Buckingham’s П theorem
The statement of the theorem is as follows : If a relation among n parameters exists in the for
then the n parameters can be grouped into n – m independent dimensionless ratios or P parameters, expressed in the form
where m is the number of dimensions required to specify the dimensions of all the parameters, q1, q2, …. qn. It is also possible to form new dimensionless P parameters as a discrete function of the (n – m) parameters. For example if there are four dimensionless parameters P1, P2, P3 and P4 it is possible to obtain P5, P6 etc. as
The limitation of this exercise is that the exact functional relationship in equation 8.3.1 cannot be obtained from the analysis. The functional relationship is generally arrived at through the use of experimental results.
8.3.1 Determination of π Groups
Irrespective of the method used the following steps will systematize the procedure.
Step 1. List all the parameters that influence the phenomenon concerned.
This has to be very carefully done. If some parameters are left out, π terms may be formed but experiments then will indicate these as inadequate to describe the phenomenon. If unsure the parameter can be added. Later experiments will show that the π term with the doubtful parameters as useful or otherwise. Hence a careful choice of the parameters will help in solving the problem with least effort. Usually three type of parameters may be identified in fluid flow namely fluid properties, geometry and flow parameters like velocity and pressure.
Step 2. Select a set of primary dimensions, (mass, length and time), (force, length and time), (mass, length, time and temperature) are some of the sets used popularly.
Step 3. List the dimensions of all parameters in terms of the chosen set of primary dimensions. Table 8.3.1. Lists the dimensions of various parameters involved.
Table 8.3.1. Units and Dimensions of Variables
Step 4. Select from the list of parameters a set of repeating parameters equal to the number of primary dimensions. Some guidelines are necessary for the choice. (i) the chosen set should contain all the dimensions (ii) two parameters with same dimensions should not be chosen. say L, L2, L3, (iii) the dependent parameter to be determined should not be chosen.
Step 5. Set up a dimensional equation with the repeating set and one of the remaining parameters, in turn to obtain n – m such equations, to determine P terms numbering n – m. The form of the equation is,
As the LHS term is dimensionless, an equation for each dimension in terms of a, b, c, d can be obtained. The solution of these set of equations will give the values of a, b, c and d. Thus the P term will be defined.
Step 6. Check whether P terms obtained are dimensionless. This step is essential before proceeding with experiments to determine the functional relationship between the P terms.
Example 8.2. The pressure drop ÄP per unit length in flow through a smooth circular pipe is found to depend on (i) the flow velocity, u (ii) diameter of the pipe, D (iii) density of the fluid ñ, and (iv) the dynamic viscosity ì.
(a) Using P theorem method, evaluate the dimensionless parameters for the flow.
(b) Using Rayleigh method (power index) evaluate the dimensionless parameters.
Choosing the set mass, time and length as primary dimensions, the dimensions of the parameters are tabulated.
There are five parameters and three dimensions. Hence two P terms can be obtained. As ΔP is the dependent variable D, r and µ are chosen as repeating variables.
Using the principle of dimensional homogeneity, and in turn comparing indices of mass, length and time.
Substituting the value of indices we obtain
This represents the ratio of pressure force and inertia force.
Check the dimension :
MODEL AND PROTOTYPE
In the engineering point of view model can be defined as the representation of physical system that may be used to predict the behavior of the system in the desired aspect. The system whose behavior is to be predicted by the model is called the prototype. The discussion in this chapter is about physical models that resemble the prototype but are generally smaller in size. These may also operate with different fluids, at different pressures, velocities etc. As models are generally smaller than the prototype, these are cheaper to build and test. Model testing is also used for evaluating proposed modifications to existing systems. The effect of the changes on the performance of the system can be predicted by model testing before attempting the modifications. Models should be carefully designed for reliable prediction of the prototype performance.
CONDITIONS FOR SIMILARITY BETWEEN MODELS AND PROTOTYPE
Dimensional analysis provides a good basis for laying down the conditions for similarity. The PI theorem shows that the performance of any system (prototype) can be described by a functional relationship of the form given in equation 9.2.1.
The PI terms include all the parameters influencing the system and are generally ratios of forces, lengths, energy etc. If a model is to be similar to the prototype and also function similarly as the prototype, then the PI terms for the model should also have the same value as that of the prototype or the same functional relationship as the prototype. (eqn. 9.2.1)
For such a condition to be satisfied, the model should be constructed and operated such that simultaneously
Equation 9.2.3 provides the model design conditions. It is also called similarity requirements or modelling laws.
9.2.1 Geometric Similarity
Some of the PI terms involve the ratio of length parameters. All the similar linear dimension of the model and prototype should have the same ratio. This is called geometric similarity. The ratio is generally denoted by the scale or scale factor. One tenth scale model means that the similar linear dimensions of the model is 1/10 th of that of the prototype. For complete similarity all the linear dimensions of the model should bear the same ratio to those of the prototype. There are some situations where it is difficult to obtain such similarity. Roughness is one such case. In cases like ship, harbor or dams distorted models only are possible. In these cases the depth scale is different from length scale. Interpretation of the results of the tests on distorted models should be very carefully done. Geometric scale cannot be chosen without reference to other parameters. For example the choice of the scale when applied to the Reynolds number may dictate a very high velocity which may be difficult to achieve at a reasonable cost.
9.2.2 Dynamic Similarity
Similitude requires that ð terms like Reynolds number, Froude number, Weber number etc. be equal for the model and prototype. These numbers are ratios of inertia, viscous gravity and surface tension forces. This condition implies that the ratio of forces on fluid elements at corresponding points (homologous) in the model and prototype should be the same. This requirement is called dynamic similarity. This is a basic requirement in model design. If model and prototype are dynamically similar then the performance of the prototype can be predicted from the measurements on the model. In some cases it may be difficult to hold simultaneously equality of two dimensionless numbers. In such situations, the parameter having a larger influence on the performance may have to be chosen. This happens for example in the case of model tasting of ships. Both Reynolds number and Froude number should be simultaneously
held equal between the model and prototype. This is not possible as this would require either fluids with a very large difference in their viscosities or the use of very large velocities with the model. This is illustrated in problem 9.14.
9.2.3 Kinematic Similarity
When both geometric and dynamic similarities exist, then velocity ratios and acceleration ratios will be the same throughout the flow field. This will mean that the streamline patterns will be the same in both cases of model and prototype. This is called kinematic similarly. To achieve complete similarity between model and prototype all the three similarities – geometric, dynamic and kinematic should be maintained.