Explained: Turbojet Thrust Equation

Let's derive the thrust equation for a turbojet engine! In this video I'll show you how to derive the thrust equation for a single-inlet, single-outlet air breathing engine (turbojet is kind of just a buzzword here, because the engine doesn't necessarily need to only be a turbojet).
Yes, this video is heavy on the math, but I'm planning on starting a video series called 'In a Nutshell', which will take some of my math-heavy videos and break them down in to shorter, more easily understandable videos that focus on the big picture. Stay tuned for those.
===== RELEVANT VIDEOS/LINKS=====
→ Mass Conservation Derivation
goo.gl/Qx7PZ6
→ 1D Mass Conservation
goo.gl/hM46LU
→ Momentum Conservation Derivation
goo.gl/DuYHcG
→ 1D Momentum Conservation
goo.gl/8uo6VM
→ Surface Area Blog Post
goo.gl/1s2z2X
→ Turbofan Thrust Equation Derivation
www.joshtheengi...
===== EXTRA LINKS =====
► This is the Wikipedia page for Turbojets, and while I don't tend to like Wiki pages for math, take a look at the 'Net Thrust' section.
en.wikipedia.o...
===== ASSUMPTIONS =====
1) Flow is reversible external to the engine
2) Steady state
3) No viscous forces
4) No heat addition
5) No body forces
6) Velocity only has X-direction component
7) Momentum from fuel flow rate is negligible
===== THUMBNAIL PICTURE ATTRIBUTION =====
By Jeff Dahl [GFDL (www.gnu.org/cop...) or CC BY-SA 4.0-3.0-2.5-2.0-1.0 (creativecommons...)], via Wikimedia Commons
Modified slightly by me

[@smferreiro2610]

Now I see what Einstein meant with "time compression"... You're videos are extremely interesting! You can record the hour long topic, in a whole hour without the risk of loosing anyone... I almost got choked with my coffee! Excellent work, dude! I'll continue with the others. Thank you!

[@abdullahhba4638]

fantabulous explanation . never expected understanding this would be such easy

[@JoshTheEngineer]

Thank you!

[@danielmercier24]

Great video thank you ! just unlocked me in a calculation. It now becomes clear that a ducted fan actually pulls air from a larger area (Aa) than its exit area (Ae), accelerating that air along that path (Ue>Ua).

[@Yasuookuya]

I really appreciate all the time you spent explaining this kind of information, I really like when I understand the reason of the actual equation, I mean, why is it like that... Thank you very much!

[@JoshTheEngineer]

Well thanks for taking the time to watch!

[@guysbeingdudesbeingbrosinc4248]

Great video for a single stream engine! I would like to see how to extend this thrust equation into a 2-stream or 3-4-(infinitely many) streams.

[@enotdetcelfer]

am I the only one bugged that the incoming cold air is red and the post combusted air is blue? nice vid tho! *eye twitch

[@JoshTheEngineer]

You're right, they're switched. But to be fair, it still would have annoyed you since the ambient outgoing flow would then have been red as well.

[@0623kaboom]

@@JoshTheEngineer lol actually if you stayed on form the blue on the left solid ... and red only for the exit portion on the right with blue top and bottom ... and your calculations dont account for the coriollis effect ... any surface will have the effect you never covered if it was a factor or not ... as you get in tight spaces the airflow there and its ability to do work are radically different than more open space in the same engine ... mind you this does get one more into actual flow dynamics and fuel air mixture performance as well

[@JoshTheEngineer]

@@0623kaboom The point of the color is not to show the hot or cold streams, but instead to make it clear what side of the control volume the calculations are taking place. The reason for this is that the surface normal vector points out of the control volume on every side. This means it points in the opposite direction of the incoming flow velocity on the left side, and in the same direction of the outgoing flow velocity on the right side. Also, I'm not quite sure what your second comment is referring to. This is a control volume analysis. You don't need to worry about the internals of the engine when we're simply trying to account for mass/momentum/energy at the volume boundaries.

[@0623kaboom]

@@JoshTheEngineer the body of the aircraft also adds more effect to the exit not very often does one fly an engine with out a plane to put it in ...

[@BradleyG01]

It isn't your fault, but this video brought me to tears

[@nikan4now]

Thanks for the great video. I think if you picked your CV to "hug" the turbojet so that it would follow the contours of the body on the outside, the derivation would simply considerably.

[@JoshTheEngineer]

You're welcome! There are definitely multiple ways you can go about deriving these equations, and I just happened to pick this one because I've found it to be the easiest to understand and explain. If you look in the propulsion book by Hill and Peterson, they actually show how you can choose two different control volumes and end up at the same result. The one thing I will say is that a lot of the derivations you may find in books or in notes online will skip many steps in the overall derivation. The reason I do the derivation this way is so that it is continuous from my other videos' base derivations of mass and momentum conservation. Even in the derivations that just say the momentum in equals the momentum out, they started with the same equations I'm using, made some assumptions, and then wrote down their simplified equation. I'm just backing up a few steps to show where those simple equations come from.

[@mono7891]

Great explanation ..... Thank you

[@karunbolli1873]

thank you so much..., u helped me in preparing for my exams tq

[@JoshTheEngineer]

That's great!

[@deetimeless5836]

Ahh very rigorous. Good work, thanks! It is an assumption, though, that the input velocity will be uniform. A jet engine at rest will have a non-zero inflow to Ai.

[@harshitkumar9672]

Amazing videos. Thank you❤️

[@JoshTheEngineer]

No problem!

[@berfinylmaz3667]

great explanation!! ty very much

[@JoshTheEngineer]

Thank you!

[@gerardoespino4382]

Great work Josh. Thank you very much, it's very helpful.

[@JoshTheEngineer]

No problem!

[@RockeybhaifromKGF]

Q.2 Turbojet engine flies with velocity of 270 m/s at the altitude where ambient temperature is -15°C. Gas temperature at nozzle exit is 873tK and fuel air ratio is 0.019. Corresponding enthalpy values for air and gas at inlet and exit are 260 kJ/kg and 912 kJ/kg respectively. Combustion efficiency is 95% and calorific value of fuel is 44.5 MJ/kg. For the heat losses from engine amounting to 21 kJ/kg of air. Determine the velocity of gas jet at exit. Please solve it & post here

[@tbpom1003]

wow awesome explanation! thanks!

[@JoshTheEngineer]

You're welcome!

[@MrjordanObelisk2u]

Hey Josh, very good video extremely helpful, ive watched many of your videos now. Do you think you could just give me a couple sentences further explaining what the spillage mass flow rate is?

[@JoshTheEngineer]

Thanks Jordan. That term is pretty much just there to link the two conservation equations together, so it's sort of a dummy variable in that sense. We know both the mass and momentum conservation equations will include an mdot_s term because it's crossing the control volume boundary, so we express the mass conservation as an expression for mdot_s, and then plug it into the momentum conservation to solve for the thrust. There are other ways to do this derivation, but I just happen to like this one. For instance, one thing to think about is that you could make the top and the bottom boundaries of the control volume streamlines of the flow, which means the velocity is tangent to those sides, and thus you have no flow across the top or bottom.

[@KacperTeo]

Nice! In the final thrust equation I presume u_e (outlet velocity) is dependent on the combustion and engine parameters itself, how would we calculate u_e, as it's an essential part of calculating the thrust?

[@michaelgarcia812]

Good job, is “spillage” exhaust?

[@kazansky22]

Great video, However, I challenge you to make a similar one for ducted fan engines, similar to the UL-39 or the PJ-II dreamer. Now those are things that have some sparse information and are hard to learn about.

[@JoshTheEngineer]

Thanks! Yea those two were difficult to find much information on. In a general, simplified sense though, the thrust equation boils down to a momentum balance between the inlet and the outlet, so you can get a pretty good estimate based on knowing mass flow rates and velocities.

[@jaldo7364]

Good video, thanks !

[@prerakvaishnav7512]

If I am not incorrect, the pressure term Pa(A-Ae) for the outlet should be negative as well.

[@JoshTheEngineer]

The sign comes from the outward normal in the dS (vector) term. For the outlet, the outward normal points in the positive X direction, so that outlet term is positive.

[@j.n3267]

Congratulation!! Very nice video

[@JoshTheEngineer]

Thank you!

[@محمديحيى-ت2س]

Great video Josh. However, I was wondering if it is possible to derive a thrust equation based on a control volume analysis for the combustion chamber only. Also, if I don't have values for air mass flow rate (only have those of fuel flow rate), would it be possible to perform such analysis for a combustion chamber ?

[@francescoindolfo]

Why the conditions at the inlet of the engine are not equal to the freestream ??

[@JoshTheEngineer]

It's because that's what is called the "inlet capture area". Imaging you're standing right in front of an engine that's running on the ground, but not moving. If the conditions in front of the inlet plane of the engine were equal to the freestream, you wouldn't feel anything, since the freestream is still air. In reality, you would probably get sucked into the engine. The same general logic can be applied to the engine in flight.

[@mehmetsarkaya7211]

I can't understand why mdot_f is negligable for moment equation ?

[@yubarajghimire3205]

does heat increases the thrust of an engine, if it does than by how much percentage. thanks

[@amrudhaa7834]

in the term Pa(-Ai) . why the area is not in vector quantity? isnt it supposed to be (Ai )i^?

[@aidealzohary3512]

Life Saviour!

[@JoshTheEngineer]

Glad I could help!

[@stephenmounioloux115]

Nice vid Josh, ever considered of putting together a similar for (electric) ducted fans thrust equations?

[@ceoof601]

I would assume it's the same right?

[@ceoof601]

I mean the simplified form at least

[@subhashsshankar7665]

very good video.....!

[@JoshTheEngineer]

Thank you!

[@stopthephilosophicalzombie9017]

Is this theory basically what Whittle came up with in his senior thesis?

[@pandacorn7964]

Takeoff Thrust?

[@openskies11]

You speak very, very fast.