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

Energy dissipation

2

*

*

*3

2

21

U

Ee

LU

Ee

LUeL

LMLT

Vol

EnE

2*

2

2

1

2

2

1

2

1

2

1UegzVPgzVP

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

Pimp working point

Q

H

• Ver sebenta (capítulo IV), White (capítulos 5 e 6)• Problemas Aula Prática 9