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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.