Dimensional analysis is a method of reducing the number of variables required to describe a given physical situation by making use of the information implied by the units of the physical quantities involved. It is also known as the "theory of similarity".
In observing the physical world, we make use of different physical concepts such as size, distance, time, temperature, etc., and to make quantitative deductions, we must adopt independent and reproducible methods of measuring these physical quantities. "Measurement" in essence means the comparison of the unknown quantity with a known reference, as in the measurement of distance using a ruler, or the balancing of an unknown weight against known weights on a chemical balance. For this purpose, agreement on a known "unit" for each physical quantity is needed; for example, the French Bureau of Standards keeps a block of platinum whose mass is defined to be one kilogram, and known "weights" used for chemical balances are ultimately "calibrated" by comparison with this primary reference—usually by a series of intermediate comparisons.
In fact, it is not necessary to have an independent primary reference for every physical quantity of interest since some physical quantities are defined in terms of others. For example, "velocity" is defined as rate of change of position, or as length travelled in a given time; thus if length is measured in meters and time in seconds, velocity can be measured in meters per second. Units based directly on primary references are called fundamental or basic units, while those defined in terms of other units are called compound or derived units. (See also SI Units.)
Given such a definition of one quantity in terms of others, a choice of basic primary references is thus possible. In the above case for example, astronomers prefer to take the velocity of light and the terrestrial year as primary references, so that length is measured in the derived units "light-years."
Some physical quantities are defined through physical laws. Such a law is simply the expression of a general conclusion from a set of experimental observations—often expressed in the form of a mathematical equation. For example, Fourier's Law of heat conduction: "The heat flux due to conduction through a solid is proportional to the temperature gradient," may be written:
where T is the temperature; x, the distance, and the "constant" of proportionality λ is called the Thermal Conductivity. Experiment shows that λ is not, in fact, a constant but varies with both temperature and the nature of the solid; it is, in fact, a new physical variable defined by the physical law, just as velocity was defined above.
There are situations where the new physical variable defined by a physical law does turn out to be constant. For example. Joule's experiments on the conversion of mechanical work (W) into heat (Q) are summarized by the equation:
which defines J as the "mechanical equivalent of heat". Joule found that J was independent of the particular conversion process involved and always had the same value (J = 4.184 × 10^{7}erg/calorie).
Such a constant is called a "universal constant" (other examples are Planck's constant and Boltzmann's constant), and when a law of this kind is available, there is further flexibility in choosing the set of fundamental units:
Keep the original system, with heat and work both based on arbitrary references (e.g., calorie and erg), and accept J as a new physical quantity (with units erg/calorie) which is relevant in any situation where work is converted to heat or vice versa.
Drop one of the original primary references (e.g., calorie) and use the law to define its units in terms of the other quantities involved by making the universal constant unity (i.e., a pure number). In this example, this implies that heat and work are the same kind of physical quantity (both are forms of energy). This is called a coherent choice of units.
It may be thought that a coherent choice is always the preferred choice, but this is not necessarily the case.
To summarize, a given physical system is described in terms of a set of physical variables of various types. To make quantitative observations, units of measurement for each variable are required. For this purpose, a subset of physical variables must be chosen, to which arbitrary reference units are assigned, then definitions or physical laws are used to derive units for the remaining variables.
The basic physical quantities for which arbitrary references are assigned are referred to as "dimensions," and the following notation is used:
Example: Heat transfer to fluid flowing through a pipe. The heat transfer coefficient (α) between the fluid and pipe-wall will possibly depend on fluid properties: density (ρ), viscosity (η), specific heat (c_{p}), thermal conductivity (λ), and also on the fluid mean velocity (u), the length (l) and diameter (D) of the pipe, and the temperature difference (ΔT) between the wall and the fluid.
It is reasonable to take mass [M], length [L], time [T], heat [H] and temperature [Θ] as basic dimensions.
Then, for example, α = [H/L^{2}TΘ] and the indices for all the variables can be set out in the form of a table called the dimensional matrix:
The value of a physical variable is always written as a number of units. For example, density Kg/m^{3} using the SI system of units. Of course, a change in the primary references, or basic units, induces a change in the corresponding numbers, , , etc., for given values of the variables. This is referred to as a unit-transformation. Thus, if two alternative sets of basic units U_{i1}, U_{i2} are available for the basic dimensions:
then for a typical physical variable v_{j}:
where m_{ij}, i = 1, 2, ... k, j = 1, 2, ... n are the entries in the dimensional matrix, and it follows that:
This equation formally defines a unit-transformation.
Clearly the outcome of any set of experiments cannot depend on the particular choice of basic units for the variables, which implies that any mathematical equation representing a valid physical law must be invariant to a unit-transformation. Such an equation is said to be dimensionally homogeneous. Thus if: f(v_{1}, v_{2}, ... v_{n}) = 0 represents a physical law, then:
where c_{1}, c_{2}, ... c_{n} satisfy Eq. (1) for some set of ratios U_{i1}/U_{i2}, i = 1, 2, ... k. This condition is a constraint on the form of an equation representing a physical law, which allows it to be expressed in terms of a reduced number of variables.
Example: Consider the simple fluid flow equation:
Using SI units, each variable can be written in the form kg/m^{3} etc., where etc. are pure numbers. Then each term in the equation has the units of pressure (kg/m.s^{2}), and the units cancel from the equation, confirming that it is indeed dimensionally homogeneous, and leaving a similar equation with replacing p_{1} etc.
Using a different set of basic units, these would also cancel giving a relation between a different set of numbers , etc. Notice that if only the two sets of numbers , , ... and , , …, were given, it would be difficult to tell which units were used; so the equation must also be invariant to changes in the variables within a given system of units, provided that the variables are scaled in the same way as if changes have been made in the system of units (i.e., are subjected to a unit transformation).
This can be seen more clearly if the equation is divided by its first term, to give:
Here, the units cancel within each bracketed group of variables, which are thus Dimensionless Groups or pure numbers, and the equation has been reduced to a relationship between five groups, rather than the nine original variables. Different physical situations giving the same values for the five groups are said to be "similar."
More generally, any dimensionally homogeneous equation can be reduced to dimensionless form and "similar" solutions can be exploited.
It is clear that the members of a given set of variables can be combined to form dimensionless groups in various different ways. In general, we have
and the indices p_{j}, j = 1, 2, ... n can be chosen to make the product ∏ dimensionless. From the dimensional matrix, , j = 1,2, ... n, so substituting this in (2) and equating the resulting index for each Di to zero yields the set of equations:
where p is the column vector with elements p_{j}. Now (3) has (n - r) linearly independent solutions p^{(1)}, p^{(2)}, ... p^{(n-r)}, where r is the rank of the matrix M, and any other solution is a linear combination of these solutions. This means that a set of (n - r) independent dimensionless groups can be formed, such that none of these can be formed by combination of the other groups in the set, but any group not in the set can be formed by combination of groups in the set. Such a set of dimensionless groups is called a complete set, and clearly any physical law must be expressible as a relation between members of this set.
In order to find the complete set, a subset of linearly independent columns of M must be found, then the columns permuted so that these are the first r columns and M can be written [M_{1}, M_{2}]. Then, using (3):
where e_{j} is a unit vector, with the jth element unity and the remaining elements zero. Then each generates one group of the complete set.
A more intuitive way to describe this procedure is to select "units" for each basic dimension as a combination of variables describing the system, using the same number of variables as there are basic dimensions. Then using these units for each of the remaining variables generates the required set of dimensionless groups.
Example: Heat transfer to fluid flowing through a pipe. Using the dimensional matrix given earlier, lengths can be measured in pipe-diameters D, and temperatures with ΔT as the unit. For mass, the mass of unit volume of the fluid, ρD^{3} can be used, and for heat, the capacity of this volume for unit temperature-rise, ρD^{3}c_{p}ΔT. Finally, for time, D/u can be used. For the remaining variables:
whence ∏_{1} = α/ρc_{p}u
Here, units for all basic dimensions (effectively finding as a solution to ) have been formed, which indicates that M is of full rank. However, note that none of the four groups obtained involve ΔT, which implies that α does not, in fact, depend on ΔT (or that other possibly relevant variables have not been considered, such as the coefficient of cubic expansion).
Had ΔT been omitted from the table, the rows for H and Θ would have been identical, except for a change of sign, showing that all variables only involve H and Θ as a ratio H/Θ, so units for H and Θ cannot be separately formed. Replacing the separate rows by a row for H/Θ (in fact identical to that for H), a unit can be defined [H/Θ] = ρD^{3}c_{p}, and the same four dimensionless groups as before can then be obtained.
In general, more complicated groupings of the original choice of basic dimensions may have to be used to obtain a matrix with linearly independent rows, which thus define a new reduced set of basic dimensions, necessary and just sufficient to define the units for all the remaining variables. Again, this is called a complete set of basic dimensions.
This leads to the famous "∏-Theorem:" "For a physical system described by n physical variables using a complete set of r basic dimensions, the laws governing the system can be expressed as mathematical relations among at most (n - r) dimensionless groups of variables."
Buckingham enunciated this theorem in 1914 without the all-important qualification that the basic dimensions form a complete set, and, of course, was unable to prove it. This remained a challenge until the correct form was enunciated and proved by Langhaar (1951).
The most obvious advantage of putting physical laws in dimensionless form is that it reduces the number of independent variables needed to describe the situation. Thus, for example, in planning an experimental investigation of heat transfer to fluid in a pipe, the form of the function: ∏_{1} = φ(∏_{2}, ∏_{3}, ∏_{4}) can be investigated, rather than α = f(D, L, u, ΔT, ρ, η, λ, c_{p}). Moreover, to vary ∏_{2}, ∏_{3} and ∏_{4}, the most convenient parameters can be chosen. Thus a rig can be built using only a single diameter pipe, and temperatures measured at several points along the length to obtain the effect of varying ∏_{2}, while varying the flow-rate (for a given fluid) gives the effect of varying ∏_{3}. It is in fact better to use ∏'_{4} = ∏_{3}/∏_{4} = c_{p}η/λ, which involves only fluid properties, rather than ∏_{4}, and to investigate variation of ∏'_{4} by choosing a range of different fluids. Note again that it is easier to find a range of fluids to cover a range of values for ∏'_{4}, rather than separate ranges for the individual properties c_{p}, η, λ, which tend to vary together.
In this example, the analysis itself indicated that one variable (ΔT) was irrelevant—or that other factors were being ignored. It also showed that the effect of varying pipe-diameter can be deduced from experiments on a single pipe.
The latter is a special case of exploiting "similar" solutions, which is perhaps better illustrated by the following example:
Example: Wind-tunnel testing of aircraft. Assume that the drag force F on the aircraft is a function of the density (ρ), viscosity (η) and speed of sound (u_{s}) of the air, and of the velocity (u), wing-span (l) and other dimensions (l_{1}, l_{2}, l_{3} ...) of the aircraft. Dimensional analysis yields:
To eliminate effects of the shape factors (l_{1}/l, ... etc.), a scale model is built, geometrically similar to the prototype aircraft, so that these have the same value for model and prototype.
To vary the other two groups independently, the air velocity u can be altered, but otherwise it is necessary either to build several models of different sizes or vary the air properties—in practice, a single model is used and either air temperature or air pressure is varied.
However, if the wind-tunnel uses only atmospheric air and only one model is available, only a partial solution is possible. If in fact u is well below us, then drag does not depend on u_{s}, and hence only one group (the Reynolds Number ulρ/η) is important. Remember however that the model size (l_{m}) is much smaller than the prototype (l_{p}), so the velocity (u_{m}) required in the wind-tunnel will be higher than the prototype velocity of interest (u_{p}); hence it is the Mach Number (u_{m}/u_{sm}) of the model which limits the range of validity of the tests. In general, of course, conditions can be chosen so that:
which yields F_{p}/u_{p}l_{p}η_{p} = F_{m}/u_{m}l_{m}η_{m}. These are then similar solutions, and conditions (4) are often called "similarity conditions."
It is quite common for similarity conditions to be incompatible, making it impossible to model actual conditions in all respects on a different scale.
Finally, the larger the number of basic dimensions for a given set of variables, the smaller the number of dimensionless groups in the complete set, and the simpler the resulting system.
Now if a given physical law is relevant, a noncoherent choice of units requires addition of the universal constant as a relevant physical variable. This is not necessary for a coherent choice, but then there is one less basic dimension and the number of groups is the same in each case.
On the other hand, if a physical law is not relevant the universal constant is not needed, but a coherent choice of units will reduce the number of basic dimensions and create an extra dimensionless group.
This is illustrated by the heat transfer example, where generation of heat by fluid friction was ignored. A coherent choice of heat unit would not have indicated that ΔT was irrelevant, and hence would have generated five groups.
REFERENCES
Buckingham, E. (1914) "On Physically Similar Systems: Illustrations of the Use of Dimensional Equations", Phys. Rev., 4, 345.
Langhaar, H. L. (1951) "Dimensional Analysis and the Theory of Models", John Wiley, New York. DOI: 10.1016/0016-0032(52)90438-9
References
- Buckingham, E. (1914) "On Physically Similar Systems: Illustrations of the Use of Dimensional Equations", Phys. Rev., 4, 345. DOI: 10.1103/PhysRev.4.345
- Langhaar, H. L. (1951) "Dimensional Analysis and the Theory of Models", John Wiley, New York. DOI: 10.1016/0016-0032(52)90438-9