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Fluid mechanics ( 流體力學 )

Fluid mechanics ( 流體力學 ). Aquacultural Engineering W 2 - 3. Fluids. Characteristics of fluid Fluid statics ( 流體靜力學 ) Fluid dynamics ( 流體動力學 ) Open channel flow ( 明渠水流 ). Fluid statics. Units of pressure intensity Pressure measurement Manometers ( 壓力計 )

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Fluid mechanics ( 流體力學 )

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  1. Fluid mechanics (流體力學) Aquacultural Engineering W 2 - 3

  2. Fluids • Characteristics of fluid • Fluid statics (流體靜力學) • Fluid dynamics (流體動力學) • Open channel flow (明渠水流) AE-5-Fluid Mechanics

  3. Fluid statics • Units of pressure intensity • Pressure measurement • Manometers (壓力計) • Burst pressure of cylindrical vessels AE-5-Fluid Mechanics

  4. Fluid dynamics • Type of flow • Conservation of mass • Conservation of energy AE-5-Fluid Mechanics

  5. Definition of fluid • a substance that has particles that move easily relative to one another without separation of the mass • a substance that deforms continuously when subjected to a shear force • shear force is a force having a component tangent to a surface AE-5-Fluid Mechanics

  6. A fluid : • May be a gas or a liquid. • Liquids • have a definite volume and when placed into a container, occupy only that volume. • nearly incompressible and can be treated as having this property without introducing appreciable error. • Gases • have no definite volume, and • when placed in a container they expand to fill the entire container. • compressible and must be treated as such to prevent the introduction of large errors. AE-5-Fluid Mechanics

  7. Water • Water is a liquid at ordinary temperatures and pressures, although it also exists in small quantities as a gas under these conditions. • can be treated as an in­compressible fluid AE-5-Fluid Mechanics

  8. physical properties • Density of a liquid (); (kg/m3) • is the mass per unit volume. • Specific weight; (N/m3) • is the weight per unit volume. •  = g (9.1) • g = acceleration of gravity (9.8 m/s2) AE-5-Fluid Mechanics

  9. Physical properties • Absolute viscosity; dynamic viscosity; viscosity; µ; pascal­seconds (Pa-s) • Property of a fluid which offers resistance to shear stress. • Kinematic viscosity; ; (m2/s) • The ratio of the absolute viscosity of a fluid to its density •  = µ/ (9.2) AE-5-Fluid Mechanics

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  12. Fluid statics • the study of fluids at rest • important concepts of fluid statics: • variation of pressure intensity throughout the fluid, and • the force exerted on surfaces by the fluid • pressure intensityat any point in a fluid is the pressure per unit area AE-5-Fluid Mechanics

  13. Fluid pressure • average pressure intensity P = F/A (9.3) whereF = normal force acting on area A (N) A = area over which force is acting (m2) P = pressure (Pa) • pressure at a point in a motionless fluid AE-5-Fluid Mechanics

  14. pressure at the bottom of the container • F = V • F = Ah • P = Ah/A • P = h • pressure at point 2 P = (h/2) • pressure is the same at all points in the liquid that lie on the same horizontal plane AE-5-Fluid Mechanics

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  16. Unit of pressure intensity • referenced to two reference planes • absolute zero pressure (a complete vacuum) and • atmospheric pressure • Standard atmospheric pressure is • the pressure used when specifying standard conditions for calculations with gases, • it is also the pressure used to approximate the difference between a gauge pressure (錶壓) and absolute pressure (絕對壓力). AE-5-Fluid Mechanics

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  18. gauge pressure • = absolute pressure - actual atmospheric pressure • Actual atmospheric pressuremay be either above or below standard atmospheric pressure • Absolute pressure is always positive • Gauge pressure may be either positive or negative. AE-5-Fluid Mechanics

  19. Pressure measurement • Manometers and • bourdon tube (栓管) pressure gauges AE-5-Fluid Mechanics

  20. Manometer (壓力計) • Ps + 1h1+ 2h2 = 3h3 + 4h4 • 2 = 3 = 4 =  • Ps+ 1h1 + h2 = h3 + h4 • 右肢開放至大氣 • 與大氣接觸點為 零錶壓 或 1  105 Pa 絕對壓力 AE-5-Fluid Mechanics

  21. Example 9.1計算容器A之錶壓及絕對壓力 • h1 = 3 m • h2 = h3 = 2 m • h4 = 6 m • 流体 1: 油 (比重 0.9) and • 流体 2: 水 (比重 1.0). • 水之單位体積重 9800 N/m3. AE-5-Fluid Mechanics

  22. (SG) (water) = oil (9.13) • PA + water (SGoil) h1 + h2water = h3water + h4water (9.14) • h2 = h3,管子 h2及 h3部分之液体相同, 9.14 式簡化為 • PA + water (SGoil) h1= h4water(9.15) • PA + (9800 N/m3)(0.9)(3 m) = (6m)(9800N/m3) • PA + 26,460 N/m2 = 58,800 N/m2 • PA = 32,340 N/m2錶壓 • 絕對壓力= Pg + 標準大氣壓力 • 絕對壓力 = 32,340 N/m2 + 100,000 N/m2 • = 132,340 N/m2 AE-5-Fluid Mechanics

  23. Example 9.2Find the gauge pressure in vessel B (Figure 9.6) • PA = 3  105 Pa • SG1= 1 • SG2= 13.6 • h1= 3m • h2= 1 m AE-5-Fluid Mechanics

  24. PA + h1water (SG1) = h2water (SG2) + PB 3  105 N/m2 + (3 m)(9800 N/m3)(1) = (1 m)(9800 N/m3) 13.6 + PB 3  105 N/m2 + 29,404 N/m2 = 133,280 N/m2 + PB 196,124 N/m2 gauge pressure = PB AE-5-Fluid Mechanics

  25. 圓筒容器爆破壓力(Burst Pressure) • 圓筒容器: 例如圓管及圓槽使用時常受很大內部壓力(internal pressure) • 需預測圓管及圓槽可承受之最大壓力 • 需推導出內部壓力與容器璧應力(stress)之關係 • 若容許應力(allowable stress )大於計算應力 (calculated stress), 容器可承受預期之壓力 AE-5-Fluid Mechanics

  26. 內部壓力與璧應利關係 • 流體產生之內部壓力 Fp = PDL (9.16) • 容器璧產生之抵抗力(Resistance force) Fp = 2Fw (9.17) Fw = wAw (9.18) (張應力) Fw = wttL (9.19) Fp = LDP = 2(wLtt) (9.20) • 求解得知璧應力, w = LDP/2Ltt (9.21) w = DP/2tt AE-5-Fluid Mechanics

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  29. 某些無法預測之因素會影響材料應力 • 這些因素設計時必須考慮而以安全因子 (safety factor SF)表示 • 安全因子為 容許應力與設計實際應力之比值 • 大部分管線系統 (plumbing systems)安全係數為 2. 水槽(tanks)及其他容器安全係數為 1 .5 - 5 • wA/SF (9.22) AE-5-Fluid Mechanics

  30. 容許工作壓力計算 • nominal diameter = 5.08 cm; schedule 80 PVC pipe; ID = 4.93 cm; OD = 6.03 cm; SF = 2 tt = (OD - ID)/2 = (6.03 – 4.93)/2 = 0.55 cm • allowable stress in PVC is 48 M Pa. wA/SF w (48 * 106 Pa)/2 w 24 * 106 Pa • w = DP/2tt w = (4.93 cm)P/[2(0.55 cm)] 24 * 106 = (0.0493 m)P/[2(0.0055 m)] P = 24 * 106 Pa (2)(0.0055 m)/0.0493 m P = 5.35 M Pa AE-5-Fluid Mechanics

  31. 流體動力學 • Type of flow • Conservation of mass • Conservation of energy AE-5-Fluid Mechanics

  32. Type of flow • 層流(Laminar flow) and紊流(Turbulent flow) • 層流中每一元素(element)以相同方向及相同速度移動 • 這些元素產生流線(streamlines)流線間以固定的關係移動 • 層流中不存在流線間內部漩渦(eddies)及穿越流線之移動 AE-5-Fluid Mechanics

  33. 紊流 之特性 • 穿越流線之移動, 內部漩渦, 元素間其他移動. • 紊流中能量損失比層流大, 因內部摩擦阻力大. • 層流在低流速, 小管徑及高黏滯度流體中較易發生. • 易於發生紊流之條件剛好相反. • 養殖工程應用之水流大部分為紊流 AE-5-Fluid Mechanics

  34. 質量守恆 • 密閉導管中, 通過某一段面之流體質量與通過其他段面者相等 • 質量平衡方程式 v1A11 = v2A22 (9.23)  = 單位體積重 • 不可壓縮流體1 = 2 (9.24) v1Al = v2A2 (9.25) • 連續方程式(continuity equation) 由密閉管線中之不可壓縮流體推導, 但亦可應用於明渠(open channel flow)之不可壓縮流體 AE-5-Fluid Mechanics

  35. Find the velocity of the flow at section 2 • 0.5 m3/s flowing past section 1 and the pipe diameter at section 2 is 30 cm. v1A1 = v2A2 v1A1 = Q1 Q2 = 0.5 m3/s = v2A2 v2A2 =  (radius)2 v2 0.5 m3/s = v2(0.30 m/2)2 0.5 m3/s = v2(0.0707) 7.07 m/s = v2 AE-5-Fluid Mechanics

  36. Conservation of energy • Total energy at any point in a fluid: • potential energy due to location, potential energy due to pressure, and kinetic energy due to motion of the fluid. • Potential energy due to its elevation (PE)e • weight Wdistance above the datum plane Z (PE)e1 = WZ1 (9.26) (PE)e2 = WZ2 (9.27) AE-5-Fluid Mechanics

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  38. Pressure energy • The pressure energy (PE)p is the weight of an element of fluid times its pressure. • The fluid pressure h = P/ (PE)p = (P/)W (9.28) • kinetic energy KE = (1/2)mv2 (9.29) m = W/g (9.30) KE = Wv2/2g (9.31) AE-5-Fluid Mechanics

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  40. The total energy of an element of fluid Ee Ee = ZW + (P/)W + (v2/2g)W(9.32) • By the law of conservation of energy Z1W + (P1/)W+ (v12/2g)W = Z2W + (P2/)W + (v22/2g)W (9.33) Z1 + (P1/)+ (v12/2g) = Z2 + (P2/) + (v22/2g) (9.34) • Bernoulli's equation, is applicable to flow of an idealized fluid that has no energy losses between points 1 and 2. AE-5-Fluid Mechanics

  41. Real fluid Z1 + (P1/)+ (v12/2g)+ external energy input = Z2 + (P2/) + (v22/2g) + minor losses + pipe friction losses (9.35) • Minor losses due to internal fluid friction • (1) fluid passes through a change in cross-sectional area of the pipe, • (2) the fluid changes direction, • (3) the fluid enters or leaves a conduit, and • (4) other changes occur that increase fluid losses. • minor losses = K(v2/2g) (9.36) AE-5-Fluid Mechanics

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  43. Calculation of head loss • nominal diameter 2.54 cm; 90o elbow; flow 60 liters/min of water; inside diameter 2.43 cm. Q = Av = 60 l/min = 1liter/s 0.001 m3/s = Av = r2v v = (110-3 m/s)/[3.14((2.43/2)10-2 m)2] v= 2.2 m/s minor losses = K(v2/2g) minor losses = [0.9(2.2 m/s)2]/[2(9.8 m/s)] minor losses = 0.22 m of water AE-5-Fluid Mechanics

  44. Frictional losses are a function of the fluid velocity, fluid density, diameter of the pipe, fluid viscosity, and pipe roughness. ff = f(v,D, , , ) (9.37) ff= friction factor v = velocity D= pipe diameter  = roughness (absolute)  = fluid density = fluid viscosity f= function of AE-5-Fluid Mechanics

  45. ff = f((vD/), (/D)) (9.38) • vD/is referred to as Reynolds number • /D is the relative roughness • Nre < 2000 : laminar flow: ff = 64/Re (9.39) • Nre > 4000: turbulent flow • /D relates the height of irregularities on the internal pipe surface with the pipe diameter • values for the dimensionless parameters : Moody's diagram AE-5-Fluid Mechanics

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  49. Once the friction factor is determined, the head loss due to pipe friction can be computed h = ff (L/D)(v2/2g) (9.40) h = head loss due to pipe friction (m) ff = friction factor (dimensionless) L = length of pipe (m) D = pipe diameter (m) v= fluid velocity (m/s) g = acceleration of gravity (m/s2) AE-5-Fluid Mechanics

  50. Head loss in a pipe • L = 100 m; nominal diameter 7.62 cm; galvanized pipe; flow 0.6 m3/min of water at 20°C; internal diameter 7.37 cm. Q = Av 0.6 m3/min = (0.0737 m/2)2 v 0.64/[3.14(0.0054)] = v 141.5 m/min = v 2.36 m/s = v Reynolds number= Re = vD/ = vD/ Kinectic viscosity of water at 20°C = 1.1  10-2 cm2/s Re =[2.36 m/s (7.37  10-2 m)]/1.1  10-6 m2/s Re = 15.8  104 AE-5-Fluid Mechanics

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