Question 14.14: Turbine Specific Speed Calculate and compare the turbine spe...
Turbine Specific Speed
Calculate and compare the turbine specific speed for both the small (A) and large (B) turbines of Example 14–13.
The turbine specific speed of two dynamically similar turbines is to be compared.
Properties The density of water at T=20^{\circ} C \text { is } \rho=998.0 kg / m ^{3} .
Analysis We calculate the dimensionless turbine specific speed for turbine A,
\begin{aligned}N_{ St , A } &=\frac{\omega_{ A }\left( bhp _{ A }\right)^{1 / 2}}{\rho_{ A }^{1 / 2}\left(g H_{ A }\right)^{5 / 4}} \\&=\frac{(12.57 rad / s )\left(242 \times 10^{6} W \right)^{1 / 2}}{\left(998.0 kg / m ^{3}\right)^{1 / 2}\left[\left(9.81 m / s ^{2}\right)(75.0 m )\right]^{5 / 4}}\left(\frac{ kg \cdot m / s ^{2}}{ W \cdot s }\right)^{1 / 2}=1.615 \cong 1 . 6 2\end{aligned}
and for turbine B,
\begin{aligned}N_{ St , B } &=\frac{\omega_{ B }\left( bhp _{ B }\right)^{1 / 2}}{\rho_{ B }^{1 / 2}\left(g H_{ B }\right)^{5 / 4}} \\&=\frac{(12.57 rad / s )\left(548 \times 10^{6} W \right)^{1 / 2}}{\left(998.0 kg / m ^{3}\right)^{1 / 2}\left[\left(9.81 m / s ^{2}\right)(104 m )\right]^{5 / 4}}\left(\frac{ kg \cdot m / s ^{2}}{ W \cdot s }\right)^{1 / 2}=1.615 \cong 1.62\end{aligned}
We see that the turbine specific speeds of the two turbines are the same. As a check of our algebra we calculate N_{ St } in Fig. 14–109 a different way using its definition in terms of C_{P} \text { and } C_{H} (Eq. 14–50). The result is the same (except for roundoff error). Finally, we calculate the turbine specific speed in customary U.S. units from the conversions of Fig. 14–107,
N_{ St , US , A }=N_{ St , US , B }=43.46 N_{ St }=(43.46)(1.615)= 7 0 . 2
Turbine specific speed: N_{ St }=\frac{C_{P}^{1 / 2}}{C_{H}^{5 / 4}}=\frac{\left( bhp / \rho \omega^{3} D^{5}\right)^{1 / 2}}{\left(g H / \omega^{2} D^{2}\right)^{5 / 4}}=\frac{\omega( bhp )^{1 / 2}}{\rho^{1 / 2}(g H)^{5 / 4}} (14–50)
Discussion Since turbines A and B operate at homologous points, it is no surprise that their turbine specific speeds are the same. In fact, if they weren’t the same, it would be a sure sign of an algebraic or calculation error. From Fig. 14–108, a Francis turbine is indeed the appropriate choice for a turbine specific speed of 1.6.
