Determine the performance envelope for a propeller driven aircraft. This is a plot of the minimum and

maximum velocity of the aircraft as a function of altitude.

Consider a plane with the following characteristics

$W=10,000N$

$S=40m^2$

$AR=7$

$c_{D_0}=0\mathrm{.}005$

$c_{L_{}}=2$

$k=\frac{1}{}AR$

${}_{\text{m}\text{i}\text{n}}=1^0,{}_{\text{s}\text{t}\text{a}\text{l}\text{l}}=12$

The plane has an electric propeller driven engine delivering power of $P_{SL}=25000W.$ The air density at sea level,

${}_{SL}=1\mathrm{.}225k\frac{g}{m^3}.$ Determine the velocity envelope for the plane, given that you can approximate the air

density at any altitude $(h)={}_{SL}{e}^{-\frac{h}{10\mathrm{.}4}},$ where $h$ is the altitude in $km.$ The power varies with altitude and is

given by$,P(h)=P_{SL}(\frac{h}{{}_{SL}}{)}^{0\mathrm{.}75}.$ To do so$,$ you are advised to follow the following steps.

First show that the ceiling height of the aircraft is $8\mathrm{.}24km.$ This you can show by comparing the

minimum required power with available power at any altitude. It is best you plot two curves: $P_{\text{m}\text{i}\text{n}\text{P}}$ versus

$h$ and $P(h)$ versus $h$ and determine the intersection point.

Plot the power curve for the aircraft at sea level. You can do this by following the example code uploaded

on moodle. The required power can be computed using the following algorithm:

a$.$ Change aoa from ${}_{\text{m}\text{i}\text{n}}$ to ${}_{\text{s}\text{t}\text{a}\text{l}\text{l}}$

b$.$ Compute $c_L=c_L,c_D=c_{Do}+k{c}_L^2$

c$.$ Compute $V=\sqrt[\phantom{2}]{\frac{2W}{S{c}_L}}$ and $D$

d$.$ Compute $P=DV$

Plot the power curve at different altitudes up to the ceiling service altitude.

Determine the intersection of the required power and available power curve.

Plot the curve for the maximum and minimum velocity as a function of altitude $-$ that is the operating

envelope. Note you must be aware of the stall limit$.$ Also plot stall velocity vs height.