Dimension | Applications and methods of Dimensional analysis

Dimension

Dimension is an arbitrary value fixed for a physical quantity.

For example, the dimension of physical quantity Length is L and that of acceleration is LT-2.

Fundamental dimensions

These are

1. M = Mass.
2. L = Length.
3. T = Time.
4. F = Force.

Area, acceleration, viscosity, Power, energy are the derived units. So their dimensions are written in terms of the fundamental.

System for fundamental dimensions

There are two systems

1. M-L-T system (In this system Force is derived).
2. F-L-T system (In this system Mass is derived).

These two systems are related by Newton’s 2^nd Law of motion.

F=ma

F=MLT^-2

M=FL^-1T²

Dimensional Homogeneity

An equation is said to be homogeneous when fundamental dimensions and their respective powers are identical on either side of sign of equality.

Q = A.V

L³ T^-1 = L² . L/T

So the above equation is dimensionally homogeneous.

Dimensional Analysis

It is a mathematical technique, which deals with the dimensions of physical quantities. It is an important tool for establishing relationship between physical quantities involved in a fluid phenomenon.

Applications of dimensional analysis

It includes,

1. To develop equation for a fluid phenomenon. 
2. Converting one system of units into another.
3. To check the dimensional homogeneity of an equation.
4. To determine the dimension and unit of a physical quantity in an equation.
5. To reduce the number of variables required in an experimental program.
6. To establish principles for hydraulics similitude for model study. 

Methods of dimensional analysis

The following two methods are important to develop the relationship between physical quantities.

1. Rayleigh’s method.
2. Buckingham’s pi-theorem.

Rayleigh’s method of dimensional analysis

This method of dimensional analysis was originally proposed by Lord Rayleigh in 1899. He used this method for determining the effect of temperature on gases.

In this method, functional relationship of variables is expressed in the form of an exponential equation. The equation must be dimensionally homogeneous.

For example if Y is some function of independent variables X1,X2,X3… etc., then functional relationship may be written as

Y = f [ X1,X2,X3…]

In this equation, Y is dependent variable and X1,X2,X3 are independent variables.

Dependent variable is one about which information is required. Independent variables are those which govern the variation of dependent variables.

Procedure

Rayleigh’s method is based on the following steps.

1. Write the functional relationship of the given data. Y = f [ X1,X2,X3…]
2. Write equation in the exponential form with exponenets a, b, c…
3. Put the dimensions of variables involved using any one system (MLT, FLT).
4. Apply dimensional homogentiy and evaluate the values of exponents a,b,c…,d.
5. Substitute the values of exponenets a,b,c… in the equation form in step number 2.
6. Simplify the equation for the required physical quantity.

Buckingham’s Pi theorem

It has been observed that the Rayleigh’s method of dimensional analysis becomes bulky when more variables are involved. In order to overcome this, Buckingham’s method may be used. It states that

If there are n variables in a dimensionally homogeneous equation and if these variables contain m fundamental dimensions such as ( M-L-T). They may be grouped into (n minus m) non dimensional independent pi-terms.

Mathematically, if a variable X1 depends upon independent variables X2,X3,X4..Xn the functional equation may be written as :

X1 = k ( X2,X3,X4…Xn).

The equation may be written in its general form as

F(X1,X2,X3,X4…Xn) = C

where C is a constant, and f represents some function. in this equation, there are n variables. If there are m fundamental dimensions, then according to buckingham’s pi-theorem.

f1(pi1,pi2,pi3…pi n-m) = constant

The Buckingham’s pi method is based on the following steps.

1. Write the functional relationship with the given data.
2. Then write the equation in its general form.
3. Now choose m repeating variables and write separate expression for each pi term. Every pi term will contain the repeating variables and one of the remaining variables. The repeating variables are written in the exponential form.
4. With the help of principle of dimensional homogeneity find out the values of a,b,c… by obtaining simultaneous equations.
5. Now substitute the values of these exponents in the pi-terms.
6. After the pi terms are determined, write the functional relation in the required form.

Selection of repeating variables

Though there is no hard and fast rule for the selection of repeating variables. Following points should be kept in mind while selecting repeating variables.

1. The variables should be such that none of them is dimensionless.
2. No two variables should have same dimensions.
3. Independent variables should as far as possible be selected as repeating variables.

Reference : A textbook of hydraulics, fluid mechanics and hydraulic machines by R.S. Khurmi