Similarity Hypothesis, Dimensional Analysis, and Dimensionless Numbers

3 Similarity Hypothesis, Dimensional Analysis, and Dimensionless Numbers 

Scale-up of apparatus and reactors is often carried out in such a way that the industrial unit and the model are geometrically similar. Geometrical similarity means that any length ratio is the same in both units. Similar pipes have the same ratio of length L based on the diameter d. Fluid motion is the result of forces acting on a fluid element. Remember that the Reynolds number is the ratio of inertia forces based on forces due to viscosity. The Navier-Stokes equations show that forces due to gravitation and pressure fields can be effective. The ratio of inertia forces based on gravitational forces is known as the Froude number  

This number is mainly important for liquids because the density of gases at low pressure is small. The rise or fall of solid or fluid particles is the result of the difference of buoyancy and gravity forces and proportional to the density difference IA a Therefore, an extended version of the Fronde number is known  

Since the velocity is present in both the Reynolds and the Fronde numbers it is practicable to combine both numbers in such a way that the velocity is eliminated. By this, the Archimedes number v4r can be derived  

The surface or interfacial tension ct is an eneigy per unit interfacial area for neighboring fluid phases with different densities. The resulting pressure depends on the two main radii and R2 of a convex or concave interface  

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