Dimensional analysis

Dimensional analysis is a conceptual tool often applied in physics, chemistry, and engineering to understand physical situations involving a mix of different kinds of physical quantities. It is routinely used by physical scientists and engineers to check the plausibility of derived equations. Only like dimensioned quantites may be added, subtracted, compared, or equated. When unlike dimensioned quantities appear opposite of the "+" or "-" or "=" sign, that physical equation is not plausible which might prompt one to correct errors before proceeding to use it. When like dimensioned quantities or unlike dimensioned quantites are multiplied or divided, their dimensions are likewise multiplied or divided. When dimensioned quantites are raised to a power or a power root, the same is done to the dimensions attached to those quantities.

The dimensions of a physical quantity is associated with symbols, such as M, L, T which represent mass, length and time, each raised to rational powers. For instance, the dimension of the physical variable, speed, is distance/time (L/T) and the dimension of a force is mass×distance/time² or ML/T². In mechanics, every dimension can be expressed in terms of distance (which physicists often call "length"), time, and mass, or alternatively in terms of force, length and mass. Depending on the problem, it may be advantageous to choose one or another other set of dimensions. In electromagnetism, for example, it may be useful to use dimensions of M, L, T, and Q, where Q represents quantity of electric charge.

The units of a physical quantity are defined by convention, related to some standard; e.g. length may have units of meters, feet, inches, miles or micrometres; but a length always has a dimension of L whether it is measured in meters, feet, inches, miles or micrometres. Dimensional symbols, such as L, form a group: there is an identity, 1; there is an inverse to L, which is 1/L, and L raised to any rational power p is a member of the group, having an inverse of 1/L raised to the power p. There are conversion factors between units; for example one meter is equal to 39.37 inches, but a meter and an inch are both associated with the same symbol, L.

In the most primitive form, dimensional analysis may be used to check the correctness of physical equations: in every physically meaningful expression, only quantities of the same dimension can be added or subtracted. Moreover, the two sides of any equation must have the same dimensions. For example, the mass of a rat and the mass of a flea may be added, but the mass of a flea and the length of a rat cannot be added. Furthermore, the arguments to exponential, trigonometric and logarithmic functions must be dimensionless numbers. The logarithm of 3 kg is undefined, but the logarithm of 3 is 0.477.

It should be noted that very different physical quantities may have the same dimensions: work and torque, for example, both have the same dimensions, M L²T^-2. An equation with torque on one side and energy on the other would be dimensionally correct, but cannot be physically correct! However, torque multiplied by an angular twist measured in (dimensionless) radians is work or energy. (The radian is the mathematically natural measure of an angle and is the ratio of arc of a circle swept by such an angle divided by the radius of the circle. That ratio of like dimensioned quantities, length over length, is dimensionless.)

The value of a dimensionful physical quantity is written as the product of a unit within the dimension and a dimensionless numerical factor. Strictly, when like dimensioned quantities are added or subtracted or compared, these dimensioned quantities must be expressed in consistent units so that the numerical values of these quantities may be directly added or subtracted. But, conceptually, there is no problem adding quantities of the same dimension expressed in different units. For example, 1 meter added to 1 foot is a length, but it would not be correct to add 1 to 1 to get the result. A conversion factor, which is a ratio of like dimensioned quantities and is equal to the dimensionless unity:

<math> 1 \ \operatorname{ft} = 0.3048 \ \operatorname{m} \ <math> is identical to saying <math> 1 = \frac{0.3048 \ \operatorname{m}}{1 \ \operatorname{ft}} <math>

The factor <math> 0.3048 \frac{\operatorname{m}}{\operatorname{ft}} <math> is identical to the dimensionless 1, so multiplying by this conversion factor changes nothing. Then when adding two quantites of like dimension, but expressed in different units, the appropriate conversion factor, which is essentially the dimensionless 1 used to convert units, is used to convert quanties to the same units so that their numrical values can be added or subtracted.

<math> 1 \operatorname{m} + 1 \operatorname{ft} <math>	<math>= 1 \operatorname{m} + 1 \operatorname{ft} \times 0.3048 \frac{\operatorname{m}}{\operatorname{ft}} <math>
	<math>=1 \operatorname{m} + 1 \operatorname{ft} \!\!\!\! / \times 0.3048 \frac{\operatorname{m}}{\operatorname{ft} \!\!\!\! /} <math>
	<math>=1 \operatorname{m} + 0.3048 \operatorname{m} <math>
	<math>=1.3048 \operatorname{m} \ <math>

Only in this manner, it is meaningful to speak of adding like dimensioned quantities of differing units, although to do so mathematically, all units must be the same. It is not meaningful, either physically or mathematically, to speak of adding unlike dimensioned physical quantities such as adding length (say, in meters) to mass (perhaps in kilograms).

The Buckingham π-theorem forms the basis of the central tool of dimensional analysis. 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 dimensions 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.

A worked example

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. We might suppose that the variables involved under some conditions to be the speed, density and viscosity of the fluid, the size of the body (expressed in terms of its frontal area <math>A<math>), and the drag 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.

Alternatively, one can derive the dimensionless parameters via direct manipulation of the underlying differential equations.

That this is so becomes obvious when the drag force <math>F<math> is expressed as part of a function of the other variables in the problem:

<math>

f(F,u,A,\rho,\nu)=0. \! <math> This rather odd form of expression is used because it does not assume a one-one relationship. Here, <math>f<math> is some function (as yet unknown) that takes five arguments. We note that the right hand side is zero in any system of units; so it should be possible to express the relationship described by <math>f<math> in terms of only dimensionless groups.

There are many ways of combining the five arguments of <math>f<math> to form dimensionless groups, but the Buckingham Pi theorem states that there will be two such groups. The most appropriate are the Reynolds number, given by

<math>

Re=\frac{u\sqrt{A}}{\nu} <math>

and the drag coefficient, given by

<math>

C_D=\frac{F}{\rho Au^2}. <math>

Thus the original law involving a function of five variables may be replaced by one involving only two:

<math>

f\left(\frac{F}{\rho Au^2},\frac{u\sqrt{A}}{\nu}\right)=0. <math> where <math>f<math> is some function of two arguments. The original law is then reduced to a law involving only these two numbers.

Because the only unknown in the above equation is <math>F<math>, it is possible to express it as

<math>

\frac{F}{\rho Au^2}=f\left(\frac{u\sqrt{A}}{\nu}\right) <math>

<math>

F=\rho Au^2f(Re). \! <math>

Thus the force is simply <math>\rho Au^2<math> times some (as yet unknown) function of the Reynolds number: a considerably simpler system than the original five-argument function given above.

Dimensional analysis thus makes a very complex problem (trying to determine the behaviour of a function of five variables) a much simpler one: the determination of the drag as a function of only one variable, the Reynolds number.

The analysis also gives other information for free, so to speak. We know that, other things being equal, the drag force will be proportional to the density of the fluid. This kind of information often proves to be extremely valuable, especially in the early stages of a research project.

To empirically determine the Reynolds number dependence, 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.

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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