Aircraft Performancedraft

Range and endurance have been mentioned in the preceding sections - and we have introduced that for maximum range the pilot should fly at $V_{md}$ , whilst to fly for maximum endurance the pilot should fly at $V_{mp}$ .

However - the above speeds are valid only for unpowered flight. So these speeds are suitable for a glider, but once an engine is introduced to an aircraft, these speeds no longer give the maximum range or the maximum endurance.

Furthermore, the propulsor type changes the speeds for both. In the following section, the best range and endurance speeds for jet and propeller-driven aircraft will be explored.

Range and Endurance

Range and endurance are intuitive concepts:

Range ( $R$ ): The maximum horizontal distance that an aircraft can cover.
Endurance ( $E$ ): The time an aircraft can remain in flight

In the following analysis, the range and endurance for certain fuel quantities will be determined. For (hopefully) obvious reasons, some further definitions of range will help us here

Range subdefinitions

Safe Range: The maximum distance between two airfields, for which an aircraft can fly a safe a reliable mission with a given payload.
- This is an involved calculation involving take-off/landing/weather/diversion allowances etc. - there isn’t an easy means to do this calculation, so tends to be performed computationally.
- For aircraft performance there are some more simple definitions of range:
Still Air Range (SAR): The maximum distance possible if an aircraft takes off, climbs to cruise altitude, and then cruises until all fuel is expended.
- Obviously not desirable to run out of fuel at altitude, but SAR gives a good indication of the influence of aircraft parameters on range.
Gross Still Air Range (GSAR): The maximum distance possible if an aircraft commences cruise at altitude and continues until all fuel is expended.
- The relationship between SAR and GSAR tends to be easy to define.
- GSAR much easier to calculate
- What will be covered here

Defining the problem

To calculate GSAR, you might think that we need to:

find out the fuel flow rate, $\dot{m}_{fuel}$ for thrust/power associated with $V_{md}$
get the endurance from the fuel mass, $m_{fuel}$ divided by the flow rate $E=\frac{m_{fuel}}{\dot{m}_{fuel}}$
hence $R=E\cdot V$

The above reasoning is effectively how range is determined, but there are some complications that you may/may not be considering:

Aircraft take-off with a lot of fuel, so the aircraft weight changes with time and accordingly so do almost all the parameters we have considered up to this point, including the aircraft speed whilst the above relationship only holds true for constant airspeed
This means that some trade-offs have to be made in efficiency to allow a reasonable cruise - and we will see these

Breguet Range Equation

The Breguet Range Equation (BRE) is named after a French aircraft designer, but was actually derived in the 1920’s by J G Coffin.

History and namesakes...

I’ve loosely read the history of the BRE and it being Coffin who actually came up with it in NACA Report 1969, but I’ve never actually delved into the legacy of this equation. If you wish to, and suggest a correct here, feel free.

The BRE allows a simple means to calculate GSAR, and can be defined in words as:

\substack{\text{"The rate of}\\\text{aircraft weight reduction"}}=\substack{\text{"The rate of}\\\text{fuel weight burned"}}

The BRE was first implemented for propeller aircraft, and is derived differently for thrust and power engines. In the following it will be derived separately for the two engines types.

Engine fuel burn

To calculate the engine fuel burn of both types of engine, two new parameters are introduced

A parameter, $c_t$ , is introduced which is:

Thrust Specific Fuel Consumption (TSFC) for a turbojet $c_t$ - mass of fuel burned per unit of thrust per second
Specific Fuel Consumption (SFC) for a turboprop $c$ - mass of fuel burned per unit of power per second

Thrust specific fuel consumption - units

In the above, the SI units are $\left\{c_t\right\}=\left\{\frac{kg}{N\cdot s}\right\}$ which has dimensions of $\left[\frac{\text{L}}{\text{T}}\right]$ . You might see this expressed as $\frac{g}{kN\,s}$ , which is actually the same units.

In US customary units, this is $\left\{c_t\right\}=\left\{\frac{lb}{lbf\cdot s}\right\}$ .

You may see some slightly different units such as $\left[\frac{kg}{kN\cdot hr}\right]$ so be sure to convert to SI or US customary base units

Specific fuel consumption - units

In the above, the SI units are $\left\{c\right\}=\left\{\frac{kg}{W\cdot s}\right\}$ which has dimensions of $\left[\frac{\text{T}^2}{\text{L}^2}\right]$ . You might see this expressed as $\frac{g}{kW\,s}$ , which is actually the same units.

In US customary units, this is $\left\{c_t\right\}=\left\{\frac{lb}{hp\cdot s}\right\}$ .

You may see some slightly different units such as $\left[\frac{kg}{kW\cdot hr}\right]$ so be sure to convert to SI or US customary base units

The analysis is slightly different for jet and propeller-driven aircraft, so jet aircraft will be explored first.

BRE - Jet Aircraft

The fundamental concept is, again:

\substack{\text{"The rate of}\\\text{aircraft weight reduction"}}=\substack{\text{"The rate of}\\\text{fuel weight burned"}}

So for a jet aircraft this is

\frac{\text{d}W}{\text{d}t}=-c_t\,T\, g

which can be rearranged for the time

\text{d}t = -\frac{\text{d}W}{c_t\,T\, g}

since the endurance/range is defined by cruise conditions, the equilibrium steady flight conditions of $T=D$ and $L=W$ can be utilised such that

T=\frac{D}{L}W=\frac{C_D}{C_L}W

which can be substituted into the BRE to give

\text{d}t = -\frac{1}{c_t\, g}\frac{C_L}{C_D}\frac{\text{d}W}{W}

for constant lift-to-drag ratio and TSFC, the equation above can be integrated with the limits $t_{0}$ and $t_{1}$ corresponding to $W_{0}$ and $W_{1}$ where 0 denotes the start of cruise, and 1 denotes the end.

E=t_{1}-t_{0}=\frac{1}{c_t\, g}\frac{C_L}{C_D}\ln\left|\frac{W_{0}}{W_{1}}\right|\tag{1}

Jet Aircraft: Maximum Endurance

For a given $c_t$ , $W_0$ , and $W_{1}$ , Equation (1) shows that the best endurance for a jet aircraft is found at the minimum drag speed. If you’re unsure why it shows this - look at the equation and consider what can be maximised.

To find the range, we’ll take a step back to the Equation (1) and substitute the aircraft speed equation.

Jet Aircraft: Range

Taking Equation (1) and multiplying Equation (1) by it yields the incremental distance, $dS$ , covered during cruise

\begin{gather} \text{d}S &= V\,\text{d}t=-\frac{V}{c_t\,g}\frac{C_L}{C_D}\frac{1}{W}\text{d}W\\ &= -\frac{1}{c_t\,g}\sqrt{\frac{2\,W}{\rho S C_L}}\frac{C_L}{C_D}\frac{1}{W}\text{d}W\\ &= -\frac{1}{c_t\,g}\sqrt{\frac{2\,W}{\rho S}}\frac{C_L^{1/2}}{C_D}\frac{1}{W}\text{d}W \end{gather}

Don’t be tempted to combine the two $W$ terms at this point - those will be dealt with in a little while, first some observations can be made about the equation above:

Interactive demo

As with all things in aeronautics (well, with science and mathematics), if you derive a relationship, you should check that what it says makes sense. The equation above says:

Range is inversely proportional to fuel burn, which makes sense
Range is inversely proportonal to weight, which makes sense (combine the two into $W^{-1/2}$ )
Range is proportional to altitude (we know aircraft cruise at altitude)

We see that the best range is given at the aerodynamic condition (that is, the velocity) corresponding to the maximum value of $\frac{C_L^{1/2}}{C_D}$ which is given by:

C_L=\sqrt{\frac{C_{D0}}{3\,K}}

For equilibrium, lift must equal weight, so for the best $C_L$ , this occurs at a single airspeed (Eq. (1)
Fuel is burned, so the aircraft gets lighter, so looking at the definition of the lift coefficient:

C_L=\frac{2\,W}{\rho\,S\,V^2}

To maintain the best $C_L$ , either the velocity has to reduce or the density has to reduce.

This yields two types of cruise:

Constant Velocity aka Cruise-Climb
Constant Altitude (where the aircraft slows down)

Constant Velocity Cruise

Looking at the definition of the lift coefficient, if velocity is constant, then in order to maintain a constant $C_L=\sqrt{\frac{C_{D0}}{3\,K}}$ , the ratio $\frac{W}{\rho}$ must be constant. Accordingly, whilst the terms $W$ and $\rho$ both change during the cruise and could be included in the integration, the ratio of them is a constant, and can be removed from the integration. If the aircraft starts at an altitude $h_0$ with a corresponding density $\rho_0$ , then

\frac{W_0}{\rho_0}=\frac{W_1}{\rho_1}=\frac{W_n}{\rho_n}

where $n$ is any intermediate value between the start and end of the cruise. Hence $\frac{W}{\rho}$ can be replaced with $\frac{W_0}{\rho_0}$ . That is

R = \int^{R}_0\text{d}S = -\frac{1}{c_t\,g}\sqrt{\frac{2\,W_0}{\rho_0 S}}\frac{C_L^{1/2}}{C_D}\int_{W0}^{W1}\frac{1}{W}\text{d}W

Similarly, the variation of velocity can be yielded from the aircraft weight to velocity ratio.

Jet BRE: Numerical Example

About units

You (that’s you, IIT student) should be able to do these calculations both in SI units and US customary units. Since I got my degree and PhD outside of America, I have an appreciation (to use the term loosely) for US customary units, but my actual application has always been to convert to SI at the start, and then convert the answer back to US customary units if I need to provide one.

For this reason, I work in SI in written examples for class but I will provide examples with US customary units used throughout. I simply don’t want to make a mistake when going through work in class, and end up a factor of 32 out, or have mixed up lb for lbf or whatever else I could have done.

I managed to find this whilst googling about the US and the metric system, if you want some further reading. https://www.nist.gov/system/files/documents/pml/wmd/metric/1136a.pdf

This is an adaptation of Example 5.19 in Anderson[Anderson:1999AP] - I claim no originality or authorship for the data provided, but I’ve used it to confirm my US customary calculation is correct before adapting.

Variation of jet range with lift coefficient, airspeed

Recall that the lift coefficient is effectively a measure of the aircraft cruise speed. The range can be plotted vs. lift coefficient and forward speed for the two different jet cruise cases over a range of starting altitudes.

You can click on the entries in the legend to hide/show different plots.

Beware of source for the plots below...

Producing the plots below is fairly simple - but in order to get the labels and legend to work correctly, there’s a bit of obscure logic flow in the way the plot is created.

That is, it makes it look more complicated than it actually is (and it probably could be done better if I knew my way around plotly better).

Notice that the maximum range is found at a considerably higher speed than both the minimum power and minimum drag speeds, for both types of cruise climb. The ratio between the best range speed and the minimum drag speed may be readily shown.

\frac{V_{mr}}{V_{md}} = \frac{\sqrt{\sqrt{3}}}{1} = 1.3161

These ratios hold across altitudes, as you should expect.

BRE - Propeller Aircraft

The BRE for propeller aircraft is similar to that for jet aircraft, but SFC is used in place of TSFC - the SI units are $\text{kg}/{\text W s}$ .

The BRE for propeller aircraft is

\frac{\text{d}W}{\text{d}t}=-c\,g\,P

where $P$ is the power delivered to the aircraft from the propeller. With $\eta$ as the propeller efficiency, the power delivered is function of the power required

\eta P=D\,V

so the BRE becomes

\frac{\text{d}W}{\text{d}t}=-c\,g\,\frac{D\,V}{\eta}

\implies \text{d}t=-\frac{\eta}{c\,g\,V}\frac{C_L}{C_D}\frac{\text{d}W}{W}

Assuming, as for the jet cruise-climb case, that $\tfrac{C_L}{C_D}$ , $f$ , and $V$ remain constant, the equation above can be integrated from $W_S$ to $W_E$ to yield the endurance, $E$ :

E_{propeller}=t_e-t_s=\frac{\eta}{c\,g}\frac{1}{V}\frac{C_L}{C_D}\ln\left|\frac{W_S}{W_E}\right|

Propeller Aircraft: Maximum Endurance

The equation above shows that for the maximum endurance for a propeller-driven aircraft, the quanity $\frac{C_L}{V\,C_D}$ must be maximised, which is different to the jet aircraft case.

\frac{C_L}{V\,C_D}=\frac{L}{V\,D}=\frac{W}{V\,D}=\frac{W}{P}

Clearly the maximum endurance is found at the minimum power condition, thus for maximum endurance a propeller-driven aircraft should fly at $V_{mp}$ .

Propeller Aircraft: Maximum Range

The increment in aircraft distance, $\text{d}S$ when flown at velocity $V$ is given by

\text{d}S = V\text{d}t=-\frac{\eta}{c\,g}\frac{C_L}{C_D}\frac{\text{d}W}{W}

Clearly the maximum endurance is found at the minimum drag condition, thus for maximum endurance a propeller-driven aircraft should fly at $V_{md}$ .

Hence the maximum range is given by the integration $R=\int\text{d}S$

R = \frac{\eta}{c\,g}\frac{C_L}{C_D}\ln\frac{W_0}{W_1}

Range and Endurance Summary

	Glider	Jet Aircraft	Propeller Aircraft
Maximum Endurance	At $V_{mp}$	At $V_{md}$	At $V_{mp}$
Maximum Range	At $V_{md}$	$\gt V_{md}$	At $V_{md}$

All of the cruise-climb scenarios are theoretical behaviour, and are reliant on the assumptions made within these models - furthermore, cruise-climb is typically not allowed by ATC. Rather, a series of stepped-climbs are made.

Nonetheless, the methods shown here allow a good estimate of range to be made - and afford the ability to look at the effect of design parameters on range and endurance.