The pi Theorem

Every physical relationship between n physical quantities can be reduced to a relationship between m = n — r mutually independent dimensionless groups, whereby r stands for the rank of the dimensional matrix, made up of the physical quantities in question and generally equal to the number of the base quantities contained in them. 

As a rule, more than two dimensionless numbers will be necessary to describe a physical technical problem and therefore they cannot be derived by the method described above. In this case the easy and transparent matrix transformation introduced by J. Pawlowski [429] is increasingly used. It will be demonstrated by the following stirring example. 

In this case, too, we first draw a sketch of the apparatus. Fig. 1.35. We consider an arbitrary stirrer in a baffled vessel filled with a liquid. Then we facilitate our work by proceeding systematically For this, the influencing parameters will be devided into three groups geometrical, material and process related parameters  

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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. 

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. 

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. 

Let us once more recall the statement of the pi-theorem (section 2.7) ... 

These six quantities contain four base dimensions [L, T, , H], wherein H means the amount of heat with calorie as measuring unit. According to the pi-theorem, a dependence between two pi-numbers will result. Rayleigh obtained the following two pi-numbers which are today named The Nusselt number Nu and the Peclet number Pe, the latter being the product of Reynolds and Prandtl numbers, Pe = RePr ... 

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. 

This step can be accomplished by means of the pi theorem which indicates that the number of pi terms is equal to m - n, where m (determined in step 1) is the number of selected variables and n (determined in step 2) is the number of basic dimensions required to describe these variables. The reference dimensions usually correspond to the basic dimensions and can be determined by a careful inspection of the variables dimensions obtained in step 2. As previously noted, the basic dimensions rarely appear combined, which results in a lower number of reference dimensions than the number of basic ones. 

The first problem is the classical example used to show the scientific force of the dimensional analysis - and especially of the pi theorem. Remember that we are interested in the pressure drop per unit length (Ap/1) along the pipe. According to the experimenter s knowledge of the problem and to step 1, we must list all the pertinent variables that are involved in this problem, it was assumed that ... 

We could also use F, L, and T as basic dimensions. Now, we can apply the pi theorem to determine the required number of pi terms (step 3). An inspection of the variable dimensions obtained in step 2 reveals that the three basic dimensions are all required to describe the variables. Since there are five (m = 5) variables (do not forget to count the dependent variable, Ap/1) and three required reference dimensions (n = 3), then, according to the pi theorem, two pi groups (5 - 3) will be required. 

We observe that four variables and three basic dimensions (M, L, T) are required to describe the variables. For this problem, one pi term (group) can be produced according to the pi theorem. This pi group can easily be expressed as ... 

I 6 Similitude, Dimensional Analysis and Modelling the application of the pi theorem yields ... 

If we analyze the case of the rectification column which loses heat by natural convection, then we change the list of variables by considering the specific ascension force, gPtAt, as an important variable and by removing the fluid velocity, w. In this case, the application of the pi theorem shows that ... 

In geometrically similar systems there is complete similarity if all necessary dimensionless criteria derived either from differential equations or by using the pi theorem are equal. In complex precipitation processes such complete similarity is impossible. Moreover, because we want to obtain identical not similar products from the systems differing in scale, we usually want to reproduce the product quality (particle size, particle morphology), mixture composition, and structure of the suspension on a larger scale. We thus use limited similarity, which means that we lose several degrees of freedom (we cannot manipulate particle size, solution composition, viscosity, and diffu-sivity), and we obtain this way a reduced number of similarity criteria. 

The pi theorem is a generalized method of dimensional analysis and detailed discussions can be found in [1-7]. Below is a brief review of the pi theorem. 

Equation 8.1 is required to be dimensionally homogeneous. The pi theorem says that if the number of distinct reference quantities required to express the dimensional formula of all n magnitudes is r, then the n magnitudes may be grouped into n — r independent dimensionless IT terms, resulting in the relation... 

The following example illustrates the key steps in the application of the Pi theorem [8]. 

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