How much energy is required to send a payload of 40 t from Earth's surface to LEO?

Question

I am confused with energy requirement equations...

Am I anywhere near with my following assumptions:

$\Delta v$ from surface to LEO is 9000 m/s typical exhaust velocity $v_e$ = 4500 m/s

and mass of payload $m_l = 40\mathrm{t}$ and $m_p$ = mass of propellant

$=> \Delta v = v_e \, \ln \frac{m_p+m_l}{m_l}$

$=> m_p = m_l (\mathrm{e}^{\frac{\Delta v}{v_e}}-1)$

$=> m_p = 40 000 (\mathrm{e}^\frac{9000}{4500}-1) = 256 \mathrm{t}$ of propellant

The energy with 100% efficiency would then be:

$E = 0.5 \, m_p \, v_e^2$

or (gets me the same result?)

$E = 0.5 \, m_l (\mathrm{e}^\frac{\Delta v}{v_e}-1) \, v_e^2$

$=> E = 2588 \mathrm{GJ}$

Am I in a right order of magnitude here...?

Answer 1

The 'sending' part of your question is what makes this a tangle. Because sending is a process, it isn't possible to speak in terms of perfect efficiency. You can't isolate the fuel it took to put the payload into orbit from the fuel it took to put the rocket carrying the payload into orbit.

It may be that what you seek is a theoretical minimum energy required to put 40 tons in orbit. In that case maybe it means something to say what the kinetic energy of 40 tons of material moving at orbital velocity is, because if there was a way to get it there with absolutely no energy losses, that is all the energy you would need. Orbital speed at 200km above the Earth, which is where low earth orbit is considered to begin, is 6.9 km/s.

0.5 x 40,000kg x 6900 (m/s) $^2$ = 952.2 GJ

Maybe if there was a space elevator to get above the atmosphere, and you had a way to make the mass of the cargo pod hauling the payload negligible compared to the payload itself, and you were able to minimize losses due to friction, and then the payload could be accelerated sideways to orbital speed with maybe a high-efficiency magnetically levitating mass driver, which had a cargo pod whose mass was also negligible, you could start to approach that amount. Exactly how close we could ever get to that ideal figure is highly debatable.

Thinking about engineering problems in terms of the theoretical ideal has real application. It is good to know exactly how much room there is for improvement sometimes. For instance, it has often been noted that building a rocket and then only using it once is awfully inefficient. But it doesn't really serve a purpose unless you can identify a place where the process can be improved, and design a method to make it so.

Failing that, the amount of energy it currently takes to get 40 tons of material to orbit, stated in terms of the energy in the fuel it takes to launch it, requires that we choose a rocket to do the job and then identify what fuel it uses, and how much. There is no rocket currently in service that can carry a payload of 40 tons. So let's imagine a Falcon 9 that is 3 times as big, since we are only going for a rough estimate anyhow and I was able to find enough data to go on. That data didn't include the oxidizer to fuel ratio the Merlin engine uses, but we are already pretty abstract here, let's use the 2.56 ratio referenced on Wikipedia for LOX/RP-1.

In that case, it would burn about 400,000 kg of RP-1 to get to LEO, which at a specific energy of roughly 46 MJ/kg is 18,400 GJ, or 19 times the theoretical minimum mentioned above.

Answer 2

Not sure, if I should "answer" or "comment", but let's put some more questions...

If I have a two stage launch and I still want to stick up with the $9000 \text {km/s} \Delta V$, should I then first calculate the fractions of propellant I need for the two stages?

$$M_f = 1 - (M_1/m_0) = 1 - e^{-\Delta V/V_e}$$

And If the first stage would be until $5000 \text {m/s}$ and second until $9000 \text {m/s}$ with $V_e = 4500 \text {m/s}$, I would get:

$$M_{f1} = 1 - e^{-5000/4500} = 67.1% \text {propellant}$$ $$M_{f2} = 1 - e^{-4000/4500} = 58.9% \text {propellant}$$

With the second one, I need to use the deltaV from $5000$ to $9000 \text {m/s}$ ($=4000 \text {m/s}$), right?

With assumption of $8%$ from the mass being construction, I end up with

First stage: - construction $8%$ - propellant $67.1%$ - Second stage $24.9%$

Second stage: - construction $8%$ ($2.0%$ from initial mass) - Propellant $58.9%$ ($14.7%$ from initial mass) - Final payload $33.1%$ $(8.3%$ initial mass)

So in TOTAL the shares are: - $8.25%$ the final payload - $81.76%$ propellant - $9.99%$ construction

As the payload needs to be $40 \text t$, I end up with:

First stage: - Construction $38.8 \text t$ - Propellant $325.2 \text t$ - Second stage $120.8 \text t$ TOTAL: $484.8 \text t$

• Second stage:
• Construction $9.7 \text t$
• Propellant $71.1 \text t$
• Final payload $40 \text t$ TOTAL $120.8 \text t$

So far so good (even though I am skipping many details as this is not really my field of science at all, I am trying to keep the focus of only getting the order of magnitude of the energy use...)

Now I am in trouble though... Which energy equation should I use:

$$E = 0.5 \times (m_0-m_1) \times Ve^2$$ ($m_0-m_1$ would be the mass of the propellant)

or

$$E = 0.5 \times m_1 \times (e^{\Delta V/V_e}-1) \times V_e^2$$

With the first equation I get $3293 \text {GJ}$ and $0.720 \text {GJ}$, so $4013 \text {GJ}$ in TOTAL.

With the second I got confused with m1. In first stage the $m_1$ is the mass of the second stage ($120.8 \text t$)? But with the second stage it must be the mass of the payload and the construction, right...? Or just the payload ($40 \text t$)? Or is this matter of definition, how loose boundaries do I want to set to these calculations? (the payload is ONLY the useful payload ($= 40 \text t$) or also the whole construction of the second stage (= useful payload + engines, empty tanks etc)

Thanks a lot, if there's still more people willing to help in this interdisciplinary research of mine...