Pi-theorem

Pitch control Pitched blade turbine Pit furnace Pi theorem Pitocin... 

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. 

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Pi theorem (Buckingham), 3 582 11 744 Pitot tubes, 11 661-662 Pit sealants, 3 334-335 PIT technique, 23 845. See also PIT (powder-in- tube) conductors Pitting corrosion... 

More promising are correlations according to the Buckingham-PI theorem [118], such as the correlation published by Vijayraghvan and Gupta [119] ... 

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. 

Here, five dimensional quantities [Eq. (1)] produce two dimensionless numbers [Eq. (4)]. This had to be expected because the dimensions in question are comprised of three basic dimensions 5—3 = 2 (see the discussion on pi theorem later in this chapter). 

From the above example, we also learn that transformation of physical dependency from a dimensional into a dimensionless form is automatically accompanied by an essential compression of the statement the set of the dimensionless numbers is smaller than the set of the quantities contained in them, but it describes the problem equally comprehensively. In the above example, the dependency between five dimensional parameters is reduced to a dependency between only two dimensionless numbers. This is the proof of the so-called pi theorem (pi after Ft, the sign used for products), which states ... 

The pi theorem is often associated with the name of Buckingham (4), because he introduced this term in 1914, but the proof of it was accomplished in the course of a mathematical analysis of partial differential equations by Federmann in 1911 see Ref. 5, Chap. 1.1, A Brief Historical Survey. 

To obtain the dimensionless groups for a specific process, the so-called Buckingham Pi theorem is frequently used. The first step in this approach is to define the variables that affect the process or assume the most important physical parameters for the specific process, if the equation that describes the process is entirely unknown. This is the weak... 

Reduction of the number of parameters required to define the problem. The pi theorem states that a physical problem can always be described in dimensionless terms. This has the advantage that the number of dimensionless groups that fully describe it is much smaller than the number of dimensional physical quantities. It is generally equal to the number of physical quantities minus the number of base units contained in them. 

Determine appropriate dimensionless parameters that can relate the mass transfer coefficient by applying the Buckingham-Pi theorem. 

The first step of Buckingham-Pi theorem is to count the total number of parameters. In this case, there are five parameters kL, D32, DAB, p(, and pc, all of which can be expressed with three principle units mass M, length L, and time T. Therefore,... 

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. 

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

Note The pi-theorem only stipulates the number of the dimensionless numbers and not their form. Their form is laid down by the user, because it must suit the physics of the process and be suitable for the evaluation and presentation of the experimental data. 

The following example has been chosen because it impressively demonstrates the scale-invariance of the pi-space. Besides this, in the matrix transformation we will encounter a reduction of the rank r of the matrix. This will enable us to understand why, in the definition of the pi-theorem (section 2.7), it was pointed out that the rank of the matrix does not always equals the number of base dimensions contained in the dimensions of the respective physical quantities. 

Reduction of the number of parameters necessary to define the problem. The pi-theorem states that a physical relationship between n physical quantities can be reduced to a relationship between m = n - r mutually independent pi-numbers. Herein, r represents the rank of the dimensional matrix which is formed by the physical quantities in question and corresponds, in most cases, to the number of base dimensions contained in their dimensions. 

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