Convergent-divergent steam nozzle design

Given: $p_1=10$ bar $=10^6$ Pa, $v_1=0.2327$ m³/kg, $n=1.3$, $p_2=1.5$ bar, $m=1.5$ kg/s.

$p_1v_1=232.7$ kJ/kg.

(a) Throat (critical) pressure

$$\frac{p_t}{p_1}=\left(\frac{2}{n+1}\right)^{\frac{n}{n-1}}=\left(\frac{2}{2.3}\right)^{4.333}=0.5457$$ $$p_t=0.5457\times10=5.46\ \text{bar}$$

Since $p_2/p_1=0.150<0.5457$, the exit pressure is below the critical pressure, so a convergent-divergent nozzle is needed.

(b) Throat velocity and area

$$V_t=\sqrt{\frac{2n}{n+1}p_1v_1}=\sqrt{\frac{2\times1.3}{2.3}\times10^6\times0.2327}=512.9\ \text{m/s}$$

Specific volume at throat:

$$v_t=v_1\left(\frac{p_1}{p_t}\right)^{1/n}=0.2327\times\left(\frac{1}{0.5457}\right)^{1/1.3}=0.3708\ \text{m}^3/\text{kg}$$ $$A_t=\frac{m\,v_t}{V_t}=\frac{1.5\times0.3708}{512.9}=10.84\ \text{cm}^2\quad\Big(d_t=37.2\ \text{mm}\Big)$$

(c) Exit conditions

$$V_2=\sqrt{\frac{2n}{n-1}p_1v_1\left[1-\left(\frac{p_2}{p_1}\right)^{\frac{n-1}{n}}\right]} =\sqrt{8.667\times10^6\times0.2327\times\left[1-0.6455\right]}=845.6\ \text{m/s}$$ $$v_2=v_1\left(\frac{p_1}{p_2}\right)^{1/n}=0.2327\times\left(\frac{10}{1.5}\right)^{1/1.3}=1.0013\ \text{m}^3/\text{kg}$$ $$A_2=\frac{m\,v_2}{V_2}=\frac{1.5\times1.0013}{845.6}=17.76\ \text{cm}^2\quad\Big(d_2=47.6\ \text{mm}\Big)$$

Area ratio $A_2/A_t=1.64$.

Answer: (a) $p_t=5.46$ bar; (b) $V_t=512.9$ m/s, $A_t=10.84$ cm²; (c) $V_2=845.6$ m/s, $A_2=17.76$ cm², $d_t=37.2$ mm, $d_2=47.6$ mm.