There are two methods of dimensional analysis used.
- Rayleigh’s method
- Buckingham π-Theorem
Rayleigh’s Method
In Rayleigh’s method, the expression is determined for a variable depending upon maximum three or four variables only. If the number of independent variables becomes more than four, it is very difficult to find the expression for the dependent variable. So, a functional relationship between variables is expressed in exponential form of equations.
Steps involved in Rayleigh’s method:
First, the functional relationship is written with the given’ data.
Consider X as a variable which depends on X1,X2,X3,…..Xn.
So, the functional equation is written by X = ƒ (X1,X2,X3,…..Xn)
Then the equation is expressed in terms of a constant with exponents such as powers of a,b,c …
Therefore, the equation is again written by X=\phi\left(X_{1}^{a}, X_{2}^{b}, X_{3}^{c}, \ldots X_{n}^{2}\right)
Here, Φ = Constant
a,b,c,…z = Arbitrary powers
The values of a,b,c,…z are determined with the help of a dimensional homogeneity. It means, the powers of the fundamental dimensions on both sides are compared to obtain the values of exponents.
Finally, these exponents/power values are substituted in the functional equation and simplified to obtain the suitable form.
Buckingham π – Theorem
As mentioned above, Rayleigh method is not helpful when the number of independent variables is more than three or four. This difficulty is eliminated in Buckingham π-Theorem.
Buckingham π-theorem states that if there are ‘n’ variables in a dimensionally homogeneous equation and if these variables contain ‘m’ fundamental dimensions (M,L,T), then they are grouped into (n-m), dimensionless independent π-terms.
Let X1,X2,X3,…..Xn are the variables involved in a physical phenomenon. Let X1 be the variables involved in a physical phenomenon. Let X1 be the dependent variables and X2,X3,…..Xn are the independent variables on which X1 depends. Then, X1 is a function of X2,X3,…..Xn and mathematically, it is expressed by X = ƒ (X2,X3,…..Xn)
can also be written by
F1 (X1,X2,X3,…..Xn) = 0
This equation is a dimensionally homogeneous equation. It contains n variables. If there are m fundamental dimensions then according to Buckingham-π-theorem, equation can be written in terms in which number of π-terms is equal to (n-m). Hence, the equation F1 (X1,X2,X3,…..Xn) = 0 becomes F (π1,π2,…..πn-m) = 0
Each of π terms is dimensionless and independent of the system. The division or multiplication by a constant does not change the character of the π term. Each of π-term contains (m+1) variables, where m is the number of fundamental dimensions and is also called repeating variables. Let m in the above case X2,X3 and X4 are repeating variables, if the fundamental dimensions (M, L, T)- 3, then, each π-term is written by
\begin{array}{l} \pi_{1}=X_{2}^{a 1} \cdot X_{3}^{b 1} \cdot X_{4}^{c 1} \cdot X^{1} \\ \pi_{2}=X_{2}^{a 2} \cdot X_{3}^{b 2} \cdot X_{4}^{c 2} \cdot X^{5} \\ & \ldots \ldots \ldots \ldots \ldots \ldots \\ & \ldots \ldots \ldots \ldots \ldots . . \\\pi_{n-m}&=X_{2}^{a n-m} \cdot X_{3}^{b n-m} \cdot X_{4}^{c n-m} \cdot X^{n} \end{array}Each equation is solved by the principle of dimensional homogeneity and values of a1 ,b1 c1 etc. are obtained. These values are substituted in equation and values of π1, π2, … πn-m are obtained. These values of π‘s are substituted in equation. The final equation for the phenomenon is obtained by expressing any one of the π-terms as a function of others which can be expressed by
\begin{array}{l} \pi_{1}=\phi\left[\pi_{2}, \pi_{3}, \ldots, \pi_{n-m}\right] \\ \pi_{2}=\phi\left[\pi_{1}, \pi_{3}, \ldots,, \pi_{n-m}\right] \end{array}Selection of repeating variables
There is no a separate rule for selecting repeating variables. But the number of repeating variables is equal to the fundamental dimensions of the problem. Generally, ρ, ν, ι or ρ, ν, D are chosen as repeating variables. It means, one refers the fluid property (ρ), the other one refers the flow property (ν) and last one refers the geometric property (ι or D).
In addition to this, the following points should be kept in mind while selecting the repeating variables.
- No variables should be dimensionless.
- The selected two repeating variables should not have the same dimensions.
- The selected repeating variables should be independent as far as possible.
Steps to be followed in Buckingham π-method
- First, the variables involved in a given analysis are listed to study about the given phenomenon thoroughly.
- Then, these variables are expressed in terms of primary dimensions.
- Next, the repeating variables are chosen according to the hint given in the selection of repeating variables. Once, the repeating variables should be checked either those are independent or dependent variables because all should be independent variables.
- Then the dimensionless parameters are obtained by adding one at a time with repeating variables.
- The number of π-terms involved in dimensional analysis is calculated by using n-m = Number of π-terms. where n = Total number of variables involved in a given analysis. m = Number of fundamental variables.
- Finally, each equation in exponential form is solved which means the coefficients of exponents are found by comparing both side exponents. Then these dimensionless parameters are suitably recombined and arranged.
Limitations of dimensional analysis
- Dimensional analysis does not give any rule regarding the selection of variables.
- The complete information is not provided by dimensional analysis. It only indicates that there are some relationships between parameters.
- The values of co-efficient and the nature of function can be obtained only by experiments or from mathematical analysis.
State the Buckingham π theorem
It states that if there are ‘n’ variable in a dimensionally homogeneous equation and if these variable contain ‘m’ fundamental dimensions (M, L, T) then they are grouped into (n – m), dimensionless independent π-terms.
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