# 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.

## 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).

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.