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Formula Sheet CHE 312 Fluid Mechanics (Fall 2010)Final Exam, December 20, 9-11 am
Print this document on a single sheet of paper and bring it to the exam; there will be no spare sheets at the exam. You are
allowed to add information on the side of the sheet you printed on; the reverse side should be blank
Newtonian fluids: x xy
dv
dyτ µ = xy
τ shear stress, µ viscosity, xdv
dy velocity gradient
Hydrostatics
p g z
ρ ∂ = −∂
p hydrostatic pressure, ρ density, g gravitational acceleration, z points up
z f F V g ρ = z
F buoyancy force, V displaced fluid volume, f
ρ fluid density
Total mass balance:, ,m in m out
dm
dt φ φ = − total mass m is a conserved quantity
Steady state mechanical energy balance (Bernoulli equation)
21
2
nf
fr
m
W pv gz e
ρ φ
∆ + + = −
ɺ
∆ is “out minus in”; nf W ɺ work rate not related to flow;
fr e frictional loss.
Frictional loss
In a straight pipe or channel with length L and (hydraulic) diameter 4
h
A D
W
≡ :2
2 fr
h
Le fU
D
= .
V U A
φ = : average velocity; A: cross sectional area; W : wetted perimeter.
f : Fanning friction factor, f is a function of Re hUD ρ
µ = , and the relative wall roughness
h D
ε .
Special cases: laminar flow in a round tube:16
Re f = ; turbulent flow with smooth walls:
1/ 44 0.316Re f
−= .
Loss coefficients K :2
2 fr
U e K = .
Momentum balance
( ) , ,m in m out d mdt
φ φ = − + ∑in outv v v F v is the velocity vector ; ∑ F is the sum of forces acting on the system.
Drag force D
F on a particle moving relative to fluid
1
2 D f
C A ρ ⊥
=D
F ∆v ∆v D
C drag coefficient; A⊥
area ‘seen’ by the flow; = −f p∆v v v .
DC depends on the shape of the particle and Re
f pd ρ
µ =
∆v,
pd is the particle size.
Spherical particles (diameter pd , and therefore
2
4 p
A d π
⊥ = )
For Re<1 Stokes law applies: 3 pd π µ =DF ∆v , which is the same as
24
Re DC = .
A more general correlation: ( )0.687241 0.15Re if Re 1000; 0.44 if Re 1000
Re D D
C C = + < = ≥ .
Ergun equation (frictional loss in a packed bed with porosity ε made of spheres all having diameter pd )
( ) ( )2
sup2
sup3 2 3
11 11.75 150 fr
p p
ve L v L
d d
µ ε ε
ε ρ ε
−−= + sup
V v A
φ = ; L is the bed length; A its cross sectional area.
Darcy equation & permeability for flow in a porous medium
( )d p gzv
dx k
ρ µ += − k is permeability (units m
2); v is superficial velocity.