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[SAS~Storebror]

Dear Friends,

I'm in need of your assistance, if you'd be so kind. I'm currently casting my gaze upon the flight model of the Me 163 "Comet". In this endeavour, I've delved into numerous books to gain a more precise understanding of the rocket engine's burn duration and to better comprehend its exact operational profile.

The latter is more or less identically described in all the documents, so for simplicity's sake, I'll quote Wikipedia, where the launch and subsequent climb are described as follows:

At a speed of over 320 km/h (200 mph) the aircraft would take off, in a so-called "scharfer Start" ("sharp start", with "Start" being the German word for "take-off") from the ground, from its two-wheeled dolly. The aircraft would be kept at level flight at low altitude until the best climbing speed of around 676 km/h (420 mph) was reached, at which point it would jettison the dolly, retract its extendable skid using a knob-topped release lever just forward of the throttle⁴⁸ (as both levers were located atop the cockpit's portside 120-litre T-Stoff oxidizer tank) that engaged the aforementioned pneumatic cylinder,³¹ and then pull up into a 70° angle of climb, to a bomber's altitude. It could go higher if required, reaching 12,000 m (39,000 ft) in an unheard-of three minutes.

This raises some questions for me. The Me 163, when fully fuelled and armed, has a take-off weight of 4309 kg. The fuel weighs a total of 2018 kg and according to the majority of sources, the burn duration is 7 1/2 minutes, or 450 seconds. This means that after 450 seconds, the Me 163 still weighs 2291 kg, or in other words, it loses about 4 1/2 kg of weight per second at full thrust. The engine has a thrust of 14.71 kN. At a climb angle of 70°, the (decreasing over time) weight force of the Me 163 counteracts this with the factor sin(70°). We thus obtain an accelerated motion with a starting speed of 676 km/h, a climb angle of 70°, and a linearly decreasing mass by about 4 1/2 kg with a starting weight of 4309 kg.

I'll spare you the detailed calculations (function for acceleration, speed (integral of acceleration), and distance (integral of speed)). The result is that at a climb angle of 70°, the speed reaches 0 after 32 1/2 seconds, and the Me 163 has climbed to just under 2900m during this time. And that's without even considering air resistance, which would actually further reduce the climb performance!

How on earth do all the documents on the Me 163 come up with the idea that the climb could be carried out at a constant 70°?

In order not to lose speed during the climb - again without considering air resistance, which would also significantly worsen the values here - the Me 163 would in reality only be allowed to climb at an angle of 20° at the beginning and 40° at the end of the burn duration (thanks to the then lower weight). An optimal climb to combat altitude would thus, taking into account air resistance, start at about 15° climb angle and slowly increase to about 30°.

This is because the engine has a thrust of 14.71 kN, which corresponds to a thrust force of 1500 kg. This 1500 kg must not be exceeded by the current mass of the Me 163 during the climb at the respective angle. This doesn't even work out with the empty weight of the Me 163 at a climb angle of 70°...

Question to the physicists among you: Have I overlooked something or have the documents on the Me 163 simply all copied the same nonsense from each other?


Mike

[Kopfdorfer]

Hi Storebror ,
Sorry no technical expertise to be expected of me , however one detail in the insert you used as the basis of your
model caught my eye as odd.
It struck me as strange that while accelerating to climb velocity the aircraft would keep attached the take-off dolly
and only jettison it AFTER achieving the climb velocity. For an aircraft where fuel was so precious it would seem
to me to be more sensible to jettison the dolly once safe operating speed was achieved , then jettison the dolly
for the acceleration to climb speed.
This would at least save some fuel for the rest of the operation.
There are technically approved Pilot's Guide Procedural manuals for most Allied aircraft that are available for download .
I presume there are the same sort of thing for LW aircraft . Is this so ? Are you using one for reference ?
I just had a quick look online (anglophile search , though ).
The book that comes up is the following :
                                           The Me 163 Komet: A Detailed Guide to the Luftwaffe's Rocket-Powered Interceptor
                                           (Airframe Album #10). Franks, Richard. Published by Valiant Wings
Apologies if this is way back in your search process and you are well past this point.

Kopfdorfer

[Seb]

Hi Mike
Maybe this film will add something to the discussion.
Regards
Seb

[video][/video]

[Matz]

Hi Mike,
This intrigued me and I make no claim to technical expertise.

I have the words of of one of, if not the most experienced test pilots ever - Eric "Winkle" Brown (Wings On My Sleeve).

He tested a captured Me163B in the hectic days soon after the German surrender. I have summarised his record of the flight details as
 - take off at 280kmh, (u/c jettison speed not noted), then accelerating to 725kmh to then climb at 45 degrees to 32,000ft in 2.5 minutes.

He then still had fuel left as he thottled back "to prevent compressability issues". 

Given that he planned his flights based on the actual Me163B Pilot Notes, maybe this is closer to what the aircraft was actually operated at rather than the "possible" climb angle of 70 degress?

[Ol Willy]

The first version of the rocket engine - HWK 109-509A-0 - had 14.7 kN of max thrust, "serial" Me.163B-1a had HWK 109-509A-1 with 15.7 kN.

Wikipedia says that B-1A had HWK 109-509A-2 but this looks wrong. A-2 had 16.68 kN of max thrust and was notable for having two chambers, climb and cruise ones for fuel economy. It was tested on V6 and VI8

[SAS~Storebror]

A little math before it gets lost...

Be:
F = The (static) thrust of our rocket engine (14.71kN or 15.7kN, whichever)
m_t = The takeoff mass, fully armed and fuel tanks filled (4309kg)
m_f = The mass of our fuel (2018kg)
t_f = The rocket engine burn time, i.e. time to consume all fuel (450s)
\( \alpha \) = The climb angle
v₀ = The speed at the time when the plane starts climbing, i.e. after accelerating to climb speed right above ground

And leaving all aerodynamical effects like drag, lift etc. behind, then...

The current mass m(t) at any given time t < t_f is:
\( m\left(t\right)=m_t-\frac{t}{t_f}\cdot m_f \)

The acceleration a(t) at any given time t < t_f is:
\( a\left(t\right)=\dfrac{F}{m\left(t\right)}-g\cdot\sin\left(\alpha\right)=\dfrac{F}{m_t-\frac{t}{t_f}\cdot m_f}-g\cdot\sin\left(\alpha\right) \)

The Indefinite integral of the acceleration a(t) which is used to calculate the speed (see below) is:
\( \int a\left(t\right)\,dt=C -\dfrac{F\cdot t_f\cdot\ln\left(\left|t\cdot m_f-m_t\cdot t_f\right|\right)}{m_f}-g\cdot\sin\left(\alpha\right)\cdot t \)
(C is just a constant of any scale, as this integral is indefinite)

The speed v(t) at any given time t < t_f is:
\( v\left(t\right)=v_0 + \int_0^t a\left(t\right)\,dt=v_0+\dfrac{F\cdot t_f}{m_f}\cdot\left(\ln\left(\left|m_t\cdot t_f\right|\right)-\ln\left(\left|t\cdot m_f-m_t\cdot t_f\right|\right)\right)-g\cdot\sin\left(\alpha\right)\cdot t \)

As I said: Just some math, only to make sure it doesn't get lost.


Mike

[Dimlee]

What catches my eye in Wiki article:
"The aircraft would be kept at level flight at low altitude until the best climbing speed of around 676 km/h (420 mph) was reached, at which point it would jettison the dolly, retract its extendable skid..."
Jettisoning the dolly at 676 km/h would probably cause the dolly to be destroyed after crashing on the ground. But it was supposed to be reusable. I also wonder what could happen with an extendable skid at this speed.

Two more thoughts.
70-degree angle claim was probably an error based on anecdotical and incomplete evidence or even on the parts of documentaries showing the test flights (I do remember some films where Me 163 climbed very steep, probably up to 70, immediately after the take-off).
I assume that the pilot would not keep the maximum thrust during all ascend but regulate the thrust, saving the fuel and prolonging the flight.

It was a pity I have donated most of my military history books. There was a memoir of the Me 163 pilot, Wolfgang Spate, probably.

[Dimlee]

Hi Mike
Maybe this film will add something to the discussion.
Regards
Seb

[video][/video]

Good find, Seb.
The take-off procedure is described from about 4:10. Speeds of the dolly release and the "best climbing speed" are mentioned (and they are different, of course). Probably, that was a source of the Wiki's article error.

Some more videos.
Hanna Reich, Heini Dittmar. Take-offs, dollies separations, very steep climbs (but for how long?)


"Famous" Greg about Me 163. At 19:50 there is the table with "best angle of climb ... 40-45". It makes sense to me.

[SAS~Storebror]

One more math, sorry for not getting back to your feedback yet gents, it's much appreciated and I will get back to it later, but for now I just want to save some things before they get lost.
Here's the height formula:

The height h(t) at any given time t < tf is:
\( h\left(t\right)=\sin\alpha\left(v_0\cdot t + \frac{{t_f}^{2}\cdot m_t\cdot F}{{m_f}^{2}}\cdot\left(\ln\left(|t\cdot m_f-t_f\cdot m_t|\right)-\ln\left(t_f\cdot m_t\right)\right)+\frac{t\cdot t_f\cdot F}{m_f}\cdot\left(1+\ln\left(t_f\cdot m_t\right)-\ln\left(t\cdot m_f -t_f\cdot m_t\right)\right)-\frac{t^2}{2}\cdot g\cdot\sin\alpha\right) \)


Mike

[Dimlee]

sin, ln... warm nostalgia. Where is my plywood computer... 

[SAS~Storebror]

For those who want to play around with different values, here's a Google Spreadsheet illustrating the math shown above.



https://docs.google.com/spreadsheets/d/1URo8UpgB6kgQ4MLuO5UkbcVvpdwnvs2IL-fa_Jk8dRc/edit?usp=sharing


Mike

[Dimlee]

Thank you, Mike!
A cool graph is worth a thousand formulas, isn't it. At least for those who haven't worked with complex formulas since Win 3.1. 
According to this sheet and with an initial speed of 676 km/h, if controls remain effective until 129 km/h (Wiki says), we can climb at an angle of 70 degrees until about 2,900 m and feel fine. Climbing at a more "intuitive" angle of 45 brings us to about 3,700 m before we need to level, gain speed and climb again. It looks like a typical pattern in the game.
And 25 degrees of climb is practical when the opposition is near. We reach the altitude of B-17s in less than 2 min, all the time keeping the safe speed. Unless some higher-flying P-47 dives on us.
Really cool graph, thumbs up.