The jet engine is tested on the ground at standard...

The jet engine is tested on the ground at standard atmospheric pressure of $101.3 \mathrm{kPa}$. If the fuel-air mixture enters the inlet of the 300 -mm-diameter nozzle at $250 \mathrm{~m} / \mathrm{s}$, with an absolute pressure of $300 \mathrm{kPa}$ and temperature of $800 \mathrm{~K}$, and exits with supersonic flow, determine the velocity of the exhaust developed by the engine. Take $k=1.4$ and $R=249 \mathrm{~J} / \mathrm{kg} \cdot \mathrm{K}$. Assume isentropic flow.

Key Concepts

Nozzle Flow Dynamics

Nozzle flow dynamics involve the study of how a fluid accelerates as it flows through a nozzle that typically converges and/or diverges. For compressible flows, such as those in jet engines, the nozzle is designed to convert internal energy into kinetic energy, creating high-speed exhaust. The design includes areas where flow may become sonic at the throat and then supersonic in the diverging section when expanded further, and these dynamics are essential to predict the engine's performance and exhaust velocity.

Mach Number and Speed of Sound

The Mach number, defined as the ratio of the flow velocity to the local speed of sound, is a fundamental parameter in compressible flow. The speed of sound itself in a gas is related to its temperature (via the relation a = ?(kRT)), and knowing the Mach number helps in determining if the flow is subsonic, sonic, or supersonic. In nozzle analyses, the Mach number is used together with isentropic relations to link the static temperature, pressure, and velocity of the flow from the inlet to the exit.

Isentropic Flow

Isentropic flow refers to a thermodynamic process that is both adiabatic and reversible, meaning there is no heat transfer and no entropy change in the system. In the context of compressible flow through nozzles, assuming the process is isentropic allows engineers to use specific relations between pressure, temperature, and density to relate inlet (stagnation) conditions to the exit conditions. These relations simplify the analysis and help determine properties like the exit Mach number and velocity without accounting for dissipative effects such as friction or shock waves.

Energy Conservation in Compressible Flows

In compressible flows, the conservation of energy principle (often expressed as the conservation of total or stagnation enthalpy) is used to relate the static temperature and kinetic energy of the flow. As the fluid accelerates through the nozzle, its static temperature decreases due to the conversion of thermal energy to kinetic energy, thereby increasing its velocity. This principle, combined with isentropic relations, provides a way to calculate the exhaust velocity, an essential aspect of determining the thrust produced by the engine.

Transcript

00:01 Okay, in this problem, we're asked to consider a jet engine tested on the ground at standard atmospheric pressure 1 .1 .3 kilopascals.

00:09 If the fuel air mixture is given as follows, we're given the inlet of 300 millimeters at 250 meters per second at a pressure of 300 kilopascals and a temperature of 800k.

00:21 In this question, we're doing our second question, determine the velocity of the exhaust developed by the engine.

00:30 I'm going to erase those.

00:33 And then i'm going to go to my next page here, right here.

00:46 Okay, so we've got all of our values here.

00:49 Let's go ahead and figure out our, i think i can still go back and do one part here.

01:04 The part that i'm about to highlight is also the same.

01:08 Whoops, a green won't be a good color on that.

01:16 So using this, i can get my area, ratio as if i need that.

01:24 I don't think i need that.

01:25 Do i need that? i need this.

01:38 And that will be the same calculation.

01:44 And this will be the same calculation.