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FLUID PROPERTIES Chapter 2 CE319F: Elementary Mechanics of Fluids 1 Fluid Properties • Define “characteristics” of a specific fluid •Properties expressed by basic “dimensions” – length, mass (or force), time, temperature • Dimensions quantified by basic “units” We will consider systems of units, important fluid properties (not all), and the dimensions associated with those properties. 2 Systeme International (SI) • Length = meters (m) • Mass = kilograms (kg) • Time = second (s) • Force = Newton (N) – Force required to accelerate 1 kg @ 1 m/s2 – Acceleration due to gravity (g) = 9.81 m/s2 – Weight of 1 kg at earth’s surface = W = mg = 1 kg (9.81 m/s2) = 9.81 kg-m/s2 = 9.81 N • Temperature = Kelvin (oK) – 273.15 oK = freezing point of water – oK = 273.15 + oC 3 Système International (SI) • Work and energy = Joule (J) J = N*m = kg-m/s2 * m = kg-m2/s2 • Power = watt (W) = J/s • SI prefixes: G = giga = 109 c = centi = 10-2 M = mega = 106 m = milli = 10-3 k = kilo = 103 m = micro = 10-6 4 English (American) System • Length = foot (ft) = 0.3048 m • Mass = slug or lbm (1 slug = 32.2 lbm = 14.59 kg) • Time = second (s) • Force = pound-force (lbf) – Force required to accelerate 1 slug @ 1 ft/s2 • Temperature = (oF or oR) – oRankine = oR = 460 + oF • Work or energy = ft-lbf Banana Slug • Power = ft-lbf/s Mascot of UC Santa Cruz – 1 horsepower = 1 hp = 550 ft-lbf/s = 746 W 5 Density • Mass per unit volume (e.g., @ 20 oC, 1 atm) – Water rwater = 1,000 kg/m3 (62.4 lbm/ft3) – Mercury rHg = 13,500 kg/m3 – Air rair = 1.205 kg/m3 • Densities of gases = strong f (T,p) = compressible • Densities of liquids are nearly constant (incompressible) for constant temperature • Specific volume = 1/density = volume/mass 6 Example: Textbook Problem 2.8 • Estimate the mass of 1 mi3 of air in slugs and kgs. Assume rair = 0.00237 slugs/ft3, the value at sea level for standard conditions 7 Example • A 5-L bottle of carbon tetrachloride is accidentally spilled onto a laboratory floor. What is the mass of carbon tetrachloride that was spilled in lbm? 8 Specific Weight • Weight per unit volume (e.g., @ 20 oC, 1 atm) gwater = (998 kg/m3)(9.807 m2/s) = 9,790 N/m3 [= 62.4 lbf/ft3] gair = (1.205 kg/m3)(9.807 m2/s) = 11.8 N/m3 [= 0.0752 lbf/ft3] 9 Specific Gravity • Ratio of fluid density to density of water @ 4oC Water SGwater = 1 Mercury SGHg = 13.55 Note: SG is dimensionless and independent of system of units 10 Example • The specific gravity of a fresh gasoline is 0.80. If the gasoline fills an 8 m3 tank on a transport truck, what is the weight of the gasoline in the tank? 11 Ideal Gas Law (equation of state) P = absolute (actual) pressure (Pa = N/m2) V = volume (m3) n = # moles Ru = universal gas constant = 8.31 J/oK-mol T = temperature (oK) R = gas-specific constant R(air) = 287 J/kg-oK (show) 12 Example • Calculate the volume occupied by 1 mol of any ideal gas at a pressure of 1 atm (101,000 Pa) and temperature of 20 oC. 13 Example • The molecular weight of air is approximately 29 g/mol. Use this information to calculate the density of air near the earth’s surface (pressure = 1 atm = 101,000 Pa) at 20 oC. 14 Example: Textbook Problem 2.4 • Given: Natural gas stored in a spherical tank – Time 1: T1=10oC, p1=100 kPa – Time 2: T2=10oC, p2=200 kPa • Find: Ratio of mass at time 2 to that at time 1 • Note: Ideal gas law (p is absolute pressure) 15 Viscosity 16 Some Simple Flows • Flow between a fixed and a moving plate Fluid in contact with plate has same velocity as plate (no slip condition) u = x-direction component of velocity y Moving plate u=V V B Fluid x Fixed plate u=0 17 Some Simple Flows • Flow through a long, straight pipe Fluid in contact with pipe wall has same velocity as wall (no slip condition) u = x-direction component of velocity r R x V Fluid 18 Fluid Deformation • Flow between a fixed and a moving plate • Force causes plate to move with velocity V and the fluid deforms continuously. y Moving plate u=V t0 t1 t2 Fluid x Fixed plate u=0 19 Fluid Deformation For viscous fluid, shear stress is proportional to deformation rate of the fluid (rate of strain) y dL Moving plate u=V+dV t da t+dt dy dx Fluid x Fixed plate u=V 20 Viscosity • Proportionality constant = dynamic (absolute) viscosity • Newton’s Law of Viscosity V+d • Viscosity v V • Units • Water (@ 20oC): m = 1x10-3 N-s/m2 • Air (@ 20oC): m = 1.8x10-5 N-s/m2 Kinematic viscosity: m2/s • Kinematic viscosity 1 poise = 0.1 N-s/m2 1 centipoise = 10-2 poise = 10-3 N-s/m2 21 Shear in Different Fluids • Shear-stress relations for different fluids • Newtonian fluids: linear relationship • Slope of line = coefficient of proportionality) = “viscosity” Shear thinning fluids (ex): toothpaste, architectural coatings; Shear thickening fluids = water w/ a lot of particles, e.g., sewage sludge; Bingham fluid = like solid at small shear, then liquid at greater shear, e.g., flexible plastics 22 Effect of Temperature Gases: greater T = greater interaction between molecules = greater viscosity. Liquids: greater T = lower cohesive forces between molecules = viscosity down. 23 24 Typical Viscosity Equations T = Kelvin S = Sutherland’s constant Gas: Air = 111 oK +/- 2% for T = 170 – 1900 oK C and b = empirical constants Liquid: 25 Flow between 2 plates Force is same on top and bottom Thus, slope of velocity profile is constant and velocity profile is a st. line y Moving plate u=V V Fluid Force acting B ON the plate x Fixed plate u=0 26 Flow between 2 plates Shear stress anywhere between plates y Moving plate u=V V t Shear B on fluid t x Fixed plate u=0 27 Flow between 2 plates • 2 different coordinate systems r B x V y x 28 Example: Textbook Problem 2.33 Suppose that glycerin is flowing (T = 20 oC) and that the pressure gradient dp/dx = -1.6 kN/m3. What are the velocity and shear stress at a distance of 12 mm from the wall if the space B between the walls is 5.0 cm? What are the shear stress and velocity at the wall? The velocity distribution for viscous flow between stationary plates is 29 30 Example: Textbook Problem 2.34 A laminar flow occurs between two horizontal parallel plates under a pressure gradient dp/ds (p decreases in the positive s direction). The upper plate moves left (negative) at velocity ut. The expression for local velocity is shown below. Is the magnitude of the shear stress greater at the moving plate (y = H) of at the stationary plate (y = 0)? 31 32 Elasticity (Compressibility) • If pressure acting on mass of fluid increases: fluid contracts • If pressure acting on mass of fluid decreases: fluid expands • Elasticity relates to amount of deformation for a given change in pressure Ev = bulk modulus of elasticity How does second part of equation come about? Small dV/V = large modulus of elasticity 33 Example: Textbook Problem 2.45 • Given: Pressure of 2 MPa is applied to a mass of water that initially filled 1000-cm3 (1 liter) volume. • • Find: Volume after the pressure is applied. • Ev = 2.2x109 Pa (Table A.5) 34 Example • Based on the definition of Ev and the equation of state, derive an equation for the modulus of elasticity of an ideal gas. 35 Surface Tension • Below surface, forces act equal in all directions Interface air • At surface, some forces are missing, pulls molecules down and together, like membrane exerting tension on the surface water Net force inward • Pressure increase is balanced by surface tension, s No net force • surface tension = magnitude of tension/length • s = 0.073 N/m (water @ 20oC) 36 Surface Tension • Liquids have cohesion and adhesion, both involving molecular interactions – Cohesion: enables liquid to resist tensile stress – Adhesion: enables liquid to adhere to other bodies • Capillarity = property of exerting forces on fluids by fine tubes or porous media – due to cohesion and adhesion – If adhesion > cohesion, liquid wets solid surfaces at rises – If adhesion < cohesion, liquid surface depresses at pt of contact – water rises in glass tube (angle = 0o) – mercury depresses in glass tube (angle = 130-140o) • See attached information 37 Example: Capillary Rise • Given: Water @ 20oC, d = 1.6 mm • Find: Height of water W 38 Example: Textbook Problem 2.51 q Find: Maximum capillary rise between two vertical s s glass plates 1 mm apart. h t 39 Examples of Surface Tension 40 Example: Textbook Problem 2.48 Given: Spherical soap bubble, inside radius r, film thickness t, and surface tension s. Find: Formula for pressure in the bubble relative to that outside. Pressure for a bubble with a 4-mm radius? Should be soap bubble 41 Vapor Pressure (Pvp) • Vapor pressure of a pure liquid = equilibrium partial pressure of the gas molecules of that species above a flat surface of the pure liquid – Concept on board – Very strong function of temperature (Pvp up as T up) – Very important parameter of liquids (highly variable – see attached page) • When vapor pressure exceeds total air pressure applied at surface, the liquid will boil. • Pressure at which a liquid will boil for a given temperature – At 10 oC, vapor pressure of water = 0.012 atm = 1200 Pa – If reduce pressure to this value can get boiling of water (can lead to “cavitation”) • If Pvp > 1 atm compound = gas • If Pvp < 1 atm compound = liquid or solid 42 Example • The vapor pressure of naphthalene at 25 oC is 10.6 Pa. What is the corresponding mass concentration of naphthalene in mg/m3? (Hint: you can treat naphthalene vapor as an ideal gas). 43 Vapor Pressure (Pvp) - continued Vapor pressure of water (and other liquids) is a strong function of temperature. 44 Vapor Pressure (Pvp) - continued Pvp,H2O = Pexp(13.3185a – 1.9760a2 – 0.6445a3 – 0.1299a4) P = 101,325 Pa a = 1 – (373.15/T) T = oK valid to +/- 0.1% accuracy for T in range of -50 to 140 oC Equation for relative humidity of air = percentage to which air is “saturated” with water vapor. What is affect of RH on drying of building materials, and why? Implications? 45 Example: Relative Humidity The relative humidity of air in a room is 80% at 25 oC. (a) What is the concentration of water vapor in air on a volume percent basis? (b) If the air contacts a cold surface, water may condense (see effects on attached page). What temperature is required to cause water condensation? 46 47 Saturation Vapor Pressure 48 49

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