1.3 Analysis of Fluid Behavior

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1 1.3 Analysis of Fluid Behavior Fluid Statics : When the fluid is at rest. Fluid Dynamics : When the fluid is moving. Governing equations : Conservation of Conservation of Conservation of mass momentum (Newtons second law) energy (First law of themodynamics)

2 1.4 Measures of Fluid Mass and Weight Density, : Mass of a fluid per unit volume [slug/ft3, kg/m3] lim VV m V For water at 5 o C, water = slugs/ft 3 = 1000 kg/m 3 For air at standard pressure and at 20 o C, air = slugs/ft 3 = kg/m 3

3 Specific volume, : Volume per unit mass. 1

4 Specific weight, : Weight per unit volume g [lb/ft 3, N/m 3 ] For water at 5 o C, water = 62.4 lb/ft 3 = 9.8 kn/m 3 For air at standard pressure and at 20 o C, air = lb/ft 3 = N/m 3

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6 Specific gravity, SG : The ratio of the density of the given fluid to the density of water at some specified temperature, usually at 4 o C(39.2 o F). SG H 2 O at 4 o C 1.94 slugs / ft 3, or1000 kg / m 3 Specific gravity of gases is usually based on dry air as the reference fluid.

7 Pressure : the normal compressive force per unit area acting on a real or imaginary surface in the fluid. p lim AA F n A Microscopically, pressure represents molecular momentum and intermolecular forces within the fluid. Gage pressure: Pressure measured relative to local atmosphere pressure Absolute pressure: Pressure measured (or defined) relative to zero pressure

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9 Absolute pressure = Gage pressure + Atmosphere pressure in vicinity of gage Vacuum pressure : pressure below local atmosphere pressure Vacuum pressure = Atmosphere pressure Absolute pressure = - Gage pressure The subscripts g and a indicates whether the pressure is gage or absolute. (ex. 10 psig = 10 pounds per square inch, gage ; 10 psia = 10 pounds per square inch, absolute)

10 Standard value of atmospheric pressure is kpa ( psia, in. Hg, abs) [Pascal ; Pa=N/m2]

11 Temperature : defined as a measure of (not equal to) the energy contained in the molecular motion of the fluid T (Rankine) = T (Fahrenheit) T (Kelvin) = T (Celsius) T (Rankine) = 1.80 T (Kelvin) Internal energy (U) : Energy contained in random molecular motions and intermolecular forces, U = U(T) Specific internal energy ( u ) : Internal energy per unit mass

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13 Specific heat at constant volume, Specific heat at cont. pressure, For an incompressible fluid, all processes are constant specific volume and So c p = c v ( for incompressible fluid) v v T u c p p T h c pv u p u h ; enthalpy specific 0 T v p T pv p p

14 1.5 Ideal (Perfect) Gas Law (=Equation of State for an ideal gas) p RT RT v R is the specific gas constant and is equal to the universal gas constant (R 0 ) devided by the molecular weight (MW) of the gas : R R 0 MW 8314 MW N.m / kg.k 1545 MW ft.lb / lbm. O R

15 Liquids exhibit slight variation of density with temperature and pressure. No simple, exact equations are available for properties of liquids. For most practical purposes, liquids are treated as incompressible fluids.

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18 1.6 Viscosity

19 Newton s law of viscosity du dy where the constant of proportionality,, is called the absolute viscosity, dynamic viscosity, and simply viscosity of the fluid. Dimension = [lb.s/ft2], [N.s/m2]

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22 viscosity [Pa s] liquid 77K acetone* methanol* benzene* water ethanol* mercury* nitrobenzene* propanol* Ethylene glycol sulfuric acid* olive oil glycerol* castor oil* corn syrup* HFO viscosity [cp] viscosity [cp] honey 2,000 10,000 molasses 5,000 10,000 molten glass 10,000 1,000,000 chocolate syrup 10,000 25,000 molten chocolate * 45, ,000 [19] ketchup * 50, ,000 peanut butter ~250,000 shortening * ~250,000 Viscosity of Liquids at 25 o C pitch

23 Sutherland equation : (for gases) T S where C and S are empirical constants and T is absolute temperature. De CT 3 / 2 B / T Andrade s equation : (for liquids) where D and B are constants. T is absolute temperature. Kinematic viscosity : [ft2/s, m2/s] In CGS (centimeter-gram-second) unit, the dynamic viscosity has the unit of dyne.s/cm 2 (=poise, abbreviated as P). The kinematic viscosity has the unit of cm 2 /s (=stoke, St) ** 1 dyne = (1g) x (1cm/s 2 )

24 Newtonian and Non-Newtonian Fluid Newtonian fluid : Fluids that obey the Newton s law of viscosity. (The shearing stress is linearly related to the rate of shearing strain) Most common fluids, both liquids and gases, are Newtonian. Non-Newtonian fluid : Fluids for which the shearing stress is not linearly related to the rate of shearing strain

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26 Shear-thinning fluid The coefficient of resistance decreases with increasing strain rate. Ex. Ketchup (It all comes out of the bottle at once) Colloidal suspensions Polymer solutions, Latex paint (It does not drip from the brush because the shear rate is small and the shear stress is large. However, it flows smoothly onto the wall because the thin layer of paint between the wall and brush causes a large shear rate (large du/dy) and a small shear stress.)

27 Shear-thickening fluid Fluids having the characteristics that the shear stress increases with increasing the shear strain. The harder the fluid is sheared, the more viscous it becomes. Ex. Water-corn starch mixture Water-sand mixture (quicksand): The difficulty in removing an object from quicksand increases dramatically as the speed of removal increases.

28 Bingham plastic This is neither a fluid nor a solid. This material can withstand a finite shear stress without motion ( hence, not a fluid), but once the yield stress is exceeded it flows like a fluid (i.e., not a solid). Ex. Toothpaste, Mayonnaise

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31 No-slip condition : Whenever a fluid is in contact with a solid surface, the velocity of the fluid at the surface is equal to the velocity of the surface; that is, the fluid sticks to the surface and does not slip relative to it. This condition is true regardless of the type of the fluid, type of surface, or surface roughness, so long as the continuum hypothesis is valid. Inviscid fluid : the fluid with zero viscosity, i.e., = 0. Consequently, = 0. The assumption of an inviscid fluid is often useful for analyzing flow remote from the solid boundaries. Ideal fluid : = 0 and = constant(incompressible)

32 1.7 Compressibility of Fluids Bulk modulus, E v : measure of the compressibility of fluid E v dp dv V [psi, Pa] Large values of the bulk modulus indicate that the fluid is relatively incompressible-that is, it takes a large pressure change to create a small change in volume. Common liquids have large value of E v, For example, at atmospheric pressure and a temperature of 60 o F it would require a pressure of 3120 psi to compress a unit volume of water 1%. i.e., E v =3.12x10 5 psi (=2.15x10 9 Pa) for water. For most practical engineering problems, we consider the liquids are incompressible. m V dp d

33 p For isothermal process, RT =constant, E v dp d / dprtd RTd d / RT p p For isentropic process, constant k E dp d / v k 1 dp(const)k d (const)k d / k1 d (const)k k k p k k kp where k c p c v (for air k=1.4) and R=c p -c v

34 For air under standard atmospheric conditions with p=14.7 psi and k=1.4, the isentropic bulk modulus is 20.6 psi. Comparing this value with that of water (E v,water=312,000 psi), the air is approximately 15,000 times as compressible as water.

35 Speed of Sound Speed of sound : defined as c dp d Since the disturbance is small, there is negligible heat transfer and the process is assumed to be isentropic. c dp d Ev for gasundergoing isentropic process kp ideal gas krt Thus, for ideal gases the speed of sound is proportional to the square root of the absolute temperature.

36 For example, for air at 60 o F with k=1.4 and R=1716 ft.lb/slug. o R, c=1117 ft/s(340 m/s). For water at 20 o C, E v =2.19 gn/m 2 and =998.2 kg/m 3 so that c=1481 m/s or 4860 ft/s. The speed of sound in water is much higher than in air. If a fluid is truly incompressible (E v =), the speed of sound would be infinite.

37 Speed of sound as a function of depth at north Hawaii

38 Sound during the Day Sound in the Evening Cold Warm Warm Sound during the Day Cold Sound in the Evening

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42 Mach Number Mach number, Ma : defined as Ma= V c Subsonic flow regime : Ma < 1.0 Sonic flow : Ma = 1.0 Supersonic flow regime : Ma > 1.0 Transonic flow regime : 0.7~0.8 < Ma <1.2~1.5 (depends on the configuration of flying object)

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47 1.8 Vapor Pressure Evaporation takes place because some liquid molecules at the surface have sufficient momentum to overcome the intermolecular cohesive forces and escape into the atmosphere. When the saturation is reached, the pressure that exerts on the liquid surface is termed the Vapor Pressure.

48 Since the development of a vapor pressure is closely associated with molecular activity, the value of vapor pressure for a particular liquid depends on temperature (because the molecular activity (internal energy) depends on temperature). Generally, as the temperature increases, the vapor pressure of a fluid also increases.

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50 Boiling is initiated when the absolute pressure in the fluid reached the vapor pressure. Water at standard atmospheric pressure will boil when the temperature reaches 212 o F (100 o C)-that is, the vapor pressure of water at 212 o F is 14.7 psi abs. However, if at a higher elevation, say 10,000 ft above sea level, where the atmospheric pressure is 10.1 psi abs, the boiling will start at about 193 o F. At this temperature the vapor pressure is 10.1 psi abs. Thus, boiling occurs at a given pressure acting on the fluid by raising the temperature, or at a given fluid temperature by lowering the pressure. Cavitation phenomena in the pump, valve, marine propeller, etc.

51 (mmhg) 760 mmhg 14.7 psia

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55 Cavitation

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57 Erosion by Cavitation Bubble

58 Erosion by Cavitation Bubble

59 Supercavitating Torpedo

60 VA-111 Shkval Torpedo Length: 8.2 m (27 feet) Diameter: 533 mm Weight: 2700 kg (5940 pounds) Warhead weight: 210 kg Speed Launch Speed: 50 kt (93 km/h) Maximum Speed: 200+ kt (370 km/h) Range: Around 7000 m to m (New version)

61 Research is on going by PNU CFD lab.

62 German Barracuda Western countries are not far behind though, with Germany currently developing the "Barracuda", which is guided and has been offically stated as being capable of 360Km/h, but has been rumoured to travel at up to 800km/h. It looks like the Russians have been them to the punch again though, with the Shkval-II already deployed and rumoured of being cable of at least 720km/h whilst also being guided.

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64 Iran

65 Underwater Express DARPA (Defense Advanced Research Projects Agency) / ATO (Advanced Technology Office) Period : April 2006 ~ August 2009 Technology development (Model scale=1/4~1/2) and demonstration program Demonstrate stable and controllable high-speed underwater transport through supercavitation for future littoral missions Speed ~100 knots Size : 8 ft diameter, 60 tones for super-fast submerged transport (SST) - comparable in size to current special purpose craft such as the MK V Special Operations Craft and the Advanced Seal Delivery Vehicle - Mark V: 82 feet long aluminum monohull surface craft, 40 knots

66 Research is on going by PNU CFD lab.

67 RAMICS (RAPID AIRBORNE MINE CLEARANCE SYSTEM)

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69 AHSUM (Adaptable High-Speed Munitions)

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71 1.9 Surface Tension The surface tension is due to the unbalanced cohesive forces acting on the liquid molecules at the fluid surface. Molecules in the interior of the fluid mass are surrounded by the molecules that are attracted to each other equally. However, molecules along the surface are subjected to a net force toward the interior. This unbalanced force along the surface creates the membrane. The tensile force along the surface is called the surface tension. Unit = [lf/ft], [N/m]

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75 2R pr 2 p p i p e 2 R

76 Capillary : In fig (a), the attractive(adhesive) force between the wall of the tube and liquid molecules is strong enough to overcome the mutual attractive (cohesive) force of the molecules. The liquid is said to wet the solid surface. R 2 h 2R cos h 2 cos h

77 Note that the height in a capillary tube is inversely proportional to the tube radius, and thus the rise of a liquid becomes increasingly pronounced as the tube radius is decreased. If adhesion of molecules to the solid surface is weak compared to the cohesion between molecules, the liquid will not wet the surface and the level in a tube placed in a nonwetting liquid will be depressed as shown in fig.1.8(c). Mercury is nonwetting liquid when it is contact with the glass, 130 o.

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79 Surface Tension

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81 Application: - Detergent - Washing in hot water

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