Consider two versions of a jet engine. The first is a standard turbojet engine run with a turbine inlet stagnation temperature of $1200$ K $.$ The second is identical, except that its turbine is cooled by bleeding air from the compressor; in this way the allowable turbine inlet stagnation temperature may be raised to $1600$ K $.$ To make an estimate of the effect of this modification on the thrust, suppose that the engine is flying at Mach $2$ at an altitude where the ambient temperature is $T_0=200K$ and that the compressor stagnation pressure ratio is ${}_c=9.$

Suppose also that for the second engine, $10\%$ of the airflow is bled from the compressor at a point where the stagnation pressure ratio is $9.$ This bleed air is used to cool the turbine blades at a low speed, so that there is no appreciable velocity of this bleed air in the exhaust. For simplicity assume all components of both engines to be ideal. Let $=1\mathrm{.}4$ and $c_p=1\mathrm{.}0k\frac{J}{k}g-K,$ and assume the heating value $Q_R$ is equal to $45,000k\frac{J}{k}g.$ Note here that in your analysis you could employ a simplification associated with $\frac{Q_R}{T_0{c}_p}{}_{}.$

a$)$ Determine the thrust of the two engines per unit mass flow of air entering the compressor, i$.$e$.,\frac{T}{m_0^{}}$ with units $(\frac{N*sec}{kg}).$