Question 13.6: A preliminary design of the low-pressure axial compressor, w......

On the basis of the suggested inlet Mach numbers, assume that the relative tip Mach number M_{\text{1tip}} = 1.1 and the axial Mach number M_{\text{a}} = 0.5. The procedure is to calculate the static temperature and pressure and then the density. Next, the axial, relative, and rotational speeds are calculated. Finally, the tip radius is calculated, from which the hub-to-tip ratio is calculated. If the tip diameter is far from the turbine outer diameter, then the axial Mach number is to be changed within its design range. The results for the example is arranged in Table 13.6 where the axial Mach number varies from 0.5 to 0.6.
Since from turbine calculations

N=180\text{ rps} \quad \quad \text{and} \quad \quad (r_{\text{t}})_{\text{turbine}}=0.285 \text{ ms}

Choose M_1 = 0.6, where \zeta = 0.5 and r_{\text{t}}=0.283 m, which is close to the tip radius of the turbine.
The corresponding blade rotational speed at tip is U_{\text{t}} = 320 m/s.
The mean radius is r_{\text{m}}=((1+\zeta)/2)r_{\text{t}}=0.213 \text{ m}. The hub radius is 0.143 m. The mean rotational speed U_m , or simply U, is 241 m/s. Here, a decision has to be made concerning the type of variation of the compressor passage at its successive stages. One has to decide whether the compressor will have a constant mean, tip, or hub radius through all stages.
Now, to determine the outlet dimensions of the compressor, assume that the stage efficiency (or the polytropic efficiency) of the compressor is \eta_{\text{s}} ; the outlet temperature of the compressor is calculated as

\frac{T_{02}}{T_{01}} =\left(\frac{P_{02}}{P_{01}} \right) ^{(n-1)/n}

where(n − 1)/n = (1/η_{\text{s}}) ((γ − 1)/γ ).
Assume that air leaves the stator of the last stage of the compressor axially. Thus C_2 = C_{\text{a}} = C_1 , which was previously calculated based on the assumed axial Mach number. The static pressure and temperature will be

T_2=T_{02}-\frac{C_2^2}{2Cp} , \quad P_2=P_{02}\left(\frac{T_2}{T_{02}} \right) ^{\gamma/(\gamma-1)}

The density and the annulus area are \rho_2=P_2/RT_2,A_2=\dot{m}/\rho_2C_{\text{a2}} .
If the constant mean radius option is selected, then the mean radius is already known and the blade height at outlet is calculated as follows: h=(A_2/ 2\pi r_{\text{m}}), the tip and root radii at outlet are calculated from the relations

r_{\text{t}}=r_{\text{m}}+\frac{h}{2} , \quad r_{\text{r}}=r_{\text{m}}-\frac{h}{2}

The results at the compressor outlet are

T_{02}=460.3 \text{ K}, \quad P_{02}=423 \text{ kPa}, \quad T_2=423.6 \text{ K}, \quad P_2=357.22 \text{ kPa}, \\ \rho_2=2.8378 \text{ kg/m}^3, \quad r_{\text{t2}}=0.244 \text{ m}, \quad r_{\text{h2}}=0.182 \text{ m}, \quad h=0.062 \text{ m}