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EF 152 Lec 4-5 States of Matter Density Gas Solid In a gas, the molecules are far apart and the forces between them are very small In a solid, the molecules are very close together, and the form of the solid depends on the details of the forces between them; that form is often a lattice. Solids resist changes in shape. Liquid In a liquid, the molecules are also close together and resist changes in density, but not in shape. Pressure is force per unit area: F P= A The pressure in a static fluid at any point must be the same in all directions. Pressure increases with depth, due to the force of gravity: P1 = Po + ρgh ρ= m V Varies slightly with temperature and pressure. Specific gravity of a substance is the density of that substance divided by the density of water at 4.0oC. SG = ρ ρWater Fluids (gasses and liquids), cannot support a tensile or shear load. Pressure Density is mass per unit volume: Buoyancy and Archimedes’ Principle Assume block is in equilibrium. Then upward forces must equal downward forces. Upward force: pressure from fluid Downward force: atmospheric pressure plus weight Therefore Fup = Fdown P2 − P1 = − ρg ( y2 − y1 ) 1 EF 152 Lec 4-5 Buoyancy and Archimedes’ Principle Archimedes’ Principle: The buoyant force on an immersed object equals the weight of displaced fluid. FBouyant = ρ w gVsubmerged Hydrodynamics – Fluids in Motion Assumptions Continuity Equation Conservation of Mass For a given time t. the amount of “stuff” (mass) that passes by someone at 1 must equal the amount of “stuff” mass that masses by someone at two. m1 = m2 ρA1Δl1 = ρA2 Δl2 ρA1v1t = ρA2 v2t ρA1v1 = ρA2 v2 v1 A1 = v2 A2 The fluid is incompressible. The temperature does not vary. The flow is steady, so that the velocity and pressure do not depend on time. The flow is laminar rather than turbulent. The flow is irrotational, so there is no circulation. There is no viscosity in the fluid. Bernoulli’s Equation If a fluid is incompressible and has no viscosity, energy is conserved. Therefore, the work done on a “piece” of fluid is equal to the change in its kinetic energy. 1 1 P1 + ρv12 + ρgy1 = P2 + ρv22 + ρgy2 2 2 1 ⇒ P + ρv 2 + ρgy = constant 2 v=0 P2 − P1 = − ρg ( y2 − y1 ) y1 = y2 1 1 P1 + ρv12 = P2 + ρv22 2 2 2 EF 152 Lec 4-5 Lift Real Fluids 1 1 Pb − Pt = ρvt2 − ρvb2 2 2 Viscosity Viscosity acts like a drag or frictional force; it slows flow speed relative to a surface. v is the average speed or slipstream speed. If a wing has a width w and tip-to-tip span S L = wSρKv 2 K is a constant << 1 Venturi Tube / Meter ⎡⎛ A ⎞ 2 ⎤ 1 P2 − P1 = ρv12 ⎢⎜⎜ 1 ⎟⎟ − 1⎥ 2 ⎢⎣⎝ A2 ⎠ ⎥⎦ There are 8 problems worth 12 points each and graded on a 0, 3, 6, 8, 10 12 scale. There are two questions per page and they MAY or MAY NOT be related to each other. If they are related to each other, you can use any and all the work that you write out for your solution to the first problem for the solution of the second problem on that page. You DO NOT need to redraw Coordinate Systems, FBDs, etc. You can bring one sheet of notes (i.e., a “cheat sheet”) Must be on Engineering paper Front side ONLY In YOUR handwriting Put your name at top Anything you want to write You are to turn it in with the exam NO other books, notes, etc. are allowed. Q= πR 4 (P1 − P2 ) 8ηL Viscous (Poiseuille) Flow Reynolds Number ρ Re = vL η Exam 4 You must show ALL work on ALL problems - Coordinate system on all problems - - - - Final answers must have the correct SF Vector symbols and answers when required - - Final answers must have correct and consistent units Watch unit conversions and unit consistency Significant Figures (SF) - - Provide generic versions of the equations Units - - Make sure they are easily read, everything is labeled and the are complete Equations - - When needed and not provided FBD on all problems that need them - ηvA y Turbulence. Exam 4 – Friday, Oct 26 Top: non-viscous velocity profile. Bottom: same flow with viscosity. F= Non – Viscous Flow Use over bars or over arrows (hats for unit vectors) Vector answers must be in either I j notation or magnitude and direction format. Some questions may specify which format. etc. A cover sheet with Material Data, Useful Constants and Formulas will be provided. 3