Aircraft$-$Engine Matching: In this problem, we will estimate the performance of the

GEnx$-1$B$70$ turbofan engine that powers the Boeing $787-8$ commercial transport aircraft.

Assume the $787-8$ has a simplified drag polar of the form

$C_D=C_{D0}+K{C}_L^2$

where $C_{D0}=0\mathrm{.}014$ and $K=0\mathrm{.}052.$ Also assume the engine thrust lapses with altitude and

flight Mach number according to the following correlation proposed by Mattingly, Heiser,

and Pratt $[3],$

$\frac{J}{T_{SLS}}=\frac{p_a}{p_{\text{r}\text{e}\text{f}}}(1+\frac{-1}{2}{M}^2{)}^{\frac{}{-1}}(1-0\mathrm{.}49\sqrt[\phantom{2}]{M})$

where $T$ is the maximum thrust at a given altitude and speed, $J_{SLS}$ is the maximum engine

thrust at sea level static conditions, $p_{\text{r}\text{e}\text{f}}=1$atm is sea level atmospheric pressure, and the air

can be treated as an ideal gas with $R=287\frac{J}{k}\frac{g}{K}$ and $=1\mathrm{.}4.$

$($a$)$ Determine values of the following design and performance parameters for the $787-8$:

Maximum takeoff weight $W_{\text{M}\text{T}\text{O}},$

Wing area $s,$

Maximum operating altitude.

$($b$)$ Calculate the aircraft drag $D$ at the maximum operating altitude, assuming a flight

Mach number of $M=0\mathrm{.}8$ at $90\%$ of the maximum takeoff weight. Divide by two

to determine the thrust $T$ per engine at these conditions. $($Hint: you can look up the

ambient pressure and temperature, $p_a$ and $T_a,$ and speed of sound $a$ as functions of

altitude in Appendix III of Hill & Peterson $[1]$ or Appendix A of Mattingly $[2].)$

$($c$)$ Estimate the engine sea level static thrust $T_{SLS}$ based on the thrust calculated in part $($b$)$

and the assumed thrust lapse correlation.

$($d$)$ How does your answer to part $($c$)$ compare to the takeoff thrust of the GEnx$-1$B$70?$

What is the relative $($percent$)$ error of your estimation? How does your result compare

with the approximation discussed in class of $0\mathrm{.}3W_{\text{M}\text{T}\text{O}}?$

$($e$)$ Calculate the aircraft drag $D$ and maximum thrust $T$ at a cruising altitude of $30,000ft$

and a flight Mach number of $M=0\mathrm{.}78$ at $90\%$ of of the maximum takeoff weight.

$($f$)$ Based on your answers to part $($e$),$ calculate the maximum rate of climb, in $f\frac{t}{m}in$ at the

"top of climb" condition before the start of cruise.