Question 5.20: Define critical pressure ratio of a nozzle and discuss why a...
Define critical pressure ratio of a nozzle and discuss why attainment of sonic velocity determines the maximum mass rate of flow through steam nozzle.
The critical pressure ratio is given by
\frac{p_{2} }{ p_{1}} = (\frac{2}{n + 1}) ^{\frac{n}{n – 1}}The velocity at throat
C_{2} = \sqrt{2 \frac{n}{n – 1} ( p_{1} v_{1} – p_{2} v_{2})}
[where p_{2} and C_{2} are the pressure and velocity at the throat]
= \sqrt{2 (\frac{n}{n – 1} ) p_{2} v_{2} [ \frac{p_{1} v_{1} }{p_{2} v_{2}} – 1]} = \sqrt{ 2 (\frac{n}{n – 1} ) p_{2} v_{2} [ ( \frac{p_{2} }{ p_{1}}) ^{ \frac{1 – n}{n} } – 1] }
Substituting the value of \frac{p_{2} }{ p_{1} }, we get,
C_{2} = \sqrt{ 2 (\frac{n}{n – 1} ) p_{2} v_{2} [ \left\{( \frac{2 }{ n + 1}) ^{ \frac{n}{n – 1} } \right\} ^{ \frac{1 – n}{n} } – 1] }= \sqrt{2 ( \frac{n}{n – 1} ) p_{2} v_{2} [\frac{ n + 1}{2 } – 1] }
= \sqrt{2 p_{2} v_{2} ( \frac{n}{n – 1} ) × (\frac{ n – 1}{2 })} = \sqrt{n p_{2} v_{2}}
Velocity of sound is given by
a² = – v² (\frac{\partial p}{\partial v} ) _{s}For constant entropy process, it is assumed that
p v^{n} = constant
Differentiating this and on substitution, we have,
a² = npv
∴ a = \sqrt{n p_{2} v_{2}} = C_{2}
This proves that under the conditions of maximum discharge the velocity of fluid at throat is equal to the sonic velocity at the throat conditions.