Chapter 3 2

PT Nng lềng Chẽng 3: ẻng lác hc lu chòt Ph¶n 2: Ph˜Ïng trình n´ng l˜Ịng cho dịng l˛ t˜ng dịng th¸c Bài gi£ng cıa TS Nguyπn Qc fi nguyenquocy@hcmut.edu.vn Ngày tháng 10 n´m 2015 / 19 PT Nng lềng Nẻi dung cản nm Phẽng trỡnh nng l˜Ịng cho dịng l˛ t˜ng Ph˜Ïng trình n´ng l˜Ịng cho dịng th¸c Các ˘ng dˆng cÏ b£n cıa PT n´ng l˜Òng: bÏm, turbine, o v™n tËc, l˜u l˜Òng / 19 PT N´ng l˜Ịng cho dịng l˛ t˜ng: Ph˜Ïng trình Bernoulli Bernoulli equation Conservation of Energy for Inviscid Flows A cylindrical particle of inviscid fluid , A streamline with coordinates shown Newton’s 2nd law: m~a F external forces: pressure, and weight m dv dt dA p ds s mg cos ✓ / 19 PT N´ng l˜Òng dA ds v cho dịng l˛ t˜ng: Ph˜Ïng trình Bernoulli d–V v s, t dv dt for steady flows cos ✓ v v ds v dt dt t s dt t dv dv p dp : v , dt ds s ds v v s dz ds dv ds dp d– V ds mv dv dp d– V mg dz dp d– V mg mv mg dz ds Integrate along the streamline m v dv dz / 19 PT N´ng l˜Òng V2 for incompressible fluids m m dp d– V , d– V V2 pd– V per unit area /volume p ⇢gz ⇢gz p V2 ⇢ ⇢ V2 mgz const const.: mgz const per unit weight const : hydrostatic pressure : static pressure : cho dịng l˛ t˜ng: Ph˜Ïng trình Bernoulli dynamic pressure z p ⇢g V2 2g z H p ⇢g V2 2g H const : pressure head : velocity head : potential head : total head / 19 PT N´ng l˜Òng cho dịng l˛ t˜ng: Ph˜Ïng trình Bernoulli Be reminded: of a fluid particle z p ⇢g V2 2g H const or along a streamline, from Point to Point 2: z1 p1 ⇢g V12 2g z2 p2 ⇢g V22 2g Bernoulli equation only VALID for: Inviscid fluids Steady flows Along streamlines Incompressible flows / 19 PT N´ng l˜Ịng cho dịng l˛ t˜ng: Ph˜Ïng trình Bernoulli Bernoulli equation Across the streamline ˜Ìng dịng thØng: R Ÿng dˆng: »ng o áp z p const.: qui lu™t thu tænh / 19 PT N´ng l˜Ịng cho dịng l˛ t˜ng: Ph˜Ïng trình Bernoulli Bernoulli equation Example of stagnation points Stagnation point Stagnation streamline Stagnation point (a) V2 = (2) (b) V1 = V0 (1) z1 z1 p1 ⇢g V12 2g z2 p2 ⇢g V22 2g z2 , V2 0, p2 p1 ⇢V1 2 (1) Áp st d¯ng = Áp st tỉnh + p suòt ẻng / 19 PT Nng lềng cho dịng l˛ t˜ng: Ph˜Ïng trình Bernoulli Bernoulli equation Exchange of kinetic, Potential, and Pressure Energy A2 A1 v2 v1 p2 p1 / 19 PT N´ng l˜Ịng cho dịng l˛ t˜ng: Ph˜Ïng trình Bernoulli Bernoulli effect 10 / 19 PT N´ng l˜Ịng cho dịng l˛ t˜ng: Ph˜Ïng trình Bernoulli Application of Bernoulli Equation Ventury tube for measuring flow rate z1 V12 2g p1 Assume z1 p2 z2 V22 2g z2 : horizontally v2 v1 p1 p2 2g v1 Q v2 A2 A1 , C A2 v2 p1 p2 H A2 C A2 A1 2gH C :(emperical) coe due to energy loss 11 / 19 PT N´ng l˜Ịng cho dịng l˛ t˜ng: Ph˜Ïng trình Bernoulli Application of Bernoulli Equation Pitot tube for measuring flow velocity V1 = 100 mi/hr (2) (1) Pitot-static tube pA ⇢g vA Pitot’s 1st exp 2g pB some loss: vA vA 2g pA ⇢g pB ⇢g 2g pB pC ⇢g 2gH Cv 2gH 12 / 19 PT N´ng l˜Ịng cho dịng l˛ t˜ng: Ph˜Ïng trình Bernoulli Application of Bernoulli Equation Flow through a small hole VA zA for large tank, zB H VB 2gH due to some loss VB Cv 2gH due to contraction of the jet at exit: ac Cc a actual flow rate Q zA patm VA2 2g zB patm VB2 2g Cc aCv VB Q Cc Cv a 2gH Ca 2gH C: Coe of discharge 13 / 19 PT N´ng l˜Ịng cho dịng l˛ t˜ng: Ph˜Ïng trình Bernoulli Application of Bernoulli Equation Flow through a small hole Coe of contraction dj dh CC = 0.61 CC = 1.0 CC = A j /A h = (dj /dh)2 CC = 0.61 CC = 0.50 14 / 19 PT N´ng l˜Ịng cho dịng l˛ t˜ng: Ph˜Ïng trình Bernoulli Application of Bernoulli Equation Measuring water flow rate by WEIRs Consider a minute area b.dz as an orifice: v 2gz dQ C b.dz 2gz H Q Cb 2g zdz Cb 2g H 3 15 / 19 PT N´ng l˜Ịng cho dịng th¸c Energy Equation Energy Equation for Viscous Flows Bernoulli equation to be modified for real incompressible fluid: introducing a term to account energy loss, hloss : energy loss by a unit weight of fluid, due to: viscous friction, turbulent shear stress, local loss at valves, fittings, correcting velocity head for real velocity distribution on a wetted area flows through hydraulic machines: PUMPS, TURBINES 16 / 19 PT N´ng l˜Ịng cho dịng th¸c Kinetic energy correction factor Nonuniform distribution: m KE mg ⇢dA v t mv KE A ↵ ↵ V2 2g ⇢ v dA t ↵ ⇢V A t V : averaged velocity at the section, hence: ↵ A A v V dA 17 / 19 PT N´ng l˜Ịng cho dịng th¸c Modified energy equation for flows through PUMPS z1 p1 ↵1 V12 Hb 2g z2 p2 ↵2 V22 2g hloss Hp is the energy supplied to a unit weight of fluid, or Pump head Cơng st bÏm Nb QHb Nb Cơng st Îng cÏ N c ⌘b Outlet Pipe Pump Elbow Tee Valve b : Hiêu suòt bẽm (%) Inlet 18 / 19 PT N´ng l˜Ịng cho dịng th¸c Modified energy equation for flows through TURBINES Ht is the energy taken from a unit weight of fluid, or Turbine head z1 p1 ↵1 V12 Ht 2g z2 p2 ↵2 V22 2g hloss Cụng suòt turbine Nt Cụng suòt ẻng cẽ N QHt c Nt t t : Hiêu suòt turbine (%) 19 / 19