The Importance of Dimensional Analysis
How does the diameter of a tube effect the flow of fluid through it? In order to answer this question we may try running experiments for several different tube diameters. Another approach would be to only test one diameter and recognize that the flow will be similar, but scaled to the pipe diameter. This principle of flow similarity is the basis of dimensional analysis. We can understand a lot about a flow by testing scale models that exhibit geometric, kinematic, and dynamic similarity with the actual article to be used.
- Geometric Similarity: The physical dimensions of the flow should all be scaled by a constant factor.
- Kinematic Similarity: Velocity of the flow at given points should all be scaled by a constant factor.
- Dynamic Similarity: Forces within the flow at given points should all be scaled by a constant factor.
Buckingham Pi Theorem
The Buckingham Pi theorem allows a method for determining non-dimensional parameters. Non-dimensional parameters are useful for establishing flow similarity, for instance two flows with the same Reynold’s numbers will have some dynamic similarity. Buckingham Pi theorem solves for the possible non-dimensional parameters that can be made given a set of dimensional properties. A set of n dimensional properties may be related in some form and can be expressed as:
f(q_1,q_2,…,q_n)=0
This relationship can be transformed into n-k dimensionless groups, or Pi groups. The number k is the number of unique dimensions within the set of properties.
F(\Pi_q,\Pi_2,…,\Pi_{n-k})=0
The process for determining dimensionless parameters is as follows:
- Choose a set of k dimensional parameters that includes at least one of each dimension.
- Create all possible combinations with the above set of dimensional parameters and all remaining dimensional parameters.
- Determine what exponents are needed to non-dimensionalize each combination.
Important Dimensionless Parameters
- Reynold’s Number: \frac{\rho VL}{\mu}
- Prandtl Number: \frac{\nu}{\alpha}
- Strouhal Number: \frac{fL}{V}
- Nusselt Number: \frac{hL}{k}
- Stanton Number: \frac{h}{\rho C_pV}
- Mach Number: \frac{V}{c}