In engineering, applied mathematics, and physics, the Buckingham π theorem is a key theorem in dimensional analysis. It is a formalization of Rayleigh's method of dimensional analysis. Loosely, the theorem states that if there is a physically meaningful equation involving a certain number n of physical variables, then the original equation can be rewritten in terms of a set of p = n − k dimensionless parameters π1, π2, ..., πp constructed from the original variables. (Here k is the number of physical dimensions involved; it is obtained as the rank of a particular matrix.)
The theorem provides a method for computing sets of dimensionless parameters from the given variables, or nondimensionalization, even if the form of the equation is still unknown.
Although named for Edgar Buckingham, the π theorem was first proved by French mathematician Joseph Bertrand in 1878. Bertrand considered only special cases of problems from electrodynamics and heat conduction, but his article contains, in distinct terms, all the basic ideas of the modern proof of the theorem and clearly indicates the theorem's utility for modelling physical phenomena. The technique of using the theorem (“the method of dimensions”) became widely known due to the works of Rayleigh. The first application of the π theorem in the general case to the dependence of pressure drop in a pipe upon governing parameters probably dates back to 1892, a heuristic proof with the use of series expansions, to 1894.
Formal generalization of the π theorem for the case of arbitrarily many quantities was given first by A. Vaschy in 1892, then in 1911—apparently independently—by both A. Federman and D. Riabouchinsky, and again in 1914 by Buckingham. It was Buckingham's article that introduced the use of the symbol "πi" for the dimensionless variables (or parameters), and this is the source of the theorem's name.
More formally, the number of dimensionless terms that can be formed, p, is equal to the nullity of the dimensional matrix, and k is the rank. For experimental purposes, different systems that share the same description in terms of these dimensionless numbers are equivalent.
In mathematical terms, if we have a physically meaningful equation such as
where the qi are the n physical variables, and they are expressed in terms of k independent physical units, then the above equation can be restated as
where the πi are dimensionless parameters constructed from the qi by p = n − k dimensionless equations — the so-called Pi groups — of the form
where the exponents ai are rational numbers (they can always be taken to be integers by redefining πi as being raised to a power that clears all denominators).
The Buckingham π theorem provides a method for computing sets of dimensionless parameters from given variables, even if the form of the equation remains unknown. However, the choice of dimensionless parameters is not unique; Buckingham's theorem only provides a way of generating sets of dimensionless parameters and does not indicate the most "physically meaningful".
Two systems for which these parameters coincide are called similar (as with similar triangles, they differ only in scale); they are equivalent for the purposes of the equation, and the experimentalist who wants to determine the form of the equation can choose the most convenient one. Most importantly, Buckingham's theorem describes the relation between the number of variables and fundamental dimensions.
It will be assumed that the space of fundamental and derived physical units forms a vector space over the rational numbers, with the fundamental units as basis vectors, and with multiplication of physical units as the "vector addition" operation, and raising to powers as the "scalar multiplication" operation: represent a dimensional variable as the set of exponents needed for the fundamental units (with a power of zero if the particular fundamental unit is not present). For instance, the standard gravity g has units of (distance over time squared), so it is represented as the vector with respect to the basis of fundamental units (distance, time).
Making the physical units match across sets of physical equations can then be regarded as imposing linear constraints in the physical-units vector space.
Given a system of n dimensional variables (with physical dimensions) in k fundamental (basis) dimensions, write the dimensional matrix M, whose rows are the fundamental dimensions and whose columns are the dimensions of the variables: the (i, j)th entry is the power of the ith fundamental dimension in the jth variable. The matrix can be interpreted as taking in a combination of the dimensions of the variable quantities and giving out the dimensions of this product in fundamental dimensions. So
is the units of
A dimensionless variable is a quantity with fundamental dimensions raised to the zeroth power (the zero vector of the vector space over the fundamental dimensions), which is equivalent to the kernel of this matrix.
By the rank–nullity theorem, a system of n vectors (matrix columns) in k linearly independent dimensions (the rank of the matrix is the number of fundamental dimensions) leaves a nullity, p, satisfying (p = n − k), where the nullity is the number of extraneous dimensions which may be chosen to be dimensionless.
The dimensionless variables can always be taken to be integer combinations of the dimensional variables (by clearing denominators). There is mathematically no natural choice of dimensionless variables; some choices of dimensionless variables are more physically meaningful, and these are what are ideally used.
This example is elementary but serves to demonstrate the procedure.
Suppose a car is driving at 100 km/h; how long does it take to go 200 km?
This question considers three dimensioned variables: distance d, time t, and velocity v, and we are seeking some law of the form t = Duration(v, d) . These variables admit a basis of two dimensions: time dimension T and distance dimension D. Thus there is 3 − 2 = 1 dimensionless quantity.
The dimensional matrix is
in which the rows correspond to the basis dimensions D and T, and the columns to the considered dimensions D, T, and V, where the latter stands for the velocity dimension. The elements of the matrix correspond to the powers to which the respective dimensions are to be raised. For instance, the third column (1, −1), states that V = D0T0V1, represented by the column vector , is expressible in terms of the basis dimensions as , since .
For a dimensionless constant , we are looking for vectors such that the matrix-vector product Ma equals the zero vector [0,0]. In linear algebra, the set of vectors with this property is known as the kernel (or nullspace) of (the linear map represented by) the dimensional matrix. In this particular case its kernel is one-dimensional. The dimensional matrix as written above is in reduced row echelon form, so one can read off a non-zero kernel vector to within a multiplicative constant:
If the dimensional matrix were not already reduced, one could perform Gauss–Jordan elimination on the dimensional matrix to more easily determine the kernel. It follows that the dimensionless constant, replacing the dimensions by the corresponding dimensioned variables, may be written:
Since the kernel is only defined to within a multiplicative constant, the above dimensionless constant raised to any arbitrary power yields another (equivalent) dimensionless constant.
Dimensional analysis has thus provided a general equation relating the three physical variables:
or, letting denote a zero of function ,
which can be written as
The actual relationship between the three variables is simply . In other words, in this case has one physically relevant root, and it is unity. The fact that only a single value of C will do and that it is equal to 1 is not revealed by the technique of dimensional analysis.
We wish to determine the period T of small oscillations in a simple pendulum. It will be assumed that it is a function of the length L, the mass M, and the acceleration due to gravity on the surface of the Earth g, which has dimensions of length divided by time squared. The model is of the form
(Note that it is written as a relation, not as a function: T isn't written here as a function of M, L, and g.)
There are 3 fundamental physical dimensions in this equation: time , mass , and length , and 4 dimensional variables, T, M, L, and g. Thus we need only 4 − 3 = 1 dimensionless parameter, denoted π, and the model can be re-expressed as
where π is given by
for some values of a1, ..., a4.
The dimensions of the dimensional quantities are:
The dimensional matrix is:
(The rows correspond to the dimensions , and , and the columns to the dimensional variables T, M, L and g. For instance, the 4th column, (−2, 0, 1), states that the g variable has dimensions of .)
We are looking for a kernel vector a = [a1, a2, a3, a4] such that the matrix product of M on a yields the zero vector [0,0,0]. The dimensional matrix as written above is in reduced row echelon form, so one can read off a kernel vector within a multiplicative constant:
Were it not already reduced, one could perform Gauss–Jordan elimination on the dimensional matrix to more easily determine the kernel. It follows that the dimensionless constant may be written:
In fundamental terms:
which is dimensionless. Since the kernel is only defined to within a multiplicative constant, if the above dimensionless constant is raised to any arbitrary power, it will yield another equivalent dimensionless constant.
This example is easy because three of the dimensional quantities are fundamental units, so the last (g) is a combination of the previous. Note that if a2 were non-zero, there would be no way to cancel the M value; therefore a2 must be zero. Dimensional analysis has allowed us to conclude that the period of the pendulum is not a function of its mass. (In the 3D space of powers of mass, time, and distance, we can say that the vector for mass is linearly independent from the vectors for the three other variables. Up to a scaling factor, is the only nontrivial way to construct a vector of a dimensionless parameter.)
The model can now be expressed as:
Assuming the zeroes of f are discrete, we can say gT2/L = Cn, where Cn is the nth zero of the function f. If there is only one zero, then gT2/L = C. It requires more physical insight or an experiment to show that there is indeed only one zero and that the constant is in fact given by C = 4π2.
For large oscillations of a pendulum, the analysis is complicated by an additional dimensionless parameter, the maximum swing angle. The above analysis is a good approximation as the angle approaches zero.
A simple example of dimensional analysis can be found for the case of the mechanics of a thin, solid and parallel-sided rotating disc. There are five variables involved which reduce to two non-dimensional groups. The relationship between these can be determined by numerical experiment using, for example, the finite element method.
The year 1914 in science and technology involved some significant events, listed below.Affinity laws
The affinity laws (Also known as the "Fan Laws" or "Pump Laws") for pumps/fans are used in hydraulics, hydronics and/or HVAC to express the relationship between variables involved in pump or fan performance (such as head, volumetric flow rate, shaft speed) and power. They apply to pumps, fans, and hydraulic turbines. In these rotary implements, the affinity laws apply both to centrifugal and axial flows.
The laws are derived using the Buckingham π theorem. The affinity laws are useful as they allow prediction of the head discharge characteristic of a pump or fan from a known characteristic measured at a different speed or impeller diameter. The only requirement is that the two pumps or fans are dynamically similar, that is the ratios of the fluid forced are the same. It is also required that the two impellers' speed or diameter are running at the same efficiency.
Law 1. With impeller diameter (D) held constant:
Law 1a. Flow is proportional to shaft speed:
Law 1b. Pressure or Head is proportional to the square of shaft speed:
Law 1c. Power is proportional to the cube of shaft speed:
Law 2. With shaft speed (N) held constant:
Law 2a. Flow is proportional to the impeller diameter:
Law 2b. Pressure or Head is proportional to the square of the impeller diameter:
Law 2c. Power is proportional to the cube of impeller diameter (assuming constant shaft speed):
These laws assume that the pump/fan efficiency remains constant i.e. , which is rarely exactly true, but can be a good approximation when used over appropriate frequency or diameter ranges (i.e., a fan will not move anywhere near 1000 times as much air when spun at 1000 times its designed operating speed, but the air movement may be increased by 99% when the operating speed is only doubled). The exact relationship between speed, diameter, and efficiency depends on the particulars of the individual fan or pump design. Product testing or computational fluid dynamics become necessary if the range of acceptability is unknown, or if a high level of accuracy is required in the calculation. Interpolation from accurate data is also more accurate than the affinity laws. When applied to pumps the laws work well for constant diameter variable speed case (Law 1) but are less accurate for constant speed variable impeller diameter case (Law 2).
For radial flow centrifugal pumps, it is common industry practice to reduce the impeller diameter by "trimming", whereby the outer diameter of a particular impeller is reduced by machining to alter the performance of the pump. In this particular industry it is also common to refer to the mathematical approximations that relate the volumetric flow rate, trimmed impeller diameter, shaft rotational speed, developed head, and power as the "affinity laws". Because trimming an impeller changes the fundamental shape of the impeller (increasing the specific speed), the relationships shown in Law 2 cannot be utilized in this scenario. In this case the industry looks to the following relationships, which is a better approximation of these variables when dealing with impeller trimming.
With shaft speed (N) held constant and for small variations in impeller diameter via trimming:
The volumetric flow rate varies directly with the trimmed impeller diameter:
The pump developed head (the total dynamic head) varies to the square of the trimmed impeller diameter:
The power varies to the cube of the trimmed impeller diameter:
Buckingham is a surname. Notable people with the surname include:
A. David Buckingham, British physical chemist
Catharinus P. Buckingham, American Civil War general
Celeste Buckingham, Slovak recording artist of Swiss-American origins
Des Buckingham, English football manager
Edgar Buckingham, creator of the Buckingham π theorem, a key theorem in dimensional analysis
Edward Taylor Buckingham, III, former CNMI Attorney General
James Silk Buckingham, oriental traveller
John Buckingham (chemist), British chemist
John Buckingham (cricketer), English cricketer
John Buckingham (jockey) (1940–2016), English horse racing jockey
Leicester Silk Buckingham, playwright
Lindsey Buckingham, American rock musician and member of Fleetwood Mac
Owen Buckingham, (1674 – 1720) English politician
Sir Owen Buckingham, (c.1649 – 1713) English MP and Lord Mayor of London
Vic Buckingham, British football player and coach
William Alfred Buckingham, former governor of Connecticut
Steve Buckingham (record producer), American record producer and musicianDebye–Hückel equation
The chemists Peter Debye and Erich Hückel noticed that solutions that contain ionic solutes do not behave ideally even at very low concentrations. So, while the concentration of the solutes is fundamental to the calculation of the dynamics of a solution, they theorized that an extra factor that they termed gamma is necessary to the calculation of the activity coefficients of the solution. Hence they developed the Debye–Hückel equation and Debye–Hückel limiting law. The activity is only proportional to the concentration and is altered by a factor known as the activity coefficient . This factor takes into account the interaction energy of ions in solution.Dimensional analysis
In engineering and science, dimensional analysis is the analysis of the relationships between different physical quantities by identifying their base quantities (such as length, mass, time, and electric charge) and units of measure (such as miles vs. kilometers, or pounds vs. kilograms) and tracking these dimensions as calculations or comparisons are performed. The conversion of units from one dimensional unit to another is often somewhat complex. Dimensional analysis, or more specifically the factor-label method, also known as the unit-factor method, is a widely used technique for such conversions using the rules of algebra.The concept of physical dimension was introduced by Joseph Fourier in 1822. Physical quantities that are of the same kind (also called commensurable) have the same dimension (length, time, mass) and can be directly compared to each other, even if they are originally expressed in differing units of measure (such as yards and meters). If physical quantities have different dimensions (such as length vs. mass), they cannot be expressed in terms of similar units and cannot be compared in quantity (also called incommensurable). For example, asking whether a kilogram is larger than an hour is meaningless.
Any physically meaningful equation (and any inequality) will 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.Dimensionless quantity
In dimensional analysis, a dimensionless quantity is a quantity to which no physical dimension is assigned. It is also known as a bare number or pure number or a quantity of dimension one and the corresponding unit of measurement in the SI is one (or 1) unit and it is not explicitly shown. Dimensionless quantities are widely used in many fields, such as mathematics, physics, chemistry, engineering, and economics. Examples of quantities to which dimensions are regularly assigned are length, time, and speed, which are measured in dimensional units, such as metre, second and metre per second. This is considered to aid intuitive understanding. However, especially in mathematical physics, it is often more convenient to drop the assignment of explicit dimensions and express the quantities without dimensions, e.g., addressing the speed of light simply by the dimensionless number 1.Drag equation
In fluid dynamics, the drag equation is a formula used to calculate the force of drag experienced by an object due to movement through a fully enclosing fluid. The equation is:
The equation is attributed to Lord Rayleigh, who originally used L2 in place of A (with L being some linear dimension).
The reference area A is typically defined as the area of the orthographic projection of the object on a plane perpendicular to the direction of motion. For non-hollow objects with simple shape, such as a sphere, this is exactly the same as a cross sectional area. For other objects (for instance, a rolling tube or the body of a cyclist), A may be significantly larger than the area of any cross section along any plane perpendicular to the direction of motion. Airfoils use the square of the chord length as the reference area; since airfoil chords are usually defined with a length of 1, the reference area is also 1. Aircraft use the wing area (or rotor-blade area) as the reference area, which makes for an easy comparison to lift. Airships and bodies of revolution use the volumetric coefficient of drag, in which the reference area is the square of the cube root of the airship's volume. Sometimes different reference areas are given for the same object in which case a drag coefficient corresponding to each of these different areas must be given.
For sharp-cornered bluff bodies, like square cylinders and plates held transverse to the flow direction, this equation is applicable with the drag coefficient as a constant value when the Reynolds number is greater than 1000. For smooth bodies, like a circular cylinder, the drag coefficient may vary significantly until Reynolds numbers up to 107 (ten million).Dynamic scraped surface heat exchanger
The dynamic scraped surface heat exchanger (DSSHE) was designed to face some problems found in other types of heat exchangers. They increase heat transfer by:
removing the fouling layers, increasing turbulence in case of high viscosity flow, and avoiding the generation of ice and other process by-products. DSSHEs incorporate an internal mechanism which periodically removes the product from the heat transfer wall.Edgar Buckingham
Edgar Buckingham (July 8, 1867 in Philadelphia, Pennsylvania – April 29, 1940 in Washington DC) was an American physicist.
He graduated from Harvard with a bachelor's degree in physics in 1887. He did additional graduate work at the University of Strasbourg and the University of Leipzig, where he studied under chemist Wilhelm Ostwald. Buckingham received a PhD from Leipzig in 1893. He worked at the USDA Bureau of Soils from 1902 to 1906 as a soil physicist. He worked at the (US) National Bureau of Standards (now the National Institute of Standards and Technology, or NIST) 1906-1937. His fields of expertise included soil physics, gas properties, acoustics, fluid mechanics, and blackbody radiation. He is also the originator of the Buckingham π theorem in the field of dimensional analysis.
In 1923, Buckingham published a report which voiced skepticism that jet propulsion would be economically competitive with prop driven aircraft at low altitudes and at the speeds of that period.
Buckingham's first work on soil physics is on soil aeration, particularly the loss of carbon dioxide from the soil and its subsequent replacement by oxygen. From his experiments he found that the rate of gas diffusion in soil was not dependent significantly on the soil structure, compactness or water content of the soil. Using an empirical formula based on his data, Buckingham was able to give the diffusion coefficient as a function of air content. This relation is still commonly cited in many modern textbooks and used in modern research. The outcomes of his research on gas transport were to conclude that the exchange of gases in soil aeration takes place by diffusion and is sensibly independent of the variations of the outside barometric pressure.
Buckingham then worked on soil water, research for which he is now renowned. Buckingham’s work on soil water is published in Bulletin 38 USDA Bureau of Soils: Studies on the movement of soil moisture, which was released in 1907. This document contained three sections, the first of which looked at evaporation of water from below a layer of soil. He found that soils of various textures could strongly inhibit evaporation, particularly where capillary flow through the uppermost layers was prevented. The second section of Bulletin 38 looked at the drying of soils under arid and humid conditions. Buckingham found evaporative losses were initially higher from the arid soil, then after three days the evaporation under arid conditions became less than under humid conditions, with the total loss ending up greater from the humid soil. Buckingham believed this occurred due to the self-mulching behaviour (he referred to it as the soil forming a natural mulch) exhibited by the soil under arid conditions.
The third section of Bulletin 38 contains the work on unsaturated flow and capillary action for which Buckingham is famous. He firstly recognized the importance of the potential of the forces arising from interactions between soil and water. He called this the capillary potential, this is now known as the moisture or water potential (matric potential). He combined capillary theory and an energy potential in soil physics theory, and was the first to expound the dependence of soil hydraulic conductivity on capillary potential. This dependence later came to be known as relative permeability in petroleum engineering. He also applied a formula equivalent to Darcy's law to unsaturated flow.Grashof number
The Grashof number (Gr) is a dimensionless number in fluid dynamics and heat transfer which approximates the ratio of the buoyancy to viscous force acting on a fluid. It frequently arises in the study of situations involving natural convection and is analogous to the Reynolds number. It's believed to be named after Franz Grashof. Though this grouping of terms had already been in use, it wasn't named until around 1921, 28 years after Franz Grashof's death. It's not very clear why the grouping was named after him.Index of physics articles (B)
The index of physics articles is split into multiple pages due to its size.
To navigate by individual letter use the table of contents below.Nondimensionalization
Nondimensionalization is the partial or full removal of units from an equation involving physical quantities by a suitable substitution of variables. This technique can simplify and parameterize problems where measured units are involved. It is closely related to dimensional analysis. In some physical systems, the term scaling is used interchangeably with nondimensionalization, in order to suggest that certain quantities are better measured relative to some appropriate unit. These units refer to quantities intrinsic to the system, rather than units such as SI units. Nondimensionalization is not the same as converting extensive quantities in an equation to intensive quantities, since the latter procedure results in variables that still carry units.
Nondimensionalization can also recover characteristic properties of a system. For example, if a system has an intrinsic resonance frequency, length, or time constant, nondimensionalization can recover these values. The technique is especially useful for systems that can be described by differential equations. One important use is in the analysis of control systems.
One of the simplest characteristic units is the doubling time of a system experiencing exponential growth, or conversely the half-life of a system experiencing exponential decay; a more natural pair of characteristic units is mean age/mean lifetime, which correspond to base e rather than base 2.
Many illustrative examples of nondimensionalization originate from simplifying differential equations. This is because a large body of physical problems can be formulated in terms of differential equations. Consider the following:
List of dynamical systems and differential equations topics
List of partial differential equation topics
Differential equations of mathematical physicsAlthough nondimensionalization is well adapted for these problems, it is not restricted to them. An example of a non-differential-equation application is dimensional analysis; another example is normalization in statistics.
Measuring devices are practical examples of nondimensionalization occurring in everyday life. Measuring devices are calibrated relative to some known unit. Subsequent measurements are made relative to this standard. Then, the absolute value of the measurement is recovered by scaling with respect to the standard.Pi (letter)
Pi (; uppercase Π, lowercase π and ϖ; Greek: πι [pi]) is the sixteenth letter of the Greek alphabet, representing the sound [p]. In the system of Greek numerals it has a value of 80. It was derived from the Phoenician letter Pe (). Letters that arose from pi include Cyrillic Pe (П, п), Coptic pi (Ⲡ, ⲡ), and Gothic pairthra (𐍀).Similitude (model)
Similitude is a concept applicable to the testing of engineering models. A model is said to have similitude with the real application if the two share geometric similarity, kinematic similarity and dynamic similarity. Similarity and similitude are interchangeable in this context.
The term dynamic similitude is often used as a catch-all because it implies that geometric and kinematic similitude have already been met.
Similitude's main application is in hydraulic and aerospace engineering to test fluid flow conditions with scaled models. It is also the primary theory behind many textbook formulas in fluid mechanics.
The concept of similitude is strongly tied to dimensional analysis.Wilhelm Nusselt
Ernst Kraft Wilhelm Nußelt (Nusselt in English; born November 25, 1882 in Nuremberg – died September 1, 1957 in München) was a German engineer. Nusselt studied mechanical engineering at the Munich Technical University (Technische Universität München), where he got his doctorate in 1907. He taught in Dresden from 1913 to 1917.
During this teaching tenure he developed the dimensional analysis of heat transfer, without any knowledge of the Buckingham π theorem or any other developments of Lord Rayleigh.
In so doing he opened the door for further heat transfer analysis. After teaching and working in Switzerland and Germany between 1917 and 1925, he was appointed to the Chair of Theoretical Mechanics in München. There he made important developments in the field of heat exchangers. He held that position until 1952, being succeeded in the job by another important figure in the field of heat transfer, Ernst Schmidt.
The Nusselt number used in Fluid Mechanics and Heat Transfer is named in his honour.Wind-turbine aerodynamics
The primary application of wind turbines is to generate energy using the wind. Hence, the aerodynamics is a very important aspect of wind turbines. Like most machines, there are many different types of wind turbines, all of them based on different energy extraction concepts.
Though the details of the aerodynamics depend very much on the topology, some fundamental concepts apply to all turbines. Every topology has a maximum power for a given flow, and some topologies are better than others. The method used to extract power has a strong influence on this. In general, all turbines may be grouped as being either lift-based, or drag-based; the former being more efficient. The difference between these groups is the aerodynamic force that is used to extract the energy.
The most common topology is the horizontal-axis wind turbine (HAWT). It is a lift-based wind turbine with very good performance. Accordingly, it is a popular choice for commercial applications and much research has been applied to this turbine. Despite being a popular lift-based alternative in the latter part of the 20th century, the Darrieus wind turbine is rarely used today. The Savonius wind turbine is the most common drag type turbine. Despite its low efficiency, it remains in use because of its robustness and simplicity to build and maintain.