Relations between stagnation and characteristics state

The stagnation speed and characteristics speed of sound may be written as,

(4.3.11)

Rewrite Eq. (4.3.7) for stagnation conditions as given below;

(4.3.12)

Equate Eqs. (4.3.8) and (4.3.12),

(4.3.13)

More useful results may be obtained for Eqs. (4.3.4) & (4.3.6), when we define

(4.3.14)

With (for air), the Eqs (4.3.13) & (4.3.14) reduces to constant value.

(4.3.15)

Critical speed and Maximum speed

The critical speed of the gas is same as that speed of sound at sonic state i.e. . A gas can attain the maximum speed when it is hypothetically expanded to zero pressure. The static temperature corresponding to this state is also zero. The maximum speed of the gas represents the speed corresponding to the complete transformation of kinetic energy associated with the random motion of gas molecules into the directed kinetic energy. Rearranging Eq. (4.3.3), one can obtain the following equation;

(4.3.16)

Now, the Eqs (4.3.13) & (4.3.16) can be simplified to obtain the following relation;

(4.3.17)