Dimensional analysis

Dimensional analysis is a mathematical tool often applied by engineering sciences to simplify a problem by reducing the number of variables to the smallest number of "essential" parameters. Systems which share these parameters are called similar and do not have to be studied separately.

The dimension of a physical quantity is the type of units needed to express it. For instance, the dimension of a speed is length/time, the dimension of a force is mass*length/time². In mechanics, every dimension can be expressed in terms of length, time, and mass, or alternatively in terms of force, length and mass. Depending on the problem, it may be advantageous to chose one or the other set of fundamental units. Every unit is a product of powers of the fundamental units, and the units form a group.

In the most primitive form, dimensional analysis is used to check the correctness of algebraic derivations: in every physically meaningful expression, only quantities of the same dimension can be added or subtracted. The two sides of any equation must have the same dimensions. Furthermore, the arguments to exponential, trigonometric and logarithmic functions must be dimensionless numbers, which is often achieved by multiplying a certain physical quantity by a suitable constant of the inverse dimension.

The above mentioned reduction of variables uses the Buckingham Pi theorem as its central tool. This theorem describes how every physically meaningful equation involving n variables can be equivalently rewritten as an equation of n-m dimensionless parameters, where m is the number of fundamental units used. Furthermore, and most importantly, it provides a method for computing these dimensionless parameters from the given variables, even if the form of the equation is still unknown.

Two systems for which these parameters coincide are called similar; they are equivalent for the purposes of the equation, and the experimentalist who wants to determine the form of the equation can choose the most convenient one.

A typical application of dimensional analysis occurs in fluid dynamics. If a moving fluid meets an object, it exerts a force on the object, according to a complicated (and not completely understood) law. The variables involved are: the speed, density and viscosity of the fluid, the size of the body, and the force. Using the algorithm of the pi theorem, one can reduce these five variables to two dimensionless parameters: the drag coefficient and the Reynolds number. The original law is then reduced to a law involving only these two numbers. To empirically determine this law, instead of experimenting on huge bodies with fast flowing fluids, one may just as well experiment on small models with slow flowing, more viscous fluids, because these two systems are similar.

References

• Barenblatt, G. I., "Scaling, Self-Similarity, and Intermediate Asymptotics", Cambridge University Press, 1996

• Bridgman, P. W., "Dimensional Analysis", Yale University Press, 1937

• Langhaar, H. L., "Dimensional Analysis and Theory of Models", Wiley, 1951

• Murphy, N. F., Dimensional Analysis, Bull. V.P.I., 1949, 42(6)

• Porter, "The Method of Dimensions", Methuen, 1933

• Boucher and Alves, Dimensionless Numbers, Chem. Eng. Progress, 1960, 55, pp.55-64

• Buckingham, E., On Physically Similar Systems: Illustrations of the Use of Dimensional Analysis, Phys. Rev, 1914, 4, p.345

• Klinkenberg A. Chem. Eng. Science, 1955, 4, pp. 130-140, 167-177

• Rayleigh, Lord, The Principle of Similitude, Nature 1915, 95, pp. 66-68

• Silberberg, I. H. and McKetta J. J., Jr., Learning How to Use Dimensional Analysis, Petrol. Refiner, 1953, 32(4), p179; (5), p.147; (6), p.101; (7), p. 129

• Van Driest, E. R., On Dimensional Analysis and the Presentation of Data in Fluid Flow Problems, J. App. Mech, 1946, 68, A-34, March

• Perry, J. H. et al., "Standard System of Nomenclature for Chemical Engineering Unit Operations", Trans. Am. Inst. Chem. Engrs., 1944, 40, 251

• Moody, L. F., "Friction Factors for Pipe Flow", Trans. Am. Soc. Mech. Engrs., 1944, 66, 671

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