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## Dimensional analysis
The 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 π-theorem as its central tool. This theorem describes how every physically meaningful equation involving
Two systems for which these parameters coincide are called The π-theorem uses linear algebra: the space of all possible physical units can be seen as a vector space over the rational numbers if we represent a unit as the set of exponents needed for the fundamental units (with a power of zero if the particular fundamental unit is not present). Multiplication of physical units is then represented by vector addition within this vector space. The algorithm of the π-theorem is essentially a Gauss-Jordan elimination carried out in this vector space. 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 π-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 (such as real-size airplanes in wind-tunnels), one may just as well experiment on small models with slow flowing, more viscous fluids, because these two systems are similar. ## Worked exampleConsider Einstein's well-known equation E = mc². As stated above, the two sides of any equation must have the same dimensions. We can check this as follows.
- E is energy, which has units of mass × length² / time². (This is because energy = force × length, and force = mass × acceleration, and acceleration = length / time².)
- m is mass, which is a unit on its own.
- c is speed, which has units of length / time.
- The left-hand side, E, therefore has units of mass × length² / time².
- The right-hand side, mc², has units of mass × (length / time)² = mass × length² / time².
- The two sides therefore have the same dimensions.
## 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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