USEFUL FORMULAE - FLUID MECHANICS
GENERAL CONSERVATION LAWS FOR STATIONARY, FINITE CONTROL VOLUMES

Mass conservation:

0=∀

∂ρ + ∂t

out flows

∑ (ρVA)

out

−

in flows

∑ (ρVA )

in

Energy conservation:  V2 ∂ρ  u + gz +  2  +W = ∀   Q ∂t     +

  V2  ρVA  u + gz + ∑   2 out flows  

   −   out

  V2  ρVA  u + gz + ∑  2 in flows  

     in

Linear momentum conservation:

∑F = ∀

∂ρ V + ∂t

out flows

∑ (ρVA V )

out

−

in flows

∑ (ρVA V )

in

Angular momentum conservation:

∑T = ∀

∂ρ (r × V ) + ∂t

out flows

∑ (ρVA (r × V ))

out

−

in flows

∑ (ρVA (r × V ))

in

Steady incompressible fluid flow in systems with one inlet and one outlet:

Conservation law Mass Energy Linear Momentum Angular Momentum

Equation

(p

) (  ∑ F = m (V − V )
1

  m1 = m2 or ρ1V1 A1 = ρ 2V2 A2

  ρ + 1 V1 2 + gz1 − p 2 ρ + 1 V2 2 + gz 2 + u1 − u 2 +q + w=0 2 2
2

)

 ∑ T = m (r Vθ
2

,2

− r1Vθ ,1 )

1

DIMENSIONAL ANALYSIS AND PHYSICAL SIMILARITY Dynamic similarity requirement for flow systems governed by Π-groups

Π 1 and Π 2 :

(Π 1 )M = (Π 1 )P

and

(Π 2 )M =(Π 2 )P

Useful dimensionless groups in fluid dynamics: Name Euler number, Eu Reynolds number, Re Π-group p ρV 2 ρVL µ Interpretation pressure/inertia forces inertia/viscous forces

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Froude number, Fr Weber number, We Mach number, M Force coefficient: F ρV 2 L2

V

gL

inertia/gravity forces inertia/surface tension inertia/elastic force

ρV 2 L σ V a

Rotating fluid machinery:

Power Q gH , , 2 2 3 5 3 ρN D ND N D

TURBOMACHINERY

Pumps T = ρQ(r2Vθ 2 −r1Vθ 1 ) Power =ωT gH E =u 2Vθ 2 −u1Vθ 1

Turbines T = ρQ(r1Vθ 1 −r2Vθ 2 ) Power =ωT gH E =u1Vθ 1 −u 2Vθ 2

η = gH gH E
Pumps and turbines:

η = gH E gH all machines U = ωr Q radial machines Vr = 2πrb Q axial machines Va = 2 π (rt − rh2 )

PIPE FLOWS

Extended Bernoulli’s theorem:

p1

ρ

+

V12 p V2 + gz1 + gh pump = 2 + 2 + gz 2 + ghL 2 2 ρ

Darcy-Weisbach equation: Power requirement:

L V2 ghL = f D 2
 Power =m gh pump η

System characteristic:

 p  Q2  1 p 1   2 − 2  + ghL  gh pump =  2 + gz 2  −  1 + gz1  +  ρ  ρ  A  2  2 A1    

Transition from laminar to turbulent flows: Re= ρVD µ =2000

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