Chapter 12

A rocket engine has a propellant mass flow rate of \(1000 \mathrm{~kg} / \mathrm{s}\) and an effective exhaust speed of \(c=3500 \mathrm{~m} / \mathrm{s}\). Calculate (a) rocket thrust \(F\) in \(\mathrm{kN}\) (b) specific impulse \(I_{\mathrm{s}}\) in seconds

A rocket engine has a chamber pressure of \(p_{\mathrm{c}}=1000\) psia and the throat area is \(A_{\mathrm{th}}=1.5 \mathrm{ft}^{2}\). Assuming that the nozzle is perfectly expanded with the gas ratio of specific heats \(\gamma=1.2\) and the ambient pressure of \(p_{0}=14.7 \mathrm{psia}\), calculate (a) optimum thrust coefficient \(C_{\mathrm{F}, \mathrm{opt}}\) (b) thrust \(F\) in lbf (c) nozzle exit Mach number \(M_{2}\) (d) nozzle area expansion ratio \(A_{2} / A_{\mathrm{th}}\)

A rocket is vertically launched and operates for \(60 \mathrm{~s}\) and has a mass ratio of \(0.05\). The (mean) rocketspecific impulse is \(375 \mathrm{~s}\). Assuming the average gravitational acceleration over the burn period is \(9.70 \mathrm{~m} / \mathrm{s}^{2}\), calculate the terminal velocity of the rocket with and without gravitational effects. Neglect the effect of aerodynamic drag in both cases.

A rocket has a mass ratio of \(\mathrm{MR}=0.10\) and a mean specific impulse of \(365 \mathrm{~s}\). The flight trajectory is described by a constant dynamic pressure of \(q_{0}=50 \mathrm{kPa}\). The mean drag coefficient is approximated to be \(0.25\), the vehicle initial mass is \(m_{0}=100,000 \mathrm{~kg}\), and the vehicle (maximum) frontal cross- sectional area \(A_{\mathrm{f}}\) is \(5 \mathrm{~m}^{2}\). For a burn time of \(100 \mathrm{~s}\), calculate the rocket terminal speed while neglecting gravitational effect.

In comparing the flight performance of a singlestage with a two-stage rocket, let us consider the two rockets have the same initial mass \(m_{0}\), the same payload mass \(m_{\mathrm{L}}\), and the same overall structural mass \(m_{s}\). The structural mass fraction \(\varepsilon\), which is defined as the ratio of stage structural mass to the initial stage mass, is also assumed to be the same for the single-stage rocket and each of the two stages of the twostage rocket. For the effective exhaust speed of \(3500 \mathrm{~m} / \mathrm{s}\) be constant for the single-stage and each stage of the two-stage rocket, calculate the terminal velocity for the two rockets in zero gravity and vacuum flight, for $$ \begin{aligned} m_{0} &=100,000 \mathrm{~kg} \\ m_{\mathrm{L}} &=500 \mathrm{~kg} \\ m_{\mathrm{s}} &=10,000 \mathrm{~kg} \\ \varepsilon_{\text {single-stage }} &=0.1 \\ \varepsilon_{\text {stage-1 }} &=\varepsilon_{\text {stage- } 2}=0.1 \end{aligned} $$

A liquid propellant rocket uses a hydrocarbon fuel and oxygen as propellant. The heat of reaction for the combustion is \(Q_{\mathrm{R}}=18.7 \mathrm{MJ} / \mathrm{kg}\). The specific impulse is \(335 \mathrm{~s}\) and the flight speed is \(2500 \mathrm{~m} / \mathrm{s}\). Neglecting the propellant kinetic power at the injector plate, calculate (a) effective exhaust speed \(c\) in \(\mathrm{m} / \mathrm{s}\) (b) propulsive efficiency \(\eta_{\mathrm{p}}\) (c) overall efficiency \(\eta_{\circ}\)

An injector plate uses an unlike impingement design. The fuel and oxidizer orifice discharge coefficients are \(C_{\mathrm{df}}=0.80\) and \(C_{\mathrm{do}}=0.75 .\) The static pressure drop across the injector plate for both oxidizer and fuel jets is the same, \(\Delta p_{\mathrm{f}}=\Delta p_{\mathrm{o}}=180 \mathrm{kPa}\). The fuel and oxidizer densities are \(\rho_{\mathrm{f}}=325 \mathrm{~kg} / \mathrm{m}^{3}\) and \(\rho_{\mathrm{o}}=1200 \mathrm{~kg} / \mathrm{m}^{3}\) and the oxidizer-fuel mass ratio is \(r=3.0\). Calculate (a) oxidizer-to-fuel orifice area ratio \(A_{\mathrm{o}} / A_{\mathrm{f}}\) (b) oxidizer and fuel jet velocities \(v_{\mathrm{o}}\) and \(v_{\mathrm{f}}\)

A solid rocket motor has a design chamber pressure of \(10 \mathrm{MPa}\), an end-burning grain with \(n=0.4\) and \(r=3\) \(\mathrm{cm} / \mathrm{s}\) at the design chamber pressure and design grain temperature of \(15^{\circ} \mathrm{C}\). The temperature sensitivity of the burning rate is \(\sigma_{\mathrm{p}}=0.002 /{ }^{\circ} \mathrm{C}\), and chamber pressure sensitivity to initial grain temperature is \(\pi_{\mathrm{K}}=0.005 /{ }^{\circ} \mathrm{C}\). The nominal effective burn time for the rocket is \(120 \mathrm{~s}\), i.e., at design conditions. Calculate (a) the new chamber pressure and burning rate when the initial grain temperature is \(75^{\circ} \mathrm{C}\) (b) the corresponding reduction in burn time \(\Delta t_{\mathrm{b}}\) in seconds

A regeneratively cooled rocket thrust chamber has its maximum heat flux of \(15 \mathrm{MW} / \mathrm{m}^{2}\) near its throat. The hot gas stagnation temperature is \(3000 \mathrm{~K}\) and the local gas Mach number is assumed to be \(\sim 1.0\). The gas mean molecular weight is \(\mathrm{MW}=23 \mathrm{~kg} / \mathrm{kmol}\) and the ratio of specific heats is \(\gamma=1.24\). Calculate (a) gas static temperature \(T_{g}\) in \(\mathrm{K}\) (b) gas speed near the throat in \(\mathrm{m} / \mathrm{s}\) (c) gas-side film coefficient \(h_{\mathrm{g}}\) for \(T_{\mathrm{wg}} \sim 1000 \mathrm{~K}\)

A rocket combustion chamber is designed for a chamber pressure of \(p_{\mathrm{c}}=50 \mathrm{MPa}\). The combustion gas has a ratio of specific heats \(\gamma=1.25\). If this rocket is to operate between sea level and \(200,000 \mathrm{ft}\) altitude, calculate the range of area ratios in the nozzle that will lead to perfect expansion at all altitudes. Assume isentropic flow in the nozzle.

The propellant flow rate in a chemical nozzle is \(10,000 \mathrm{~kg} / \mathrm{s}\), the nozzle exhaust speed is \(2200 \mathrm{~m} / \mathrm{s}\), and the nozzle exit pressure is \(p_{2}=0.01 \mathrm{~atm}\). Assuming the nozzle exit diameter is \(D_{2}=2 \mathrm{~m}\), calculate (a) the pressure thrust (in MN) at sea level (b) the effective exhaust speed \(c\) (in \(\mathrm{m} / \mathrm{s}\) ) at sea level

A solid propellant rocket motor uses a composite propellant with \(16 \%\) aluminum. The same propellant with \(18 \%\) aluminum enhances the combustion temperature by \(5.7 \%\). Assuming in both cases that the solid particles are fully accelerated (i.e., \(V_{\mathrm{s}}=V_{\mathrm{g}}\) ) in the nozzle but the solid temperature remains constant (i.e., \(T_{\mathrm{s}}=\) constant), calculate the ratio of specific impulse in the two cases. Aluminum specific heat is \(\mathrm{c}_{\mathrm{s}}=903 \mathrm{~J} / \mathrm{kg} \cdot \mathrm{K}\), and the specific heat at constant pressure for the gas is \(c_{\mathrm{pg}}=2006 \mathrm{~J} / \mathrm{kg} \cdot \mathrm{K}\).

The coefficient of linear thermal expansion for a solid propellant grain is \(1.5 \times 10^{-4} /{ }^{\circ} \mathrm{C}\). Calculate the change of length \(\Delta L\) for a \(1 \mathrm{~m}\) long propellant grain that experiences a temperature change from \(-30^{\circ} \mathrm{C}\) to \(+70^{\circ} \mathrm{C}\).

A ramjet has a maximum temperature \(T_{\mathrm{t} 4}=\) \(2500 \mathrm{~K}\). The inlet total pressure recovery \(\pi_{\mathrm{d}}\) varies with flight Mach number according to $$ \pi_{\mathrm{d}}=1-0.075\left(\mathrm{M}_{0}-1\right)^{1.35} $$ The ramjet burns hydrogen fuel with \(Q_{\mathrm{R}}=120,000 \mathrm{~kJ} / \mathrm{kg}\) and combustor efficiency and total pressure ratio are \(\eta_{\mathrm{b}}=0.99\) and \(\pi_{\mathrm{b}}=0.95\), respectively. The nozzle is perfectly expanded and has a total pressure ratio \(\pi_{\mathrm{n}}=0.90\). Assuming a calorically perfect gas with \(\gamma=1.4\) and \(R=287 \mathrm{~J} / \mathrm{kg} \cdot \mathrm{K}\), use a spreadsheet to calculate the ramjet (fuel)-specific impulse, propulsive, thermal, and overall efficiencies over a range of flight Mach number starting at \(M_{0}=3\) up until ramjet ceases to produce any thrust. Altitude pressure and temperature are \(15 \mathrm{kPa}\) and \(250 \mathrm{~K}\), respectively.