Question 5.4: Figure 5.14 shows the rotor-inlet velocity diagram for an ad......

We begin by locating the stage operating conditions on the stage map:

\theta_{i n}=\theta_{0}={\frac{T_{t\ 0}}{T_{S T P}}}=1.0

\delta_{in}=\delta_{0}=\frac{p_{t\ 0}}{p_{S T P}}=1.0

It follows that

\dot{m}_{C}={\frac{\dot{m}\sqrt{\theta_{i n}}}{\delta_{i n}}}=7.0\,\mathrm{kg/s}

Referring to the rotor-inlet velocity triangle in Figure 5.14, we get

U_{m}=W_{1}\ \sin\beta_{1}+V_{1}\ \sin\alpha_{1}=304.6\,\mathrm{m/s}

V_{z}=V_{1}\ \cos\alpha_{1}=131.6\mathrm{\,m/s}

V_{\theta\ 1}=V_{1}\ \sin\alpha_{1}=76.0\ \mathrm{m/s}

N={\frac{U_{m}}{r_{m}}}{\frac{60}{2\pi}}\approx34,000\ \mathrm{rpm}

\frac{N}{\sqrt{\theta_{i n}}}=34,000\,\mathrm{rpm}

Now, referring to the stage map, we get the following variables:

{\frac{p_{t\ 2}}{p_{t\ 1}}}=1.85

\eta_{t-t}=0.8

Using this efficiency magnitude, and referring to expression (3.60), we have

T_{t\ r}=T_{t}+\frac{W^{2}-V^{2}}{2c_{p}} (3.60)

T_{t\ 2}=357.2\,\mathrm{K}

Now, applying Euler’s equation,

w_{s}=c_{p}(T_{t\ 2}-T_{t\ 1})=U_{m}(V_{\theta\ 2}-V_{\theta\ 1})

which yields

V_{\theta\ 2}=304.2\ {\mathrm{m/s}}

Pursuing the solution procedure,

V_{2}={\sqrt{V_{\theta\ 2}{}^{2}+V_{z}{}^{2}}}=331.5\ {\mathrm{m/s}}

V_{c r_{2}}={\sqrt{\left({\frac{2\gamma}{\gamma+1}}\right)R T_{t\ 2}}}=345.8\ {\mathrm{m/s}}

M_{c r_{2}}=\frac{V_{2}}{V_{c r_{2}}}=0.96

\rho_{2}=\left(\frac{p_{t\ 2}}{R T_{t\ 2}}\right)\left[1-\frac{\gamma-1}{\gamma+1}(M_{cr_{2}})^{2}\right]^{\frac{1}{\gamma-1}}=1.19\ \mathrm{kg/m^{3}}

At this point, we are prepared to compute the stage specific speed (N_{s}), by substituting the foregoing variables in expressions (5.28) and (5.29),

N_{s}={\frac{N{\sqrt{\frac{\dot m}{\rho_{e x}}}}}{(\Delta h_{t\ i d})^{\frac{3}{4}}}}     (5.28)

\Delta h_{t\ i d}=c_{p}\,T_{t\ i n}\left[\left(\frac{p_{t\ e x}}{p_{t\ i n}}\right)^{\frac{\gamma-1}{\gamma}}-1\right]   (5.29)

N_{s} = 2.38 radians

Finally, we see (by reference to Figure 5.8) that this specific-speed magnitude places this stage well within the axial-flow compressor-stage “dome.” This clearly justifies the axial-flow choice for this compressor stage.