Buckingham s pi 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. 

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-... 

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. 

Thus the above Dimension matrix has a rank of 3. The mathematical statement for your analysis contains five variables therefore, this Dimensional Analysis will generate two dimensionless parameters, per Buckingham s Pi Theorem, which is... 

Thus the rank of the above Dimension matrix is 3. From Buckingham s Pi Theorem... 

The rank for this Dimension matrix is 3. Therefore, from Buckingham s Pi Theorem, the number of dimensionless parameters Np for fluid flow through a column of material particles is... 

The quantities( ), (g), and (W) are dimensionally independent, hence there will be one rexpression in this problem (by Buckingham s Pi Theorem 4-3 = 1). This single Pi quantity will be found to be... 

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The determining parameters of the system are (p, S, tr) according to the Buckingham pi theorem (j), we can then write the formal function... 

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