Chapter 3 2

Bernoulli equationConservation of Energy for Inviscid Flows Acylindrical particle of inviscid fluid , A streamline with coordinates shown Newton’s 2nd law: external forces: pressure, and

Ch˜Ïng 3: Îng l¸c hÂc l˜u chßt

Ph¶n 2: Ph˜Ïng trình n´ng l˜Òng cho dòng l˛ t˜ng và dòng th¸c

Bài gi£ng cıa TS Nguyπn QuËc fi nguyenquocy@hcmut.edu.vn

Ngày 1 tháng 10 n´m 2015

NÎi dung c¶n n≠m

Ph˜Ïng trình n´ng l˜Òng cho dòng l˛ t˜ng

Ph˜Ïng trình n´ng l˜Òng cho dòng th¸c

Các ˘ng dˆng cÏ b£n cıa PT n´ng l˜Òng: bÏm, turbine, o v™n tËc, l˜u l˜Òng

Bernoulli equation

Conservation of Energy for Inviscid Flows

Acylindrical particle of inviscid fluid ,

A streamline with coordinates shown

Newton’s 2nd law:

external

forces: pressure, and weight

m dv

p

sds mg cos ✓

dA ds d–V

v v s, t

dv dt

v

tdt

v s

ds dt

v

tdt v

v s for steady flows : dv

dt v

dv

ds,

p s

dp ds cos ✓ dz ds

mv dv ds

dp

dsd–V mg

dz ds

mv dv dp d–V mg dz Integrate along the streamline

2

2 dp d–V mgz const.

for incompressible fluids , d–V const.:

mV

2

2 pd–V mgz const.

per unit area /volume

⇢gz p ⇢V2

2 const.

⇢gz : hydrostatic pressure

p : static pressure

⇢V2

2 : dynamic pressure

per unit weight

⇢g

V2 2g H const. p

⇢g : pressure head

V2 2g : velocity head

z : potential head

H : total head

Be reminded: of a fluid particle

⇢

V2

or along a streamline, from Point 1 to Point 2:

z1 p1

⇢

V2 1

p2

⇢

V2 2

2g

Bernoulli equation only VALID for:

Inviscid fluids

Steady flows

Along streamlines

Incompressible flows

Bernoulli equation

Across the streamline

˜Ìng dòng thØng: R z p const.: qui lu™t thu tænh

Ÿng dˆng: »ng o áp

Bernoulli equation

Example of stagnation points

Stagnation point

(a)

Stagnation streamline

Stagnation point (b)

V2 = 0 V1= V0

(1) (2)

z1 p1

⇢g

V12 2g z2

p2

⇢g

V22 2g

z1 z2, V2 0, p2 p1 ⇢V1 2

(1)

Áp sußt d¯ng = Áp sußt tænh + Áp sußt Îng

Bernoulli equation

Exchange of kinetic, Potential, and Pressure Energy

A2 A1

v2 v1

p2 p1

Bernoulli effect

Application of Bernoulli Equation

z1 p1 V12

2g z2

p2 V22 2g Assume z1 z2: horizontally

v2 v1

2g

p1 p2

v1 v2A2 A1, p1 p2 H

Q CA2v2 C A2

1 A2 A1 2

2gH

C :(emperical) coe due to energy loss

Application of Bernoulli Equation

Pitot’s 1st exp

V1 = 100 mi/hr (2) (1) Pitot-static tube

pA

⇢g

vA2

2g

pB

⇢g

vA 2gpB pA

⇢g 2gpB pC

some loss: vA Cv 2gH

Application of Bernoulli Equation

Flow through a small hole

zA patm V2

A

2g zB

patm V2

B

2g

VA 0 for large tank,

zA zB H

due to some loss VB Cv 2gH due to contraction of the jet at exit:

ac Cc a actual flow rate

Q CcaCvVB CcCva 2gH

Q Ca 2gH

C: Coe of discharge

Application of Bernoulli Equation

Flow through a small hole

Coe of contraction

d h

d j

C C = 0.61

C C = 1.0

C C = A j /A h = (d j /d h) 2

Application of Bernoulli Equation

Measuring water flow rate by WEIRs

Consider a minute area b.dz as an orifice:

dQ C b.dz 2gz

3Cb 2g H

3 2

Energy Equation

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 headfor real velocity distribution on a wetted area flows throughhydraulic machines: PUMPS, TURBINES

Kinetic energy correction factor ↵

Nonuniform distribution:

m ⇢dA v t

2mv

2 A

1

2⇢ v

↵1

2⇢V

V : averaged velocity at the

section, hence:

A A

v V

3

dA

KE

V2

2g

Modified energy equation

for flows through PUMPS

z1 p1

↵1V12 2g Hb z2

p2

↵2V22 2g hloss

Hp is the energy supplied to a unit weight of fluid, orPump head

Công sußt bÏm Nb QHb

Công sußt Îng cÏ N c Nb

⌘b

⌘b: Hiªu sußt bÏm (%)

Pump Tee Valve Outlet Elbow

Inlet

Pipe

Modified energy equation

for flows through TURBINES

Ht is the energy taken from a unit weight of fluid, orTurbine head

z1 p1

↵1V12 2g Ht z2

p2

↵2V22 2g hloss

Công sußt turbine Nt QHt

Công sußt Îng cÏ N c Nt⌘t

⌘t: Hiªu sußt turbine (%)