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Jet Aircraft Range and Endurance

Issue No 41, 28 August 2023

By: Anthony O. Ives

Range and endurance for jet powered aircraft is different from that for propeller powered and rotary wing aircraft. The difference is down to the fact that for jet powered aircraft fuel consumption is related to thrust. For propeller powered or rotary wing aircraft fuel consumption is related to power. While VTOL (Vertical Take Off Landing) aircraft powered directly by jet thrust do exist they are fairly rare and generally used for specialised military aircraft but with recent interest at the time of writing in advanced air mobility (AAM) some of the proposed designs for AAM such as the Lilium Jet [1] might be considered a jet thrust powered aircraft.

Personally I cannot see there is a market or a need for AAM or what is more commonly refered to as air taxis by the general public. Even if there is a market for air taxis I would think it will cause more congestion in the air which would then really do away any advantage of going by an air taxi instead of a ground taxi. With all the new concepts being proposed at the time of writing such as air taxis, drone parcel deliver services and existing aviation also expanding its seems like the airspace is going to get more and more congested and complicated to mange with increases in the amount of restricted and controlled airspace. There is also the question of cost with air taxis, can they be made inexpensive enough that the general public will have access to them. Also some (well in my opinion most of them) of the designs proposed for air taxis could be unsafe designs with no autorotation capability [2]. Over the years NASA (National Aeronautics and Space Administration) has investigated various different designs for VTOL [3] but the conventional helicopter was generally considered the most practical and safest.

However, currently at the time of writing there is a lot of investment in AAM or the air taxi concept only time will tell whether it will ever really amount to anything. The Lilium Jet is one design of air taxi I can think would perform similar to a jet for range and endurance, but jet range and endurance mostly applies to fixed wing aircraft powered by a turbojet or turbofan engine so this article is really just for completeness so you are aware there are different equations for jet aircraft. In a separate later article I will go into more detail on the different types of jet engines but references [4], [5] and [6] will also give you an explanation.

First we will start with the jet aircraft range equation below:

\[R=\frac{2}{c_t} \sqrt{\frac{2}{\rho S}} \frac {C_L^{\frac12}}{C_D} \left( W_0^\frac12 - W_1^\frac12 \right) \]

Where R is range, c_t is the thrust specific fuel consumption, ρ is air density, S is wing Area, C_L is lift coefficient, C_D is drag coefficient, W₀ is the intial weight of the aircraft including the fuel required and W₁ is the final weight of the aircraft which is really W₀ with the spent fuel subtracted. Remember weight is different from mass, weight = mass × gravity, gravity is typically 9.81 metres per second squared (ms^-2) . As you can see the range is a maximum when the value of \(\frac {C_L^{\frac12}}{C_D}\) is highest which is not the same ratio for a propeller powered aircraft [7] to give its maximum range.

The below equation gives the equation for jet aircraft endurance:

\[E=\frac{1}{c_t} \frac {C_L}{C_D} ln \frac{W_0}{W_1} \]

Where most of notation has been already defined and E is simply the aircraft endurance. The maximum endurance is achieved for the highest ratio of \(\frac {C_L}{C_D}\) which is the lift to drag ratio which is also different from that for a propeller powered aircraft [8]. In the case of a propeller powered aircraft the highest lift to drag ratio gives maximum range. Its the highest value of the ratio \(\frac {C_L^{\frac32}}{C_D}\) the give maximum endurance of a propeller powered aircraft.

The difference is due to the fact that Power = Thrust × Speed. The square root of lift coefficient inverted can be used to represent speed in a non-dimensional sense if that makes sense. As the jet aircraft equations depend on thrust if you divide them by non-dimensional speed which is the square root of lift coefficient inverted you will get the propeller powered aircraft equations. Range and endurance is inversely proportional to thrust or power, thrust or power has to increase with drag so that is why an increase in thrust or power will give you less endurance and range. Its also explains why you divide the equations by speed hence multiply them by the square of lift coefficient to go from jet powered (thrust) to propeller powered (power) aircraft equations. In theory if you multiply the equations for jet aircraft by the equation below you should get the equations for propeller powered aircraft:

\[\frac{1}{V} = \sqrt {C_L} \times \sqrt {\left(\frac {\rho S}{2W}\right)} \]

Where V is speed, and W is general weight because the equations require integration 'W' will make the equations look different but it should help you understand why there are differences between the two sets equation for jet aircraft and propeller powered aircraft. Integration is something we will look at in a later article. A later article will also explain how you determine the highest values of the three ratios \(\frac {C_L}{C_D}\), \(\frac {C_L^{\frac32}}{C_D}\) and \(\frac {C_L^{\frac12}}{C_D}\) and the corresponding speeds to get the highest values of the ratios.

Another important point is that equation for jet aircraft range assumes constant density which is essentially constant altitude however, if it assumes constant speed the equation changes and the maximum range is for highest the lift to drag ratio, \(\frac {C_L}{C_D}\). For propeller powered aircraft its endurance that has two different equations one which assumes constant density and another that assumes constant speed, with constant speed assumed endurance equation is maximum for highest lift to drag ratio. However, in most cases constant altitude hence constant density is assumed hence the equation for range in this article is the one more commonly used but at least you are aware there is a variation on the equations for range for jet aircraft and endurance for propeller powered aircraft. The equations given in this article and references [7] and [8] are technically approximate there are integral versions of the equations for aircraft aircraft range and endurance which I will discuss in later articles.

As an action point you could try to determine the equivalent jet aircraft range and endurance using the information for the aircraft which range and endurance was determined in articles [7] and [8], additional information can also be found in articles [9] to [12]. Although this may be easier to do using the later article on the three lift to drag ratios. Let me know in the comments how you get on or drop me an email, you can also let know if you are able to partially derive the equations in articles [7] and [8] from those for jet aircraft as described in this article.

Please leave a comment on my facebook page or via email and let me know if you found this blog article useful and if you would like to see more on this topic. Most of my blog articles are on:

Mathematics
Helicopters
VTOL UAVs (RC Helicopters)
Sailing and Sailboat Design

If there is one or more of these topics that you are specifically interested in please also let me know in your comments this will help me to write blog articles that are more helpful.

References:

[1] https://lilium.com/jet

[2] http://www.eiteog.com/EiteogBLOG/No20EiteogBlogAutorotation.html

[3] The X-Planes X-1 to X-45, Jay Miller, 2nd Edition, 2001, Midland Publishing

[4] Aircraft Performance & Design, John D. Anderson Jr., 1999, McGraw Hill

[5] Civil Jet Aircraft Design, Lloyd R. Jenkinson, Paul Simpkin, Darren Rhodes, 1999, Butterworth Heinemann

[6] Aircraft Performance (Cambridge Aerospace Series, Series Number 5), W. Austyn Mair, David L. Birdsall, 1996, Cambridge University Press

[7] http://www.eiteog.com/EiteogBLOG/No6EiteogBlogRange.html

[8] http://www.eiteog.com/EiteogBLOG/No7EiteogBlogEndurance.html

[9] http://www.eiteog.com/EiteogBLOG/No1EiteogBlogLiftCL.html

[10] http://www.eiteog.com/EiteogBLOG/No2EiteogBlogDragCD.html

[11] http://www.eiteog.com/EiteogBLOG/No3EiteogBlogMass.html

[12] http://www.eiteog.com/EiteogBLOG/No4EiteogBlogThrust.html

Disclaimer: Eiteog makes every effort to provide information which is as accurate as possible. Eiteog will not be responsible for any liability, loss or risk incurred as a result of the use and application of information on its website or in its products. None of the information on Eiteog's website or in its products supersedes any information contained in documents or procedures issued by relevant aviation authorities, manufacturers, flight schools or the operators of aircraft, UAVs.