Buckingham s theorem

In terms of linear vector space, Buckingham s theorem (Theorem 2) simply states that the null space of the dimensional matrix has a fixed dimension, and Van Driest s rule (Theorem 3) then specifies the nullity of the dimensional matrix. The problem of finding a complete set of B-numbers is equivalent to that of computing a fundamental system of solutions of equation 13 called a complete set of B-vectors. For simplicity, the matrix formed by a complete set of B-vectors will be called a complete B-matrix. It can also be demonstrated that the choice of reference dimensions does not affect the B-numbers (22). 

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For a correct dimensional analysis, it is necessary to consider Buckingham s theorem, which may be stated as follows (3,4) ... 

According to Buckingham s theorem, the following dimensionless groups can be identified ... 

There are nine variables and three primary dimensions, and therefore by Buckingham s theorem, Equation 7-1 can be expressed by (9-3) dimensionless groups. Employing dimensional analysis, Equation 7-1 in terms of the three basic dimensions (mass M, length L, and time T) yields Power = ML2T 3. 

In previous paragraphs, we obtained groups of dimensionless vaiiables by means of Buckingham s theorem. In the development of Reynolds, Froude, and Weber numbers we utilized the concept of force ratios although the same numbers can be produced by means of Buckingham s theorem. 

The theorem provides a method for computing sets of dimensionless parameters from the given variables, even if the form of the equation is still unknown. However, the choice of dimensionless parameters, using this non-dimensionalization scheme, is not unique Buckingham s theorem only provides a way of generating sets of dimensionless parameters, and will not choose the most physically meaningful dimensionless parameters. 

If two dimensions occur only in a specific ratio, then they are not independent and must be treated as one dimension. Suppose that our list of >l s and 5 s consists of two velocities and and two forces Fj and F. By simple application of Buckingham s theorem we conclude that n -f-1 equals 4 and that k equals 3 (length, time, force) so there should be one tt. But to conclude that there is only one tt here is incorrect. Since length and time appear in our list of variables only in the combination length/time, there are really only two independent dimensions, force and length/time so F is 2,... 

Buckingham s theorem, also provides an algorithm for selecting tt s ... 

There are two classical methods in dimensional analysis, Buckingham s pi theorem and the method of indices by Lord Rayleigh. Here we will briefly explain the more common of the two Buckingham s theorem. 

Buckingham s theorem. This theorem can be divided into three steps as shown below (Sonin 2001). Step 1 define the dependent variable and find all the relevant independent variables. This is a critical and, normally, difficult task. A simple example will be to relate the distance covered (dependent variable) by a body in fi-ee fall before hitting the ground. Assuming that the air resistance is negligible, the two independent variables are time (t) and acceleration of gravity (g) ... 

It is seen that there is a total of six variables, including the dependent variable k. Hence, by Buckingham s theorem, there are 6-3 = 3 dimensionless groups to be established. 

This result was first discussed by Buckingham (8) and stated in its present form by Langhaar (23). It states in effect that an equation is dimensionally homogeneous if and only if it can be reduced to a relationship among a complete set of B-numbers. Buckingham s result (8) was originally stated as Theorem 2. 

The need for dimensional consistency imposes a restraint in respect of each of the fundamentals involved in the dimensions of the variables. This is apparent from the previous discussion in which a series of simultaneous equations was solved, one equation for each of the fundamentals. A generalisation of this statement is provided in Buckingham s n theorem(4) which states that the number of dimensionless groups is equal to the number of variables minus the number of fundamental dimensions. In mathematical terms, this can be expressed as follows ... 

This list includes seven variables and there are three fundamentals (M, L, T). By Buckingham s n theorem, there will be 7 — 3 = 4 dimensionless groups. 

A theorem known as Buckingham s it theorem is very pertinent in the context of dimensionless groups. According to this theorem the number of dimensionless groups is equal to the difference between the number of variables and the number of dimensions used to express them. Any physical equation can be expressed in the form... 

In the example under discussion the number of variables is 6 and the number of dimensions is 3, so that the number of dimensionless groups should be (6 - 3) = 3 according to Buckingham s it theorem. 

A technique which can assist in the scale-up of commercial plants designs is the use of scale models. A scale model is an experimental model which is smaller than the hot commercial bed but which has identical hydrodynamic behavior. Usually the scale model is fluidized with air at ambient conditions and requires particles of a different size and density than those used in the commercial bed. The scale model relies on the theory of similitude, sometimes through use of Buckingham s pi theorem, to design a model which gives identical hydrodynamic behavior to the commercial bed. Such a method is used in the wind tunnel testing of small model aircraft or in the towing tank studies of naval vessels. 

There are three techniques of developing the dimensionless similarity parameters. The use of Buckingham s pi theorem can be found in most fluid mechanics books, where the variables of importance are used to determine the number of dimensionless parameters that should describe an application and help to identify these parameters. One difficulty with Buckingham s pi theorem is the unspecified form of the dimensionless numbers, which can result in unusual combinations of parameters. 

These five variables (considering Apflas a single variable), m = 5, may be expressed in terms of n = 3 fundamental dimensions, i.e. mass, length and time. Using Buckingham s n theorem, these five variables may be rearranged into m — n = 5 — 3 = 2 new dimensionless variables ... 

Using M, L and T as fundamentals, there are five variables and three fundamentals and therefore by Buckingham s n theorem, there will be two dimensionless groups. Choosing D, N and p as the recurring set, dimensionally ... 

Buckinghams 7r-theorem [i] predicts the number of -> dimensionless parameters that are required to characterize a given physical system. A relationship between m different physical parameters (e.g., flux, - diffusion coefficient, time, concentration) can be expressed in terms of m-n dimensionless parameters (which Buckingham dubbed n groups ), where n is the total number of fundamental units (such as m, s, mol) required to express the variables. For an electrochemical system with semiinfinite linear geometry involving a diffusion coefficient (D, units cm2 s 1), flux at x = 0 (fx=o> units moles cm-2 s 1), bulk concentration (coo> units moles cm-3) and time (f, units s), m = 4 (D, fx=0, c, t) and n - 3 (cm, s, moles). Thus m-n - 1 therefore only one dimensionless parameter can be constructed and that is fx=o (t/Dy /coo. Dimensional analysis is a powerful tool for characterizing the behavior of complex physical systems and in many cases can define relationships... 

Dimensionless analysis — Use of dimensionless parameters (-> dimensionless parameters) to characterize the behavior of a system (- Buckinghams n-theorem and dimensional analysis). For example, the chronoampero-metric experiment (-> chronoamperometry) with semiinfinite linear geometry relates flux at x = 0 (fx=o, units moles cm-2 s-1), time (t, units s-1), diffusion coefficient (D, units cm2 s-1), and concentration at x = oo (coo, units moles cm-3). Only one dimensionless parameter can be created from these variables (-> Buckingham s n-theorem and dimensional analysis) and that is fx=o (t/D)1/2/c0C thereby predicting that fx=ot1 2 will be a constant proportional to D1/,2c0O) a conclusion reached without any additional mathematical analysis. Determining that the numerical value of fx=o (f/D) 2/coo is 1/7T1/2 or the concentration profile as a function of x and t does require mathematical analysis [i]. 

To express these five variables only three fundamental units are needed. According to Buckingham s it theorem these variables can be combined in 5-3 = 2 dimensionless parameters (see for example Ref. [61]) as follows... 

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