A Rotating Detonation Rocket Engine $($RDRE$)$ is deployed in a hypersonic vehicle for flight in $10$ km altitude. The ambient pressure and temperature are $P_0=26\mathrm{.}5$kPa and $T_0=225K,$ respectively.

Using the $0-$D energy conservation law across the detonation wave, derive the temperature ratio, $\frac{T_1}{T_0},$ in terms of: Mach number of the incoming flow, $M_0,$ ratio of specific heats, $,$ specific heat at constant pressure, $c_p,$ specific heat release per unit mass $q,$ and initial propellant temperature, $T_0.$ Assume ideal gas law with constant specific heats. In addition, assume the incoming flow velocity, $u_0$ is significantly higher relative to the velocity of the burned gas, $u_1.$

Assume the above temperature ratio is:

$\frac{T_1}{T_0}=1+\frac{-1}{2}{M}_0^2+\frac{q}{c_p{T}_0}$

Evaluate the pressure ratio, $\frac{P_1}{P_0},$ given the density ratio is $\frac{{}_1}{{}_0}=0\mathrm{.}1$ for $=1\mathrm{.}22,c_p=2\mathrm{.}5k\frac{J}{k}g,$ and $M_0=3.$ The vehicle operates using liquid methane$-$oxygen propellants with $q=50M\frac{J}{k}g.$

$3.$ Calculate the thrust specific impulse of the RDRE for a mass flow rate of propellants $m^{}=5k\frac{g}{s},$ exit velocity of $u_e=3000\frac{m}{s},$ exit pressure $P_e=50$kPa, exit area of $A_e=0\mathrm{.}1m^2,$ and gravitational acceleration $g_0=9\mathrm{.}81\frac{m}{s^2}.$

$4.$ Provide three advantages and three disadvantages of RDREs over traditional rocket propulsion. For your justification, use a single sentence for each.