Dimensional analysis

In dimensional analysis, Rayleigh's method is a conceptual tool used in physics, chemistry, and engineering. It expresses a functional relationship of some variables in the form of an exponential equation. It was named after Lord Rayleigh.

The method involves the following steps:

1. Gather all the independent variables that are likely to influence the dependent variable.
2. If R is a variable that depends upon independent variables R₁, R₂, R₃, ..., R_n, then the functional equation can be written as R = F(R₁, R₂, R₃, ..., R_n).
3. Write the above equation in the form R = C R₁^a R₂^b R₃^c ... R_n^m, where C is a dimensionless constant and a, b, c, ..., m are arbitrary exponents.
4. Express each of the quantities in the equation in some base units in which the solution is required.
5. By using dimensional homogeneity, obtain a set of simultaneous equations involving the exponents a, b, c, ..., m.
6. Solve these equations to obtain the value of exponents a, b, c, ..., m.
7. Substitute the values of exponents in the main equation, and form the non-dimensional parameters by grouping the variables with like exponents.

As a drawback, Rayleigh's method does not provide any information regarding number of dimensionless groups to be obtained as a result of dimensional analysis.

Percentages are dimensionless quantities, since they are ratios of two quantities with the same dimensions. In other words, the % sign can be read as "hundredths", since 1% = 1/100.

Taking a derivative with respect to a quantity divides the dimension by the dimension of the variable that is differentiated with respect to. Thus:

• position (x) has the dimension L (length);
• derivative of position with respect to time (dx/dt, velocity) has dimension T⁻¹L—length from position, time due to the gradient;
• the second derivative (d²x/dt² = d(dx/dt) / dt, acceleration) has dimension T⁻²L.

Likewise, taking an integral adds the dimension of the variable one is integrating with respect to, but in the numerator.

• force has the dimension T⁻²LM (mass multiplied by acceleration);
• the integral of force with respect to the distance (s) the object has travelled (${\displaystyle \textstyle \int F\ ds}$, work) has dimension T⁻²L²M.

In economics, one distinguishes between stocks and flows: a stock has a unit (say, widgets or dollars), while a flow is a derivative of a stock, and has a unit of the form of theis unit divided by one of time (say, dollars/year).

In some contexts, dimensional quantities are expressed as dimensionless quantities or percentages by omitting some dimensions. For example, debt-to-GDP ratios are generally expressed as percentages: total debt outstanding (dimension of currency) divided by annual GDP (dimension of currency)—but one may argue that, in comparing a stock to a flow, annual GDP should have dimensions of currency/time (dollars/year, for instance) and thus debt-to-GDP should have the unit year, which indicates that debt-to-GDP is the number of years needed for a constant GDP to pay the debt, if all GDP is spent on the debt and the debt is otherwise unchanged.

A simple application of dimensional analysis to mathematics is in computing the form of the volume of an n-ball (the solid ball in n dimensions), or the area of its surface, the n-sphere: being an n-dimensional figure, the volume scales as ${\displaystyle x^{n},}$ while the surface area, being ${\displaystyle (n-1)}$-dimensional, scales as ${\displaystyle x^{n-1}.}$ Thus the volume of the n-ball in terms of the radius is ${\displaystyle C_{n}r^{n},}$ for some constant ${\displaystyle C_{n}.}$ Determining the constant takes more involved mathematics, but the form can be deduced and checked by dimensional analysis alone.

In finance, economics, and accounting, dimensional analysis is most commonly referred to in terms of the distinction between stocks and flows. More generally, dimensional analysis is used in interpreting various financial ratios, economics ratios, and accounting ratios.

• For example, the P/E ratio has dimensions of time (unit: year), and can be interpreted as "years of earnings to earn the price paid".
• In economics, debt-to-GDP ratio also has the unit year (debt has a unit of currency, GDP has a unit of currency/year).
• Velocity of money has a unit of 1/years (GDP/money supply has a unit of currency/year over currency): how often a unit of currency circulates per year.
• Annual continuously compounded interest rates and simple interest rates are often expressed as a percentage (adimensional quantity) while time is expressed as an adimensional quantity consisting of the number of years. However, if the time includes year as the unit of measure, the dimension of the rate is 1/year. Of course, there is nothing special (apart from the usual convention) about using year as a unit of time: any other time unit can be used. Furthermore, if rate and time include their units of measure, the use of different units for each is not problematic. In contrast, rate and time need to refer to a common period if they are adimensional. (Note that effective interest rates can only be defined as adimensional quantities.)
• In financial analysis, bond duration can be defined as (dV/dr)/V, where V is the value of a bond (or portfolio), r is the continuously compounded interest rate and dV/dr is a derivative. From the previous point, the dimension of r is 1/time. Therefore, the dimension of duration is time (usually expressed in years) because dr is in the "denominator" of the derivative.

In fluid mechanics, dimensional analysis is performed to obtain dimensionless pi terms or groups. According to the principles of dimensional analysis, any prototype can be described by a series of these terms or groups that describe the behaviour of the system. Using suitable pi terms or groups, it is possible to develop a similar set of pi terms for a model that has the same dimensional relationships.[8] In other words, pi terms provide a shortcut to developing a model representing a certain prototype. Common dimensionless groups in fluid mechanics include:

• Reynolds number (Re), generally important in all types of fluid problems:
  $${\displaystyle \mathrm {Re} ={\frac {\rho \,ud}{\mu }}.}$$

  
• Froude number (Fr), modeling flow with a free surface:
  $${\displaystyle \mathrm {Fr} ={\frac {u}{\sqrt {g\,L}}}.}$$

  
• Euler number (Eu), used in problems in which pressure is of interest:
  $${\displaystyle \mathrm {Eu} ={\frac {\Delta p}{\rho u^{2}}}.}$$

  
• Mach number (Ma), important in high speed flows where the velocity approaches or exceeds the local speed of sound:
  $${\displaystyle \mathrm {Ma} ={\frac {u}{c}},}$$

  
  where c is the local speed of sound.

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? That period is the solution for T of some dimensionless equation in the variables T, m, k, and g. The four quantities have the following dimensions: T [T]; m [M]; k [M/T²]; and g [L/T²]. From these we can form only one dimensionless product of powers of our chosen variables, ${\displaystyle G_{1}}$ = ${\displaystyle T^{2}k/m}$ [T² · M/T² / M = 1], and putting ${\displaystyle G_{1}=C}$ for some dimensionless constant C gives the dimensionless equation sought. The dimensionless product of powers of variables is sometimes referred to as a dimensionless group of variables; here the term "group" means "collection" rather than mathematical group. They are often called dimensionless numbers as well.

Note that the variable g does not occur in the group. It is easy to see that it is impossible to form a dimensionless product of powers that combines g with k, m, and T, because g is the only quantity that involves the dimension L. This implies that in this problem the g is irrelevant. 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 from this argument we can conclude that the period of the mass on the spring is independent of g: it is the same on the earth or the moon. The equation demonstrating the existence of a product of powers for our problem can be written in an entirely equivalent way: ${\displaystyle T=\kappa {\sqrt {\tfrac {m}{k}}}}$, for some dimensionless constant κ (equal to ${\displaystyle {\sqrt {C}}}$ from the original dimensionless equation).

When faced with a case where dimensional analysis rejects a variable (g, here) that one intuitively expects to belong in a physical description of the situation, another possibility is that the rejected variable is in fact relevant, but that some other relevant variable has been omitted, which might combine with the rejected variable to form a dimensionless quantity. That is, however, not the case here.

When dimensional analysis yields only one dimensionless group, as here, there are no unknown functions, and the solution is said to be "complete" – although it still may involve unknown dimensionless constants, such as κ.

Consider the case of a vibrating wire of length ℓ (L) vibrating with an amplitude A (L). The wire has a linear density ρ (M/L) and is under tension s (LM/T²), and we want to know the energy E (L²M/T²) in the wire. Let π₁ and π₂ be two dimensionless products of powers of the variables chosen, given by

${\displaystyle {\begin{aligned}\pi _{1}&={\frac {E}{As}}\\\pi _{2}&={\frac {\ell }{A}}.\end{aligned}}}$

The linear density of the wire is not involved. The two groups found can be combined into an equivalent form as an equation

${\displaystyle F\left({\frac {E}{As}},{\frac {\ell }{A}}\right)=0,}$

where F is some unknown function, or, equivalently as

${\displaystyle E=Asf\left({\frac {\ell }{A}}\right),}$

where f is some other unknown function. Here the unknown function implies that our solution is now incomplete, but dimensional analysis has given us something that may not have been obvious: the energy is proportional to the first power of the tension. Barring further analytical analysis, we might proceed to experiments to discover the form for the unknown function f. But our experiments are simpler than in the absence of dimensional analysis. We'd perform none to verify that the energy is proportional to the tension. Or perhaps we might guess that the energy is proportional to ℓ, and so infer that E = ℓs. The power of dimensional analysis as an aid to experiment and forming hypotheses becomes evident.

The power of dimensional analysis really becomes apparent when it is applied to situations, unlike those given above, that are more complicated, the set of variables involved are not apparent, and the underlying equations hopelessly complex. Consider, for example, a small pebble sitting on the bed of a river. If the river flows fast enough, it will actually raise the pebble and cause it to flow along with the water. At what critical velocity will this occur? Sorting out the guessed variables is not so easy as before. But dimensional analysis can be a powerful aid in understanding problems like this, and is usually the very first tool to be applied to complex problems where the underlying equations and constraints are poorly understood. In such cases, the answer may depend on a dimensionless number such as the Reynolds number, which may be interpreted by dimensional analysis.

Dimensional analysis and numerical experiments for a rotating disc

Consider the case of a thin, solid, parallel-sided rotating disc of axial thickness t (L) and radius R (L). The disc has a density ρ (M/L³), rotates at an angular velocity ω (T⁻¹) and this leads to a stress S (T⁻²L⁻¹M) in the material. There is a theoretical linear elastic solution, given by Lame, to this problem when the disc is thin relative to its radius, the faces of the disc are free to move axially, and the plane stress constitutive relations can be assumed to be valid. As the disc becomes thicker relative to the radius then the plane stress solution breaks down. If the disc is restrained axially on its free faces then a state of plane strain will occur. However, if this is not the case then the state of stress may only be determined though consideration of three-dimensional elasticity and there is no known theoretical solution for this case. An engineer might, therefore, be interested in establishing a relationship between the five variables. Dimensional analysis for this case leads to the following (5 − 3 = 2) non-dimensional groups:

demand/capacity = ρR²ω²/S

thickness/radius or aspect ratio = t/R

Through the use of numerical experiments using, for example, the finite element method, the nature of the relationship between the two non-dimensional groups can be obtained as shown in the figure. As this problem only involves two non-dimensional groups, the complete picture is provided in a single plot and this can be used as a design/assessment chart for rotating discs[20]

The dimensions that can be formed from a given collection of basic physical dimensions, such as T, L, and M, form an abelian group: The identity is written as 1; L⁰ = 1, and the inverse of L is 1/L or L⁻¹. L raised to any rational power p is a member of the group, having an inverse of L^−p or 1/L^p. The operation of the group is multiplication, having the usual rules for handling exponents (Lⁿ × L^m = L^n+m).

This group can be described as a vector space over the rational numbers, with the dimensional symbol TⁱL^jM^k corresponding to the vector (i, j, k). When physical measured quantities (be they like-dimensioned or unlike-dimensioned) are multiplied or divided by one other, their dimensional units are likewise multiplied or divided; this corresponds to addition or subtraction in the vector space. When measurable quantities are raised to a rational power, the same is done to the dimensional symbols attached to those quantities; this corresponds to scalar multiplication in the vector space.

A basis for such a vector space of dimensional symbols is called a set of base quantities, and all other vectors are called derived units. As in any vector space, one may choose different bases, which yields different systems of units (e.g., choosing whether the unit for charge is derived from the unit for current, or vice versa).

The group identity, the dimension of dimensionless quantities, corresponds to the origin in this vector space.

The set of units of the physical quantities involved in a problem correspond to a set of vectors (or a matrix). The nullity describes some number (e.g., m) of ways in which these vectors can be combined to produce a zero vector. These correspond to producing (from the measurements) a number of dimensionless quantities, {π₁, ..., π_m}. (In fact these ways completely span the null subspace of another different space, of powers of the measurements.) Every possible way of multiplying (and exponentiating) together the measured quantities to produce something with the same unit as some derived quantity X can be expressed in the general form

${\displaystyle X=\prod _{i=1}^{m}(\pi _{i})^{k_{i}}\,.}$

Consequently, every possible commensurate equation for the physics of the system can be rewritten in the form

${\displaystyle f(\pi _{1},\pi _{2},...,\pi _{m})=0\,.}$

Knowing this restriction can be a powerful tool for obtaining new insight into the system.

The dimension of physical quantities of interest in mechanics can be expressed in terms of base dimensions T, L, and M – these form a 3-dimensional vector space. This is not the only valid choice of base dimensions, but it is the one most commonly used. For example, one might choose force, length and mass as the base dimensions (as some have done), with associated dimensions F, L, M; this corresponds to a different basis, and one may convert between these representations by a change of basis. The choice of the base set of dimensions is thus a convention, with the benefit of increased utility and familiarity. The choice of base dimensions is not entirely arbitrary, because they must form a basis: they must span the space, and be linearly independent.

For example, F, L, M form a set of fundamental dimensions because they form a basis that is equivalent to T, L, M: the former can be expressed as [F = LM/T²], L, M, while the latter can be expressed as [T = (LM/F)^1/2], L, M.

On the other hand, length, velocity and time (T, L, V) do not form a set of base dimensions for mechanics, for two reasons:

• There is no way to obtain mass – or anything derived from it, such as force – without introducing another base dimension (thus, they do not span the space).
• Velocity, being expressible in terms of length and time (V = L/T), is redundant (the set is not linearly independent).

Depending on the field of physics, it may be advantageous to choose one or another extended set of dimensional symbols. In electromagnetism, for example, it may be useful to use dimensions of T, L, M and Q, where Q represents the dimension of electric charge. In thermodynamics, the base set of dimensions is often extended to include a dimension for temperature, Θ. In chemistry, the amount of substance (the number of molecules divided by the Avogadro constant, ≈ 6.02×10²³ mol⁻¹) is also defined as a base dimension, N. In the interaction of relativistic plasma with strong laser pulses, a dimensionless relativistic similarity parameter, connected with the symmetry properties of the collisionless Vlasov equation, is constructed from the plasma-, electron- and critical-densities in addition to the electromagnetic vector potential. The choice of the dimensions or even the number of dimensions to be used in different fields of physics is to some extent arbitrary, but consistency in use and ease of communications are common and necessary features.

Scalar arguments to transcendental functions such as exponential, trigonometric and logarithmic functions, or to inhomogeneous polynomials, must be dimensionless quantities. (Note: this requirement is somewhat relaxed in Siano's orientational analysis described below, in which the square of certain dimensioned quantities are dimensionless.)

While most mathematical identities about dimensionless numbers translate in a straightforward manner to dimensional quantities, care must be taken with logarithms of ratios: the identity log(a/b) = log a − log b, where the logarithm is taken in any base, holds for dimensionless numbers a and b, but it does not hold if a and b are dimensional, because in this case the left-hand side is well-defined but the right-hand side is not.

Similarly, while one can evaluate monomials (xⁿ) of dimensional quantities, one cannot evaluate polynomials of mixed degree with dimensionless coefficients on dimensional quantities: for x², the expression (3 m)² = 9 m² makes sense (as an area), while for x² + x, the expression (3 m)² + 3 m = 9 m² + 3 m does not make sense.

However, polynomials of mixed degree can make sense if the coefficients are suitably chosen physical quantities that are not dimensionless. For example,

${\displaystyle {\tfrac {1}{2}}\cdot (\mathrm {-9.8~m/s^{2}} )\cdot t^{2}+(\mathrm {500~m/s} )\cdot t.}$

This is the height to which an object rises in time t if the acceleration of gravity is 9.8 metres per second per second and the initial upward speed is 500 metres per second. It is not necessary for t to be in seconds. For example, suppose t = 0.01 minutes. Then the first term would be

${\displaystyle {\begin{aligned}&{\tfrac {1}{2}}\cdot (\mathrm {-9.8~m/s^{2}} )\cdot (\mathrm {0.01~min} )^{2}\\[10pt]={}&{\tfrac {1}{2}}\cdot -9.8\cdot \left(0.01^{2}\right)(\mathrm {min/s} )^{2}\cdot \mathrm {m} \\[10pt]={}&{\tfrac {1}{2}}\cdot -9.8\cdot \left(0.01^{2}\right)\cdot 60^{2}\cdot \mathrm {m} .\end{aligned}}}$

The value of a dimensional physical quantity Z is written as the product of a unit [Z] within the dimension and a dimensionless numerical factor, n.[21]

${\displaystyle Z=n\times [Z]=n[Z]}$

When like-dimensioned quantities are added or subtracted or compared, it is convenient to express them in the same unit so that the numerical values of these quantities may be directly added or subtracted. But, in concept, there is no problem adding quantities of the same dimension expressed in different units. For example, 1 metre added to 1 foot is a length, but one cannot derive that length by simply adding 1 and 1. A conversion factor, which is a ratio of like-dimensioned quantities and is equal to the dimensionless unity, is needed:

${\displaystyle 1\ {\mbox{ft}}=0.3048\ {\text{m}}\ }$ is identical to ${\displaystyle 1={\frac {0.3048\ {\text{m}}}{1\ {\text{ft}}}}.\ }$

The factor ${\displaystyle 0.3048\ {\frac {\text{m}}{\mbox{ft}}}}$ 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 the same unit so that their numerical values can be added or subtracted.

Only in this manner is it meaningful to speak of adding like-dimensioned quantities of differing units.

Some discussions of dimensional analysis implicitly describe all quantities as mathematical vectors. (In mathematics scalars are considered a special case of vectors; vectors can be added to or subtracted from other vectors, and, inter alia, multiplied or divided by scalars. If a vector is used to define a position, this assumes an implicit point of reference: an origin. While this is useful and often perfectly adequate, allowing many important errors to be caught, it can fail to model certain aspects of physics. A more rigorous approach requires distinguishing between position and displacement (or moment in time versus duration, or absolute temperature versus temperature change).

Consider points on a line, each with a position with respect to a given origin, and distances among them. Positions and displacements all have units of length, but their meaning is not interchangeable:

• adding two displacements should yield a new displacement (walking ten paces then twenty paces gets you thirty paces forward),
• adding a displacement to a position should yield a new position (walking one block down the street from an intersection gets you to the next intersection),
• subtracting two positions should yield a displacement,
• but one may not add two positions.

This illustrates the subtle distinction between affine quantities (ones modeled by an affine space, such as position) and vector quantities (ones modeled by a vector space, such as displacement).

• Vector quantities may be added to each other, yielding a new vector quantity, and a vector quantity may be added to a suitable affine quantity (a vector space acts on an affine space), yielding a new affine quantity.
• Affine quantities cannot be added, but may be subtracted, yielding relative quantities which are vectors, and these relative differences may then be added to each other or to an affine quantity.

Properly then, positions have dimension of affine length, while displacements have dimension of vector length. To assign a number to an affine unit, one must not only choose a unit of measurement, but also a point of reference, while to assign a number to a vector unit only requires a unit of measurement.

Thus some physical quantities are better modeled by vectorial quantities while others tend to require affine representation, and the distinction is reflected in their dimensional analysis.

This distinction is particularly important in the case of temperature, for which the numeric value of absolute zero is not the origin 0 in some scales. For absolute zero,

−273.15 °C ≘ 0 K = 0 °R ≘ −459.67 °F,

where the symbol ≘ means corresponds to, since although these values on the respective temperature scales correspond, they represent distinct quantities in the same way that the distances from distinct starting points to the same end point are distinct quantities, and cannot in general be equated.

For temperature differences,

1 K = 1 °C ≠ 1 °F = 1 °R.

(Here °R refers to the Rankine scale, not the Réaumur scale). Unit conversion for temperature differences is simply a matter of multiplying by, e.g., 1 °F / 1 K (although the ratio is not a constant value). But because some of these scales have origins that do not correspond to absolute zero, conversion from one temperature scale to another requires accounting for that. As a result, simple dimensional analysis can lead to errors if it is ambiguous whether 1 K means the absolute temperature equal to −272.15 °C, or the temperature difference equal to 1 °C.

Similar to the issue of a point of reference is the issue of orientation: a displacement in 2 or 3 dimensions is not just a length, but is a length together with a direction. (This issue does not arise in 1 dimension, or rather is equivalent to the distinction between positive and negative.) Thus, to compare or combine two dimensional quantities in a multi-dimensional space, one also needs an orientation: they need to be compared to a frame of reference.

This leads to the extensions discussed below, namely Huntley's directed dimensions and Siano's orientational analysis.

Huntley has pointed out that a dimensional analysis can become more powerful by discovering new independent dimensions in the quantities under consideration, thus increasing the rank ${\displaystyle m}$ of the dimensional matrix.[22]

He introduced two approaches:

• The magnitudes of the components of a vector are to be considered dimensionally independent. For example, rather than an undifferentiated length dimension L, we may have L_x represent dimension 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 the quantity of matter is to be considered dimensionally independent from mass as a measure of inertia.

As an example of the usefulness of the first approach, suppose we wish to calculate the distance a cannonball travels when fired with a vertical velocity component ${\displaystyle V_{\mathrm {y} }}$ and a horizontal velocity component ${\displaystyle V_{\mathrm {x} }}$, assuming it is fired on a flat surface. Assuming no use of directed lengths, the quantities of interest are then R, the distance travelled, with dimension L, ${\displaystyle V_{\mathrm {x} }}$, ${\displaystyle V_{\mathrm {y} }}$, both dimensioned as T⁻¹L, and g the downward acceleration of gravity, with dimension T⁻²L.

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

${\displaystyle R\propto V_{\text{x}}^{a}\,V_{\text{y}}^{b}\,g^{c}.\,}$

Or dimensionally

${\displaystyle {\mathsf {L}}=\left({\frac {\mathsf {L}}{\mathsf {T}}}\right)^{a+b}\left({\frac {\mathsf {L}}{{\mathsf {T}}^{2}}}\right)^{c}\,}$

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

However, if we use directed length dimensions, then ${\displaystyle V_{\mathrm {x} }}$ will be dimensioned as T⁻¹L_x, ${\displaystyle V_{\mathrm {y} }}$ as T⁻¹L_y, R as L_x and g as T⁻²L_y. The dimensional equation becomes:

${\displaystyle {\mathsf {L}}_{\mathrm {x} }=\left({\frac {{\mathsf {L}}_{\mathrm {x} }}{\mathsf {T}}}\right)^{a}\left({\frac {{\mathsf {L}}_{\mathrm {y} }}{\mathsf {T}}}\right)^{b}\left({\frac {{\mathsf {L}}_{\mathrm {y} }}{{\mathsf {T}}^{2}}}\right)^{c}}$

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

Huntley's concept of directed length dimensions however has some serious limitations:

• 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 directed length dimensions to real problems.

In Huntley's second approach, he holds that it is sometimes 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 the quantity of matter. Quantity of matter is defined by Huntley as a quantity (a) proportional to inertial mass, but (b) not implicating inertial properties. No further restrictions are added to its definition.

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:

Symbol	Variable	Dimension
${\displaystyle {\dot {m}}}$	mass flow rate	T−1M
${\displaystyle p_{\text{x}}}$	pressure gradient along the pipe	T−2L−2M
ρ	density	L−3M
η	dynamic fluid viscosity	T−1L−1M
r	radius of the pipe	L

There are three fundamental variables, so the above five equations will yield two independent dimensionless variables:

${\displaystyle \pi _{1}={\frac {\dot {m}}{\eta r}}}$

${\displaystyle \pi _{2}={\frac {p_{\mathrm {x} }\rho r^{5}}{{\dot {m}}^{2}}}}$

If we distinguish between inertial mass with dimension ${\displaystyle M_{\text{i}}}$ and quantity of matter with dimension ${\displaystyle M_{\text{m}}}$, then mass flow rate and density will use quantity of matter as the mass parameter, while the pressure gradient and coefficient of viscosity will use inertial mass. We now have four fundamental parameters, and one dimensionless constant, so that the dimensional equation may be written:

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

where now only C is an undetermined constant (found to be equal to ${\displaystyle \pi /8}$ by methods outside of dimensional analysis). This equation may be solved for the mass flow rate to yield Poiseuille's law.

Huntley's recognition of quantity of matter as an independent quantity dimension is evidently successful in the problems where it is applicable, but his definition of quantity of matter is open to interpretation, as it lacks specificity beyond the two requirements (a) and (b) he postulated for it. For a given substance, the SI dimension amount of substance, with unit mole, does satisfy Huntley's two requirements as a measure of quantity of matter, and could be used as a quantity of matter in any problem of dimensional analysis where Huntley's concept is applicable.

Angles are, by convention, considered to be dimensionless quantities. As an example, consider again the projectile problem in which a point mass is launched from the origin (x, y) = (0, 0) at a speed v and angle θ above the x-axis, with the force of gravity directed along the negative y-axis. It is desired to find the range R, at which point the mass returns to the x-axis. Conventional analysis will yield the dimensionless variable π = R g/v², but offers no insight into the relationship between R and θ.

Siano has suggested that the directed dimensions of Huntley be replaced by using orientational symbols 1_x 1_y 1_z to denote vector directions, and an orientationless symbol 1₀.[23] Thus, Huntley's L_x becomes L1_x with L specifying the dimension of length, and 1_x specifying the orientation. Siano further shows that the orientational symbols have an algebra of their own. Along with the requirement that 1_i⁻¹ = 1_i, the following multiplication table for the orientation symbols results:

	${\displaystyle \mathbf {1_{0}} }$	${\displaystyle \mathbf {1_{\text{x}}} }$	${\displaystyle \mathbf {1_{\text{y}}} }$	${\displaystyle \mathbf {1_{\text{z}}} }$
${\displaystyle \mathbf {1_{0}} }$	${\displaystyle 1_{0}}$	${\displaystyle 1_{\text{x}}}$	${\displaystyle 1_{\text{y}}}$	${\displaystyle 1_{\text{z}}}$
${\displaystyle \mathbf {1_{\text{x}}} }$	${\displaystyle 1_{\text{x}}}$	${\displaystyle 1_{0}}$	${\displaystyle 1_{\text{z}}}$	${\displaystyle 1_{\text{y}}}$
${\displaystyle \mathbf {1_{\text{y}}} }$	${\displaystyle 1_{\text{y}}}$	${\displaystyle 1_{\text{z}}}$	${\displaystyle 1_{0}}$	${\displaystyle 1_{\text{x}}}$
${\displaystyle \mathbf {1_{\text{z}}} }$	${\displaystyle 1_{\text{z}}}$	${\displaystyle 1_{\text{y}}}$	${\displaystyle 1_{\text{x}}}$	${\displaystyle 1_{0}}$

Note that the orientational symbols 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 z-direction has the orientation of 1_z. For angles, consider an angle θ that lies in the z-plane. Form a right triangle in the z-plane with θ 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. Since (using ~ to indicate orientational equivalence) tan(θ) = θ + ... ~ 1_y/1_x we conclude that an angle in the xy-plane must have an orientation 1_y/1_x = 1_z, which is not unreasonable. Analogous reasoning forces the conclusion that sin(θ) has orientation 1_z while cos(θ) has orientation 1₀. These are different, so one concludes (correctly), for example, that there are no solutions of physical equations that are of the form a cos(θ) + b sin(θ), where a and b are real scalars. Note that an expression such as ${\displaystyle \sin(\theta +\pi /2)=\cos(\theta )}$ is not dimensionally inconsistent since it is a special case of the sum of angles formula and should properly be written:

${\displaystyle \sin \left(a\,1_{\text{z}}+b\,1_{\text{z}}\right)=\sin \left(a\,1_{\text{z}})\cos(b\,1_{\text{z}}\right)+\sin \left(b\,1_{\text{z}})\cos(a\,1_{\text{z}}\right),}$

which for ${\displaystyle a=\theta }$ and ${\displaystyle b=\pi /2}$ yields ${\displaystyle \sin(\theta \,1_{\text{z}}+[\pi /2]\,1_{\text{z}})=1_{\text{z}}\cos(\theta \,1_{\text{z}})}$. Siano distinguishes between geometric angles, which have an orientation in 3-dimensional space, and phase angles associated with time-based oscillations, which have no spatial orientation, i.e. the orientation of a phase angle is ${\displaystyle 1_{0}}$.

The assignment of orientational symbols to physical quantities and the requirement that physical equations be orientationally homogeneous can actually be used in a way that is similar to dimensional analysis to derive more information about acceptable solutions of physical problems. In this approach, one solves the dimensional equation 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, putting it into normal form. The orientational equation is then solved to give a more restrictive condition on the unknown powers of the orientational symbols. The solution is then more complete than the one that dimensional analysis alone gives. Often, the added information is that one of the powers of a certain variable is even or odd.

As an example, for the projectile problem, using orientational symbols, θ, being in the xy-plane will thus have dimension 1_z and the range of the projectile R will be of the form:

${\displaystyle R=g^{a}\,v^{b}\,\theta ^{c}{\text{ which means }}{\mathsf {L}}\,1_{\mathrm {x} }\sim \left({\frac {{\mathsf {L}}\,1_{\text{y}}}{{\mathsf {T}}^{2}}}\right)^{a}\left({\frac {\mathsf {L}}{\mathsf {T}}}\right)^{b}\,1_{\mathsf {z}}^{c}.\,}$

Dimensional homogeneity will now correctly yield a = −1 and b = 2, and orientational homogeneity requires that ${\displaystyle 1_{x}/(1_{y}^{a}1_{z}^{c})=1_{z}^{c+1}=1}$. In other words, that c must be an odd integer. In fact, the required function of theta will be sin(θ)cos(θ) which is a series consisting of odd powers of θ.

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

Siano's orientational analysis is compatible with the conventional conception of angular quantities as being dimensionless, and within orientational analysis, the radian may still be considered a dimensionless unit. The orientational analysis of a quantity equation is carried out separately from the ordinary dimensional analysis, yielding information that supplements the dimensional analysis.

The dimensionless constants that arise in the results obtained, such as the C in the Poiseuille's Law problem and the ${\displaystyle \kappa }$ in the spring problems discussed above, come from a more detailed analysis of the underlying physics and often arise from integrating some differential equation. Dimensional analysis itself has little to say about these constants, but it is useful to know that they very often have a magnitude of order unity. This observation can allow one to sometimes make "back of the envelope" calculations about the phenomenon of interest, and therefore be able to more efficiently design experiments to measure it, or to judge whether it is important, etc.

Paradoxically, dimensional analysis can be a useful tool even if all the parameters in the underlying theory are dimensionless, e.g., lattice models such as the Ising model can be used to study phase transitions and critical phenomena. Such models can be formulated in a purely dimensionless way. As we approach the critical point closer and closer, the distance over which the variables in the lattice model are correlated (the so-called correlation length, ${\displaystyle \xi }$ ) becomes larger and larger. Now, the correlation length is the relevant length scale related to critical phenomena, so one can, e.g., surmise on "dimensional grounds" that the non-analytical part of the free energy per lattice site should be ${\displaystyle \sim 1/\xi ^{d}}$ where ${\displaystyle d}$ is the dimension of the lattice.

It has been argued by some physicists, e.g., M. J. Duff,[4][24] that the laws of physics are inherently dimensionless. The fact that we have assigned incompatible dimensions to Length, Time and Mass is, according to this point of view, just a matter of convention, borne out of the fact that before the advent of modern physics, there was no way to relate mass, length, and time to each other. The three independent dimensionful constants: c, ħ, and G, in the fundamental equations of physics must then be seen as mere conversion factors to convert Mass, Time and Length into each other.

Just as in the case of critical properties of lattice models, one can recover the results of dimensional analysis in the appropriate scaling limit; e.g., dimensional analysis in mechanics can be derived by reinserting the constants ħ, c, and G (but we can now consider them to be dimensionless) and demanding that a nonsingular relation between quantities exists in the limit ${\displaystyle c\rightarrow \infty }$, ${\displaystyle \hbar \rightarrow 0}$ and ${\displaystyle G\rightarrow 0}$. In problems involving a gravitational field the latter limit should be taken such that the field stays finite.

If c = ħ = 1, where c is the speed of light and ħ is the reduced Planck constant, and a suitable fixed unit of energy is chosen, then all quantities of time T, length L and mass M can be expressed (dimensionally) as a power of energy E, because length, mass and time can be expressed using speed v, action S, and energy E:[27]

${\displaystyle t={\frac {S}{E}},\quad L={\frac {Sv}{E}},\quad M={\frac {E}{v^{2}}}}$

though speed and action are dimensionless (v = c = 1 and S = ħ = 1) – so the only remaining quantity with dimension is energy. In terms of powers of dimensions:

${\displaystyle {\mathsf {E}}^{n}={\mathsf {T}}^{p}{\mathsf {L}}^{q}{\mathsf {M}}^{r}={\mathsf {E}}^{-p-q+r}}$

This is particularly useful in particle physics and high energy physics, in which case the energy unit is the electron volt (eV). Dimensional checks and estimates become very simple in this system.

However, if electric charges and currents are involved, another unit to be fixed is for electric charge, normally the electron charge e though other choices are possible.

• Covariance and contravariance of vectors
• Exterior algebra
• Geometric algebra
• Quantity calculus

Dimensional correctness as part of type checking has been studied since 1977.[28] Implementations for Ada[29] and C++[30] were described in 1985 and 1988. Kennedy's 1996 thesis describes an implementation in Standard ML,[31] and later in F#.[32] There are implementations for Haskell,[33] OCaml,[34] and Rust,[35] Python,[36] and a code checker for Fortran.[37][38]
Griffioen's 2019 thesis extended Kennedy's Hindley–Milner type system to support Hart's matrices.[39][40] McBride and Nordvall-Forsberg show how to use dependent types to extend type systems for units of measure.[41]

• Unicalc Live web calculator doing units conversion by dimensional analysis
• Math Skills Review
• U.S. EPA tutorial
• A Discussion of Units
• Short Guide to Unit Conversions
• Canceling Units Lesson
• Chapter 11: Behavior of Gases Chemistry: Concepts and Applications, Denton independent school District
• Air Dispersion Modeling Conversions and Formulas
• www.gnu.org/software/units free program, very practical