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Weighting it up

Issue 3, 1 August 2022

By: Anthony O. Ives

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Estimating aircraft wing area is the first thing that allows you to define the rest of the geometry of the aircraft. However, you generally need an estimate of mass to determine your wing area. If you are building functional UAV which has a specific purpose this article makes an intial estimate of aicraft total mass based on your payload, endurance and range requirements.

The simplist way of estimating the mass of an aircraft is by considering a similar type of existing aircraft. For example if you are building a small 2 seat aircraft assume it will weight similar to an existing 2 seat aircraft.

However if you want a more stratagic mass estimation around payload, aircraft's range and endurance then you could use the following equation:

\[w_0={w_{UL} \over 1-\frac{w_f}{w_0} -\frac{w_e} {w_0}} \]

w_UL is the useful load including crew and payload in the case of a manned aircraft. The payload is what the aircraft is built to carry such as cargo, passengers or camera equipment, etc. The onboard remote control systems could be considered in place of the 'crew' on a remoted controlled aircraft so its probably a good idea to include this as part of your useful load.

Typically w_e/w₀=0.62, this is the ratio of aircraft empty weight to total weight

Typically w_f/w₀=0.16, this is the ratio of aircraft fuel weight to total weight. There is a more accurate way of estimating this ratio based on range and endurance.

The equations used to calculate fuel to mass ratio are depend on whether the aircraft has jet engine or is propeller driven and also whether is has expendable fuel such as an aircraft powered by jet fuel or gasoline. An aircraft powered by expendable fuel will reduce in weight as the fuel is used up. A battery powered aircraft will generally will remain the same weight throughout its flight. The range and endurance equations for battery power aircraft will be introduced first as they are the most applicable to small UAVs or Remote Controlled Aircraft. The equations for propeller driven aircraft using an expendable fuel will be briefly introduced as these may apply to small UAVs. The equations for a jet powered will be discussed in a future article as they may apply to some small UAVs.

The fuel mass ratio for the range of a battery powered aircraft can be calculated using the following equation:

\[\frac{w_f}{w_0} = \frac{T_{req}}{w_0} \frac{1}{ 3.6 C_{sb}} R\]

T_req/w₀ is the thrust to weight ratio which is typically the inverse of the lift to drag ratio which can be between 4 and 12, on average it is about 8. Hence T_req/w₀=(1/8). C_sb is the specific battery capacity typically 21000 mAhVN^-1 for Lipo batterys. R is the range or just simply the maximum distance the aircraft can travel. For simplicity the equations for battery powered aircraft assumes that batteries discharge at a linear rate while in reality batteries will discharge at an exponential rate. This should not really make big difference to the initial mass estimate for a small remote controlled aircraft, a later article will get into more detail on batteries. The table below gives some characteristics for different types and sizes of batteries:

Battery Type	LiPo	LiPo	NiMH	NiMH	NiCd	NiCd
Voltage(V_req) /V	11.1	11.1	4.8	8.4	4.8	8.4
Capacity (C_b)/mAh	2200 (3s)	1200 (3s)	800	3000	1100	600
mass (m_b) /kg	0.120	0.060	0.040	0.450	0.080	0.100
Specific Capacity (C_sb)/mAhVN^-1	20744	22629	9785	5708	6728	5138

The masses in this table are rough estimates, the manufacturer will usually give a more accurate mass for their battery and you can calculate the specific battery capacity, C_sb using:

\[C_{sb} ={C_b V_{req} \over m_b g} \]

Where C_b is battery capacity, V_req is the required battery voltage, m_b is the battery mass and g is gravity normally 9.81ms^-1. The table also overviews how LiPo have made high performance electric UAVs possible, without them only an internal combustion engine could give the same performance. For endurance the equation would be like this:

\[\frac{w_f}{w_0} = \frac{P_{req}}{w_0} \frac{1}{ 3.6 C_{sb}} E\]

P_req/w₀ is the power to weight ratio which can be crudely worked out by mutiplying T_req/w₀ by the aircraft velocity. C_sb is the specific battery capacity as defined previously. E is the endurance or just simply the maximum time the aircraft can fly for.

A worked example demonstrates how to use the equations. To get an intial estimate of the aircraft mass you need to define a payload mass, range and possibly an endurance. So let our example use a surveying UAV which has to carry a camera weighting approximately 0.5kg a specific distance 28km, remain in the area for 30 mins to carry out a survey. We will assume a cruise speed of 15ms^-1 and use typical values for all other properties. See table below which defines all the properties along with the calculations:

The following table shows how to calculate the total aircraft mass.

Symbol	Property	Example Value	Units
w_UL	Useful load	0.1+0.5=0.6	kg
w_e/w₀	Aircraft empty mass to total mass ratio	0.62	None
C_sb	Specific battery capacity	21000	mAhVN^-1
T_req/w₀	Aircraft thrust to weight ratio	1/8=0.125	None
V	Aircraft cruise speed	15	ms^-1
P_req/w₀	Aircraft power to weight ratio	(1/8)x15=1.875	ms^-1
R	Aircraft range	28x1000=28000	m
E	Aircraft endurance	30x60=1800	s
w_f/w₀	fuel to weight ratio for range	(0.125x28x1000)/(3.6x21000)=0.046	None
w_f/w₀	fuel to weight ratio for endurance	(1.875x30x60)/(3.6x21000)=0.044	None
w_f/w₀	Total fuel to weight ratio	0.046+0.044=0.09	None
w₀	Total aircraft weight	0.6/(1-0.62-0.09)=2.07	kg

The equation for fuel mass ratio needed for range of a propeller driven aircraft with an expendable fuel is:

\[ \frac{w_f}{w_0} =1-\frac{1}{exp \left( R \frac{T_{req}}{w_0} \frac{c_{fuel}} {\eta} \right) } \]

Where c_fuel is specific fuel consumption and η is propulsion efficiency assumed to be 1 for initial mass estimate.

exp(a)=e^a where e=2.718281828.... which is a mathematical constant number associated with natural occuring systems. The e and exp refers to exponential. This may seem a bit random, most people will wander where does e come from. I will explain it in more detail in a later article. Using an example if a=3 then:

exp(3)=e³=e×e×e

=2.71828×2.71828×2.71828=20.1

The equation for fuel mass ratio needed for endurance for a propeller driven aircraft with an expendable fuel is:

\[ \frac{w_f}{w_0} =1-\frac{1}{exp \left( E \frac{P_{req}}{w_0} \frac{c_{fuel}} {\eta} \right)} \]

Please leave a comment on my facebook page or via email and let me know how helpful these the equations are at estimating initial aircraft mass. You could also have a go and see how good these equations are at estimating the total mass of existing aircraft using data from an aircraft directory publication such as Ref [3]. Aircraft mass estimation is a trial and error type of task and the estimate will get more accurate as you decide on more details of your aircraft design. Ref [1] and [2] get into more detail on ways of estimating aircraft mass using similar equations as in this article.

References:

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

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

[3] The Encyclopedia of World Aircraft, David Donald, 1997, Bookmart Ltd

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