Dimensional Analysis

In engineering and science, dimensional analysis of different physical quantities is the analysis of their physical dimension or quantity dimension, defined as a mathematical expression identifying the powers of the base quantities involved (such as length, mass, time, etc.), and tracking these dimensions as calculations or comparisons are performed. The concepts of dimensional analysis and quantity dimension were introduced by Joseph Fourier in 1822. Commensurable physical quantities have the same dimension and are of the same kind, so they can be directly compared to each other, even if they are expressed in differing units of measurement; e.g., metres and feet, grams and pounds, seconds and years. Incommensurable physical quantities have different dimensions, so can not be directly compared to each other, no matter what units they are expressed in, e.g. metres and grams, seconds and grams, metres and seconds. For example, asking whether a gram is larger than an hour is meaningless. Any physically meaningful equation or inequality must have the same dimensions on its left and right sides, a property known as dimensional homogeneity. Checking for dimensional homogeneity is a common application of dimensional analysis, serving as a plausibility check on derived equations and computations.…

In engineering and science, dimensional analysis of different physical quantities is the analysis of their physical dimension or quantity dimension, defined as a mathematical expression identifying the powers of the base quantities involved (such as length, mass, time, etc.), and tracking these dimensions as calculations or comparisons are performed. The concepts of dimensional analysis and quantity dimension were introduced by Joseph Fourier in 1822. Commensurable physical quantities have the same dimension and are of the same kind, so they can be directly compared to each other, even if they are expressed in differing units of measurement; e.g., metres and feet, grams and pounds, seconds and years. Incommensurable physical quantities have different dimensions, so can not be directly compared to each other, no matter what units they are expressed in, e.g. metres and grams, seconds and grams, metres and seconds. For example, asking whether a gram is larger than an hour is meaningless. Any physically meaningful equation or inequality must have the same dimensions on its left and right sides, a property known as dimensional homogeneity. Checking for dimensional homogeneity is a common application of dimensional analysis, serving as a plausibility check on derived equations and computations. It also serves as a guide and constraint in deriving equations that may describe a physical system in the absence of a more rigorous derivation. \operatorname{dim} Q = \mathsf{T}^a \mathsf{L}^b \mathsf{M}^c \mathsf{I}^d \mathsf{\Theta}^e \mathsf{N}^f \mathsf{J}^g where , , , , , , are the dimensional exponents. Other physical quantities could be defined as the base quantities, as long as they form a basis – for instance, one could replace the dimension (I) of electric current of the SI basis with a dimension (Q) of electric charge, since . A quantity that has only (with all other exponents zero) is known as a geometric quantity. A quantity that has only both and is known as a kinematic quantity. A quantity that has only all of , , and is known as a dynamic quantity. A quantity that has all exponents null is said to have dimension one. The unit chosen to express a physical quantity and its dimension are related, but not identical concepts. The units of a physical quantity are defined by convention and related to some standard; e.g., length may have units of metres, feet, inches, miles or micrometres; but any length always has a dimension of L, no matter what units of length are chosen to express it. Two different units of the same physical quantity have conversion factors that relate them. For example, ; in this case 2.54 cm/in is the conversion factor, which is itself dimensionless and equal to 1. Therefore, multiplying by that conversion factor does not change either the dimensions nor the value of the physical quantity. There are also physicists who have cast doubt on the very existence of incompatible fundamental dimensions of physical quantity, although this does not invalidate the usefulness of dimensional analysis. \operatorname{dim}v = \frac{\text{length}}{\text{time}} = \frac{\mathsf{L}}{\mathsf{T}} = \mathsf{T}^{-1}\mathsf{L} . The dimension of the physical quantity acceleration is \operatorname{dim}a = \frac{\text{velocity}}{\text{time}} = \frac{\mathsf{T}^{-1}\mathsf{L}}{\mathsf{T}} = \mathsf{T}^{-2}\mathsf{L} . The dimension of the physical quantity force is \operatorname{dim}F = \text{mass} \times \text{acceleration} = \mathsf{M} \times \mathsf{T}^{-2}\mathsf{L} = \mathsf{T}^{-2}\mathsf{L}\mathsf{M} . The dimension of the physical quantity pressure is \operatorname{dim}P = \frac{\text{force}}{\text{area}} = \frac{\mathsf{T}^{-2}\mathsf{L}\mathsf{M}}{\mathsf{L}^2} = \mathsf{T}^{-2}\mathsf{L}^{-1}\mathsf{M} . The dimension of the physical quantity energy is \operatorname{dim}E = \text{force} \times \text{displacement} = \mathsf{T}^{-2}\mathsf{L}\mathsf{M} \times \mathsf{L} = \mathsf{T}^{-2}\mathsf{L}^2\mathsf{M} . The dimension of the physical quantity power is \operatorname{dim}P = \frac{\text{energy}}{\text{time}} = \frac{\mathsf{T}^{-2}\mathsf{L}^2\mathsf{M}}{\mathsf{T}} = \mathsf{T}^{-3}\mathsf{L}^2\mathsf{M} . The dimension of the physical quantity electric charge is \operatorname{dim}Q = \text{current} \times \text{time} = \mathsf{T}\mathsf{I} . The dimension of the physical quantity voltage is \operatorname{dim}V = \frac{\text{power}}{\text{current}} = \frac{\mathsf{T}^{-3}\mathsf{L}^2\mathsf{M}}{\mathsf{I}} = \mathsf{T^{-3}}\mathsf{L}^2\mathsf{M} \mathsf{I}^{-1} . The dimension of the physical quantity capacitance is \operatorname{dim}C = \frac{\text{electric charge}}{\text{electric potential difference}} = \frac {\mathsf{T}\mathsf{I}}{\mathsf{T}^{-3}\mathsf{L}^2\mathsf{M}\mathsf{I}^{-1}} = \mathsf{T^4}\mathsf{L^{-2}}\mathsf{M^{-1}}\mathsf{I^2} . == (commensurability)== The most basic rule of dimensional analysis is that of dimensional homogeneity. However, the dimensions form an abelian group under multiplication, so: For example, it makes no sense to ask whether 1 hour is more, the same, or less than 1 kilometre, as these have different dimensions, nor to add 1 hour to 1 kilometre. However, it makes sense to ask whether 1 mile is more, the same, or less than 1 kilometre, being the same dimension of physical quantity even though the units are different. On the other hand, if an object travels 100 km in 2 hours, one may divide these and conclude that the object's average speed was 50 km/h. The rule implies that in a physically meaningful expression only quantities of the same dimension can be added, subtracted, or compared. For example, if , and denote, respectively, the mass of some man, the mass of a rat and the height of that man, the dimensionally homogeneous expression is meaningful, but the heterogeneous expression is meaningless. However, is fine. Thus, dimensional analysis may be used as a sanity check of physical equations: the two sides of any equation must be commensurable or have the same dimensions. Even when two physical quantities have identical dimensions, it may nevertheless be meaningless to compare or add them. For example, although torque and energy share the dimension , they are fundamentally different physical quantities. To compare, add, or subtract quantities with the same dimensions but expressed in different units, the standard procedure is first to convert them all to the same unit. For example, to compare 32 metres with 35 yards, use to convert 35 yards to 32.004 m. A related principle is that any physical law that accurately describes the real world must be independent of the units used to measure the physical variables. For example, Newton's laws of motion must hold true whether distance is measured in miles or kilometres. This principle gives rise to the form that a conversion factor between two units that measure the same dimension must take multiplication by a simple constant. It also ensures equivalence; for example, if two buildings are the same height in feet, then they must be the same height in metres. For example, if one is calculating a speed, units must always combine to L/T; if one is calculating an energy, units must always combine to ML 2 /T 2 , etc. For example, the following formulae could be valid expressions for some energy: E_\text{k} = \frac 12 m v^2 ;~~ E = m c^2 ;~~ E = p v ; ~~ E = \frac{hc}{\lambda} if m is a mass, v and c are velocities, p is a momentum, h is the Planck constant, λ a length. On the other hand, if the units of the right hand side do not combine to [mass][length] 2 /[time] 2 , it cannot be a valid expression for some energy. Being homogeneous does not necessarily mean the equation will be true, since it does not take into account numerical factors. For example, could be or could not be the correct formula for the energy of a particle of mass m traveling at speed v, and one cannot know if hc/λ should be divided or multiplied by 2π. Reynolds number ( ), generally important in all types of fluid problems: \mathrm{Re} = \frac{\rho u d}{\mu}. Froude number ( ), modeling flow with a free surface: \mathrm{Fr} = \frac{u}{\sqrt{gL}}. Euler number ( ), used in problems in which pressure is of interest: \mathrm{Eu} = \frac{\Delta p}{\rho u^2}. Mach number ( ), important in high speed flows where the velocity approaches or exceeds the local speed of sound: \mathrm{Ma} = \frac{u}{c_s}, where is the local speed of sound. \begin{align} \pi_1 &= \frac{E}{As} \\ \pi_2 &= \frac{\ell}{A}. \end{align} The linear density of the wire is not involved. The two groups found can be combined into an equivalent form as an equation F\left(\frac{E}{As}, \frac{\ell}{A}\right) = 0 , where is some unknown function, or, equivalently as E = As f{\left(\frac{\ell}{A}\right)} , where 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  . 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  , and so infer that . 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. demand/capacity = thickness/radius or aspect ratio = 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. 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 f(\pi_1,\pi_2, ..., \pi_m)=0\,. Knowing this restriction can be a powerful tool for obtaining new insight into the system. 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/T 2 ], 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 ( ), is redundant (the set is not linearly independent). \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  if the acceleration of gravity is 9.8 and the initial upward speed is 500 . It is not necessary for to be in seconds. For example, suppose . Then the first term would be \begin{align} &\tfrac{1}{2} \cdot (\mathrm{-9.8~m/s^2}) \cdot (\mathrm{0.01~min})^2 \\[10pt] 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: \mathrm{1\,ft} = \mathrm{0.3048\,m} ~~\text{ is identical to }~~ 1 = \frac{\mathrm{0.3048\,m}}{\mathrm{1\,ft}}. The factor 0.3048 m/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. for = 5 m/s, where and may be expressed in any units, converted if necessary. In contrast, a corresponding numerical-value equation would be: where is the numeric value of when expressed in seconds and is the numeric value of when expressed in metres. Generally, the use of numerical-value equations is discouraged. −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. 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 corresponding to the Celsius temperature −272.15 °C, or a temperature difference equal to 1 °C. R \propto v_\text{x}^a\,v_\text{y}^b\,g^c . Or dimensionally \mathsf{L} = \left(\mathsf{T}^{-1}\mathsf{L}\right)^{a+b} \left(\mathsf{T}^{-2}\mathsf{L}\right)^c from which we may deduce that a + b + c = 1 and , 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 v_\mathrm{x} will be dimensioned as T −1 L , v_\mathrm{y} as T −1 L , as L and as T −2 L . The dimensional equation becomes: \mathsf{L}_\mathrm{x} = \left({\mathsf{T}^{-1}}{\mathsf{L}_\mathrm{x}}\right)^a \left({\mathsf{T}^{-1}}{\mathsf{L}_\mathrm{y}}\right)^b \left({\mathsf{T}^{-2}}{\mathsf{L}_\mathrm{y}}\right)^c and we may solve completely as , and . 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 , L , L , 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. \pi_1 = \frac{\dot{m}}{\eta r} \pi_2 = \frac{p_\mathrm{x}\rho r^5}{\dot{m}^2} If we distinguish between inertial mass with dimension M_\text{i} and quantity of matter with dimension 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: C = \frac{p_\mathrm{x}\rho r^4}{\eta \dot{m}} where now only is an undetermined constant (found to be equal to \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 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. 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 . For angles, consider an angle that lies in the z-plane. Form a right triangle in the z-plane with being one of the acute angles. The side of the right triangle adjacent to the angle then has an orientation and the side opposite has an orientation . Since (using to indicate orientational equivalence) we conclude that an angle in the xy-plane must have an orientation , which is not unreasonable. Analogous reasoning forces the conclusion that has orientation while 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 , where and are real scalars. An expression such as \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: \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 a = \theta and b = \pi/2 yields . 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 . 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, , being in the xy-plane will thus have dimension and the range of the projectile will be of the form: 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 and , and orientational homogeneity requires that . In other words, that must be an odd integer. In fact, the required function of theta will be which is a series consisting of odd powers of . It is seen that the Taylor series of and are orientationally homogeneous using the above multiplication table, while expressions like and 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.

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