Buckingham pi method

In general, the velocity in the flow (it) is a function of x, v, l, it x, f and /i. Seven variables and parameters are identified in the problem p, u, uQQ, x, v, l and p (viscosity). Since three dimensions appear, M (mass), L (length) and T (time), three variables or parameters can be eliminated by forming four dimensionless groups IIi, II2, II3, II4. 

Three repeating variables are selected, as it x, l and p, to form the II s from the other variables. Let IIj = w, lb, pc, p. Therefore by dimensional analysis IIi must have no dimensions. Equating powers for each dimension gives 

It is found that IIi = Uaalp/p = Re, the Reynolds number. (Note that p ML 1T. ) The other II s can be determined in a like manner, or simply by inspection  

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Example Buckingham Pi Method—Heat-Transfer Film Coefficient It is desired to determine a complete set of dimensionless groups with which to correlate experimental data on the film coefficient of heat transfer between the walls of a straight conduit with circular cross section and a fluid flowing in that conduit. The variables and the dimensional constant believed to be involved and their dimensions in the engineering system are given below ... 

The same dimensionless functionality is apparent as from the Buckingham pi method. 

The Buckingham method is based on the Buckingham Pi Theorem, which states... 

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 generation of such dimensionless groups in heat transfer (known generally as dimensional analysis) is basically done (1) by using differential equations and their boundary conditions (this method is sometimes called a differential similarity) and (2) by applying the dimensional analysis in the form of the Buckingham pi theorem. 

The method of obtaining the important dimensionless numbers from the b sic differential equations is generally the preferred method. In many cases, however, we are not able to formulate a differential equation which clearly applies. Then a more general procedure is required, which is known as the Buckingham method. In this method the listing of the important variables in the particular physical problem is done first. Then we determine the number of dimensionless parameters into which the variables may be combined by using the Buckingham pi theorem. 

Safoniuk et al. (1999) presented a method of scaling three-phase fluidized beds based upon achieving geometric and dynamic similitude with the aid of the Buckingham Pi theorem, using the following dimensionless groups ... 

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 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. 

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