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Going the Distance for Range

Issue 6, 5 September 2022

By: Anthony O. Ives

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Range is the usually the maximum distance an aircraft is designed to fly. It is important for most aircraft, as generally aircraft are designed to carry a payload a certain distance. The range equation is different for propeller driven aircraft with expendable fuel such as gasoline to that of a battery powered propeller driven aircraft. Jet powered aircraft also use a different form of the range equation it will be looked at in a future article.

Generally small remote controlled aircraft or UAVs use electric motors so will we look battery powered aircraft first and in more detail. The range equation for battery powered aircraft is as follows:

\[R={3.6 C_b V_{req} \over T_{req} }\]

Where R is range which the maximum distance the aircraft can fly. T_req is the required thrust to overcome the drag in cruise. V_req is the battery voltage required which depends on the type battery you want to use could assume a 3 cell lipo which would give 11.1V.

Generally when you are designing an aircraft you how far you want it travel hence range is known. What you really want to known is how much fuel or what size of battery is needed. So the equation can be rearranged to give battery capacity required for specific range which would give the following equation:

\[C_b={R T_{req} \over 3.6 V_{req} } \]

The following table shows how to use the range equation for battery powered aircraft to calculate the required battery capacity using some assumed and calculated values from previous articles.

Symbol	Property	Example Value	Units
R	Aircraft range	28x1000=28000	m
E	Aircraft endurance	30x60=1800	s
V_Req	Battery voltage required	11.1	N
T_Req	Thrust required	6.292	N
P	Propulsive power required	134	W
C_b	Battery capacity required for Range	1800134/(3.611.1)=6036	mAhr

In most cases when you are designing an aircraft you may not know the thrust required intially, you may just know a typical lift to drag ratio for your type of aircraft. The range for battery powered can be take a more general form as below:

\[R={\frac{w_f}{w_0} \frac{L}{D} 3.6 C_{sb} } \]

Where L/D is the lift to drag ratio. C_sb is the specific battery capacity. w_f/w₀ is the ratio of fuel mass to total aircraft mass, in the case of battery powered aircraft the fuel mass is the battery mass. This equation is also discussed in different form to determine the fuel mass ratio in Ref [1]. Specific battery capacity is also discussed in Ref [1].

For an propeller driven aircraft with expendable fuel such as gasoline jet fuel the equation a similar but slightly different form:

\[R={\frac{L}{D} \frac{\eta}{c_{fuel} } log_e \frac{1}{1- \frac{w_f}{w_0}}} \]

log_e is natural logarithm something a future article will explain in more detail. You can work out natural logarithm on a scientific calculator or you could use a computer such microsoft excel. c_fuel is specific fuel consumption and η is propulsion efficiency. It is worth note the main difference between propeller driven aircraft and jet aircraft is how fuel consumption is defined. For propeller driven aircraft fuel consumption is related to power were as with jet aircraft it is thrust.

The one advantage of an aircraft with expendable fuel is that it gets lighter as fuel is used up so that is why the equation has log_e this is to account for it. Range for an propeller aircraft driven is higher with a larger L/D ratio, commerical airliners and gliders typically have high L/D ratios to maximise their range. Obivously lower fuel burn helps along with a high propulsion efficiency. The equations for range apply to helicopters or rotary aircraft as maybe with slight differences. Jet and rotary aircraft range will be the topic of future articles. The range equation should help summarise the main factors influencing aircraft range. Ref [2],[3] and [4] discuss aircraft in more detail.

Please leave a comment on my facebook page or via email and let me know how if you understand what factors influence aircraft range and what the main difference between jet and propeller aircraft in determining range.

References:

[1] http://www.eiteog.com/No1EiteogBlogLiftCL.html

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

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

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

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