Page 65 - 48Fundamentals of Compressible Fluid Mechanics

3.3. SPEED OF SOUND IN IDEAL AND PERFECT GASES 27

An expression is needed to represent the right hand side of equation (3.5). For
is a function of two independent variables. Here, it is considered that
where is the entropy. The full differential of the pressure can be
expressed as follows:
ideal gas

(3.6)

In the derivations for the speed of sound it was assumed that the flow is isentropic,

therefore it can be written

(3.7)

Note that the equation (3.5) can be obtained by utilizing the momentum

equation instead of the energy equation.

Example 3.1:
Demonstrate that equation (3.5) can be derived from the momentum equation.
SOLUTION
The momentum equation written for the control volume shown in Figure (3.2) is
(3.8)
Neglecting all the relative small terms results in

(3.9)

"

!

(3.10)

This yields the same equation as (3.5).

3.3 Speed of sound in ideal and perfect gases
The speed of sound can be obtained easily for the equation of state for an ideal
gas (also perfect gas as a sub set) because of a simple mathematical expression.
The pressure for ideal gas can be expressed as a simple function of density, , and
a function “molecular structure” or ratio of specific heats, namely
(3.11)

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