Buckingham’s n theorem

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. 

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 ... 

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]. 

Based on the dimensional analysis and the Buckingham s n theorem, a set of scale factors is... 

To implement Buckingham s n theorem, we retain the three steps we formulated earlier but amplify Step 3 in the following fashion ... 

The use of If as the symbol dimensionless variables was inlroduced 1 Buckingham, and this method has been called Buckingham s n theorem since then. This is a very general and useftil method, but it will not guarantee the most physical meaningful correlation. 

To derive dimensionless parameters (n-,), Buckingham s pi-theorem needs to be utilized. To reiterate, the number of pi-numbers (j) is equal to the number of physical quantity considered (n = 4) minus the rank (r = 2) of the matrix (Li, 1983,1986a). Thus, there will be two pi-numbers, denoted n and ni-... 

By Buckingham s jt theorem, there will be 8 — 3 = 5 dimensionless groups. Since n is already a dimensionless parameter, there will be four more jr-groups. 

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... 

The dimensionless equation describing the transfer phenomena may be obtained either by direct reference to the ratios of the physical quantities or by recourse to the classical techniques of dimensional analysis, i.e., the Buckingham n Theorem or Rayleigh s method of indices. In addition, the basic differential equations governing the process may be reduced to dimensionless form and the coefficients identified. In general, the dimensionless equation for heat transfer through the combined film is... 

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,... 

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