Climbing Flight

In climbing flight, \(T>D\), so the aircraft cannot maintain equilibrium in straight and level flight.

The aircraft ascends with climb angle \(\bar{\gamma}\), with the horizontal component of the aircraft weight opposing the thrust.

Fig. 23 Forces on aircraft in a climb

What’s actually happening in a climb?

Using the same reasoning as before, you can see that the aircraft is converting vertical kinetic energy into gravitational potential energy.

Climb Angle

As for the glide angle, the climb angle can be determined by resolving forces perpendicular to the flight path

\[L=W\cos\bar{\gamma}\]

and parallel to the flight path

\[T-D=W\sin\bar{\gamma}\]

and from trigonometry, the climb angle is simply

\[\sin\bar{\gamma}=\frac{T-D}{W}\]

the rate of climb is \(V_{climb}\) and is

\[V_{climb}=V\sin\bar{\gamma}=\frac{V\left(T-D\right)}{W}\]

which gives the rate of increase of GPE

\[\begin{split}\underbrace{W\cdot V_{climb}}_{\substack{\text{Rate of increase}\\\text{of potential energy}}} = \underbrace{T\,V}_{\substack{\text{Thrust}\\\text{Power}}} - \underbrace{D\,V}_{\substack{\text{Drag}\\\text{Power}}}\end{split}\]

Climb Performance

The maximum climb angle requires the maximum excess thrust

The maximum rate of climb requires the maximum excess power

This should feel intuitively correct to you, based upon what we know about glide angle/rate and \(D_{min}/P_{min}\) and - obviously - these do not occur at the same speed. These depend on the powerplant type, and individual engine characteristics.

For propeller and turbojet engines, there are some simplifications that can be made about the propulsor that allows easy determination of one of the parameters - excess thrust for a turbojet, and excess power for a turboprop.

Powerplant assumptions

For turbojet aircraft and low bypass ratio turbofan aircraft is is assumed that thrust remains constant with speed, and accordingly power increases with speed.

For turboprop aircraft and high bypass ratio turbofan aircraft is is assumed that power remains constant with speed, and accordingly thrust falls with speed.

Climb Curves - Turbojet

For a turbojet aircraft with a drag equation described by:

\[C_D = C_{D0} + K\cdot C_L^2\]

with \(C_{D0}\)=0.010, and \(K\)=0.030, a wing area of 75\(\text{m}^2\), a weight of 140.000kN, and a maximum lift coefficient of 1.750, capable of producing a thrust of 140.000kN at a given altitude, the climb rate and angles can be taken from the difference between the thrust available/required curves.

These plots are sensible to produce in EAS, for hopefully obvious reasons (if you’re unsure why, ask on Slack).

Try and reproduce the plot below (ON YOUR OWN)

The source for this plot isn’t included on the website (but it is included on GitHub). You should try and reproduce the plot using some computational method - Excel, MATLAB, Python using matplotlib, plotly, or whatever.

If you just download the source, you might trick yourself that you know what’s going on - but you’ll not actually learn anything.

It would be better to do some work to learn some coding and equation manipulation now, than lose your first graduate job when they find out that you bluffed your way through your undergraduate degree. That is, I don’t care whether you’ve done your own homework when it comes in - but I promise that cheating on any of my homeworks, graded or not, will not be worth the trade-off you’ll make for time now vs. being a shit engineer as a graduate.

Rant aside, I’ve also chosen not to make the source easily-available for some of these plots because I chose to use plotly to enable you guys to pan/zoom/interact with them online. I am far less confident with plotly than I am with matplotlib, so my code probably isn’t anything to learn from technique-wise.

You can hover over the plots and check the values to see if you get the same answers as the ones I’ve produced.

Do your best to reproduce these plots as they may help you with a future homework - discuss on Slack and help each other if in doubt how to complete.

Climb Curves - Turboprop

For the same aircraft, with a turboprop capable of producing a constant power of 2.3MW, the thrust and power curves can be shown similarly:

Climb Performance: Summary

The maximum climb rate is given by the maximum excess power. You can think of this as the exchange of thrust energy to GPE.

The maximum climb angle is given by the maximum excess thrust. This is where the least horizontal resistance is experienced.

Look at the table below. Confirm using the plots that this is correct.

	Propeller Aircraft	Jet Aircraft
Maximum Climb Rate	At \(V_{mp}\)	\(>V_{mp}\)
Maximum Climb Angle	\(<V_{md}\)	At \(V_{md}\)

Turboprops tend to have superior climb performance - but occurs at a lower speed.