Question 12.4: The compressor and turbine maps of a turbojet engine are sho......

Let us first calculate the compressor-inlet total properties:

T_{t\ 2}=T_{t\ 1}=T_{1}\biggl[1+\biggl({\frac{\gamma-1}{2}}\biggr)M_{1}{}^{2}\biggr]=246.6\mathrm{~K}

p_{t\ 2}=p_{t\ 1}=p_{1}\biggl[1+\biggl({\frac{\gamma-1}{2}}\biggr)M_{1}{}^{2}\biggr]^{\frac{\gamma}{\gamma-1}}=0.323\,\mathrm{bars}

We can also calculate the compressor corrected speed as follows:

N_{C2}\equiv\frac{N}{\sqrt{\theta_{2}}}=32,000\;\mathrm{rpm}

The turbine map, on the other hand, produces the following:

(\Delta \dot{m})_{C4}=2.9\,\mathrm{kg/s}

{\frac{p_{t\ 4}}{p_{t\ 5}}}=8.5

Let us now establish a functional relationship between (\dot m)_{C2} and π_{C} as follows:

({\dot{m}})_{C2}=\left[({\dot{m}})_{C4}\left({\frac{1}{1+f}}\right){\sqrt{\frac{T_{t\ 2}}{T_{t\ 4}}}}\left({\frac{p_{t\ 4}}{p_{t\ 3}}}\right)\right]{\frac{p_{t\ 3}}{p_{t\ 2}}}

which, upon substitution, reduces to

(\dot m)_{C2}=1.18\left(\frac{p_{t\ 3}}{p_{t\ 2}}\right)

This relationship represents a straight line, shown on the compressor map in Fig. 12.16, intersecting the corrected-speed line (where N_{C2} = 32,000 rpm) at a point where

(\dot m)_{C2}=11.2\;{\mathrm{kg/s}}

{\frac{p_{t\ 3}}{p_{t\ 2}}}=9.49

\eta_{C}=78%

Let us now calculate the “physical” magnitudes of speed and mass-flow rate:

N=\sqrt{\theta_{2}}N_{C2}=29,611\,\mathrm{rpm}

\dot{m}_{2}=\frac{\delta_{2}}{\sqrt{\theta_{2}}}(\dot m)_{C2}=3.33\mathrm{~kg/s}

Finally, we can calculate the torque transmitted to the compressor as follows:

\tau_{c o m p.}=\frac{\dot m_{2}c_{p}T_{t\ 2}\biggl[\left(\frac{p_{t\ 3}}{p_{t\ 2}}\right)^{\frac{\gamma-1}{\gamma}}-1\biggr]}{\eta_{C}\omega}=307.6\,\mathrm{N\,m}