8 julio, 2024

Dimensional analysis: concept, techniques, principle, exercises

what is the dimensional analysis?

He dimensional analysis It is a tool widely used in different branches of science and engineering to better understand the phenomena that imply the presence of different physical magnitudes. Magnitudes have dimensions and from these the different units of measurement are derived.

The origin of the concept of dimension is found in the French mathematician Joseph Fourier, who was the one who coined it. Fourier also understood that for two equations to be comparable, they must be homogeneous in terms of their dimensions. That is, you cannot add meters with kilograms.

Thus, dimensional analysis is responsible for studying the magnitudes, dimensions and homogeneity of physical equations. For this reason it is frequently used to check relationships and calculations, or to build hypotheses about complicated issues that can later be verified experimentally.

In this way, dimensional analysis is a perfect tool to detect errors in calculations by checking the consistency or inconsistency of the units used in them, focusing especially on the units of the final results.

In addition, dimensional analysis is used to design systematic experiments. It allows to reduce the number of necessary experiments, as well as to facilitate the interpretation of the results obtained.

One of the fundamental bases of dimensional analysis is that it is possible to represent any physical magnitude as a product of the powers of a smaller quantity, those known as fundamental magnitudes from which the others derive.

Fundamental quantities and dimensional formula

In physics, fundamental magnitudes are considered to be those that allow the others to be expressed in terms of these. By convention, the following have been chosen: length (L), time (T), mass (M), electric current (I), temperature (θ), light intensity (J) and light intensity. amount of substance (N).

On the contrary, the rest are considered derived magnitudes. Some of these are: area, volume, density, velocity, acceleration, among others.

A dimensional formula is defined as the mathematical equality that presents the relationship between a derived quantity and the fundamental ones.

Dimensional analysis techniques

There are various techniques or methods of dimensional analysis. Two of the most important are the following:

Rayleigh method

Rayleigh, who together with Fourier was one of the precursors of dimensional analysis, developed a direct and very simple method that allows dimensionless elements to be obtained. In this method the following steps are followed:

The potential character function of the dependent variable is defined.
Change each variable to its corresponding dimensions.
The homogeneity condition equations are established.
The np unknowns are fixed.
Substitute the exponents that have been calculated and fixed into the potential equation.
The groups of variables are moved to define the dimensionless numbers.

Buckingham method

This method is based on Buckingham’s theorem or pi theorem, which states the following:

If there is a homogeneous dimensional relationship between a number «n» of physical quantities or variables where «p» different fundamental dimensions are included, there is also a dimensionally homogeneous relationship between n-p, independent dimensionless groups.

Principle of dimensional homogeneity

Fourier’s principle, also known as the principle of dimensional homogeneity, affects the proper structuring of expressions that algebraically link physical magnitudes.

It is a principle that has mathematical consistency and affirms that the only option is to subtract or add to each other physical quantities that are of the same nature. Therefore, it is not possible to add a mass with a length, nor a time with a surface, etc.

In the same way, the principle affirms that, for the physical equations to be correct at the dimensional level, the total of the terms of the members of the two sides of the equality must have the same dimension. This principle allows to guarantee the coherence of the physical equations.

principle of similarity

The principle of similarity is an extension of the character of homogeneity at the dimensional level of physical equations. It is stated as follows:

Physical laws remain unchanged when faced with changes in the dimensions (size) of a physical fact in the same system of units, whether they are changes of a real or imaginary nature.

The clearest application of the principle of similarity occurs in the analysis of the physical properties of a model made on a smaller scale, to later use the results in the object in real size.

This practice is fundamental in fields such as the design and manufacture of aircraft and ships and in large hydraulic works.

Dimensional analysis applications

Among the many applications of dimensional analysis, the following can be highlighted.

Locate possible errors in the operations performed
Solve problems whose resolution presents some insurmountable mathematical difficulty.
Design and analyze models on a reduced scale.
Make observations about how possible modifications influence a model.

Furthermore, dimensional analysis is used quite frequently in the study of fluid mechanics.

The relevance of dimensional analysis in fluid mechanics is due to the difficulty in establishing equations in certain flows as well as the difficulty in solving them, making it impossible to obtain empirical relationships. For this reason, it is necessary to resort to the experimental method.

solved exercises

First exercise

Find the dimensional equation of velocity and acceleration.

Solution

Given that v = s / t, it is true that: [v] = L / T = L ∙T-1

Similarly:

a = v/t

[a] = L / T2 = L ∙T-2

second exercise

Determine the dimensional equation of momentum.

Solution

Since the amount of movement is the product between the mass and the velocity, it is fulfilled that p = m ∙ v

Therefore:

[p] = M ∙ L / T = M ∙ L ∙T-2

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