pressure vessels and pipes for modest internal pressures are calculated using the thinwalled approximation The buoyant force exerted by a fluid on a floating or submerged body is equal to the weight of the fluid displaced Most pressuremeasuring devices either balance the pressure against the weight of a column of fluid in which case the height of the fluid column is the reading or let the pressure act on some area compressing a spring in which case the deflection of the spring is the reading PE LEI l mi l G Pete 314 1 Problems A large petroleum storage tank is 100 ft in diameter The free surface is really a very small part of a sphere with radius the radius of the earth 4000 mi If one drew an absolutely straight line from the liquid surface at one side ofthe tank to the liquid surface directly across the diameter on the other side how deep into the fluid would that line go In most uid mechanics problems we ignore the curvature of the earth Does this calculation support that simpli cation A new submarine can safely resist an external pressure of 1000 psig How deep in the ocean can it safely dive The tank in the Fig contains gasoline SG081 and water What is the absolute pressure at the bottom Sketch the curve of gauge pressure versus depth for this tank Gasoline Water Pete 314 4 01 0 l An oilstorage tank D 120 ft has a vent to the atmosphere to allow airto move in or out as the tank is lled or emptied This vent is plugged by snow in a blizzard while the oil is being pumped out of the tank and the gauge pressure in the tank falls to 1 psig What is the net force on the roof ofthe tank Hoover Dam is approximately 230 m high and 76 m wide at the top Consider it to be a rectangle only approximately true When the water is up to the top what is the pressure at the bottom What is the net force tending to move the dam We want to select a pipe with an inside diameter of 1 ft that will withstand an internal pressure of 1000 psig The steel to be used has a maximum allowable tensile stress of 40000 psi but to allow for a safety factor of4 we design for a maximum stress of 10000 psi How thick must the pipe walls be An oil well is 10000 ft deep The pressure ofthe oil at the bottom is equal to the pressure of a column of seawater 1 0000 ft deep This is typical of oil fields most ofthem at the time ofdiscovery have about the pressure of a hydrostatic column of seawater SG1 05 of equal depth there are exceptions The density of the oil is 55 lbm ft3 What is the gauge pressure ofthe oil at the wellhead at the surface iii39no LEu ENGINEER G Pete 8 The fluid shown shaded in the manometer of Fig 225 is ethyl iodide SG 193 The heights are h1 44 in Tankt llll and h2 8 in at 33 a What is the gauge pressure in the tank b What is the absolute pressure in the tank ii 1 lilfll39RIZ 225 S imp be n irmometer 9 Recalculate more precisely Problem 8 if you have the additional iformation gas specific gravity is 1 and temperature is normal Pete 314 F2 10 What is maximum force F2 that can be I l 610m diameter supported if F1 200 N Assume the two pistons have the same weight I Vertical F1 gt1 4 4cm diameter Oil S 085 L 22m gt 11 What is the pressure at the bottom of a water tank of 6 ft depth of an airplane currently falling free quotmomma mm Pete 314 239 12 A block of wood 36 096 is oating GM 56 3 between a layer ofgasoline and water a What fraction of the wood is below the interface Hint try not to use the actual numerical value Nd of the densities in one or other system of units 56 096 b Various woods have various specific gravities water A wood or other material ofwhat speci c gravity would oat 23 in the water and 13 in the gasoline Pete 314 240 13 A common scheme for measuring the Compiessed 39u39r 6ft Pressure gauge liquid depth in tanks is shown in the Fig Compressed air or nitrogen flows slowly through a dip tube into the liquid The gasflow rate is so low that the gas may be considered a static fluid The pressure gauge is 6 ft above the end of the dip tube The fluid density is 60 lbmft3 pgas0075 Ibmft3 quote nt 6in I a If the pressure gauge reads 2 psig and the dip tube is 6 in from the bottom of the tank what is the depth of the liquid in the tank b Customarily engineers read these gauges as if pgaswere zero How much error is made by such a simplification Note the book s result is not quite correct Pete 314