The Buckingham n Theorem

Given p variables, Xi, Xy Xp are related to a physical phenomenon that can be expressed in terms of r fimdamental dimensions. Then these variables, which include the dependent variable, can be gathered intop-r dimensionless groups 7t and cast in the fimctional form  

In other words, we have managed to replace a functional relation, which involves p variables, with one that involves only p - r variables. This constitutes a considerable saving. 

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

Step 3a Select a set of variables that equals the number of dimensions r and does not include the dependent variable. Raise each variable to some unknown power a, P. and form a product of the result. This product is placed in the denominator of each dimensionless group n. Step 3b Place the remaining (p - r) variables, which now include the dependent variable, in the numerators of the ip - r) dimensionless groups. This results in the set of n terms shown in Equation 5.10. 

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The Buckingham n theorem is a key theorem in dimensional analysis. The theorem states that if we have a physically meaningful equation involving a certain number of physical variables (e.g., n), and these variables are expressible in terms of k independent fundamental physical variables (such as length, mass, time, etc.), then the original equation is equivalent to an equation involving at set of p = n - k dimensionless variables constructed from the original variables. 

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... 

Although this list of variables is not inclusive by any means, it is adequate to help understanding the basic physics of the problem as we will show in the subsequent sections. For an angled injection, one could add another parameter to include the effect of the tilt of the nozzle. Using the Buckingham n theorem, we can form four nondimensional groups of parameters out of the mentioned seven parameters. So, a particular characteristic of the LJICF (such as the jet trajectory or droplet size distribution) can be written in the form... 

Consider a fluidic system with surface tension effects. The characteristic scales for length and velocity are L and U. The physical parameters are density p, viscosity v, gravity g, and surface tension y. By using the Buckingham n theorem, one can obtain three independent nondimensional groups from these six variables. An option for a set of three independent nondimensional numbers is... 

The concept of similitude is further formalized by the Buckingham n theorem that states that an equation containing n variables can be rewritten as an equation containing n m dimensionless numbers, where m is the overall number of independent dimensions in the model. 

Dimensionless numbers are needed whenever nonlinear equations are encountered because transcendental and polynomial functions cannot have units. Taking the logarithm of 10 moles or raising e to the power of 20 minutes is a meaningless calculation. This does not mean that there cannot be units within these functions but it does mean that those units must cancel out. For example, it is perfectly acceptable to raise e to kt where t is time and the rate constant, k, has the dimension of reciprocal time so that the product kt has no units. Recasting a variable into a dimensionless number for use in a nonlinear equation can be done in various ways, but all those ways are subject to the restriction of the Buckingham n theorem. 

Dimensionless numbers are commonly used in diffusion and advection models because they simplify the scaling of the models from laboratory experiments to practical dimensions. This approach also takes advantage of the Buckingham n theorem (Barenblatt, 2003), which states that n independent variables with k independent dimensions can be expressed in terms of p independent dimensionless numbers. 

Mass Transfer in Turbulent Flow Dimensional Analysis and the Buckingham n Theorem... 

The power consumption by impeller P in geometrically similar fermenters is a function of the diameter Dl and speed N of impeller, density p and viscosity p. of liquid, and acceleration due to gravity g. Determine appropriate dimensionless parameters that can relate the power consumption by applying dimensional analysis using the Buckingham-Pi theorem. 

Dimensional analysis, often referred to as the II-theorem is based on the fact that every system that is governed by m physical quantities can be reduced to a set of m - n mutually independent dimensionless groups, where n is the number of basic dimensions that are present in the physical quantities. The II-theorem was introduced by Buckingham [1] in 1914 and is therefore known as the Buckingham II-theorem. The II-theorem is a procedure to determine dimensionless numbers from a list of variables or physical quantities that are related to a specific problem. This is best illustrated by an example problem. 

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]. 

We can observe that all the basic dimensions (also specific to moment and mass transfer) are required to define the six variables, taking into consideration that, according to the Buckingham pi theorem, three pi terms will be needed (six variables minus three basic dimensions, m - n = 6 - 3). 

When there are no governing differential equations available, the Buckingham pi theorem can be used instead. This theorem states that any complete physical relationship (describing a process) can be expressed in terms of a set of independent dimensionless groups as discussed earlier and that the number of dimensionless groups i used to describe a process involving n variables is given by... 

This theorem provides a method to obtain the dimensionless groups which affect a process. First, it is important to obtain an understanding of the variables that can influence the process. Once you have this set of variables, you can use the Buckingham Pi Theorem. The theorem states that the number of dimensionless groups (designated as n, ) is equal to the number (n) of independent variables minus the number (m) of dimensions. Once you obtain each n, you can then write an expression ... 

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 ... 

Heat transfer and its counterpart diffusion mass transfer are in principle not correlated with a scale or a dimension. On a molecular level, long-range dimensional effects are not effective and will not affect the molecular carriers of heat. One could say that physical processes are dimensionless. This is essentially the background of the so-called Buckingham theorem, also known as the n-theorem. This theorem states that a product of dimensionless numbers can be used to describe a process. The dimensionless numbers can be derived from the dimensional numbers which describe the process (for example, viscosity, density, diameter, rotational speed). The amount of dimensionless numbers is equal to the number of dimensional numbers minus their basic dimensions (mass, length, time and temperature). This procedure is the background for the development of Nusselt correlations in heat transfer problems. It is important to note that in fluid dynamics especially laminar flow and turbulent flow cannot be described by the same set of dimensionless correlations because in laminar flow the density can be neglected whereas in turbulent flow the viscosity has a minor influence [144], This is the most severe problem for the scale-up of laminar micro results to turbulent macro results. 

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... 

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,... 

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

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-... 

The basic theorem of the dimensional analysis is the tt theorem of Buckingham which states that the general dependence between n values at m basic dimension units can be presented as a function of (n-m) dimensionless ratios of these values, and in case of similarity, of (n-m) criteria of similarity. 

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