hydrostatics - lth fluids at rest: • the pressure at any point in a fluid is the same in every...
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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: