The ratio of areas at different sections of the stream tube in isentropic flow

It is necessary to introduce the mass flow (m) and the equation of continuity, Eqn

(6.14) . Thus m = puA for the general section, i. e. without suffix. Introducing again the reservoir or stagnation conditions and using Eqn (6.1):

The ratio of areas at different sections of the stream tube in isentropic flow(6.21)

The ratio of areas at different sections of the stream tube in isentropic flow

Now the energy equation (6.17) gives the pressure ratio (6.18) above, which when referred to the appropriate sections of flow is rearranged to

Substituting VtPo/Po for ao and introducing both into Eqn (6.21), the equation of continuity gives

The ratio of areas at different sections of the stream tube in isentropic flow(6.22)

Now, if the general section be taken to be the particular section at the throat, where in general usage conditions are identified by an asterisk (*), the equation of continuity (6.22) becomes

The ratio of areas at different sections of the stream tube in isentropic flow(6.23)

But from Eqn (6.18b) the ratio p*/po has the explicit value

p* _ [7 + 11-7/(7-1)

Po [ 2

The ratio of areas at different sections of the stream tube in isentropic flow

converted to kinetic energy of linear motion. It follows from the definition that this state has zero pressure and zero temperature and thus is not practically attainable. Again applying the energy Eqn (6.17) between reservoir and ultimate conditions

7 Po_„T _<?

7-ІЙ) 2

The ratio of areas at different sections of the stream tube in isentropic flow Подпись: (6.27)

so the ultimate, or maximum possible, velocity

The ratio of areas at different sections of the stream tube in isentropic flow

Expressing the velocity as a ratio of the ultimate velocity and introducing the Mach number:

or

The ratio of areas at different sections of the stream tube in isentropic flow Подпись: (6.28)

and substituting Eqn (6.20a) for T/T0:

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