Problem $4(30$ points$)$

A crude resistojet thruster is constructed by injecting gaseous $N{H}_3($ammonia$)$ into a constant$-$

area duct, where the gas is heated before it is expanded adiabatically through a converging$-$

diverging nozzle. This is depicted in Figure $1,$ where the heat is added between stations $(1)$

and $(2).$ At the entrance to the duct, the gas has a static temperature of $T_1=300K,$ a static

pressure of $p_1=10\mathrm{.}0$ bar and is flowing at $u_1=2\mathrm{.}150\frac{m}{s}.$ The flow is choked at the nozzle

throat, shown in the figure as station $(3).$

Figure $1$: Geometry for Problem $4($not drawn to scale$)$

The diameter of the heating duct is $1$ cm $,$ the diameter of the nozzle throat is $0\mathrm{.}75$ mm $,$ and

the nozzle expansion ratio is $190.$

Ammonia has a ratio of specific heats of $=1\mathrm{.}3$ and a molar mass of $\frac{?}{\text{b}\text{a}\text{r}}(M)=17\mathrm{.}03\frac{g}{m}ol.$

$($a$)$ Calculate the flow properties $M_2,T_2,p_2,T_{02},$ and $p_{02}$ at the end of the duct. What is

the heat added $($per unit mass$)q?$

$($b$)$ What ambient back pressure corresponds to a normal shock sit at the nozzle exit?

$($c$)$ If the thruster operates in vacuum, what is the flow velocity $u_e=u_4$ at the nozzle exit?

Is the flow underexpanded, overexpanded, or perfectly expanded?