Debunking Apollo 11 based on the fuel usage

Question

I have checked the NASA docs and it says they needed hundred thousand liters of liquid hydrogen for the various stages of the Apollo 11 mission. For example stage 2 was calculated to need at least 984000 litres of liquid hydrogen as fuel and hundred thousand litres of liquid oxygen as oxidizer. Now in Physics, matter is anything that occupies space and has mass or weight. This stage takes place after the space craft is in the earth's orbit and needs to burn to dash towards the moon. Now the question here is the volume that 984 000 liters of liquid hydrogen would need to occupy, when you convert you get 984 cubic meters. But that is way bigger than the actual size of the Apollo 11 spacecraft which had a documented volume of 172 cubic metres. Why does the volume needed to store the fuel for stage 2 exceeds the size of the whole space craft?

Answer 1

The Saturn V was a big rocket that carried people to the moon. There is lots of evidence that it happened, but if you are still sceptical we can go through the numbers.

The Saturn V first stage is called S-IC. The S-IC has 5 F-1 engines. According to Wikipedia, it has a height of about 42 meters and a diameter of about 10 meters. To have a better overview of the data, I made this chart:

S-IC	Numbers
Engines	5x F-1
Gross mass	5,030,000 lb (2,280,000 kg)
Empty mass	290,000 lb (130,000 kg)
Height	42 meters
Diameter	10 meters
Maximum Thrust	34,500 kN
ISP sea level	263
ISP vacuum	304
Burntime	150s
Propellant	RP-1/LOX

source: https://en.wikipedia.org/wiki/S-IC

You might be thinking, well how do we know that these numbers are right. Well we can calculate certain things and compare them to actual events. So the first thing we would want to calculate all numbers related the to F-1 engines to see if those match up. Here is some data from the F-1 engines that we need to know

F-1 engines	Numbers
ISP Sea level	263s
ISP Vacuum	304s
Mass flow (LOX)	3,945 lb/s (1,789 kg/s)
Mass flow (RP-1)	1,738 lb/s (788kg/s)
Thrust Sea level	1,522,000 lbf (6,770 kN)
Thrust Vacuum	1,746,000 lbf (7,770 kN)
Height	5.6 m

source https://en.wikipedia.org/wiki/Rocketdyne_F-1

So the first thing we can now calculate is the ISP of the F1 engines since we have all the data for it.

$$Isp = \frac{force}{Mass\ flow \ rate \cdot gravitational\ acceleration}$$

The gravitational acceleration on the surface of Earth is 9,81 m/s² and it does not really change that much in LEO (low Earth orbit) either.

The Mass flow rate is the total amount of fuel being brought into the rocket engine per second. In other words, the amount of fuel used each second.

$$Mass\ flow\ rate = Mass\ flow\ rate\ (LOX) + Mass\ flow\ rate(RP-1)$$

Now lets add some numbers. The Mass flow rate (LOX) is 1,789 kg/s and the Mass flow rate RP-1 is 788kg/s. FYI I will be using the metric system from now on since it is much easier to deal with.

$$Mass\ flow\ rate = 1,789kg/s + 788kg/s$$ $$Mass\ flow\ rate = 2577kg/s$$

Now having the mass flow rate we can calculate the ISP. Now remember the ISP is going to be slightly different since it changes when you are in a vacuum or in 1 bar air pressure. So I will calculate both, however just be aware the the number will be rounded and not exact using this formula.

So as showed earlier, we need to know the Mass flow rate, force and gravitational acceleration. The Force at Sea level of the F1 engine is 6,770 kN. The force at a vacuum is 7,770 kN.

$$Isp\ sea\ level = \frac{6,770 kN}{2577 \cdot 9,81m/s²} = 267,79s $$ $$Isp\ vacuum = \frac{7,770 kN}{2577 \cdot 9,81m/s²} = 307,35s $$

The numbers on Wikipedia said it was 263 at sea level and 304 in a vacuum, however how I mentioned earlier, these numbers I calculated here are just rounded so it is definitely within range.

The Saturn V weighed 2,822,000 kg and out of those 2,822,000kg, the first stage was 2,280,000 kg. Since only 130,000 kg were dry mass of the first stage, it had 2,150,000 kg of fuel.

To calculate the delta V of the first stage, we need to know the Isp, gravitational acceleration, mass before lift off and mass after it used all its fuel.

$$\Delta v = Isp\cdot\ g \cdot\ln\left(\frac{m_0}{m_1}\right)$$

Parts of the equation	values
ISP	263s
gravitational acceleration (g)	9,81m/s²
m0	2,822,000kg
m1	2,822,000kg - 2,150,000kg = 672,000kg

$$\Delta v = 263s \cdot\ 9.81m/s\cdot\ln\left(\frac{2,822,000kg}{672,000kg}\right) = 3702,195 m/s $$

If you think that the Saturn V first stage tank couldn't hold 2,150,000 kg of propellant, then we can also go through those numbers.

The first stage (S-IC) was 42 meters high and out of those 42 meters, the F1 engines were 5.6 meters. That means that no more than 36,4 meters of the height were part of the tanks. Then there is also the intertank structure and forward skirt. Overall, lets round the height of the fuel tanks to about 30 meters

With a diameter of 10 meters and a height of about 30 meters, we can easily calculate the volume the fuel required.

$$volume= r²\cdot\ pi\cdot\ height$$ $$volume= 5²\cdot\ 3.1415 \cdot\ 30 = 2356 m³$$

In total there was about 2,150,000 kg of fuel. That means: $$density= 2,150,000kg\ per\ 2356m³$$ $$density= 912.38kg\ per\ 1m³$$ $$density= 0,91kg\ per\ 1L$$

That means that every dm³ also the same size of 1L has about 0,91kg. That is really no problem. Water has about 1kg per Liter.

Now moving on to the second stage of the Saturn V (S-II), it had 5x J-2 rocket engines. To make things simpler, I will make a chart with all the import numbers.

S-II	Numbers
Engines	5x J-2
Gross mass	480,000 kg (1,058,000 lb)
Empty mass	36,200 kg (79,700 lb)
Height	24,9 meters
Diameter	10 meters
Maximum Thrust	4,400 kN
ISP vacuum	421
Burntime	367s
Propellant	LH2 / LOX

source: https://en.wikipedia.org/wiki/S-II

This time I did not mention the Sea level Isp because this stage was only used in space and therefor it is unnecessary to know the Isp of it at Sea level.

Here are some important facts about the J-2 engines it used

J-2 engines	Numbers
ISP Sea level	200s
ISP Vacuum	421s
Mass flow (LOX)	203.663kg/s
Mass flow (LH2)	37.058kg/s
Thrust Sea level	1,522,000 lbf (486.2 kN)
Thrust Vacuum	1,746,000 lbf (1,033.1 kN)
Height	3.4 m

source: https://www.nasa.gov/centers/marshall/pdf/499245main_J2_Engine_fs.pdf

Just like with the F-1 engines, we can now calculate the mass flow rate, Isp and delta V of this stage.

$$ Mass\ flow\ rate\ = 203.663kg/s + 37.058kg/s = 240.721kg/s$$

$$Isp\ vacuum = \frac{1,033.1 kN}{240.721kg/s \cdot 9,81m/s²} = 437s $$ $$Isp\ sea\ level = \frac{486.2 kN}{240.721kg/s \cdot 9,81m/s²} = 205s $$

Even though the Isp is not the same as mentioned in the chart at the top, I already explained earlier that these calculations are only fully accurate in the perfect condition and in reality other factors come into play making the actual Isp slightly different.

To calculate the delta V like we did with the first stage, we need to know m0 and m1 (The mass before and after it uses its fuel). Without the first stage, there was only 672,000kg of mass left. That number would be the m0 here. Then we need to know how much fuel the second stage has.

Total weight of the second stage(Gross Mass): 480,000kg Weight of the S-II without fuel: 36,200kg

$$ fuel\ mass = 480,000kg - 36,200kg = 443,800kg$$ $$ m_1 = 672,000kg - 443,800kg = 228,200kg$$

Now since we have m1 and m0, we can calculate the delta V of the second stage.

$$\Delta v = Isp\cdot\ g \cdot\ln\left(\frac{m_0}{m_1}\right)$$ $$\Delta v = 421s\cdot\ 9.81m/s² \cdot\ln\left(\frac{672,000kg}{228,200kg}\right) = 4460.56 m/s$$

You mentioned in your question that you were unsure if the S-II could hold so much fuel.

Why does the volume needed to store the fuel for stage 2 exceeds the size of the whole space craft?

It actually does not exceed the size of the whole spacecraft. On the S-II there was 443,800kg of fuel. It has a volume of about 980 m³. Now lets go through the math:

$$density= 443,800kg\ per\ 980m³$$ $$density= 452.04kg\ per\ 1m³$$ $$density= 0.4528kg\ per\ 1L$$

At about 0.4 kg/L it is even less than water has. That means there was plenty of space in the S-II. It was not even close to exceeding the size of the Saturn V.

Then there is also a third stage:

It has 1x J-2 rocket engine. We already went through the calculations of the Isp of the J-2 engine, so we can skip ahead and directly go to the delta V calculation.

S-IVB has according to Wikipedia a Gross mass of 271,000 pounds (123,000 kg) and a propellant mass of 241,300 lb (109,000 kg). This source: https://history.nasa.gov/afj/ap12fj/pdf/a12_sa507-flightmanual.pdf says it had about 160,000kg when it started its burn.

$$m_1= 160,000kg - 109,000kg = 51,000 kg$$

$$\Delta v = 421s\cdot\ 9.81m/s² \cdot\ln\left(\frac{160,000kg}{51,000kg}\right) = 4722m/s$$

Now knowing the delta V, we can calculate the trajectory to see how much delta V is required. To do so we need this formula:

$$\Delta v\ required = squareroot (\ 2 \cdot\ Gravitional\ Constant \cdot\ Earth's\ Mass\cdot\ (\left(\frac{1}{radius\ of \ perigee}\right)\ -\ \left(\frac{1}{distance \ from \ perigee \ to\ Apogee}\right)\ )\ )\ $$

This might seem like a long confusing formula, but this video is a great tutorial that explains how it works.

G would be 6,673 * 10'-11 M would be 5,972 * 10'24 kg radius of Earth is about 6,300 km

periapis would be around 114 km (Apollo 11) lets round the Apoapsis to about 380,000km

Putting those numbers in the formula:

$$\Delta v\ required = squareroot (\ 2 \cdot\ 6,673 * 10'-11 \cdot\ 5,972 * 10'24 kg \cdot\ (\left(\frac{1}{114km + 6300km}\right)\ -\ \left(\frac{1}{114km + 6300km + 300,000km}\right)\ )\ )\ = 11,030.04558m/s =~ 11km/s$$

So, we would need about 11km/s to get to the moon. Now lets add up the delta V from each stage:

$$\Delta v= 3702,195m/s + 4460.56m/s + 4722m/s = 12884,755 m/s$$

This is more than the 11 km/s it needed and it is safe to say that the Saturn V had enough fuel to get to the moon and that the fuel all did fit in its fuel tanks.

Answer 2

You are comparing apples and oranges, and using bad data to do it.

This is the second stage:

It is 30 meters long and has a diameter of 10 meters, for an internal volume of 2356 m^3.

The "Apollo 11 spacecraft" consists of the Command Module and Service Module. The Service Modules has an internal volume of about 56 m^3, not 172. The Service Module carried the fuel for lunar orbit injection and lunar orbit escape.

The second stage was used to put the stack into Earth orbit. The third stage (S-IVB) finished that job, and it performed the translunar injection burn.