Aula Teórica 18 & 19
Adimensionalização. Nº de Reynolds e Nº de
Froude. Teorema dos PI’s , Diagrama de
Moody, Equação de Bernoulli Generalizada e
Coeficientes de perda de carga.
Why Dimensionless Equations?
•
•
•
•
Finite Volumes,
Partial Differential Equations,
Laboratory Models.
How to extrapolate from the model to the
prototype?
Escalas
• Equação de Navier-Stokes:
ui
ui
P


 u j


t
x j
xi x j
• Escalas:
x
 x  x * L
L
u
ui*  i  ui  ui* U
U
t
t* 
 t  t * L
U
L
U
x* 
 ui

 x
j





Replacing
* 
*

u
U * u
1 P U   ui 
z
U

uj

 2 *  *  g *
*
t
L
x
L xi L x j  x j 
xi
*
i
2
*
i
*
j
*
* 
*


u

u
L ui*
1

P


gL

z
 i 

 u *j *i  

U t
x j
U 2 xi* UL x*j  x*j  U 2 xi*
*
* 
*
*

ui*

u

u

P
1

1

z
*
i
i 


u




j
*
*
*
* 
* 
t
x j
xi R e x j  x j  Fr xi*
• The same non-dimensional geometry and the
same Reynolds and the same Froude
guarantee the same non-dimensional solution
ui
ui
p


 u j


t
x j
xi x j
 ui

 x
j

 gz

 x
i

*
* 
*
*

ui*

u

u

P
1

1

z
*
i
i 


u




j
*
*
*
* 
* 
t
x j
xi R e x j  x j  Fr xi*
Re 
UD

U2
Fr 
gL
U
Fr 
gL
Meaning of Reynolds and Froude
*
* 
*
*

ui*

u

u

P
1

1

z
*
i
i 


u




j
*
*
*
* 
* 
t
x j
xi R e x j  x j  Fr xi*
•
•
•
•
Reynolds: Inertia forces/viscous forces
Froude: Inertia forces/gravity forces.
We can’t guarantee both numbers…..
What to do?
What is the Reynolds Number?
Reynolds: Inertia forces/viscous forces…
* 
*

u
P
1   ui  1 z
* u
uj
 * 

* 
* 
t
x
xi R e x j  x j  Fr xi*
*
i
*
*
i
*
j
*
• When it is high, the diffusive term becomes
less important in the equation and can be
neglected. Then the Reynolds number looses
importance, i.e. the non-dimensional solution
becomes independent of Re (see next slide)
What is the Froude Number?
• The Froude number is the square of the ratio
between the flow velocity and the velocity of a free
surface wave in a Free surface flow.
• The geometry is similar only if the free surface wave
velocity propagation is similar in the model and in
the prototype. So the Froude number must be the
same in the model and in the prototype.
• How to calculate the period of the waves in the
model and in the prototype (using the nondimensional time): The non-dimensional periods
must be equal.
Wave Channel Experiments
• Real wave: T=10s
• Model Scale: 1/10
U
t t
D
U  gD
*
 gD 
 gD 
  tP 

t M 

 D 
D

M

P
1
DP
tM
DM


1
tP
DP
DM
The ππ’s Theorem
• We can study a process with N independent
variables and M dimensions building (N-M)
non-dimensional groups.
• M Primary variables are chosen for building
one non-dimensional group using the
remaining variables.
• Primary variables must include all the problem
dimensions and it must be impossible to build
a non-dimensional group with them.
Shear stress in a pipe
• Shear stress depends on:
• Velocity gradient, fluid properties and pipe
material (roughness) . The velocity gradient
depends on the average velocity and pipe
diameter. Fluid properties are the specific mass
and the viscosity.
• The variables involved are:
( w ,  , ,U , D,  )
• We have 3 dimensions are: Length, Mass, Time)
Primary Variables and nondimensional groups
•
•
•
•
•
We need 3 primary variables:
Mass: ρ
Length: D
Time: U
How to build the non-dimensional groups?
 
w
*
1
1 
1
 U L

*
    
 U L

*
    
 U L
2
3
2
3
2
3
1
  LT 
L   ML  LT  L
   ML  LT  L
1 
1
2
2  2
3
*
3  3
  U L
*
2
*

 1  31  1   1
 2   1
3  2
1  2
2
1  3
3
3  3
*
 LT 
 3 1
MLT L   ML
1  1
2
 3 1
1  1
MLT L   ML
   U L
*
*
2 2
w   U L
*
1  1
L 1
L 1
 f 
*
w
1
U 2
2
The 3 non dimensional groups are
 f 
*
 
*
* 

D
w
1
U 2
2

UD

1
Re
3 groups can be represented in a X-Y graph with several
curves….
Advantages of dimensional analysis
• Permits the use of the solution in a system to
obtain the solution in other geometrically
similar systems,
• It is independent of the fluid. It depends on
non-dimensional parameters,
• It permits the reduction of the number of
independent variables because the
independent variables became nondimensional groups.
Equação de Bernoulli Generalizada
• É a equação que mais uso faz dos resultados de
laboratório e da análise adimensional.
1
1




2
2
P


V


gz

P


V


gz



  E
2
2

1 
2
• É útil se podermos prever a dissipação de
energia.
• A energia dissipada em cada região do
escoamento pode ser adimensionalizada e
determinada a partir de ensaios de laboratório.
Energy dissipation
En MLT 2 L
* 
 
E 


e

U
L
3
Vol
L
E
*
e    
 U L
E
*
e 
1
2
U
2
1
1




2
2
* 1
2
P


V


gz

P


V


gz

e

U




 2
2
2

1 
2
Coeficiente de perda de carga num
tubo
1
1




2
2
* 1
2
 P  V  gz    P  V  gz    e U
2
2
2

1 
2
P 1  P 2
 e
*
Pipe
1
U 2
2
• Fazendo um balanço de força e de quantidade
de movimento:
P1  P2 
D 2
  wDL1 2
4
4 w L1 2
4 L1 2
1
P1  P2  

f
U 2
D
D
2
but :
P1  P2   e
*
Pipe
Then :
e
*
Pipe
4 fL1 2

D
or
*
ePipe
f ' L1 2

D
1
U 2
2
Equação de uma instalação
1
1




2
2
 P  U  gz 
 P  U  gz 
1
2
2

1 w 
2


  U 2 ki
g
g
g
i 2g
 P

 P

1 2
1 2
1


U  z   H  

U  z    U 2 ki
 g 2 g
1
 g 2 g
2 i 2 g


P P
1
U 2 2  U1 2  z2  z1    1 U 2 ki
H   2 1 
2g
i 2g
 g

U
Q
Q

A D 2
4
se :
k  C te
H  h  KQ 2
Ponto de funcionamento de uma
bomba
H
Q
• Ver sebenta (capítulo IV), White (capítulos 5 e 6)
• Problemas Aula Prática 9