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 : Scales:

Replacing The same non-dimensional geometry and the same Reynolds and the same Froude guarantee the same non-dimensional solution

Meaning of Reynolds and Froude Reynolds: Inertia forces/viscous forces Froude: Inertia forces/gravity forces. We can’t guarantee both numbers….. What to do?

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…

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

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)

Primary Variables and non- dimensional groups We need 3 primary variables: Mass: ρ Length: D Time: U How to build the non-dimensional groups?

The 3 non dimensional groups are 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.

Energy dissipation

Head losses in a pipe fully developed flow Performing a momentum budget one gets:

Installation equation

Pimp working point Q H

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