In cases where temperature is a basic physical quantity and it is preferable to avoid using an extra fundamental dimension such as θ, the gas constant is usually lumped together with the temperature, and the combined variable RT = p/ρ (from the equation of state) will have the dimensions
1.2.4.2 Buckingham Pi (π) Theorem
The theory states that ‘A relationship between m different variables can be reduced to a relationship between m − n dimensionless groups in terms of the n fundamental units’. If
or
(1.36)
then
or
(1.37)
Consider the steady flow of an incompressible fluid through a long, smooth, horizontal pipe. The pressure drop per unit length Δp caused by friction can be written as a general mathematical function:
where
D: pipe diameter, m
ρ: fluid density, kg/m3
μ: dynamic viscosity of the fluid, kg/m.s
C: average fluid velocity, m/s
Experimental solution of this problem would require changing one variable while keeping the other three constant and plotting the results on four graphs. The downside of this approach to solving this problem is that the plots are valid only for a specific fluid and pipe, and the obtained results are difficult to fit to a general functional relationship.
According to the Buckingham theorem, for five variables (m = 5) and three fundamental units M, L, and T (n = 3), there will be m − n = 2 nondimensional Π groups. The dimensions of the independent variables are
| Independent variable | Δp | D | C | μ | ρ |
| Dimensions | ML −2 T −2 | L | LT −1 | ML −1 T −1 | ML −3 |
First, n repeating variables that are dimensionally the simplest are selected (three in this case) – D, ρ, and C – and form the first Π group as
or as
For M: 0 = 1 + c
For L: 0 = − 2 + a + b − 3c
For T: 0 = − 2 − b
Solving these simultaneous equations, we obtain a = 1, b = − 2, c = − 1 and
The second Π group is
or
For M: 0 = 1 + c
For L: 0 = − 1 + a + b − 3c
For T: 0 = − 1 − b
Solving these simultaneous equations, we obtain a = − 1, b = − 1, c = − 1 and
The final functional form can be written as
or
The Reynolds number Re = DCρ/μ; hence, the functional relationship can be written as
Dimensional analysis will not provide the forms of the functions f and ψ. These can be obtained only from carefully set experiments.
This methodology is used in turbomachinery to develop functions for compressor and turbine performance characteristics, as will be discussed in Chapter 14. The function that was found reasonable for compressors is
(1.38)
where D is the impeller diameter, N is the rotational speed (usually rpm),
(1.39a)
