Hydrostatics - LTH Fluids at rest: • the pressure at any point in a fluid is the same in every...

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1 Review Of Basic Fluid Mechanics Hydromechanics VVR090 Hydrostatics Fluids at rest: the pressure at any point in a fluid is the same in every direction in a continuous fluid with constant density the pressure increases linearly with depth and the pressure is the same along horizontal planes dp g dz = −γ = −ρ 1 1 ( ) p p z z h = −γ Constant density:

Transcript of Hydrostatics - LTH Fluids at rest: • the pressure at any point in a fluid is the same in every...

Page 1: Hydrostatics - LTH Fluids at rest: • the pressure at any point in a fluid is the same in every direction • in a continuous fluid with constant density the pressure ...

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Review Of Basic Fluid Mechanics

Hydromechanics VVR090

Hydrostatics

Fluids at rest:

• the pressure at any point in a fluid is the same in every direction

• in a continuous fluid with constant density the pressure increases linearly with depth and the pressure is the same along horizontal planes

dp gdz

= −γ = −ρ

1 1( )p p z z h− = −γ − = γ

Constant density:

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(pressure may be expressed in height of a fluid column; e.g., mm Hg, m H2O)

ph =γ

Pressure head:

p z+γ

Static head (piezometric head):

Hydrostatic pressure distribution: .p z const+ =γ

Pressure Definitions

Force on a plane area

Total force: cF h A= γ

Center of pressure: cp c

c

Iy yy A

= +

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Force on Curved Surfaces

Look at horizontal (Fx) and vertical components (Fz) separately.

2 2tot x zF F F= +Total force:

Flowing Fluid

Basic equations for conservation of:

• mass (continuity equation)

• energy (involves potential and kinetic energy + work)

• momentum (involves momentum fluxes + forces)

Analysis through control volumes

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Classification of flow types

• 1-D, 2-D, and 3-D

• real and ideal fluid

• incompressible - compressible

• steady – unsteady

• laminar – turbulent

• established – unestablished

• uniform – non-uniform

• subcritical – supercritical

• subsonic - supersonic

Continuity Equation

1-D, steady, compressible:

1-D, steady, incompressible:

1 1 1 2 2 2AV A Vρ = ρ

1 1 2 2AV A V=

stream tube

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1 1 2 21 2

1 22 2M Lp V p Vz h z h

g g⎛ ⎞ ⎛ ⎞

+ + + = + + + +⎜ ⎟ ⎜ ⎟γ γ⎝ ⎠ ⎝ ⎠

Energy Equation

No losses:

.2

p Vz constg

⎛ ⎞+ + =⎜ ⎟γ⎝ ⎠

Bernoulli’s equation

With energy losses (hL) and energy input (hM):

Definitions in Fluid Flow

• Pressure head (p/g)

• Elevation head (z)

• Static head (piezometric head) (p/g + z)

• Velocity head (V2/2g)

• Total (energy) head (p/g + z + V2/2g)

• Hydraulic grade line

• Energy line

.p z const+ =γ

Plane and parallell streamlines

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Energy and Hydraulic Grade Lines

exit loss

flow between reservoirs

Momentum Equations

Newton’s second law (vector relationship)

2 1

2 1

2 1

( )

( )

( )

x x x

y y y

z z z

F Q V V

F Q V V

F Q V V

= ρ −

= ρ −

= ρ −

∑∑∑

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Pipe Flow

Laminar – turbulent flow

Characterized by Reynolds number:

Re DV DVρ= =

μ ν

Critical Re-value for transition to turbulence: 2000 (pipe flow)

Head Loss in Pipes

Moody’s diagram

Wall shear stress

2 2

2 4 2LL U L Uh f fD g R g

= =

ARP

= (hydraulic radius)

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Minor Losses in Pipelines

2

2L LVh K

g=General expression:

• expansion (e.g., exit)

• contraction (e.g., entrance)

• bends

• fittings (e.g., valves)

Pipe Configurations

Pipes in parallell:

1 2 3

1 2 3

......L L L L

Q Q Q Qh h h h= + + += = = =

Pipes in series:

1 2

1 2 3

......L L L L

Q Q Qh h h h= = == + + +

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Pumps and Turbines in the System

Pumps supply the system with energy

Turbines extract energy from the system

Pumping Between Two Reservoirs

Pump and system curve

2P P PH z C Q= Δ +

System curve: