Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Aerodynamics wikipedia , lookup
Reynolds number wikipedia , lookup
Coandă effect wikipedia , lookup
Navier–Stokes equations wikipedia , lookup
Hydraulic machinery wikipedia , lookup
Derivation of the Navier–Stokes equations wikipedia , lookup
Bernoulli's principle wikipedia , lookup
Environmental Fluid Dynamics Instructor: Han Seung Kim Office: A 1411 TEL: 450-4092 E-mail: [email protected] Fluid Dynamics and Env. Eng. What is fluid? What is the fluid dynamics (mechanics)? Why is it an important subject in Environmental Engineering? Basic media (air, water, groundwater, sludges, organic solvents-NAPLs) Mass transport (물질전달, macro/micro transport-advection, dispersion, diffusion) Reaction engineering and reactor/system design (반응공학 및 반응조 설계) Dimensional analysis (차원해석) Water management – water resources and quality (수계관리 - 수량, 수질) Drinking water engineering (상수도공학 - 취수, 도수, 정수, 배수, 관망) Wastewater engineering (하수도공학 - 배수로, 하수처리장, 배출) Air quality management (대기관리 - 오염배출, 확산, 배출정화시설, 실내공기정화, 덕트) Soil and groundwater management (토양, 지하수 관리 및 정화) What is Fluids? Phase of materials in nature – Soilds, Liquids, and Gases “Fluids” Differences from the solids Solids – fixed distances btw. the component molecules → rigid body (lattice structure) and resistant to shear stress Fluids – flexible (varied) mol.-mol. distance and structure → no defined shape of body and continuously deformed by shear stress, and they can be equilibrated with shear stress only at motion (운동상태에서만 전단응 력에 대해 평형) Differences bwt. gases and liquids Gases – large and varied mol.-mol. distance, subject to compression/expansion → varied density (compressible fluids) Liquids – relatively constant mol.-mol. distance, almost no compression/expansion → no density changes (incompressible fluids) Fluids vs. Solids Fluids Solids Gas or liquid Solid A substance can deform A substance resists a shear continuously under the action stress in a static condition of a shear stress Irregular or relatively constant spacing btw. component molecules Fixed spacing btw. component molecules Free or weakly limited movement of molecules Restricted movement of molecules – lattice structured Various shapes depending on containers Own shapes Shear stress time rate of shear strain Shear stress shear strain Differences btw. Ideal and Non-ideal (real) fluids Real – resistance (shear stress) generated in real fluids due to their viscosity (viscous fluids, 점성 유체) Ideal – no viscous effects (inviscid fluid, 비점성 유체), incompressible (비 압축성), useful for the theoretical analysis of fluids Plastic materials (소성체) – properties of solids and fluids inherent (jellies, paint, polymeric solutions, paraffin, etc.) What is “Fluid Mechanics”? Science that describes physical actions and effects given by the forces applied to the fluids in motion or no motion. Classification of Fluid Mechanics Upon target fluids 1. 2. 3. Hydrodynamics (동수역학) – incompressible fluids Gas dynamics (기체동역학) – compressible fluids Aerodynamics (항공역학) – gases (air) flowing over aircrafts, rockets, etc. Upon forces applied 1. 2. 3. Fluid statics (유체정역학) – fluids in no motion, no shear stress but pressure Fluid kinematics (유체운동학) – fluid elements in motion Fluid dynamics (유체동역학) – fluids in motion Fluids as a Continuum Fluids composed of individual molecules can be regarded as a hypothetical homogenous continuum (hypothetically continuous substance) for their mathematical analysis. A way of describing the behavior of fluid in a given field of flow by considering the average effects of the molecules in a given volume. The number of molecules in the air at 1 atm and 0oC = ~ 107/mm3 Fluid element: Very small pieces of fluids that posses the characteristics of fluids, Not fluid molecules Rationale (e.g., gases) Very short mol.-mol. distance (molecular mean free path ~ 10-5cm) Time scale for the mol.-mol. collisions << one for the system on which fluids work (e.g., forces) force force time time Mass, force, weight Mass - a property of physical objects that measures the amount of matter they contain, the property of a body that causes it to have weight in a gravitational field. Constant anywhere in space. (g, kg, slug, lbm) Force – a physical property that gives the movement of a static object or changes in velocity or direction of an object in motion (F = ma, N, kgf, dyne, lbf) Weight - the vertical force exerted by a mass as a result of gravity (W = mg) Class quiz A substance of which mass is 10kg weighs 8.9kgf on a planet. What is the acceleration of gravity on this planet? (질량이 10kg인 물체를 저울로 달 았더니 8.9kgf이었다. 이곳의 중력가속도는?) A substance weighs 100kgf on the earth. What are the mass and weight of this substance on a planet of which acceleration of gravity is 1/5 of that of the earth? (지구상에서 100kgf인 물체를 중력가속도가 지구의 1/5인 위 성으로 가져가면 질량과 중량은 각각 얼마인가?) What is the weight of a pound mass (lbm) on the earth’s surface, where the acceleration due to gravity is 32.2 ft/s2, and on the moon’s surface, where the acceleration is 5.31 ft/s2? Basic fluid properties Mass density (중량밀도, 밀도, ) = m/vol. kg/m3, lbm(or slug)/ft3 Specific weight (단위중량, 비중량, g ) = W/vol. = g, N/m3, lbf/ft3 Specific volume (단위체적, 비체적, Vs) = 1/ , 1/ g (in weight unit, 중력단위계) Specific gravity (비중, S, dimensionless) = for liquids, fluid fluid water water for gases, gas gas M .W .gas air air M .W .air where, water (or water): (or ) of water at 4oC and air (or air): (or ) of air at 1atm and 0oC. Ideal gas law pV = nRuT where, Ru: universal gas constant (8.31 kJ/kmol-oK, 1545 ft-lbf/lbmol-oR) → In fluid mechanics, p= RT where, R: gas constant ( ~J/kg-K, ft-lbf/slug-oR) Density (mass) of gas, p RT Class quiz) Density of air at standard sea-level pressure (atmospheric pressure) and 0, 4, and 20oC? Specific gravity and density of Helium at 1 atm, 0oC? Specific gravity of mercury? What are the specific weight, specific volume, and density of carbon dioxide gas (CO 2) at 101.3 Kpa and 100oC? Specific heat (비열, c): , kJ/kg-oC 정적비열 (cv), 정압비열(cp), 비열비 (specific heat ratio, k) = cp / cv Specific internal energy (비내부 에너지, u): J/kg = f(temp., pressure) for real gas = f(temp.) for ideal gas Elasticity (탄성) – compressibility (압축성) Bulk modulus of elasticity (체적탄성계수, Ev) ~N/m2 = -dP/(dV/V) = dP/(d/) Class quiz) A liquid has a volume of 0.4m3 and 0.396m3 in a container pressurized at 1000kgf/cm2 and 2000kgf/cm2, respectively. What is the bulk modulus of elasticity (Ev) of this liquid? Viscosity (점성) Fluids in motion tend to internally resist to the relative motion generated between fluid layers when external shear stress is applied. This property is called “viscosity”. Newton’s law → shear stress btw. fluid layers () relative deformation btw. fluid layers (dV/dy, shear rate) dV dy : dynamic/absolute viscosity (동역학적/절대 점성계수) (= /): kinematic viscosity (동점성계수) Viscosity of gases vs. liquids Driving forces for viscosity – molecular cohesive force (분자응집력), molecular momentum exchange (분자 운동량 교환) In gasses – primarily controlled by molecular momentum exchange – viscosity with temperature In liquids – by molecular cohesive force – viscosity with temperature For gases – Sutherland equation 3 2 T T0 S 0 T0 T S where, 0: dynamic viscosity at T0 S: Sutherland constant (see Table A.2) For liquids = Ceb/T Where, C, b: empirical constants Class quiz) A board (1m 1m, 25N weight) slides down an inclined ramp (slope=20o) with a velocity of 2 cm/s. The board is separated from the ramp by a thin film of oil with a viscosity of 0.05Ns/m2. Calculate the spacing between the board and ramp. (Neglect edge effects) Newtonian vs. Non-Newtonian fluids Newtonian fluids – shear stress is linearly related to shear rate (water, air, lowmolecular liquid) Non-Newtonian fluids - shear stress is NOT linearly related to shear rate Bingham – ketchup, toothpaste Shear-thinning (psedoplastic) – polymeric solution, slurries, sludge, pulp solution Shear-thickening (dilatent) – resin, highly heated glass, asphalt Surface tension When a liquid is in contact with different phases (e.g., gas, solids), liquid molecules at the surface exert “tension” on adjacent surface due to their greater attraction btw. the molecules at the surface than those below the surface (cohesive force, 응집 력 > adhesive force, 부착력). → interfacial tension (surface tension, : liquid-gas contact) Defined as “tension force per unit length”, ( ) – of water at room temp.= 0.073 N/m 1/temp The surface tension is typically ignored in most cases, but it must be taken into account in very small scale flow (gas/liquid droplets present, small scale models, etc.) How to measure? – capillary rise technique, ring tensiometer (Du Nouy ring method), contact angle measurement method Capillary rise technique Capillary tube (d < 1 cm) Typically, =0o for water and clean glass Vertical component of the surface tension, F,z = d cos Weight of water risen W = (h)(d2/4) At equilibrium, F,z = W h = 4 cos / d Contact angle () determines wetting/non-wetting phases ~ f (cohesive, adhesive forces) <90o air >90o air mercury water glass adhesive btw. liquid/solid > cohesive btw. liquid molecules glass adhesive btw. liquid/solid < cohesive btw. liquid molecules Examples of surface tension Wt 2F 2L F L pA F L pA 2r pr 2 2r 2 pr 2 2 p r p F F ,inside F ,outside ( Dinside Doutside) 4 r Vapor pressure (Pv, 증기압) The pressure at which a liquid boils Pv temp Ex) water boils at 100oC (212oF) and 1 atm (14.7 psia) and also boils at 10oC (50oF) and 0.178 psia. Important for cavitation (공동현상) Fluid statics (유체정역학) Deals with the fluids in no motion. Forces applied on the fluids 1. Surface forces (표면력) – Pressure in vertical direction, viscous shear stress in tangential direction 2. Body forces (체적력) – External forces with no contact (e.g., gravity), = f(mass and volume of fluids) → In fluid statics, just consider pressure and gravity (no shear stress!) Pressure? – dF/dA Gravity? – W=V Pressure at a point in a static fluid acts with the same magnitude in all directions. Pn=Px=Py=Pz Pascal’ law A pressure change produced at one point in a closed system is transmitted throughout the entire system. (p1 = p2 = p3 = …. = pn) 예제 3.1) If a force of 100N were exerted on the handle of this hydraulic jack, what load, F2, can the jack support? Neglect lifter weight. Abs. vs. Gage pressure (절대, 계기 압력) “0” pressure (vacuum) is the absolute pressure. → atmospheric pressure at sea level = 101.3 kN/m2 (kPa abs.), 14.6 psia. Gage pressure = abs. pressure + atmospheric pressure Negative gage pressure → vacuum pressure Pressure variation with elevation dp dz Pressures are constant along a horizontal path, but vary along a vertical path (gravity direction). If no density change ( constant-incompressibie fluids), P + z = const. (piezometric pressure) P/ + z = const. (piezometric head) Class quiz) Ex 3.3) Ex 3.4) Compressible fluids (기체) Assumption – ideal gas = p/RT, = pg/RT →Pressure variation = f (z, temp) dp pg dz RT In troposphere (대류권) temp with elevation T = T0 - (z-z0) T ( z z0 ) p p0 0 T0 g R In stratosphere (성층권) constant temp with elevation p p0e ( z z0 ) g R Pressure measurement Pressure gage 1. Manometer (액주계) Bourden-tube gage Pressure measurement Differential manometer (시차액주계) p2 p1 i hi i hi down up p( p1 p2 ) ( m f )h Class quiz) A differential mercury manometer is connected to two pressure taps in an inclined pipe as shown. Water at 50oF is flowing through the pipe. The deflection of mercury in the manometer is 1 inch. Find the difference in piezometric pressure and piezometric head btw. the two points. Hydrostatic forces (정수력) Forces given by the hydrostatic pressure (정수압, 정압) applied on a submerged plate in a no-motion fluid Note) The first moment of area (단면 1차 모멘트) = ydA y A A The second moment of area (단면 2차 모멘트)= 2 y dA I y A 2 A How much is the magnitude of hydrostatic force? Fhydrostatic p A Where does the hydrostatic force act on a submerged plate? ycp y I yA Class quiz) The end of pipe is closed by an elliptical shape gate (54 m) and the gate is fixed by a hinge at its top. How much of normal force is required to open the gate? Neglect the weight of the gate. Class quiz) Find the magnitude of the hydrostatic force acting on one side of the submerged vertical plate given below and also find the location of the center of pressure. Hydrostatic forces on curved surfaces Integrating pressure force along the curved surface Easier way – use free-body diagram (자유물체도) and consider force equilibrium in vertical/horizontal directions Fh FAC Fv FBC W F Fh Fv 2 tan 1 Fv Fh 2 Class quiz) Find the magnitude and line of action of the hydrostatic force acting on surface AB (the thickness of the circular AB is 1 m). Buoyant force (부력) A resultant hydrostatic force that acts on the surfaces of a body submerged or floating in fluids Net vertical forces acting on a body a body submerged or floating in fluids 1) Submerged body FB = Fup - Fdown = (Vb+ Va) - Va = Vb = VD where, Va: vol. ABCEF, Vb: vol. of the body, VD: displaced volume (배수체적) 2) Floating body 0 FB = Fup - Fdown = VD (Fdown = Patoms. A = 0) Archimedes’ principle 1. A body submerged in a fluid receives a buoyant force as much as the weight of the displaced volume of the fluid. 2. A body floating in a fluid displaces a volume of the fluid corresponding to the weight of the submerged part of the body. Apparent weight (겉보기 무게) of a body in a fluid From the equilibrium of forces, T T + FB = W Apparent weight, T = W – FB W FB ( body fluid )Vbody Hydrometer (비중계) Density of a fluid of interest → buoyancy difference → Difference in hydrometer submerging (측정하고자 하는 유체의 밀 도 → 부력차 → 가라앉는 정도의 차이) Specific weight (단위 중량), specific gravity (비중) Class quiz) What are the volume and specific gravity of a body that weighs 60N and 11N in the air and water, respectively? (Neglect the mass of air)? A floating iceberg exposes 10% of its body (by volume) above the surface of sea water of which specific gravity is 1.03. What is the specific gravity of this iceberg? A hydrometer of which weight is 20g and diameter is 6mm submerges in a fluid by 6 cm more than when it has been used for water. Find the specific gravity of this fluid. Class quiz) The woodblock (505010 mm) has a specific gravity S1 = 0.3 and the volume of the metal part is 6600 mm3. Mass of the metal part and the tension T of the cord? Free-body diagram