Steam nozzles and critical pressure ratio

A steam nozzle is a passage of varying cross-section that converts the heat (enthalpy) energy of steam into kinetic energy by expansion from a high to a low pressure. Its main use is in steam turbines (to produce jets), and also in steam injectors, ejectors, and jet condensers.

Types

Also, nozzles may be classified by cross-section (circular, rectangular) and by arrangement (single, ring).

Exit velocity (steady flow energy equation)

For adiabatic flow with no work, negligible change in height, the SFEE between inlet (1) and exit (2) is

$$h_1+\frac{V_1^2}{2}=h_2+\frac{V_2^2}{2}$$

Neglecting inlet velocity ($V_1\approx0$):

$$V_2=\sqrt{2(h_1-h_2)}\ \ \text{m/s},\ \ h\text{ in J/kg}$$

(or $V_2=44.72\sqrt{h_1-h_2}$ with $h$ in kJ/kg). For an isentropic expansion with $pv^n=\text{constant}$ ($n=1.3$ for superheated, 1.135 for dry saturated steam, Zeuner's law), $h_1-h_2=\dfrac{n}{n-1}(p_1v_1-p_2v_2)$, so

$$V_2=\sqrt{\frac{2n}{n-1}\,p_1v_1\left[1-\left(\frac{p_2}{p_1}\right)^{\frac{n-1}{n}}\right]}$$

Mass flow and maximum discharge

Mass flow per unit exit area, using $v_2=v_1(p_1/p_2)^{1/n}$:

$$\frac{m}{A}=\frac{V_2}{v_2}=\sqrt{\frac{2n}{n-1}\frac{p_1}{v_1}\left[\left(\frac{p_2}{p_1}\right)^{\frac{2}{n}}-\left(\frac{p_2}{p_1}\right)^{\frac{n+1}{n}}\right]}$$

Let $r=p_2/p_1$. The flow per unit area is a maximum when $d(m/A)/dr=0$, i.e. when the bracket is a maximum:

$$\frac{2}{n}r^{\frac{2}{n}-1}-\frac{n+1}{n}r^{\frac{1}{n}}=0\ \Rightarrow\ r=\left(\frac{2}{n+1}\right)^{\frac{n}{n-1}}$$

This is the critical pressure ratio $p_c/p_1$, reached at the throat. At it, the velocity at the throat equals the local velocity of sound, and the nozzle is "choked": lowering the back pressure further cannot increase the mass flow through a convergent nozzle.

Throat velocity: $V_t=\sqrt{\dfrac{2n}{n+1}p_1v_1}$.

Use of the result: if the back pressure is above $p_c$, a simple convergent nozzle is used, and the exit pressure equals the back pressure. If the back pressure is below $p_c$, a convergent-divergent nozzle is required to expand the steam fully to the back pressure, with a supersonic jet in the divergent part.