Day 3 Fluid Statics - ressure - forces we define fluid article: small body of fluid with finite mass but negligible dimension (note: continuum mechanics must aly, so not too small) we consider a fluid in hydrostatic equilibrium: each fluid article is in a force equilibrium condition: the net force due to ressure balances the weight of the fluid article
ressure is defined as the ratio between normal force to area at a oint = lim A F normal A It is a local quantity =(osition), e.g. = () ressure is a scalar field has a magnitude, but not a direction (as a vector, e.g. the fluid velocity) we defined ressure using a force comonent normal to the surface Let us consider a fluid element of volume dv area da and height d in hydrostatic equilibrium > (+d)da=(+d)da water level +d Weight = ρgdv = ρgda d = γda d ()da reference level
> water level +d (+d)da=(+d)da Weight = ρgdv = ρgda d = γda d ()da reference level Hydrostatic equilibrium: The sum of the forces along should be = da γdad + d da = da γdad da dda = γd d = d d = γ
Within a single fluid the ieometric or static ressure head is constant d d d d ref (reference level or datum) ref ( ref ref 2 ref const. const. Hence between two oints in The same static fluid sharing a common free surface 2 ) surf Free-surface the interface between a liquid and a gas marked 2 or 2 surf ref = reference
Let us now change the reference system and set it to the surface ref = reference.. let ussubstitute ) ( d (reference level or datum) const const d d d ref ref ieometric head ieometric ressure We can thus define h imose that The hydrostatic equation at oint At the surface h h h h Note that these definitions do not deend on the reference level -
d ref = reference At the surface h at oint h The hydrostatic equation imose that h h const. d d ref ref d d Can we change the orientation of and have a simler coordinate system? yes! we can define a deth d oriented oosite to! (reference level or datum) d d ; d d d d d d d d
Let us now consider a simle case : H surf deth = H- ref = reference h at the surface the ieometric head h H H if we imose At any oint we have that h Since the ieometric head is constant we can write that H therefore at any oint ( H ) deth in the fluid defined by The ieometric head coincides with the height of the water if we neglect and if we take the reference value at the bottom of the tank. In any case the ieometric deends on and thus on the reference level
In a single liquid (e.g. water) ressure increases linearly with deth At 2 C The weight of cubic meter of water is 9.79 kn = density(kg/m 3 )*Volume*g H If this volume is held in a tank with square cross sectionm 2 The ressure at the bottom of the tank will be o F / A o 9.79kN /m 2 o H [m]
mercury Basic ressure facts In fluids, ressure always acts normal to a solid surface. d d d If two oints on a common horiontal line are connected by a single liquid in hydrostatic conditions they have an equal ressure?? water 2 NOT= 5 what is the roblem here? we do not know what haens above 3 = 4
Basic ressure facts Pressure is a stress, i.e. Force er unit area, Units: N/m 2 = Pa (Pascal) In a single liquid (e.g. water) ressure increases linearly with deth atmosheric ressure results from the weight of a column of air on a unit surface atm =.3 kpa = 4.7 si = 76 mmhg =29.92 inches Hg = 26 sf (ound er square foot); note that the SI unit is very small ressure 5 Pa=atm high atm usually ressure readings are differential: we measure the ressure in a fluid relatively to a reference ressure as the atm. ressure (gage ressure as oosed to absolute ressure) high ressure in a balloon imlies that the working fluid balances the difference in ressure with its weight
Let us consider two oints A and B: each one will be described by a gage or an absolute ressure siabsolute sigage Point A Point B Vacuum ressure = difference between ATM ressure and absolute ressure e.g. absolute ressure =5 kpa < ATM if the gage ressure is negative = -5kPA, the Vacuum ressure is ositive = +5kPA ABS = ATM + GAGE VACUUM = ATM - ABS VACUUM = - GAGE
Liquid gases differences Let us consider the ressure in a gas and in a liquid m above and below the interface GASES The change of ressure is more sensitive to changes in temerature as comared to changes in deth, since secific weights are order of magnitude smaller. air.8( Pa) 979( Pa) o =- water = Unit cross-section m 2 T= 2 o C + = m - = -m In most engineering systems where changes in height are on the order of meters we often assume that the gas ressure does not change with vertical osition CAVEAT: At the ground however, the Earth s atmoshere is ~3 kms dee and it does exert a significant ressure At sea level kpa Roughly equivalent to m of water * 9.8kN 98.kPa BLACKBOARD 3C
Be careful with layered fluids or functionally graded fluids Sea water (salinity could change with deth) h h 2 oil water 2 Piecewise Linear change With deth sea water 2 h a = h b = h c h but not equal to h d = h e = h f water oil 9.8kN / m 8.6 kn / m 3 3 Smooth change with deth In these systems 2 2 Hence 2 hoil h2 BLACKBOARD 3D water BUT this equation does hold (valid in each fluid) d (), d () o Note deendence of
Measuring ressure: )barometer reference The barometer measures atmosheric ressure: the column of mercury rises until the weight of the mercury balances the ressure exerted by the air column (atm. ressure); It is imortant that: i) nearly vacuum is created on the to column ii) caillary (surface tension) forces are negligible iii) vaor ressure is small
2) Pieometer: measures the average ressure in the cross section of the ie; reference Note that the fluid moves because there is a (mean)ressure gradient along the ie: if d/dx<, it means that ressure on the left in larger than ressure on the right... hence it flows x
U Manometer: measure ressure in a ie using a manometer liquid with different roerties as comared to the liquid in the ie reference 3= 2 3 = 4+γl 2 = γ m Δh 4= 3 - γl = γ m Δh - γl Note that strictly I cannot yet aly the hydrostatic equation in the fluid in motion (even if it is the same fluid) that s why we talk about static ressure
Differential manometer Find h -h 2? (fluid in motion, so h h2) let us recall: h =/ γ + h -h 2 =( - 2 ) / γa + - 2 i ii reference (i)=(ii); (i)=+( Δh+ Δy) γa (ii)=2+ ( Δ+ Δy) γa + Δh γb So, +( Δh+ Δy) γa =2+ ( 2 - + Δy) γa + Δh γb -2= ( 2 - + Δy) γa + Δh γb- ( Δh+ Δy) γa = ( 2 - )γa+ Δh γb - Δh γa fig_3-4 h -h 2 =( - 2 ) / γa +( - 2 ) = 2 - +( Δh γb / γa) - Δh + - 2 = Δh(γB / γa-) 22 John Wiley & Sons, Inc. All rights reserved.
Vaor Pressure articles tend to escae from the liquid in the form of vaor (why do we smell gas when we fill the tank of our car? What haens when we seal the tank...) Vaor ressure or equilibrium vaor ressure is the ressure exerted by a vaor in thermodynamic equilibrium with its condensed hases (liquid) at a given temerature in a closed system. The vaor refers to a gas hase at a temerature where the same substance can also exist in the liquid or solid (exactly as the fuel) A substance with a high vaor ressure at normal temeratures is often referred to as volatile.
Vaor ressure (cont d) At a liquid / gas interface the liquid continuously vaories and condenses (thermodyn. equil balance). gas If the ressure in the gas above the liquid is higher than the vaor ressure, the gas is in a steady state (no more vaoriation). liquid If the ressure above the liquid is less than the vaor ressure, more vaoriation than condensation occurs and the liquid boils. If the liquid is in a closed volume subjected to a vacuum, enough liquid vaories to form a gas hase above the liquid that exerts a vaor ressure on the liquid In water at 2 C, the water vaor ressure is 234 Pa the higher the temerature, the higher the vaor ressure (more gas can be stored in a fixed volume)
vacuum?! Water barometer The main forces that determine h involve ressure and the fluid column weight (d not small, no caillarity effects) At normal atmosheric conditions, the height of the water is about m Generally, the calibration assumes that the sace in the tube above the water is a vacuum, i.e., P =. Is this true??
Water barometer and vaor ressure At a liquid / gas interface (as in the to of the barometer) the liquid continuously vaories and condenses. i.e., there is a gas hase in the tube sace. This gas hase exerts a vaor ressure, that acts downward In water at 2 C, the water vaor ressure is 234 Pa Note: a liquid gas interface in equilibrium suggest that the ressure of the gas is the vaor ressure. Note also that 234Pa is not a high ressure as comared to atm=5pa
Analogy: boiling temerature Boiling temerature = temerature at which a liquid boils Below the boiling temerature, the fluid is liquid Boiling temerature is deendent on ressure Vaor ressure Vaor ressure = ressure at which a liquid vaories Above the vaor ressure, the fluid stays liquid Vaor ressure is deendent on temerature At sea level when the water reaches the vaor ressure = atmosheric ressure = Ka In Denver the atmosheric ressure is only 95% of the above On Everest only 75%--the boiling oints are reduced accordingly Water boils at a lower temerature asta does not cook well in the mountains
Vaor ressure and engineering design - cavitation Boiling associated with low ressures often occurs in localied low-ressure ones of flowing liquids, e.g., on the suction side of a um. When this occurs, vaor bubbles start growing in local regions of very low ressure (associated with local defects) They collase in regions of higher ressure downstream. This henomenon, which is called cavitation, can cause extensive damage to fluids systems