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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.
Reduced scale Models
How do we know that to geometries are geometrically identical?
If corresponding lengths are proportional!
Why Dimensionless Equations?
• Finite Volumes,• Partial Differential Equations,• Laboratory (reduced scale) Models.• How to extrapolate from the model to the
prototype?
Scales
• Navier-Stokes Equation :
j
i
jij
ij
i
x
u
xx
P
x
uu
t
u
• Scales:
ULtt
ULt
t
UuuU
uu
LxxL
xx
**
*ii
i*i
**
Replacing
• The same non-dimensional geometry and the same Reynolds and the same Froude guarantee the same non-dimensional solution
*
*
*
*
**
*
*
**
*
*
*
*
2*
*
**2*
**
*
*
*
*
*
*2**
**
2*
11
1
1
irj
i
jeij
ij
i
ij
i
jij
ij
i
ij
i
jij
ij
i
x
z
Fx
u
xRx
P
x
uu
t
u
x
z
U
gL
x
u
xULx
P
Ux
uu
t
u
U
L
x
zg
x
u
xL
U
x
P
Lx
uu
L
U
t
uU
ij
i
jij
ij
i
x
gz
x
u
xx
p
x
uu
t
u
*
*
*
*
**
*
*
**
*
* 11
irj
i
jeij
ij
i
x
z
Fx
u
xRx
P
x
uu
t
u
gL
UFr
gL
UFr
UDRe
2
Meaning of Reynolds and Froude
• Reynolds: Inertia forces/viscous forces• Froude: Inertia forces/gravity forces.• We can’t guarantee both numbers…..• What to do?
*
*
*
*
**
*
*
**
*
* 11
irj
i
jeij
ij
i
x
z
Fx
u
xRx
P
x
uu
t
u
What is the Reynolds Number?
• 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)
Reynolds: Inertia forces/viscous forces…
*
*
*
*
**
*
*
**
*
* 11
irj
i
jeij
ij
i
x
z
Fx
u
xRx
P
x
uu
t
u
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 non-dimensional time): The non-dimensional periods must be equal.
Wave Channel Experiments
• Real wave: T=10s• Model Scale: 1/10
DU
tt*
P
P
M
M DgD
tDgD
t
gDU
10
11
1
P
M
M
P
P
M
DD
D
D
tt
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 non-dimensional groups 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:
• We have 3 dimensions are: Length, Mass, Time) ),,,,,( DUw
Primary Variables and non-dimensional groups
• We need 3 primary variables:• Mass: ρ• Length: D• Time: U• How to build the non-dimensional groups?
333
222
111
LU
LU
LU
*
*
w*
333
222
111
*
*
*
LU
LU
LUw
333
222
111
13*
13*
13*22
LLTML
LLTMLL
LLTMLLMLT
1
111
1
13*22
2
31
1
111
LLTMLLMLT
2
21U
f w*
The 3 non dimensional groups are
e
*
*
w*
RUD
D
Uf
1
21 2
3 groups can be represented in a X-Y graph with several curves….
Advantages of dimensional analysis
• Permits the use of the solution measured 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 non-dimensional groups.
Generalised Bernoulli Equation
• It is a major use of the dimensional analysis:
• It is used to quantify the energy dissipated in a flow.
• The energy dissipated is each flow region is measured and non-dimensional parameters are computed.
EgzVPgzVP
2
2
1
2
2
1
2
1
Head losses in a pipe fully developed flow
• Performing a momentum budget one gets:
2*
2
2
1
2
2
1
2
1
2
1UegzVPgzVP
2*21 2
1UePP Pipe
D
Lfe
orD
fLe
Then
UePP
but
UfD
L
D
LPP
DLD
PP
Pipe
Pipe
Pipe
w
w
21'
*
21*
2*21
2212121
21
2
21
4
:2
1
:2
1444
Installation equation
2
2
212
21
22
12
2
2
2
1
2
22
2
1
2
:4
2
1
2
1
2
1
2
1
2
1
2
12
1
2
1
KQhH
Ck
se
D
Q
A
QU
kUg
zzUUgg
PPH
kUg
zUgg
PHzU
gg
P
kUgg
gzUP
g
w
g
gzUP
te
ii
ii
ii