An airbreathing engine flies at Mach M0=2.0 at an altitude...

An airbreathing engine flies at Mach $M_{0}=2.0$ at an altitude where the ambient temperature is $T_{0}=-50^{\circ} \mathrm{C}$ and ambient pressure is $p_{0}=10 \mathrm{kPa}$. The airflow rate to the engine is $25 \mathrm{~kg} / \mathrm{s}$. The fuel flow rate is $3 \%$ of airflow rate and has a heating value of $42,800 \mathrm{~kJ} / \mathrm{kg}$. Assuming the exhaust speed is $V_{9}=1050 \mathrm{~m} / \mathrm{s}$, and the nozzle is perfectly expanded, i.e., $p_{9}=p_{0}$, calculate (a) ram drag in $\mathrm{kN}$ (b) gross thrust in $\mathrm{kN}$ (c) net (uninstalled) thrust in $\mathrm{kN}$ (d) thrust-specific fuel consumption in $\mathrm{kg} / \mathrm{h} / \mathrm{N}$ (e) engine thermal efficiency $\eta_{\text {th }}$ (f) propulsive efficiency $\eta_{p}$ (g) engine overall efficiency $\eta_{\circ}$

An airbreathing engine flies at Mach $M_{0}=2.0$ at an altitude where the ambient temperature is $T_{0}=-50^{\circ} \mathrm{C}$ and ambient pressure is $p_{0}=10 \mathrm{kPa}$. The airflow rate to the engine is $25 \mathrm{~kg} / \mathrm{s}$. The fuel flow rate is $3 \%$ of airflow rate and has a heating value of $42,800 \mathrm{~kJ} / \mathrm{kg}$. Assuming the exhaust speed is $V_{9}=1050 \mathrm{~m} / \mathrm{s}$, and the nozzle is perfectly expanded, i.e., $p_{9}=p_{0}$, calculate
(a) ram drag in $\mathrm{kN}$
(b) gross thrust in $\mathrm{kN}$
(c) net (uninstalled) thrust in $\mathrm{kN}$
(d) thrust-specific fuel consumption in $\mathrm{kg} / \mathrm{h} / \mathrm{N}$
(e) engine thermal efficiency $\eta_{\text {th }}$
(f) propulsive efficiency $\eta_{p}$
(g) engine overall efficiency $\eta_{\circ}$

Key Concepts

Ram Drag

Ram drag is the resistance force experienced by an airbreathing engine due to the forward motion of the vehicle, which forces air into the engine intake. It represents the momentum of incoming air that must be decelerated or accelerated in the engine, effectively reducing the net propulsive force available to overcome the vehicle's inertia.

Gross Thrust

Gross thrust is the total force generated by an engine as a result of the momentum change in the exhaust gases. It is calculated based on the mass flow rate and the exiting velocity of the engine exhaust, without accounting for the inlet air momentum or ram drag effects.

Net Thrust

Net thrust is the effective thrust produced after subtracting the forces opposing motion, such as ram drag, from the gross thrust. It represents the actual propulsive force available to accelerate the vehicle, taking into consideration both the engine’s production of momentum and the momentum of the incoming air.

Thrust-Specific Fuel Consumption

Thrust-specific fuel consumption (TSFC) is a performance metric that quantifies the efficiency of an engine by measuring the amount of fuel needed to produce a unit of thrust over time. Lower TSFC values indicate a more fuel-efficient engine, making it a critical parameter in engine design and operational economics.

Engine Thermal Efficiency

Engine thermal efficiency is a measure of how effectively an engine converts the chemical energy of the fuel into useful energy, typically in the form of thrust or work. It is defined as the ratio of the useful energy output to the energy input from fuel, providing insight into the combustion and energy conversion processes within the engine.

Propulsive Efficiency

Propulsive efficiency assesses how effectively an engine converts the kinetic energy of the exhaust into useful thrust that propels the vehicle. It considers the work done against the ambient air conditions and the relative velocities between the exhaust and the flight speed, indicating the effectiveness of the momentum transfer process.

Engine Overall Efficiency

Engine overall efficiency is the product of the thermal efficiency and the propulsive efficiency, providing a comprehensive measure of an engine's performance. It encapsulates both how well the engine extracts energy from the fuel and how effectively that energy is used to generate propulsive work, offering a holistic view of the engine’s energy conversion capabilities.

Transcript

00:01 We'd like to calculate each of these different values related to this problem.

00:05 We first want to find our ram drag.

00:10 Our ram drag is just our m dot, which is our airflow rate times our free stream velocity.

00:23 Our free stream velocity is just our mock number times a -0.

00:28 A -0 is our speed of sound at an ambient temperature.

00:34 So there we multiply 1 .4 our gamma times our temperature.

00:41 So give us 299 .4 meters per second.

00:46 Then we can find our free stream velocity as 598 .9.

00:52 And then we could find our ram drag.

00:57 We multiply this by 25 kilograms per second to get 14 .97 kiloons.

01:04 For baby we want our gross thrust.

01:10 Our gross thrust is the total mass flow rate times v, our exhaust speed.

01:27 But this is just equal to m .a, or pretty much our 25 plus 0 .03 times 25, which gives us, we multiply this by 1050, to give us 27 .0 kilo -neutons.

01:46 For c, we want our net thrust.

01:51 Our net thrust is our gross force minus our ram force, and so we get 12 .07 kiloons for that value.

02:03 For d, we want our thrust -specific fuel consumption.

02:08 We'll just call this t that's given as our m -dot over our net force...

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