Unless otherwise noted, use $c_p=1005\frac{J}{k}g-K,R=287\frac{J}{k}g-K,$ and $=1\mathrm{.}4.$

$[40$ pts$.]$ A single stage axial flow compressor is to be designed with mean radius conditions:

${}_1=20,\frac{c_z}{U}=0\mathrm{.}55,c_{z1}=c_{z2}=150\frac{m}{s}$

$p_{01}=101325Pa,T_{01}=288\mathrm{.}15K$

a$)$ Calculate the blade speed, $U.$

b$)$ Calculate the inlet absolute tangential velocity, $c_1.$

c$)$ Calculate the inlet absolute velocity, $c_1.$

d$)$ Calculate the inlet relative tangential velocity, $w_1.$

e$)$ Calculate the inlet relative velocity, $w_1.$

f$)$ Calculate the inlet relative flow angle, ${}_1.$

g$)$ Calculate the outlet absolute tangential velocity, $c_2,$ if the design total pressure ratio $\frac{p_{02}}{p_{01}}=1\mathrm{.}50$ and the adiabatic stage efficiency ${}_{st}=0\mathrm{.}90.$

h$)$ Calculate the outlet absolute velocity, $c_2.$

i$)$ Calculate the outlet relative tangential velocity, $w_2.$

j$)$ Calculate the outlet relative velocity, $w_2.$

k$)$ Calculate the outlet absolute flow angle, ${}_2.$

Calculate the outlet relative flow angle, ${}_2.$

$m$ Sketch the velocity triangles for the inlet and outlet of the rotor. Name and show all angles and velocities. You may assume that axial velocity is constant.

$(U,c_z,c_1,c_1,w_1,w_1,c_2,c_2,w_2,w_2,{}_1,{}_1,{}_2,{}_2)$

n$)$ Calculate the outlet total temperature, $T_{02}$ and outlet total pressure, $P_{02}.$

o$)$ Calculate the inlet and outlet static temperatures, $T_1,T_2.$

p$)$ Calculate the inlet and outlet static pressures, $p_1,p_2.$

q$)$ Calculate the inlet and outlet density, ${}_1,{}_2.$

r$)$ Calculate the work coefficient, $\frac{h_0}{U^2}.$

s$)$ Calculate the static pressure coefficient for the rotor using $C_p=\frac{p_2-p_1}{0\mathrm{.}5{}_1{w}_1^2}.$ Note that $p_1$ and $p_2$ are static pressures. Is separation of the rotor likely?

t$)$ Calculate the degree of reaction for the stage, $R.$