FUNDAMENTALS OF FLUID MECHANICS Chapter 3 Fluids in Motion - The Bernoulli Equation

Transcription

1 FUNDAMENTALS OF FLUID MECHANICS Chater 3 Fluids in Motion - The Bernoulli Equation Jyh-Cherng Shieh Deartment of Bio-Industrial Mechatronics Engineering National Taiwan University 09/8/009

2 MAIN TOPICS Newton s s Second Law F=ma Along a Streamline F=ma Normal to a Streamline Physical Interretation of Bernoulli Equation Static, Stagnation, Dynamic, and Total Pressure Alication of the Bernoulli Equation The Energy Line and the Hydraulic Grade Line Restrictions on Use of the Bernoulli Equation 推導出 Bernoulli Equation 解釋 Bernoulli equation

3 Newton s s Second Law /6 As a fluid article moves from one location to another, it exeriences an acceleration or deceleration. According to Newton s s second law of motion,, the net force acting on the fluid article under consideration must equal its mass times its acceleration. 作用在質點的合力, 等於質點質量乘以加速度 F=ma In this chater, we consider the motion of inviscid fluids.. That is, the fluid is assumed to have zero viscosity.. For such case, it is ossible to ignore viscous effects. The forces acting on the article? Coordinates used? 本章仍以非黏性流體為對象, 即假設流體沒有黏性, 忽略其黏性效應 討論 : 作用在質點上的力? 使用的座標? 3

4 Newton s s Second Law /6 作用在質點上的 Force The fluid motion is governed by F= Net ressure force + Net gravity force To aly Newton s s second law to a fluid, an aroriate coordinate system must be chosen to describe the motion.. In general, the motion will be three-dimensional and unsteady so that three sace coordinates and time are needed to describe it. 4

5 Newton s s Second Law 3/6 The most often used coordinate systems are rectangular (x,y,z) and cylindrical (r,,z),z) system. 選用合適的座標系統, 使用哪一種座標系統? 5

6 Streamline? Streamline coordinate! As is done in the study of dynamics, the motion of each fluid article is described in terms of its velocity, V, which is defined as the time rate of change of the osition of the article. The article s s velocity is a vector quantity with a magnitude and direction. As the article moves about, it follows a articular ath, the shae of which is governed by the velocity of the article. 6

7 Streamline? Streamline coordinate! The location of the article along the ath is a function of where the article started at the initial time and its velocity along the ath. If it is steady flow, each successive article that asses If it is, each successive article that asses through a given oint will follow the same ath. 7

8 Streamline? Streamline coordinate! For such cases the ath is a fixed line in the x-z lane. Neighboring articles that ass on either side of oint ( following their own aths, which may be of a different shae than the one assing through (). The entire x-z lane is filled with such aths. 8

9 Streamline? Streamline coordinate! For steady flows each article slides along its ath, and its velocity is everywhere tangent to the ath. The lines that are tangent to the velocity vectors throughout the flow field are called streamlines. 9

10 Streamline? Streamline coordinate! For many situations it is easiest to describe the flow in terms of the streamline coordinate based on the streamlines. The article motion is described in terms of its distance, s = s(t), along the streamline from some convenient origin and the local radius of curvature of the streamline, R = R(s). 0

11 Newton s s Second Law 4/6 以兩維流動為對象 In this chater, the flow is confined to be two- dimensional motion. As is done in the study of dynamics, the motion of each fluid article is described in terms of its velocity vector V. As the article moves, it follows a articular ath. The location of the article along the ath is a function of its initial osition and velocity. 質點運動, 有它特定的軌跡, 在軌跡上的 Location,, 是初始位置與速度的函數

12 5/6 Newton s s Second Law 5/6 在穩態流場下, 質點運動的速度向量與軌跡相切 For steady flows,, each article slides along its ath, and its velocity vector is everywhere tangent to the ath. The lines that are tangent to the velocity vectors throughout the flow field are called streamlines. 流場內與質點速度相切的線, 稱為 streamline For such situation, the article motion is described in terms of its distance, s=s(t), along the streamline from some convenient origin and the local radius of curvature of the streamline, R=R(s). 質點的運動, 可以由它沿著 Streamline,, 相對於某個參考原點的距離 s=s(t s(t) 來描述 該 streamline 的曲率半徑可以寫成 R=R(s R(s)

13 Newton s s Second Law 6/6 由 s=s(t s(t) 來定義質點的速度 V=ds/dtds/dt The distance along the streamline is related to the article s s seed by V=ds/dt,, and the radius of curvature is related to shae of the streamline. The acceleration is the time rate of change of the velocity of the article 由速度 V=ds/dtds/dt 來定義加速度 dv dv V a ass ann V s n 加速度的兩個分量 dt ds R The comonents of acceleration in the s and n direction a s dv V V an CHAPTER 04 ds R 先應用 CHAPTER 04 再討論 3

14 Streamlines Streamlines ast an airfoil Flow ast a biker 4

15 F=ma along a Streamline /4 Isolation of a small fluid article in a flow field. 5

16 F=ma along a Streamline /4 Consider the small fluid article of size of s s by n in the lane of the figure and y normal to the figure. For steady flow, the comonent of Newton s second law along the streamline direction s 沿著 s 方向的合力 V V FS mas mv VV s s Where F S reresents the sum of the s comonents of all the force acting on the article. 6

17 F=ma along a Streamline 3/4 The gravity force (weight) on the article in the streamline direction W s Wsin Vsin The net ressure force on the article in the streamline direction s F s s ny ny ny V F s S W s F s sin S sin s S s S V s V V s a s 單位體積 Equation of motion along the streamline direction 7

18 F=ma along a Streamline 4/4 Particle weight sin s V V s a s ressure gradient A A change in fluid article seed is accomlished by the aroriate combination of ressure gradient and article weight along the streamline. 造成質點速度改變的兩個因素 For fluid static situation,, the balance between ressure and gravity force is such that no change in article seed is roduced. sin s 0 Integration 質點速度沒有改變, 則 article weight 與 ressure gradient 相互抵銷 8

19 Integration.. V sin V a s s s dz d dv d ds ds ds d V 除非壓力與密度的關係很清楚, 否則積分不能隨便拿開 gz C Rearranged and Integrated d V dz 0 along a streamline Where C is a constant of integration to be determined by the conditions at some oint on the streamline. In general it is not ossible to integrate the ressure term because the density may not be constant and, therefore, cannot be removed from under the integral sign. 9

20 Bernoulli Equation Along a Streamline For the secial case of incomressible flow d V gz C Restrictions : Steady flow. Incomressible flow. 不可壓縮流體 V Frictionless flow. Flow along a streamline. BERNOULLI EQUATION z cons 一再提醒, 每一個結論 ( 推導出來的方程式 ), 都有它背後假設條件, 即一路走來, 是基於這些假設才有如此結果 NO 剪力 tan The Bernoulli equation is a very owerful tool in fluid mechanics, ublished by Daniel Bernoulli (700~78) in 738. t 0

21 Examle 3. Pressure Variation along A Streamline Consider the inviscid,, incomressible, steady flow along the horizontal streamline A-B A B in front of the shere of radius a, as shown in Figure E3.a. From a more advanced theory of flow ast a shere, the fluid velocity along this streamline is V a V x 已知流體沿著 streamline 的速度 Determine the ressure variation along the streamline from oint A far in front of the shere (x( A =- and V A = V 0 ) to oint B on the shere (x( B =-a and V B =0) 求沿著 streamline 的壓力變化 sin=0 s x

22 / Examle 3. Solution / The equation of motion along the streamline (sin( sin=0) s V V s () sin s V V s a s The acceleration term V V s V V x V0 a x 3 3 3V0a 4 x 3 3V 0 a x 3 3 a x 3 4 The ressure gradient along the streamline is s x 3a 3 V 0 4 x a 3 / x 3 ()

23 3 Examle 3. Examle 3. Solution Solution / / The ressure gradient along the streamline x x / a V a 3 x s () () The ressure distribution along the streamline x) (a / x a V 6 3 0

24 Examle 3. The Bernoulli Equation Consider the flow of air around a bicyclist moving through still air with velocity V 0, as is shown in Figure E3.. Determine the difference in the ressure between oints () and (). 求 oint () 與 oint () 的壓力差 4

25 Examle 3. Solution The Bernoulli s equation alied along the streamline that asses through () and () V V z z z =z () is in the free stream V =V 0 () is at the ti of the bicyclist s nose V =0 V V0 將 Bernoulli equation 應用到沿著連接 oint () 與 oint () 的 streamline 5

26 F=ma Normal to a Streamline / For steady flow, the comonent of Newton s second law in the normal direction n 法線方向的合力 mv VV Fn R R Where F n reresents the sum of the n comonents of all the force acting on the article. Hydrocyclone searator 水力旋流分璃器 6

27 F=ma Normal to a Streamline / The gravity force (weight) on the article in the normal direction W n Wcos V cos The net ressure force on the article in the normal direction n F n n V V R sy ( ) sy sy V F Normal direction n n W n F n cos n n cos n n V R n V n Equation of motion normal to the streamline 7

28 Integration.. cos n Integrated V R 除非壓力與密度的關係很清楚, 否則積分不能隨便拿開 In general it is not ossible to integrate the ressure term because the density may not be constant and, therefore, cannot be removed from under the integral sign. d Rearranged ressure gradient 造成質點速度改變的兩個因素 A change in the direction of flow of a fluid article is dz d V accomlished by the aroriate combination of ressure dn dn R gradient and article weight normal to the streamline Particle weight V R dn gz C Normal to the streamline 除非 V=V(s,n V(s,n) 與 R=R(s,n R(s,n) 很清楚, 否則積分不能隨便拿開 Without knowing the n deendent in V=V(s,n V(s,n) ) and R=R(s,n R(s,n) ) this integration cannot be comleted. 8

29 Bernoulli Equation Normal to a Streamline For the secial case of incomressible flow d V R dn gz C 不可壓縮流體 BERNOULLI EQUATION V R z 一再提醒, 每一個結論 ( 推導出來的方程式 ), 都有它背後假設條件, 即一路走來, 是基於這些假設才有如此結果 Restrictions : Steady flow. Incomressible flow. Frictionless flow. NO shear force Flow normal to a streamline. dn C 9

30 If gravity is neglected Free vortex dz dn d dn V R A larger seed or density or smaller radius of curvature of the motion required a larger force unbalance to roduce the motion. If gravity is neglected or if the flow is in a horizontal d dn V R Pressure increases with distance away from the center of curvature. Thus, the ressure outside a tornado is larger than it is near the center of the tornado. 30

31 Aircraft wing ti vortex 3

32 Examle 3.3 Pressure Variation Normal to a Streamline Shown in Figure E3.3a and E3.3b are two flow fields with circular streamlines. The velocity distributions are C V(r) Cr (a) V(r) r (b) 已知速度分佈, 求壓力場 cos n V R Assuming the flows are steady, inviscid, and incomressible with streamlines in the horizontal lane (dz/dn=0). 3

33 33 Examle 3.3 Examle 3.3 Solution Solution 0 0 r r C r V r For flow in the horizontal lane (dz/dn=0). The streamlines are circles /n=-/r The radius of curvature R=r For case (a) this gives r C r 3 r C r For case (b) this gives 0 0 r r C R V dn d dn dz

34 Physical Interreter / 基本假設 Under the basic assumtions: the flow is steady and the fluid is inviscid and incomressible. Alication of F=ma and integration of equation of motion along and normal to the streamline result in a s dv V ds a n V R 如何解讀? 作用在質點上的兩個力 : 壓合力不平衡, V V z C dn z C 力與重力 當合力不平衡 R 就會有加速度產生 To roduce an acceleration, there must be an unbalance of the resultant force, of which only ressure and gravity were considered to be imortant.. Thus, there are three rocess involved in the flow mass times acceleration (the V / term), ressure (the term), and weight (the z z term). 過程中, 有 質量 加速度 壓力 與 重量 z 34

35 Physical Interreter / 能量平衡 : 外力作的功等於質點的動能改變 The Bernoulli equation is a mathematical statement of The work done on a article of all force acting on the article is equal e to the change of the kinetic energy of the article. Work done by force : F d. Work done by weight: z Work done by ressure force: Kinetic energy: V / 外力有二, 一為壓力, 一為重力 z V C z V R dn C 你可能會對其中的單位感到困惑, 沒錯! 單位是怪怪的 但別忘了, 以上都是 Based on 單位體積 35

36 Head The Bernoulli Equation can be written in terms of heights called heads An alternative but equivalent form of the Bernoulli equation is obtained by dividing each term by 沿著 streamline P V g z 讓每一項都是長度單位 c Pressure Head Elevation Head Velocity Head 36

37 Examle 3.4 Kinetic, Potential, and Pressure Energy Consider the flow of water from the syringe shown in Figure E3.4. A force alied to the lunger will roduce a ressure greater than atmosheric at oint () within the syringe. The water flows from the needle, oint (), with relatively high velocity and coasts u to oint (3) at the to of its trajectory. Discuss the energy of the fluid at oint (), (), and (3) by using the Bernoulli equation. 停止移動 求 oint () () (3) 的三種型態的能量關係 37

38 Examle 3.4 Solution The sum of the three tyes of energy (kinetic, otential, and ressure) or heads (velocity, elevation, and ressure) must remain constant. t. V z cons tan t along the The ressure gradient between () and () roduces an acceleration to eject the water form the needle. Gravity acting on the article between () and (3) roduces a deceleration to cause the water to come to a momentary sto at the to of its flight. streamline The motion results in a change in the magnitude of each tye of energy as the fluid flows from one location to another. 38

39 Examle 3.5 Pressure Variation in a Flowing Stream Consider the inviscid,, incomressible, steady flow shown in Figure E3.5. From section A to B the streamlines are straight, while from C to D they follow circular aths. Describe the ressure variation between oints () and ()and oints(3) and (4) 求 oint ()() 與 oint(3)(4) 的壓力變化 39

40 / Examle 3.5 Solution / R=, for the ortion from A to B rz cons tan t Using =0,z =0,and z =h - Point ()~() r(z z ) rh Since the radius of curvature of the streamline is infinite, the ressure variation in the vertical direction is the same as if the fluids were stationary. 40

41 / Examle 3.5 Solution / For the ortion from C to D z 4 V 4 ( dz) rz4 3 rz3 z R 3 Point (3)~(4) With 4 =0 and z 4 -z 3 =h 4-3,this becomes z4v 3 rh43 dz z3 R 4

42 Static, Stagnation, Dynamic, and Total Pressure /5 Each term in the Bernoulli equation can be interreted as a form of ressure. 能量平衡的概念下, 每一項的單位 : 壓力 z V C Each term can be interreted as a form of ressure is the actual thermodynamic ressure of the fluid as it flows. To measure this ressure, one must move along with the fluid, thus being static relative to the moving fluid. Hence, it is termed the static ressure seen by the fluid article as it moves. 是流體流動下所量測的壓力, 量測時要緊貼著流體, 相對流動的流體而言, 它是 static, 故稱為 static ressure 4

43 Static, Stagnation, Dynamic, and Total Pressure /5 利用 wall ressure ta The static ressure is measured in a flowing fluid using a wall ressure ta,, or a static ressure robe. The static ressure z is termed the hydrostatic ressure.. It is not actually a ressure but does reresent the change in ressure ossible due to otential energy variations of the fluid as a result of elevation changes. ressure h3 3 h3 h43 ta 量測 static z 不是真的壓力, 它代表著位能 ( 高度 ) 改變所產生的壓力變化 h 43

44 Static, Stagnation, Dynamic, and Total Pressure 3/5 V / V / is termed the dynamic ressure.. It can be interreted as the ressure at the end of a small tube inserted into the flow and ointing ustream. After the initial transient motion has died out, the liquid will fill the tube to a height of H. The fluid in the tube, including that at its ti (), will be stationary. That is, V =0, or oint () is a stagnation oint. 假設 ()( ) 與 () 高度相同 Stagnation ressure / 是將管插到流場中把流體擋下來, 在管端 () 衍生出來的壓力 這個壓力等於液柱高 H 所產生的壓力 V Static ressure 兩者和, 稱為 stagnation ressure Dynamic ressure 44

45 Stagnation oint Stagnation oint flow 45

46 Static, Stagnation, Dynamic, and Total Pressure 4/5 There is a stagnation oint on any stationary body that is laced into a flowing fluid. Some of the fluid flows over and some under the object. The dividing line is termed the stagnation streamline and terminates at the stagnation oint on the body. Neglecting the elevation effects, the stagnation ressure is the largest ressure obtainable along a given streamline. 忽略高度效應下, stagnation oint 處的壓力是流場中最大的壓力 將物體固定在流場中, 會在物體上出現 stagnation oint stagnation oint 46

47 Static, Stagnation, Dynamic, and Total Pressure 5/5 The sum of the static ressure, dynamic ressure, and hydrostatic ressure is termed the total ressure. The Bernoulli equation is a statement that the total ressure remains constant along a streamline. 三者和, 稱為 Total ressure V z T cons tan t Total ressure 沿著 streamline 是一個常數! 47

48 The Pitot-static Tube /5 Knowledge of the values of the static and stagnation ressure in a fluid imlies that the fluid seed can be calculated. This is the rincile on which the Pitot- static tube is based. z 4 3 z V V V ( 3 / / 4 Static ressure ) / Stagnation ressure Pitot-static stubes measure fluid velocity by converting velocity into ressure. 利用 沿著 streamline 的 total ressure 是一個常數 概念來量測流速的量具 量具可同時量測 static 與 stagnation ressure,, 再據以計算流速 48

49 Airlane Pitot-static robe Airseed indicator 49

50 The Pitot-static Tube /5 50

51 The Pitot-static Tube 3/5 The use of itot-static tube deends on the ability to measure the static and stagnation ressure. An accurate measurement of static ressure requires that none of the fluid s s kinetic energy be converted into a ressure rise at the oint of measurement. This requires a smooth hole with no burrs or imerfections. Tas 不可出現毛邊或瑕疵, 以免造成不必要的壓升或壓降 Incorrect and correct design of static ressure tas. 5

52 The Pitot-static Tube 4/5 The ressure along the surface of an object varies from the stagnation ressure at its stagnation oint to value that may be less than free stream static ressure. It is imortant that the ressure tas be roerly located to ensure that the ressure measured is actually the static ressure. 由壓力分布或變化可以了解,tas 要放在哪一個位置是很重要的, 以免高估或低估 static ressure Tyical ressure distribution along a Pitot-static tube. 5

53 The Pitot-static Tube 5/5 Three ressure tas are drilled into a small circular cylinder, fitted with small tubes, and connected to three ressure transducers. The cylinder is rotated until the ressures in the two side holes are equal,, thus indicating that the center hole oints directly ustream. 圓柱會自行調整, 直到 P =P 3 Directional-finding Pitot-static tube. 可以自動對位的 Pitot-static tube If θ=0 P V P 3 P P 53

54 Examle 3.6 Pitot-Static Tube An airlane flies 00mi/hr at an elevation of 0,000 ft in a standard atmoshere as shown in Figure E3.6. Determine the ressure at oint () far ahead of the airlane, oint (), and the ressure difference indicated by a Pitot-static robe attached to the fuselage. 求第一點 第二點的壓力 機身下貼附 Pitot-static tube 54

55 Examle 3.6 Solution / 高度 0,000ft 處的壓力與密度 The static ressure and density at the altitude 456lb / ft (abs) 0.sia slug / ft If the flow is steady, inviscid, and incomressible and elevation changes are neglected. The Bernoulli equation V With V =00mi/hr=46.6ft/s and V =0 456lb / ft (456 ( slugs/ ft 8.9)lb / ft (abs) 3 )(46.7 ft /s ) / 55

56 Examle 3.6 Solution / In terms of gage ressure 8.9lb / ft 0.33si The ressure difference indicated by the Pitot-static tube V 0.33si 56

57 Alication of Bernoulli Equation / Bernoulli equation 的應用 The Bernoulli equation can be alied between any two oints on a streamline rovided that the other three restrictions are satisfied. The result is 沿著 streamline V V z z streamline 任意兩點 要應用 Bernoulli equation 時, 要注意問題本身有沒有符合下列限制條件 Restrictions : Steady flow. Incomressible flow. Frictionless flow. Flow along a streamline. 57

58 Alication of Bernoulli Equation / Bernoulli equation 的應用 Free jet. Confined flow. Flowrate measurement 58

59 Free Jets /3 V V z z Alication of the Bernoulli equation between oints () and () on the streamline h V V h At oint (5) gh = =0; z =h;z =0;V =0 V g(h H) = 5 =0; z =h+h;z =0;V =0 59

60 Free Jets Flow from a tank 60

61 Free Jets /3 V V z z For the horizontal nozzle,, the velocity at the centerline, V, V will be greater than that at the to V. V In general, d<<h and use the VV as average velocity. V >V ; 若 d<<h,, 則以 V 代表平均速度 For a shar-edged orifice, a vena contracta effect occurs. The effect is the result of the inability of the fluid to turn the shar 90 corner. 出現束縮效應, 流體不可能 90 轉彎 6

62 Free Jets 3/3 不同出口型態下的 flow attern 與縮流係數 Tyical flow atterns and contraction coefficients for various round exit configuration. The diameter of a fluid jet is often smaller than that of the hole from which it flows. V V z z Define Cc = contraction coefficient C c A A j h 定義縮流係數 Aj=area of the jet at the vena contracta Ah=area of the hole 6

63 Examle 3.7 Flow From a Tank- Gravity A stream of water of diameter d = 0.m flows steadily from a tank of Diameter D =.0m as shown in Figure E3.7a. Determine the flowrate,, Q, needed from the inflow ie if the water deth remains constant,, h =.0m. 求 flowrate Q 63

64 / V z The Bernoulli equation alied between oints () and () is Examle 3.7 Solution / V z V z With = = 0, z = h, and z = 0 V gh V () For steady and incomressible flow, conservation of mass requires Q = Q, where Q = AV. Thus, A V =A V, or 4 D V 4 d V V z Bernoulli equation 應用到 oint() 與 () 間 () Point() 與 () 的已知條件 加上質量守恆條件 d V ( ) V (3) D 64

65 / Examle 3.7 Solution / Combining Equation and 3 gh (d / D) (9.8m /s )(.0m) (0.m /m) V 4 4 Thus, Q A V A V (0.m) 4 (6.6m / s) V 0 0 (Q) vs. V 0 0 (Q 0 ) 6.6m /s 0.049m 3 / s Q Q V V gh /[ ( d / gh D) 4 0 D ( / ) 4 ] d D 65

66 Examle 3.8 Flow from a Tank-Pressure Air flows steadily from a tank, through a hose of diameter D=0.03m and exits to the atmoshere from a nozzle of diameter d=0.0m as shown in Figure E3.8. The ressure in the tank remains constant at 3.0kPa (gage) and the atmosheric conditions are standard temerature and ressure. Determine the flowrate and the ressure in the hose. 理想氣體 求 flowrate 與 Point() 的壓力 66

67 / Examle 3.8 Solution / For steady, inviscid,, and incomressible flow, the Bernoulli equation along the streamline Bernoulli equation 應用到 oint()() 與 (3) 間 V z V z 3 V3 z3 With z =z = z 3, V = 0, and 3 =0 Point()() () 與 (3) 的已知條件 V 3 and V The density of the air in the tank is obtained from the erfect gas law 理想氣體的密度 RT [(3.0 0)kN / m 3 0 N / kn ] (86.9N m / kg K)(5 73)K ().6kg / m 3 67

68 / Examle 3.8 Solution / Thus, 3 (3.0 0 N / m.6kg / m V3 3 ) 69.0m / s or Q A3V3 d V m 3 / s The ressure within the hose can be obtained from Eq.. and the continuity equation A V A3V3 Hence, V A3V3 / A V ( )N / m 3 N / m 963N / m 7.67m / s (.6kg / m 3 )(7.67m /s) 68

69 Examle 3.9 Flow in a Variable Area Pie Water flows through a ie reducer as is shown in Figure E3.9. The T static ressures at () and () are measured by the inverted U-tube U manometer containing oil of secific gravity, SG, less than one. Determine the manometer reading, h. reading h? h 69

70 / Examle 3.9 Solution / For steady, inviscid,, incomressible flow, the Bernoulli equation along the streamline V z V z The continuity equation Q A V A V Combining these two equations (z z ) V [ (A / A ) ] () 70

71 / Examle 3.9 Solution / This ressure difference is measured by the manometer and determine by using the ressure-deth ideas develoed in Chater. or (z z ) h SGh (z z ) ( SG) h () - + Point()() 的壓力差 ( SG) h Since V =Q/A h V Q/ A A A (A / A) g SG be indeendent of θ 7

72 Confined Flows /4 受限流 : 被限制在 device 內部, 如 nozzle ie When the fluid is hysically constrained within a device, its ressure cannot be rescribed a riori as was done for the free jet. Such cases include nozzle and ies of various diameter for which the fluid velocity changes because the flow area is different from one section to another. For such situations, it is necessary to use the concet of conservation of mass (the continuity equation) along with the Bernoulli equation. 解 受限流受限流 的工具 Tools: Bernoulli equation + Continuity equation 7

73 Confined Flows /4 Consider a fluid flowing through a fixed volume that has one inlet and one outlet. Conservation of mass requires AV AV For incomressible flow,, the continuity equation is 加上 不可壓縮 條件, 連續方程式簡化 AV AV Q Q 73

74 Confined Flows 3/4 受限流流經變化管徑的管, 當速度增加, 壓力就降低 If the fluid velocity is increased, the ressure will decrease. This ressure decrease can be large enough so that the ressure in the liquid is reduced to its vaor ressure. 當速度持續增加, 壓力可能降低到低於 vaor ressure, 導致初始空蝕現象出現 Pressure variation and cavitation in a variable area ie. Venturi channel 74

75 Confined Flows 4/4 examle of cavitation A A examle of cavitation can be demonstrated with a garden hose. If the hose is kinked, kinked, a restriction in the flow area will result. The water velocity through the restriction will be relatively large. 水管扭結, 導致管的截面積縮小, 速度增加 With a sufficient amount of restriction the sound of the flowing water will change a definite hissing sound will be roduced. The sound is a result of cavitation. 出現嘶嘶聲, 即表示空蝕現象出現 75

76 Damage from Cavitation 前進到高壓區,bubble, 破裂, 破壞器具 Cavitation from roeller 76

77 Examle 3.0 Sihon and Cavitation Water at 60 is sihoned from a large tank through a constant diameter hose as shown in Figure E3.0. Determine the maximum height of the hill, H, over which the water can be sihoned without cavitation occurring. The end of the sihon is 5 ft below the bottom of the tank. Atmosheric ressure is 4.7 sia. Max. H? 才能避免 cavitation The value of H is a function of both the secific weight of the fluid, γ,, and its vaor ressure, v. 77

78 / Examle 3.0 Solution / For ready, inviscid,, and incomressible flow, the Bernoulli equation along the streamline from () to () to (3) V z V z 3 V3 z3 () With z = 5 ft, z = H, and z 3 = -55 ft. Also, V = 0 (large tank), = 0 (oen tank), 3 = 0 (free jet), and from the continuity equation A V = A 3 V 3, or because the hose is constant diameter V = V 3. The seed of the fluid in the hose is determined from Eq.. to be ()(3) V g(z z ) (3.ft /s )[5 ( 5)]ft 35.9ft /s V 3 3 ()() 78

79 / Examle 3.0 Solution / Use of Eq.. between oint () and () then gives the ressure at the to of the hill as V z V z (z z ) V () The vaor ressure of water at 60 is 0.56 sia. Hence, for inciient cavitation the lowest ressure in the system will be = 0.56 sia. Using gage ressure: si sing gage ressure: = 0, = = -4.4 si 3 ( 4.4lb / in. )(44in. / ft ) (6.4lb / ft )(5 H)ft (.94slugs/ ft 3 )(35.9ft /s) H 8. ft 79

80 Flowrate Measurement in ies /5 Various flow meters are governed by the Bernoulli and continuity equations. Flow meters 的理論基礎 V Q A V A V V The theoretical flowrate 管流中量測流率的裝置 ( ) Q A [ (A / A ) ] Tyical devices for measuring flowrate in ies 80

81 Examle 3. Venturi Meter Kerosene (SG = 0.85) flows through the Venturi meter shown in Figure E3. with flowrates between and m 3 /s. Determine the range in ressure difference,, needed to measure these flowrates. 求維持流率所需 oint()() 壓力差 Known Q, Determine - 8

82 / Examle 3. Solution / For steady, inviscid,, and incomressible flow, the relationshi between flowrate and ressure Q [ (A A / A The density of the flowing fluid ) ] ( ) Q A [ (A / A ) ] Eq SG H O 0.85(000 kg/m 3 ) 850kg/m 3 The area ratio A /A (D / D ) (0.006m / 0.0m)

83 / Examle 3. Solution / The ressure difference for the smallest flowrate is 3 (0.005m /s) 60N/m 3 (850kg/m ).6kPa ( 0.36 ) [( / 4)(0.06m) ] The ressure difference for the largest flowrate is ( 0.36 ) (0.05 )(850 ) [( / 4)(0.06m) N/m 6 kpa ].6kPa - 6kPa 83

84 Flowrate Measurement sluice gate /5 The sluice gate is often used to regulate and measure the flowrate in an oen channel. 水閘門 : 調節與量測流率 The flowrate,, Q, is function of the water deth ustream, z, the width of the gate, b, and the gate oening, a. V z V z Q A V bv z A V bv z With = =0, the flowrate 理論值 Q z b g(z (z z / z ) ) 84

85 Flowrate Measurement sluice gate 3/5 In the limit of z >>z, this result simly becomes Q zb gz 特殊情況 :z: 遠大於 z This limiting result reresents the fact that if the deth ratio, z /z, is large, the kinetic energy of the fluid ustream of the gate is negligible and the fluid velocity after it has fallen a distance (z -z )~z is aroximately V gz Z?? 是求 Q 的關鍵 z <a( 束縮效應 ) Z = C c a 縮流係數 C c? 85

86 Flowrate Measurement sluice gate 4/5 As we discussed relative to flow from an orifice, the fluid cannot turn a shar 90 corner. A vena contracta results with a contraction coefficient, C c =z /a, less than. Tyically C c ~0.6 over the deth ratio range of 0<a/z <0.. : For large value of a/z, the value of C c increase raidly. 縮流係數 C c? 低於,, 典型值約 0.6, a/z 越大, C c 越高 86

87 Examle 3. Sluice Gate Water flows under the sluice gate in Figure E3.a. Dertermine the aroximate flowrate er unit width of the channel. 87

88 / Examle 3. Solution / For steady, inviscid, incomreesible flow, the flowerate er unit t width Q b z g(z (z z) / z ) Q z b g(z (z z / z ) ) Eq.3. With z =5.0m and a=0.80m, so the ratio a/z =0.6<0.0. Assuming contraction coefficient is aroximately C c =0.6. z =C c a=0.6(0.80m)=0.488m. The flowrate Q b (0.488m) (9.8m / s )(5.0m 0.488m) (0.488m / 5.0m) 4.6m / s 88

89 / Examle 3. Solution / If we consider z >>z and neglect the kinetic energy of the ustream fluid,, we would have Q b 5.0m 4.83m / s z gz 0.488m 9.8m /s 89

90 Flowrate Measurement weir 5/5 For a tyical rectangular, shar-crested, the flowrate over the to of the weir late is deendent on the weir height, P w, the width of the channel, b, and the head, H, of the water above the to of the weir. The flowrate Q C AV 3/ CHb gh Cb gh Where C is a constant to be determined. 堰 待由實驗來決定 90

91 Examle 3.3 Weir Water flows over a triangular weir, as is shown in Figure E3.3. Based on a simle analysis using the Bernoulli equation, determine the deendence of flowrate on the deth H. If the flowrate is Q 0 when H=H 0, estimate the flowrate when the deth is increased to H=3H 0. H 提高 3 倍, 流率增加? 9

92 Examle 3.3 Solution For steady, inviscid, and incomressible flow, the average seed of the fluid over the triangular notch in the weir late is roortional to gh The flow area for a deth of H is H[H tan( /)] The flowrate Q where C C AV 5/ CH tan ( gh) C tan gh is an unknown constant to be determined exerimentally. An increase in the deth by a factor of the three ( from H 0 results in an increase of the flowrate by a factor of Q Q / g3h 0 / gh 5/ 3H C tan 0 5/ H C tan to 3H 0 ) 9

93 EL & HGL /4 For steady, inviscid,, incomressible flow, the total energy remains constant along a streamline. P V / z / g H g V g z constan t 每一項化成長度單位 -HEAD- 沿著 streamline 的 total head 維持常數 H The head due to local static ressure (ressure energy) The head due to local dynamic ressure (kinetic energy) The elevation head ( otential energy ) The total head for the flow 93

94 EL & HGL /4 Energy Line (EL) : reresents the total head height. P V g z 沿著 streamline 的 total head 連結成一條線 Hydraulic Grade Line (HGL) height: reresents the sum of the elevation and static ressure heads. P z The difference in heights between the EL and the HGL reresents the dynamic ( velocity ) head V / g 沿著 streamline 的 壓力頭壓力頭 + 高度頭 高度頭 連結成一條線 EL 與 HGL 的差, 代表沿 streamline 各處的 速度 速度頭 94

95 EL & HGL 3/4 P V g z constan t H 壓力頭壓力頭 + 高度頭 高度頭 95

96 EL & HGL 4/4 P V g z constan t H P V g z constan t H 96

97 Examle 3.4 Energy Line and Hydraulic Grade Line Water is sihoned from the tank shown in Figure E3.4 through a hose of constant diameter. A small hole is found in the hose at location () as indicate. When the sihon is used, will water leak out of the hose, or will air leak into the hose, thereby ossibly causing the sihon to malfunction? 空氣跑進去或水溢出來? 97

98 / Examle 3.4 Solution / Whether air will leak into or water will leak out of the hose deends d on whether the ressure within the hose at () is less than or greater than atmosheric.. Which haens can be easily determined by using the energy line and hydraulic grade line concets. With the assumtion of steady, incomressible, inviscid flow it follows that the total head is constant-thus, thus, the energy line is horizontal. Since the hose diameter is constant, it follows from the continuity ity equation (AV=constant) that the water velocity in the hose is constant throughout. Thus the hydraulic grade line is constant distance, V /g, below the energy line as shown in Figure E

99 / Examle 3.4 Solution / Since the ressure at the end of the hose is atmosheric, it follows lows that the hydraulic grade line is at the same elevation as the end of the hose outlet. The fluid within the hose at any oint above the hydraulic grade line will be at less than atmosheric ressure. Thus, air will leak into the hose through the hole at oint (). 99

100 Restrictions on Use of the Bernoulli Equation comressibility effects /4 The assumtion of incomressibility is reasonable for most liquid flows. In certain instances, the assumtion introduce considerable errors for gases. To account for comressibility effects d V 考慮壓縮效應 gz C 00

101 Restrictions on Use of the Bernoulli Equation comressibility effects /4 For isothermal flow of erfect gas RT z ln z g P g For isentroic flow of erfect gas the density and ressure are related by :P P / ρ k =Ct, where k = Secific heat ratio C k d P k dp V 理想氣體等熵流 V gz gz cons tan t k P k cons tan t V V g 理想氣體等溫流 RT k gz gz k P P V V 0

102 Restrictions on Use of the Bernoulli Equation comressibility effects 3/4 To find the ressure ratio as a function of Mach number Seed of sound The ustream Mach number M a V / c V / krt Comressible flow Incomressible flow k k k Ma Ma k 0

103 Restrictions on Use of the Bernoulli Equation comressibility effects 4/4 k k k Ma Ma k 03

104 Examle 3.5 Comressible Flow Mach Number A Boeing 777 flies at Mach 0.8 at an altitude of 0 km in a standard atmoshere. Determine the stagnation ressure on the leading edge of its wing if the flow is incomressible; and if the t flow is incomressible isentroic. For incomressible flow For comressible isentroic flow k M a kpa 0.47 k M kPa a k k

105 Restrictions on Use of the Bernoulli Equation unsteady effects For unsteady flow V = V ( s, t ) a S V t V V s To account for unsteady effects t V ds d d V dz 0 + Incomressible condition 考慮非穩定效應 Along a streamline V t S V z ds V z S Oscillations in a U-tubeU 05

106 Examle 3.6 Unsteady Flow U-Tube An incomressible, inviscid liquid is laced in a vertical, constant diameter U-tube U as indicated in Figure E3.6. When released from the nonequilibrium osition shown, the liquid column will oscillate at a secific frequency. Determine this frequency. 06

107 Examle 3.6 Solution Let oints () and () be at the air-water interface of the two columns of the tube and z=0 corresond to the equilibrium osition of the interface. Hence z = 0, = = 0, z = 0, z = - z, V = V = V z = z ( t ) S S t z ds dv dt V S dz V dt d z dt dv dt g z 0 g S z ds dv dt The total length of the liquid colum Liquid oscillation g / 07