CU06997 Fluid Dynamics

Principles of fluid flow 2.5 Application of the conservations laws to fluid flows (page 25-32)

2.6 Application of the energy equation (page 32 -36)

2.8 Velocity and discharge measurement (page 42 – 48)

1

Flowing water and energy

][2

2

1111 m

g

uyzH

u1

Reference /datum [m]

Surface level [m]

Total head H [m]

P1

z1

y1

u12/2g Velocity head [m], [snelheidsh..]

y = Pressure head [m]

[drukhoogte]

z = Potential head [m]

. [plaatshoogte]

1

Bernoulli’s Equation, no energy losses

Pressure Head[m] [drukhoogte]

Potential Head[m] [plaatshoogte]

Velocity Head[m] [snelheidshoogte]

𝑦1 + 𝑧1 +𝑢1

2

2𝑔= 𝑦2 + 𝑧2 +

𝑢22

2𝑔= 𝐻 = 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡[𝑚]

y =p

ρ∙g=

𝑧 =

u2

2g=

2

Bernoulli’s law (without energy losses)

u2>0

Reference[m]

Surface Level [m]

Total Head [m]

P2

z2

y2

u22/2g Velocity head [m]

constant22

2

333

2

22211

g

uzy

g

uzyzy

P3 z3

y3

u32/2g

u3>u2 u1=0

P1

z1

y1

2

Pitot

3

Pitot

ℎ =𝑢2

2 ∙ 𝑔

3

Pitot

𝑢 = 2 ∙ 𝑔 ∙ ℎ2

𝑢 = Fluid Velocity [m/s]

𝑔 = earths gravity [m/s2]

h = Difference in pressure [m]

3

u=0

Flowing water and energy

u1

Reference / datum [m]

Surface Level [m]

Total Head [m]

P1

z1

h1

v12/2g Velocity head [m]

Prandtl buis

][2

2

1111 m

g

uyzH

Torricelli

xgu 22

g

uzy

g

uzy

22

2

222

2

111

u1=0 m/s

y2=0 m

y1+z1-z2=x

4

Small orifice [Kleine doorlaat]

4

Small orifice [Kleine doorlaat]

4

Small orifice

𝑄 = 𝐶𝑣 ∙ 𝐶𝑐 ∙ 𝐴0 ∙ 2𝑔 ∙ ℎ2

𝑄 = Flow rate [m3/s]

𝐴 = Wetted Area [m2]

𝐶𝑣 = velocity coefficient (0,97-0,99) [-]

𝐶𝑐 = contraction coefficient (0,61-0,66) [-]

𝑔 = earths gravity [m/s2]

h = Difference in pressure [m]

4

Large orifice

𝑄 =2

3∙ 𝑏 ∙ 2𝑔 ∙ (ℎ2

32 − ℎ1

32)

2

𝑄 = Flow rate [m3/s]

𝑏 = Width orifice [m2]

𝑔 = earths gravity [m/s2]

h1 = Difference in pressure from top [m]

h2 = Difference in pressure from bottom [m]

5

Bernoulli’s law, with head loss

u1

Reference [m]

Surfacelevel y +z [m]

Total Head H [m]

P1

z1

y1

u12/2g

P2 z2

y2

u22/2g

u2>u1

21

2

222

2

111

22 H

g

uzy

g

uzy

Head loss [m] ΔH

Velocity Head [m]

6

Pipe with head loss

41

2

44

2

11

22 H

g

vh

g

vh

4411 AvAvQ

Pressure

Head

Total

Head

Head loss

6

Open channel

212

2

221

2

11

22 Hz

g

uhz

g

uh

2211 AvAvQ

Head loss

Reference line

6

Bernoulli expressed in m (head)

𝐸𝑡𝑜𝑡𝑎𝑙 = 𝐸𝑝𝑜𝑡𝑒𝑛𝑡𝑖𝑎𝑙 + 𝐸𝑘𝑖𝑛𝑒𝑡𝑖𝑐

= 𝑚 ∙ 𝑔 ∙ 𝑑 + 12𝑚 ∙ 𝑢2

= 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡 [𝐽 = 𝑁𝑚]

𝐸𝑡𝑜𝑡𝑎𝑙

𝑚∙𝑔= 𝑑 +

𝑢2

2𝑔=

𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡

𝑚∙𝑔 [m]

𝑑 = 𝑧 + 𝑦 = 𝑧 +𝑝

𝜌 ∙ 𝑔 [𝑚]

𝑯 = 𝒚 + 𝒛 +𝑽𝟐

𝟐∙𝒈 [m] Total Head

7

𝐻 = 𝑦 + 𝑧 +𝑢2

2∙𝑔 [m]

You also could express the energy in Pa (N/m2) instead of m.

𝑝 = 𝜌 ∙ 𝑔 ∙ 𝑦

If you combine 1 en 2 by multiply al parameters with 𝜌 ∙ 𝑔 (they don’t

change)

𝜌 ∙ 𝑔 ∙ 𝐻 = 𝜌 ∙ 𝑔 ∙ 𝑦 + 𝜌 ∙ 𝑔 ∙ 𝑧 + 𝜌 ∙ 𝑔 ∙𝑢2

2 ∙ 𝑔

𝐻 = 𝑝 + 𝜌 ∙ 𝑔 ∙ 𝑧 + 𝜌 ∙𝑢2

2 [𝑃𝑎]

7

u1

Reference /datum [m]

Surface level

H energy [Pa]

P1

z1

p1

𝜌 ∙𝑢2

2

p = pressure [Pa]

𝜌 ∙ 𝑔 ∙ 𝑧 = “potential” [Pa]

7

Bernoulli expressed in Pa (pressure)

𝐻 = 𝑝 + 𝜌 ∙ 𝑔 ∙ 𝑧 + 𝜌 ∙𝑢2

2 [𝑃𝑎]