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 quantities 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 quantities are multiplied or divided, their dimensions are likewise multiplied or divided. When dimensioned quantities are raised to a power or a power root, the same is done to the dimensions attached to those quantities.

Introduction

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. Just as there are derived units that are derived from base units, there are primary base dimensions of physical quantity and secondary dimensions of physical quantity derived from the former. For instance, the dimension of the physical quantity, speed, is distance/time (L/T) and the dimension of a force is mass × distance/time² or ML/T². In mechanics, every dimension of physical quantity 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 base 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: the two sides of any equation must have the same dimensions, i.e., the equation must be dimensionally homogeneous. As a corollory of this requirement, it follows that in a physically meaningful expression, only quantities of the same dimension can be added or subtracted. 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. As a further corollory, scalar arguments to exponential, trigonometric and logarithmic functions must be dimensionless numbers. The logarithm of 3 kg is undefined, but the logarithm of 3 is nearly 0.477. This is essentially due to the requirement for the Taylor expansion of these functions to be dimensionally homogeneous, which means that the square of the argument must be of the same dimension as the argument itself. For scalar arguments, this means the argument must be dimensionless, but certain dimensioned tensors are dimensionally self-square (Hart, 1995) and may be used as arguments to these functions. Also, an extension of dimensional analysis called "orientational analysis" (see section below) attaches orientational elements to each physical quantity, and the square of these orientational elements are dimensionless, so that oriented physical variables may be used as arguments to the sine and cosine functions.

The value of a dimensionful physical quantity is written and thought of 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, is needed:

1\ {\mbox{ft}}=0.3048\ {\mbox{m}}\

is identical to saying

1={\frac {0.3048\ {\mbox{m}}}{1\ {\mbox{ft}}}}\

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

$1{\mbox{m}}+1{\mbox{ft}}\$	$=1{\mbox{m}}+1{\mbox{ft}}\times 0.3048{\frac {\mbox{m}}{\mbox{ft}}}\$
	$=1{\mbox{m}}+1{\mbox{ft}}\!\!\!\!/\times 0.3048{\frac {\mbox{m}}{{\mbox{ft}}\!\!\!\!/}}\$
	$=1{\mbox{m}}+0.3048{\mbox{m}}\$
	$=1.3048{\mbox{m}}\$

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).

A simple example

What is the period of oscillation $T$ of a mass $m$ attached to an ideal linear spring with spring constant $k$ suspended in gravity of strength $g$ ? The four quantities have the following dimensions: $T$ [T]; $m$ [M]; $k$ [M/T^2]; and $g$ [L/T^2]. From these we can form only one dimensionless group, $T^{2}k/m$ .

Note that there is no other group involving $g$ . Dimensional analysis can sometimes yield strong statements about the irrelevance of some quantities in a problem, or the need for additional parameters. If we have chosen enough variables to properly describe the problem, then the period of the mass on the spring is independent of g: it is the same on the earth or the moon. Indeed, dimensional analysis tells us that $T=\kappa {\sqrt {m/k}}$ , for some dimensionless constant $\kappa$ .

When faced with such a situation, we might also consider the possibility that the rejected parameter is in fact relevant, and that some other relevant variable has been omitted, which might combine with the rejected parameter to form a dimensionless quantity. (That is, however, not the case here.)

Huntley's addition

Huntley (Huntley, 1967) has claimed that it is sometimes productive to refine our concept of dimension. Two possible refinements are:

The magnitude of the components of a vector are to be considered dimensionally distinct. For example, rather than an undifferentiated length unit L, we may have $L_{x}$ represent length in the x direction, and so forth. This requirement stems ultimately from the requirement that each component of a physically meaningful equation (scalar, vector, or tensor) must be dimensionally consistent.

Mass as a measure of quantity is to be considered dimensionally distinct from mass as a measure of inertia.

As an example of the usefulness of the first refinement, suppose we wish to calculate the distance a cannon ball travels when fired with a vertical velocity component $V_{y}$ and a horizontal velocity component $V_{x}$ , assuming it is fired on a flat surface. Assuming no use of directed lengths, the quantities of interest are then $V_{x}$ , $V_{y}$ , both dimensioned as $L/T$ , R, the distance travelled, having dimension L, and g the downward acceleration of gravity, with dimension $L/T^{2}$

With these four quantities, we may conclude that the equation for the range R may be written:

R\propto V_{x}^{a}\,V_{y}^{b}\,g^{c}\,

Or dimensionally

L=(L/T)^{a+b}(L/T^{2})^{c}\,

from which we may deduce that $a+b+c=1$ and $a+b+2c=0$ which leaves one exponent undetermined. This is to be expected since we have two fundamental quantities L and T and four parameters, with one equation.

If, however, we use directed length dimensions, then $V_{x}$ will be dimensioned as $L_{x}/T$ , $V_{y}$ as $L_{y}/T$ , R as $L_{x}$ and g as $L_{y}/T^{2}$ . The dimensional equation becomes:

L=(L/T)^{a+b}(L/T^{2})^{c}\,

and we may solve completely as $a=1$ , $b=1$ and $c=-1$ . The increase in deductive power gained by the use of directed length dimensions seems apparent.

In a similar manner, it is sometimes found useful (e.g. in fluid mechanics and thermodynamics) to distinguish between mass as a measure of inertia (inertial mass), and mass as a measure of quantity (substantial mass). For example, consider the derivation of Poiseuille's Law. We wish to find the rate of mass flow of a viscous fluid through a circular pipe. Without drawing distinctions between inertial and substantial mass we may choose as the relevant variables

${\dot {m}}$ the mass flow rate with dimensions $M/T$
$p_{x}$ the pressure gradient along the pipe with dimensions $M/L^{2}T^{2}$
$\rho$ the density with dimensions $M/L^{3}$
$\eta$ the dynamic fluid viscosity with dimensions $M/LT$
$r$ the radius of the pipe with dimensions $L$

There are three fundamental variables so the above five equations will yield two dimensionless variables which we may take to be $\pi _{1}={\dot {m}}/\eta r$ and $\pi _{2}=p_{x}\rho r^{5}/{\dot {m}}^{2}$ and we may express the dimensional equation as

C=\pi _{1}\pi _{2}^{a}=\left({\frac {\dot {m}}{\eta r}}\right)\left({\frac {p_{x}\rho r^{5}}{{\dot {m}}^{2}}}\right)^{a}

where C and a are undetermined constants. If we draw a distinction between inertial mass with dimensions $M_{i}$ and substantial mass with dimensions $M_{s}$ , then the relevant parameters will be:

${\dot {m}}$ the mass flow rate with dimensions $M_{s}/T$
$p_{x}$ the pressure gradient along the pipe with dimensions $M_{i}/L^{2}T^{2}$
$\rho$ the density with dimensions $M_{s}/L^{3}$
$\eta$ the coefficient of viscoscity with dimensions $M_{i}/LT$
$r$ the radius of the pipe with dimensions $L$

These mass dimensions are chosen because the pressure and viscous forces involve inertial mass, while the mass flow rate is concerned with mass as a measure of quantity. We now have four fundamental parameters, and one dimensionless constant, so that the dimensional equation may be written:

C={\frac {p_{x}\rho r^{4}}{\eta {\dot {m}}}}

where now only C is an undetermined constant (found to be equal to $\pi /8$ ). This equation may be solved for the mass flow rate to yield Poiseuille's law.

The problem of angles

Huntley's addition has some serious drawbacks. It does not deal well with vector equations involving the cross product, nor does it handle well the use of angles as physical variables. It also is often quite difficult to assign the L, $L_{x}$ , $L_{y}$ , $L_{z}$ symbols to the physical variables involved in the problem of interest. He invokes a procedure that involves the "symmetry" of the physical problem. This is often very difficult to apply reliably: it is unclear as to what parts of the problem that the notion of "symmetry" is being invoked. Is it the symmetry of the physical body that forces are acting upon, or to the points, lines or areas at which forces are being applied? What if more than one body is involved with different symmetries? Consider the spherical bubble attached to a cylindrical tube, where one wants the flow rate of air as a function of the pressure difference in the two parts. What are the Huntley extended dimensions of the viscosity of the air contained in the connected parts? What are the extended dimensions of the pressure of the two parts? Are they the same or different? These difficulties are responsible for the limited application of Huntley's addition to real problems.

Angles are conventionally considered to be dimensionless variables, and so the use of angles as physical variables in dimensional analysis can give less meaningful results. As an example, consider the projectile problem mentioned above. Suppose that, instead of the x- and y-component of the initial velocity, we had chosen the magnitude of the velocity v and the angle $\theta$ at which the projectile was fired. The angle is conventionally considered to be dimensionless, and the magnitude of a vector has no directional quality, so that no dimensionless variable can be composed of the four variables g , v , R , and θ. Conventional analysis will correctly give the powers of g and v, but will give no information concerning the dimensionless angle θ.

Siano (Siano, 1985-I, 1985-II) has suggested that the directed dimensions of Huntley be replaced by using orientational symbols $1_{x},\;1_{y},\;1_{z}$ to specify vector directions, and an orientationless symbol $1_{0}$ . Thus, Huntley's $L_{x}$ becomes $L\,1_{x}$ with L specifying the dimension of length, and $1_{x}$ specifying the orientation. Siano further specifies that the orientational symbols have an algebra of their own. Along with the specification that $1_{i}^{-1}=1_{i}$ , the following multiplication table for the orientation symbols holds:

{\begin{matrix}&\mathbf {1_{0}} &\mathbf {1_{x}} &\mathbf {1_{y}} &\mathbf {1_{z}} \\\mathbf {1_{0}} &1_{0}&1_{x}&1_{y}&1_{z}\\\mathbf {1_{x}} &1_{x}&1_{0}&1_{z}&1_{y}\\\mathbf {1_{y}} &1_{y}&1_{z}&1_{0}&1_{x}\\\mathbf {1_{z}} &1_{z}&1_{y}&1_{x}&1_{0}\end{matrix}}

Note that the orientational sybols form a group (the Klein four-group or "viergruppe"). In this system, scalars always have the same orientation as the identity element, independent of the "symmetry of the problem." Physical quantities that are vectors have the orientation expected: a force or a velocity in the x-direction has the orientation of $1_{x}$ . For angles, consider an angle ɵ that lies in the z plane. Form a right triangle in the z plane with ɵ being one of the acute angles. The side of the right triangle adjacent to the angle then has an orientation $1_{x}$ and the side opposite has an orientation $1_{y}$ . Then, since Tan(ɵ) = ly/lx = ɵ + ... we conclude that an angle in the xy plane must have an orientation $1_{y}$ / $1_{z}$ = $1_{z}$ , which is not unreasonable. Analogous reasoning forces the conclusion that sin(ɵ) has orientation $1_{z}$ while cos(ɵ) has orientation $1_{0}$ . These are different, so one concludes (correctly), for example, that there are no solutions of physical equations that are of the form a sin(ɵ) + b cos(ɵ), where a and b are numerics.

The assignment of orientational symbols to physical quantities and the requirement that physical equations be orientationally homegeneous can actually be used in a way that is similar to dimensional analysis to derive a little more information about acceptable solutions of physical problems. In this approach one sets up the dimensional equation and solves it as far as one can. If the lowest power of a physical variable is fractional, both sides of the solution is raised to a power such that all powers are integral. This puts it into "normal form". The orientational equation is then solved to give a more restrictive condition on the unknown powers of the orientational symbols, arriving at a solution that is more complete than the one that dimensional analysis alone gives.

As an example, for the projectile problem, using orientational symbols, θ, being in the x-y plane will thus have dimension $1_{z}$ and the range of the projectile R will be of the form:

R=g^{a}\,v^{b}\,\theta ^{c}

which means

L\,1_{x}\sim \left({\frac {L\,1_{y}}{T^{2}}}\right)^{a}\left({\frac {L}{T}}\right)^{b}\,1_{z}^{c}

Dimensional homogeneity will now correctly yield a=-1 and b=2, and orientational homogeneity requires that c be an odd integer. In fact the required function of theta will be $\sin(\theta )\cos(\theta )$ which is a series of odd powers of $\theta$ .

It is seen that the Taylor series of $\sin(\theta )$ and $\cos(\theta )$ are orientationally homogeneous using the above multiplication table, while expressions like $\cos(\theta )+\sin(\theta )$ and $\exp(\theta )$ are not, and are (correctly) deemed unphysical.

It should be clear that the multiplication rule used for the orientational symbols is not the same as that for the cross product of two vectors. The cross product of two identical vectors is zero, while the product of two identical orientational symbols are the identity element.

Ultimately, it can be seen that dimensional analysis and the requirement for physical equations to be dimensionally homogeneouos reflects the idea that the laws of physics are independent of the units employed to measure the physical variables. That is, F=ma, for example, is true whether the unit system used is SI, English, or cgs, or any other consistent sytem of units. Orientational analysis and the requirement for physical equations to be orientationally homogeneous reflects the idea that the equations of physics must be independent of the coordinate system used.

Buckingham π theorem

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.

References

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