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The space shuttle ascended using two types of propulsion: (1) liquid hydrogen and oxygen stored in the external tank and burned in the main engines on the orbiter, and (2) the solid rocket boosters.

As noted in this answer, LH2/LO2 has a lot of theoretical advantages: high specific impulse (thus a higher delta v), high thrust-to-weight ratio, and low mass of the exhaust.

Suppose we redesigned the shuttle stack to remove the SRBs and expand the external tank, so all propulsion was by LH2/LO2. This would certainly save weight. However, how much volume would the new external tank occupy?

If your answer depends on increasing the number of main engines, please specify your assumptions in your answer (*1). Also, we are only replacing the propulsion for ascent; the OMS and RCS are to remain unchanged.

Bonus: I suspect such a shuttle would no longer fit in the Vehicle Assembly Building. Compare your volume to that of the VAB or another structure/building familiar to an average person. Here are a few you could use:

Also related: How would the Saturn V have differed if the first stage was also LH2/LOX?


Clarification

I should have been clearer with sentence (*1). The spacecraft certainly won't lift off with just the 3 SSMEs. In order to work, either there needs to be more engines, or different engines (or both). The tank would need to be redesigned, and probably the orbiter as well. I leave it up to the person answering to decide how much they want to bother changing to come up with an estimate. However, please include your assumptions in your answer.

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  • $\begingroup$ Compare SSME and SRB thrust... or look at Buran/Energia. If you want to change little in the shuttle itself you will have to turn the external tank into a rather powerful rocket. $\endgroup$ – OON Aug 25 '18 at 5:51
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    $\begingroup$ Or just SSME and Merlin 1D T/W. The SSME gets 73.1, the Merlin 1D gets 179.8 (as of the last available numbers). LOX/LH2 systems have relatively low T/W ratios due to having lower mass flow rates (due to the low density of LH2), which is why they almost always use boosters. $\endgroup$ – Christopher James Huff Aug 25 '18 at 14:49
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    $\begingroup$ If you remove the SRBs and expand the external tank, the Shuttle would not be able to lift off. Thrust of the 3 liquid fuel engines is much too low. But there is no space left for more engines. A very different Shuttle is needed to mount more engines. Just compare the thrust of the SRBs with that of cryogenic engines. $\endgroup$ – Uwe Aug 25 '18 at 16:12
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    $\begingroup$ @ChristopherJamesHuff T/W of the engine really doesn't matter much; T/W of the loaded stage is critical. You can use hydrogen on the first stage without solid boosters (e.g. Delta IV Heavy); it's just bulky and expensive. $\endgroup$ – Russell Borogove Aug 26 '18 at 12:41
  • $\begingroup$ @RussellBorogove: I didn't say it couldn't be done, but the fact is that LH2 fueled vehicles almost always use boosters to compensate for their poor thrust. "High thrust-to-weight ratio" is simply not a benefit of liquid hydrogen fuel, it's one of the factors LH2 designs trade off for higher specific impulse. $\endgroup$ – Christopher James Huff Aug 27 '18 at 22:20
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All numbers are from wikipedia unless remarked.

So the mass of the shuttle orbiter at takeoff is around 109 tons. The dry mass of the external tank is 26.5 tons and it carries about 735 tons of fuel and oxidiser.

So lets (very) optimistically assume that the square-cube law works perfectly and we can scale the tank up by a factor of $s$ increasing the capacity in line with $s^3$ and the tank mass in line with $s^2$.

Now a space shuttle main engine has an $I_{sp}$ of 452 seconds in vacuum. So to achieve the minimum delta-V to low Earth orbit of about 9.4 $km/s$ the rocket equation $${\displaystyle \Delta v=gI_{sp}\ln {\frac {m_{0}}{m_{f}}}}$$ tells us we need a mass ratio of at least $8.3$.

Under our very optimistic assumptions, the mass ratio is $$1 + {735 * s^3\over 26*s^2 + 109}$$ and this actually equals 8.3 when $s = 1.12$ suggesting that just a small scaling up would be enough. Such a scaling gives a lift-off mass of just under 1200 tons.

However, the sea level thrust of a space shuttle main engine is only 1.8MN, so we need at least 7 of them to actually lift this stack off the pad, probably 8 or 9 if we want any significant acceleration at all. That adds at least 20 tons of mass, just for the engines, ignoring the necessary increase in size of the orbiter, extra load-bearing structure, etc. Rerunning with the extra 20 tones gets us $s$ of 1.18, taking the total launch mass up to almost 1400 tons and requiring still more engines.

If we want a thrust to weight of 1.25 at lift-off, with $n$ engnes, we can actually say that we will need

$$180 n = 1.25*(735 s^3 + 26.5 s^2 + 97 + 4n)$$ and we already know that we need

$$ 7.3 = 735 s^3/(36.5 s^2 + 97 + 4n)$$

Which has a solution with 12 engines and $s$ about 1.25. So we have crammed 9 more engines into the shuttle somewhere, and scaled the tank up by about 25% in each direction (almost doubling its volume and mass). By this point our assumption that the orbiter structure and the structures connecting it to the tank have not increased in mass are starting to look a bit dodgy.

We have also not allowed for the fact that the $I_{sp}$ of a space shuttle main engine is much lower at sea level (366 vs 452 in vacuum), so we probably need an even higher mass ratio, especially if our launch thrust-to-weight ratio is not much over 1 (since we will spend longer in thick air).

Bottom line, is this will probably never work. hydrogen is a great fuel once you are off the ground and in vacuum, but it's not so good for the first stage of the launch.

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  • $\begingroup$ ...which makes one wonder about the feasibility of RP-1/LOX boosters in place of the SRBs, and whether SRBs had a purely technological/cost advantage or whether SRBs were chosen because it helped keep Thiokol in the booster business. $\endgroup$ – Anthony X Aug 25 '18 at 15:15
  • $\begingroup$ @Steve: If I understand your answer correctly, the volume would be 1.25^3 = 1.95 times what it actually was. Do you think you could add that in to your answer? $\endgroup$ – DrSheldon Aug 25 '18 at 16:27
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    $\begingroup$ @AnthonyX there were a number of proposals to replace the SRBs with liquid boosters. Some even were flyback boosters. One got far enough in the 90s that we were asked to give general estimates of the impacts of modeling them in the training simulators, but nothing ever came of it. $\endgroup$ – Organic Marble Aug 25 '18 at 17:51
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    $\begingroup$ @AnthonyX The thrust of one of the SRBs was about the same as a pair of the F-1 kerosene engines as used on the Saturn V. Such boosters would have been feasible, but I think the SRBs were cheaper in both development and production. The later proposals mentioned by Organic Marble assumed the use of higher-specific impulse staged-combustion kerosene engine like RD-180. $\endgroup$ – Russell Borogove Aug 25 '18 at 18:51
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    $\begingroup$ Conclusions: The ultimate issue is not volume: about twice as much is needed, and the entire stack would still fit inside the VAB. You would need either a crazy number of engines (12 SSMEs), engines not designed for LH2 (F1), or engines not available in the Shuttle era (Merlin). Despite the additional complexity, using a different fuel in the lower atmosphere (RP-1/kerosene) requires fewer engines overall. $\endgroup$ – DrSheldon Aug 26 '18 at 12:34

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