Jamil Ghojel - Fundamentals of Heat Engines

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 / m 3

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

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), is the mass flow rate of the fluid (usually air), p 1tand T 1tare the total pressure and temperature at the compressor inlet, and p 2tand T 2tare the total pressure and temperature at the compressor outlet (Saravanamuttoo et al. 2001). According to the Buckingham theorem, the number of nondimensional Π groups that can be formed is m − n = 7 − 3 = 4. These can be shown to be

(1.39a)

For a compressor with fixed size and specified gas, the terms R and D can be dropped and Eq. (1.39a) becomes

(1.39b)

The graphical form of this function is obtained by plotting one group, such as the compressor pressure ratio p 2t/ p 1t, against the mass flow rate parameter for a number of constant values of compressor speed parameter The physical interpretations of the mass flow rate and speed parameters will be explained in the following section.

Thermodynamics is the study of the interaction of heat and fluids in motion. It is closely related to work‐producing or work‐absorbing machines such as engines, refrigerators, and compressors and the working substances used in each machine. Substances are fluids that are capable of phase change (liquid to gas, gas to liquid, liquid to solid) and are expandable and compressible. The most widely used working substances are shown in the following table: