Buckingham pi theorem

A rapid check of the groups dimension, which appears in relationships (6.3) and (6.4), shows that they actually are dimensionless criteria  

With this methodology, not only has the number of variables been reduced, but also the new groups are dimensionless combinations of variables, which means that the results presented in Fig. 6.2 will be independent of the system of units used. This type of analysis is called dimensional analysis. The basis for its application to a wide variety of problems is found in the Buckingham Pi Theorem described in the next section. Dimensional analysis is also used for other applications such as  

When researchers want to use dimensional analysis of a process, the first and fundamental question they have to answer concerns the number of dimensionless groups that are required to replace the original list of process variables. The answer to this question is given by the basic theorem of dimensional analysis, which is stated as follows  

The pi theorem is based on the idea of the dimensional homogeneity of the process equations or on the relationships that characterize one particular process. From this point of view, all the coefficients of statistical models that have already been discussed in Chapter 5 have a physical dimension, because the dependent and the independent process variables have a physical dimension. Essentially, we assume that any physically meaningful equation, which characterizes one process and which involves m variables, such as yi = f(xi,X2.x j) presents, for each term contained on the right-hand side, the same dimension as for the left-hand side. This equation could be transformed into a set of dimensionless products (pi terms)  

The required number of pi terms is lower than the number of original n variables, where n is determined by the minimum number of basic dimensions required to describe the original list of variables. For common momentum and mass transfer, the basic dimensions are usually represented by M, L, and T. For heat transfer processes, four basic dimensions - M, L, T, 0 - have to be used. Moreover, in a few rare cases, the variables could be described by a combination of basic dimensions such as, for any flow processes, M/T and L. The use of the pi theorem may appear to be mysterious and complicated, although there are systematic and relatively simple procedures to develop the pi theorem for a given problem. 

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

The number of variables needed to describe the system has been reduced from m to m—n. 

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

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... 

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. 

Numerous empirical correlations for the prediction of residual NAPL dissolution have been presented in the literature and have been compiled by Khachikian and Harmon [68]. On the other hand, just a few correlations for the rate of interface mass transfer from single-component NAPL pools in saturated, homogeneous porous media have been established, and they are based on numerically determined mass transfer coefficients [69, 70]. These correlations relate a dimensionless mass transfer coefficient, i.e., Sherwood number, to appropriate Peclet numbers, as dictated by dimensional analysis with application of the Buckingham Pi theorem [71,72], and they have been developed under the assumption that the thickness of the concentration boundary layer originating from a dissolving NAPL pool is mainly controlled by the contact time of groundwater with the NAPL-water interface that is directly affected by the interstitial groundwater velocity, hydrodynamic dispersion, and pool size. For uniform... 

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 ... 

This is one important use of the Buckingham Pi Theorem. When the relationship between the various quantities is unknown (i.e., there is no equation to relate them), the dimensionless numbers provide a basis for obtaining an equation which fits the data. The exact coefficients and exponents for the dimensionless numbers are obtained from a best fit of the experimental data. 

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. 

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