ISENTROPIC, ONE-DIMENSIONAL STEADY FLOW THROUGH A NOZZLE

A nozzle with both converging and diverging section is shown in Fig. 6.11. For the control volume shown, the following relations can be written:

First law:

Property relation

Continuity equation:

 

ρAc = ṁ = const.

By logarithmic differentiation, we get

Combining Eqs (6.11) and (6.12), we have

Substituting this in Eq. (6.13), we have

Since the flow is isentropic,

and therefore,

Figure 6.11 One-dimensional isentropic flow through a nozzle

 

where c_s = velocity of sound

M = Mach number = 

γ = ratio of specific heats = c_p/c_v

This is a very significant equation, and from it, we can draw the following conclusions about the proper shape for nozzles and diffusers:

1. For a nozzle, dp < 0. Therefore,for a subsonic nozzle, M < 1, dA < 0, and the nozzle is converging;for a supersonic nozzle, M > 1, dA > 0, and the nozzle is diverging.
2. For a diffuser, dp > 0. Therefore,for a subsonic diffuser, M < 1, dA > 0, and the diffuser is diverging;for a supersonic diffuser, M > 1, dA < 0, and the diffuser is converging.
3. When M = 1, dA = 0, which means that some velocity can be achieved only at the throat of a nozzle or diffuser. These conclusions are summarised in Fig. 6.12.

Figure 6.12 Required area changes for (a) nozzles and (b) diffusers