Author Topic: Starship On-orbit refueling - Options and Discussion  (Read 818128 times)

Re: Starship On-orbit refueling - Options and Discussion
« Reply #2980 on: 01/16/2025 07:54 pm »
This has been a great discussion about absorbance, emissions, and insulation for depots. Thank you everyone for sharing your insights.

The Skylab astronauts solved their heat problem with a jury rigged sunshade.
Why is that solution not also appliucable to the depot?

Even if there is need to develop retractable shades to allow refuelling, would it not be sufficient to simply shield the depot from sun and earthshine?

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Re: Starship On-orbit refueling - Options and Discussion
« Reply #2981 on: 01/16/2025 08:02 pm »
This has been a great discussion about absorbance, emissions, and insulation for depots. Thank you everyone for sharing your insights.

Good discussion, but I haven't seen the math.

Let's take a Starship with heat shield of 6cm thick (including the tiles, the blankets, and the backup ablative), give it an R value of 30 in Imperial units (5.28 in MKS), for an Reffective of 0.32 m2K/W.  We'll give the heat shield and absorption of 1.0 (probably pessimistic) and emissivity of 0.98.

Now for the simplest scenario:  The heat shield is facing the Sun, 1300 W/m2.  What will be the equilibrium temperature and heat flux into the Starship at the boiling point of LOX, 90K?

Now we have two simultaneous equations to solve:

emitted heat = 5.67e-8 * 0.98 * Texternal4
heat flux through insulation = ΔT/R = (Texternal - 90)/0.32

That's a 4th order equation, but fortunately spreadsheets are really good at this kind of thing, and the equilibrium answer is:

outer temp: 320K
inner temp: 90K
heat flux through insulation: 726W/m2

So the heat shield is only dropping the heat flux from the sun by 44%.  Very sad.

However, the other side of the Starship is facing deep space in our simplified scenario.   So we get to solve the equation again, but this time in the other direction:

outer temp: 88K
inner temp: 90K
heat flux: -3 W/m2

So the net heat flux is 723W/m2 into the Starship.  Still very sad.

How much LOX will that heat flux boil?

Q=m⋅L

Q = heat into the system = 723 J/s
L = latent heat of vaporization for LOX = 213kJ/kg. 
m = mass of the LOX being boiled

solving for m, we get 0.00339 kg/sec

Now the fuel tank for LOX is about 20mx9m = 180m2  (this is a square cow, the real one is cylindrical), so every second we are boiling .6kg of LOX.  That's 2.2t per hour.

Is that right?  Someone check my math please.

Now this is the worst case scenario - deep space, or a highly elliptical orbit.  In VLEO the sun is occluded 1/2 the time, and so now we have to start calculating rates.  The system will heat up for half an orbit, and cool down for half an orbit.  The tiles themselves absorb heat, which we would have to take into account.  Will leave that for another post.



https://docs.google.com/spreadsheets/d/1esN27mUh7s2gFvXzmvM6x1aU_AnVjfvRYwnGs5U2plw/edit?usp=sharing
« Last Edit: 01/16/2025 08:02 pm by InterestedEngineer »

Re: Starship On-orbit refueling - Options and Discussion
« Reply #2982 on: 01/16/2025 08:16 pm »
Am surprised by these numbers,
They indicate that a shield only cuts the heat flux by about half.
That astonishes me, because the Skylab astronauts reported an immediate dramatic reduction in temperature once their sun shield was deployed.

Am also surprised by the initial calculation that the sun facing side has a temperature of only 320*K. I'd always thought that the sun facing sides baked, rather than be at 320*K, which is room temperature.

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Re: Starship On-orbit refueling - Options and Discussion
« Reply #2983 on: 01/16/2025 09:13 pm »
Am surprised by these numbers,
They indicate that a shield only cuts the heat flux by about half.
That astonishes me, because the Skylab astronauts reported an immediate dramatic reduction in temperature once their sun shield was deployed.

Am also surprised by the initial calculation that the sun facing side has a temperature of only 320*K. I'd always thought that the sun facing sides baked, rather than be at 320*K, which is room temperature.

A sun shield would be amazing, as the back side emissions aren't going to be that high.  I suspect the new fangled solar tiles are pretty much a sun shield, with fancy paint, backside insulation, and minimal coupling to the body.

Someone double check my calcs, but pretty sure they are correct.  The tiles are pretty good for everything but absorption.

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Re: Starship On-orbit refueling - Options and Discussion
« Reply #2984 on: 01/16/2025 09:35 pm »
Hard to estimate the exact R-value of the stack-up, since (most of) the tiles aren't bonded to the metal, they're only touching along the three pins.

OTOH, if we use thermal numbers for the Shuttle HRSI and assume 3.3 cm thickness (it's actually less because the tiles are hollowed out), then the tile itself only has an R-value of R-3.9.


In short, I think we're seeing why SpaceX went with white coatings and thick foam (R-6.2/inch vs R-3.0/inch) for the depot.  :o
« Last Edit: 01/16/2025 10:11 pm by Twark_Main »

Re: Starship On-orbit refueling - Options and Discussion
« Reply #2985 on: 01/16/2025 09:54 pm »
Sun screen is both light and cheap.
Is there any reason why a depot, where the very name suggests longer term duration, could not be protected more effectively and more cheaply by simply providing a large area of sun screening than by ever deeper layers of insulation on the depot tanks?

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Re: Starship On-orbit refueling - Options and Discussion
« Reply #2986 on: 01/16/2025 09:54 pm »
So the heat shield is only dropping the heat flux from the sun by 44%.  Very sad.

Interestingly, simply exposing the rolled steel to direct sunlight reduces the heat flux by 61%, since the material has a solar absorbtance of 39%. If you vapor-deposit aluminum on top the absorbtance drops to just 8%, but I doubt it would survive reentry.



However, the other side of the Starship is facing deep space in our simplified scenario.   So we get to solve the equation again, but this time in the other direction:

outer temp: 88K
inner temp: 90K
heat flux: -3 W/m2

So the net heat flux is 723W/m2 into the Starship.

Starship is a cylinder and not a pancake. This only increases the back-side radiation by a factor of pi/2 (ie negligible), but on the illuminated side you probably want to numerically integrate over the curvature, accounting for cosine loss.


Now this is the worst case scenario - deep space, or a highly elliptical orbit.  In VLEO the sun is occluded 1/2 the time, and so now we have to start calculating rates.  The system will heat up for half an orbit, and cool down for half an orbit.  The tiles themselves absorb heat, which we would have to take into account.  Will leave that for another post.

This is further pessimistic because it assumes you don't point the nose at the Sun, which is something you probably want to do. You can stuff a bunch of MLI under the nose cone (where won't freeze up on the pad, unlike trying to surround the tanks).

I think this is the minimal implementation of etudiant's "shade."  You still need the foam, however, in order to manage thermal heating and reflected sunlight from the Earth.

« Last Edit: 01/16/2025 10:15 pm by Twark_Main »

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Re: Starship On-orbit refueling - Options and Discussion
« Reply #2987 on: 01/17/2025 05:19 am »
3) Once you've committed to moving the domes forward, you might as well move them as far forward as possible for both the tanker and depot

I can't see why that would be true. Unnecessary dry mass is unnecessary.

Moving the domes doesn't increase dry mass.  Same number of ring segments, just arranged differently.

Wait, I take it back:  you have to add another meter or two to the LCH4 downcomer.  That's... what?  another 400kg?  Same thing for hot-gas piping to the LCH4 ullage space, but that's very light.

This assumes the total vehicle length must be the same, but of course that's not necessarily true. On a dedicated tanker the payload bay is entirely vestigial. As Musk has said, the mass-optimized tanker would look "weird."

Fair enough--to a point.  If you don't need more propellant in a depot than you do in a tanker, then don't move the domes more forward than you need to do to optimize prop delivered to LEO. 

But that's going to depend on your boiloff rates.  There's likely to be some lag between the final fill to a depot and when the target Starship pulls the prop out.  If more tankage ensures that you can do a full fill in the FTO, then an extra ring segment is well worth it.

Quote
Yep, that's what I'm saying tends to get over-weighted.

Currently everything is bespoke, and honestly it will be that way for a long time. The paper labels read "SN33 common dome," not "Type 1 common dome."

They already have jigs and tooling for making ring segments, and common and forward domes are the same (with both of them moving to EDomes).  To the limited extent that things are bespoke, it's because they're changing too fast to make it worth investing in tooling.  That's not going to be true for much longer, and it's certainly not going to be true at the level of the stackable subassemblies made out of ring segments, domes, and stringers.

Those suckers are big.  Maintaining separate stations and holding areas for lots of variants is expensive.  It also eats space that could be better devoted to production.  They need production, because they need to get cadence up.  And to the extent that reusability comes online, they'd much rather devote space to inspection and refurbishment than to squirrelly one-offs.

Quote
As for insulation thickness:  once the heat is past the emitting layer(s), it doesn't matter how thick the insulation is, because the heat will eventually conduct into the tank.  The time scale we're dealing with is multiple months, so everything will be in equilibrium one way or another.

Sorry, but this is not right. In practice you run the calculation iteratively, homing in on the correct exterior temperature.

You're seeking to make sure the R-value equation and the net radiant flux (absorption minus emission, as calculated above) give the same heat flux, because at equilibrium they must.

The result is that at equilibrium the outer surface will be hotter (and therefore outgoing emission greater, and therefore net heat flux lower) the thicker you make the insulation. In other words, thicker is better.

That's all true, but you're assuming that the SOFI has good emissivity.  The coatings I've seen described are optically reflective particles in an optically transparent matrix, backed by an aluminized layer that's IR-reflective.  So there are two questions that need answering:

1) Can you aluminize SOFI?

2) If you can, what happens to the heat that builds up on the outer edge of the SOFI, which is inside the aluminum?  ISTM that your coating effectively crashes whatever emissivity the SOFI had, which wasn't great in the first place. 

The coating will indeed heat up and its emissions will increase as a result, but cryo coatings really aren't designed to maximize emissivity because things don't get hot enough to make that a useful way to reject heat.  So what you're going to get is hot, low emissivity SOFI, which will conduct more heat in equilibrium than you want.

I'm betting on a tile that's a Whipple shield, with an advanced coating on the outside.  The tile form factor allows the coating to have a fairly exotic but scalable application method.  It'd be lovely if the Whipple shield could do double duty as MLI, but I suspect that making it waterproof will also make it hard to evacuate in space.

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Re: Starship On-orbit refueling - Options and Discussion
« Reply #2988 on: 01/17/2025 05:28 am »
Starship is a cylinder and not a pancake. This only increases the back-side radiation by a factor of pi/2 (ie negligible), but on the illuminated side you probably want to numerically integrate over the curvature, accounting for cosine loss.

When you integrate along the curve with the cosine losses to the irradiance, you find that the net incident radiation is proportional to the cross-section.  So a pancake works fine.

Emissivity is a different story, but to tell it you need to know a whole bunch of stuff about the orbit.  Even the night side of Earth is pretty hot in IR, and the angular width of the Earth from VLEO is large enough that considerably less than πrL will be radiating into cold space.

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Re: Starship On-orbit refueling - Options and Discussion
« Reply #2989 on: 01/17/2025 06:02 am »
Am surprised by these numbers,
They indicate that a shield only cuts the heat flux by about half.
That astonishes me, because the Skylab astronauts reported an immediate dramatic reduction in temperature once their sun shield was deployed.

The Skylab shade cooled things by having high reflectivity.  (I don't think the shade was made of MLI, though.)  SOFI has very low reflectivity, unless it has a coating.  And even then, its absorptance is much higher, so any cooling is going to occur through emissions.  Pretty different mechanisms at play here.

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Re: Starship On-orbit refueling - Options and Discussion
« Reply #2990 on: 01/17/2025 07:28 am »
Starship is a cylinder and not a pancake. This only increases the back-side radiation by a factor of pi/2 (ie negligible), but on the illuminated side you probably want to numerically integrate over the curvature, accounting for cosine loss.

When you integrate along the curve with the cosine losses to the irradiance, you find that the net incident radiation is proportional to the cross-section.  So a pancake works fine.

While writing that, I was acutely aware that this is true for a simple thermal model that ignores insulation and re-emission. What I meant is, I'm not certain whether it's still true for the more complex thermal model used above.

Emissivity is a different story, but to tell it you need to know a whole bunch of stuff about the orbit.  Even the night side of Earth is pretty hot in IR, and the angular width of the Earth from VLEO is large enough that considerably less than πrL will be radiating into cold space.

The vehicle surface is still radiating the same amount, you just also have to account for that second source of incoming 'light.'

I simplified that out of my explanation (it was already getting long-winded enough!), but I trusted that it's pretty straightforward to account for once you have the basic technique down. You just add a second 'light' source which takes up ~half the total view sphere, except you use the thermal emissivity to estimate IR absorptivity.

Same thing for reflected light from Earth when you're on the day side. You know the Earth's albedo, and it's reasonably accurate to use the same solar absorptivity number.
« Last Edit: 01/17/2025 08:16 am by Twark_Main »

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Re: Starship On-orbit refueling - Options and Discussion
« Reply #2991 on: 01/17/2025 07:57 am »
3) Once you've committed to moving the domes forward, you might as well move them as far forward as possible for both the tanker and depot

I can't see why that would be true. Unnecessary dry mass is unnecessary.

Moving the domes doesn't increase dry mass.  Same number of ring segments, just arranged differently.

Wait, I take it back:  you have to add another meter or two to the LCH4 downcomer.  That's... what?  another 400kg?  Same thing for hot-gas piping to the LCH4 ullage space, but that's very light.

This assumes the total vehicle length must be the same, but of course that's not necessarily true. On a dedicated tanker the payload bay is entirely vestigial. As Musk has said, the mass-optimized tanker would look "weird."

Fair enough--to a point.  If you don't need more propellant in a depot than you do in a tanker, then don't move the domes more forward than you need to do to optimize prop delivered to LEO. 

But that's going to depend on your boiloff rates.  There's likely to be some lag between the final fill to a depot and when the target Starship pulls the prop out.  If more tankage ensures that you can do a full fill in the FTO, then an extra ring segment is well worth it.

If you're about tank hardware commonality, then surely you would agree that "once you've committed to moving the domes forward, you might as well move them as far forward as possible for" the target Starship too, so there should be no "spare" volume.

If not, then you still have to deal with two hardware variants and the point you make next becomes moot.

Or maybe it's not a point about "spare" volume, but just a "more is better" argument. But in that case, if there's not enough propellant, then that simply means they chose the FTO wrong. "The glass is neither half-full nor half-empty, the glass is oversized by 100%."  :)


Yep, that's what I'm saying tends to get over-weighted.

Currently everything is bespoke, and honestly it will be that way for a long time. The paper labels read "SN33 common dome," not "Type 1 common dome."

They already have jigs and tooling for making ring segments, and common and forward domes are the same (with both of them moving to EDomes).  To the limited extent that things are bespoke, it's because they're changing too fast to make it worth investing in tooling.  That's not going to be true for much longer, and it's certainly not going to be true at the level of the stackable subassemblies made out of ring segments, domes, and stringers.

Those suckers are big.  Maintaining separate stations and holding areas for lots of variants is expensive.  It also eats space that could be better devoted to production.  They need production, because they need to get cadence up.  And to the extent that reusability comes online, they'd much rather devote space to inspection and refurbishment than to squirrelly one-offs.

I wouldn't describe either depots or tankers as "one-offs," and I'm not sure where you're getting "squirrelly" from.

I think SpaceX is smart enough to plan their production a bit, and not accidentally make a bunch of segments that will sit around unused because they forgot they're not making any of that variant this week. Maybe I'm giving them too much credit.


As for insulation thickness:  once the heat is past the emitting layer(s), it doesn't matter how thick the insulation is, because the heat will eventually conduct into the tank.  The time scale we're dealing with is multiple months, so everything will be in equilibrium one way or another.

Sorry, but this is not right. In practice you run the calculation iteratively, homing in on the correct exterior temperature.

You're seeking to make sure the R-value equation and the net radiant flux (absorption minus emission, as calculated above) give the same heat flux, because at equilibrium they must.

The result is that at equilibrium the outer surface will be hotter (and therefore outgoing emission greater, and therefore net heat flux lower) the thicker you make the insulation. In other words, thicker is better.

That's all true, but you're assuming that the SOFI has good emissivity. The coatings I've seen described are optically reflective particles in an optically transparent matrix, backed by an aluminized layer that's IR-reflective.

[snip]

This is hardly an exhaustive inventory. For a good overview, check out this document:

https://ntrs.nasa.gov/api/citations/19990021250/downloads/19990021250.pdf

I'll also re-post something I wrote earlier:

Since it isn't orange, we can presume they are already specc'ing SOFI covered in some sort of space-rated high-emissivity high-albedo paint.

These coatings are already widely available, even going back to the first Dragon launch:

https://phys.org/news/2013-03-white-coating-spacex-dragon-trunk.html

https://www.aztechnology.com/product/1/az-93

More details on Z-93: https://www.spectroscopyonline.com/view/optical-measurement-of-the-reflectance-behavior-of-z93-the-thermal-coating-on-the-international


I'm betting on a tile that's a Whipple shield, with an advanced coating on the outside.  The tile form factor allows the coating to have a fairly exotic but scalable application method.  It'd be lovely if the Whipple shield could do double duty as MLI, but I suspect that making it waterproof will also make it hard to evacuate in space.

Tiles are both heavier and less insulative than SOFI.

Spraying on foam is an eminently scalable application method. This is why hiring a spray foam guy isn't more expensive than hiring a bunch of brick-layers. ;)

Yes, it's quite a shame that MLI ices up on the pad if you put it right up against a cryo tank.   :(


« Last Edit: 01/17/2025 07:39 pm by Twark_Main »

Offline KWC

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Re: Starship On-orbit refueling - Options and Discussion
« Reply #2992 on: 01/17/2025 08:15 pm »
My apologies if this has been asked before, or whether it should be on a different thread.

Does on-orbit refueling have any precedent outside of SpaceX? I know Stoke Space is planning to use it for their upcoming Nova rocket, but are there any historical examples of this being used, or even just historical plans? Something akin to the myriad of alternate shuttle designs they made in the '60s and '70s?

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Re: Starship On-orbit refueling - Options and Discussion
« Reply #2993 on: 01/17/2025 08:57 pm »
My apologies if this has been asked before, or whether it should be on a different thread.

Does on-orbit refueling have any precedent outside of SpaceX? I know Stoke Space is planning to use it for their upcoming Nova rocket, but are there any historical examples of this being used, or even just historical plans? Something akin to the myriad of alternate shuttle designs they made in the '60s and '70s?

Russian space stations were, and the ISS is, refueled in orbit using hypergolic propellants. What SpaceX are planning involves cryogenic propellants, which are both more efficient in use and more complex to handle and store. For cryogenics there is no publicly-known precedent of orbital refueling; SpaceX would be the first.

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Re: Starship On-orbit refueling - Options and Discussion
« Reply #2994 on: 01/18/2025 10:45 pm »
Am surprised by these numbers,
They indicate that a shield only cuts the heat flux by about half.
That astonishes me, because the Skylab astronauts reported an immediate dramatic reduction in temperature once their sun shield was deployed.

Am also surprised by the initial calculation that the sun facing side has a temperature of only 320*K. I'd always thought that the sun facing sides baked, rather than be at 320*K, which is room temperature.

at about 47°C, not a room in which I would wish to spend any length of time!

Webb rather effectively uses a sort of thermal Whipple shield to stay cold. Forty Kelvin or so.  This suggests that, were Depot fitted with such a shield, you’d need heaters to keep both propellants from freezing.  In practice, heat and light from Earth would likely provide the necessary heat.

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Re: Starship On-orbit refueling - Options and Discussion
« Reply #2995 on: 01/19/2025 02:56 am »
This has been a great discussion about absorbance, emissions, and insulation for depots. Thank you everyone for sharing your insights.

Good discussion, but I haven't seen the math.

Let's take a Starship with heat shield of 6cm thick (including the tiles, the blankets, and the backup ablative), give it an R value of 30 in Imperial units (5.28 in MKS), for an Reffective of 0.32 m2K/W.  We'll give the heat shield and absorption of 1.0 (probably pessimistic) and emissivity of 0.98.

Now for the simplest scenario:  The heat shield is facing the Sun, 1300 W/m2.  What will be the equilibrium temperature and heat flux into the Starship at the boiling point of LOX, 90K?

Now we have two simultaneous equations to solve:

emitted heat = 5.67e-8 * 0.98 * Texternal4
heat flux through insulation = ΔT/R = (Texternal - 90)/0.32

That's a 4th order equation, but fortunately spreadsheets are really good at this kind of thing, and the equilibrium answer is:

outer temp: 320K
inner temp: 90K
heat flux through insulation: 726W/m2

So the heat shield is only dropping the heat flux from the sun by 44%.  Very sad.

However, the other side of the Starship is facing deep space in our simplified scenario.   So we get to solve the equation again, but this time in the other direction:

outer temp: 88K
inner temp: 90K
heat flux: -3 W/m2

So the net heat flux is 723W/m2 into the Starship.  Still very sad.

How much LOX will that heat flux boil?

Q=m⋅L

Q = heat into the system = 723 J/s
L = latent heat of vaporization for LOX = 213kJ/kg. 
m = mass of the LOX being boiled

solving for m, we get 0.00339 kg/sec

Now the fuel tank for LOX is about 20mx9m = 180m2  (this is a square cow, the real one is cylindrical), so every second we are boiling .6kg of LOX.  That's 2.2t per hour.

Is that right?  Someone check my math please.

Now this is the worst case scenario - deep space, or a highly elliptical orbit.  In VLEO the sun is occluded 1/2 the time, and so now we have to start calculating rates.  The system will heat up for half an orbit, and cool down for half an orbit.  The tiles themselves absorb heat, which we would have to take into account.  Will leave that for another post.



https://docs.google.com/spreadsheets/d/1esN27mUh7s2gFvXzmvM6x1aU_AnVjfvRYwnGs5U2plw/edit?usp=sharing

Just checked out the spreadsheet.

Absorption shouldn't change with exterior temperature.

Net should be equal to absorption minus emission, not absorption plus emission.

The "rows of temperature" is an interesting approach. I would have used the goal seek feature, selecting the temperature and seeking the error term to zero.
« Last Edit: 01/19/2025 03:17 am by Twark_Main »

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Re: Starship On-orbit refueling - Options and Discussion
« Reply #2996 on: 01/19/2025 05:48 am »
Just checked out the spreadsheet.

Absorption shouldn't change with exterior temperature.

Net should be equal to absorption minus emission, not absorption plus emission.

The "rows of temperature" is an interesting approach. I would have used the goal seek feature, selecting the temperature and seeking the error term to zero.

Do I have these definitions and units right?

Irradiance = incidentPower (actually it's incident normal power--multiply by cos(θ) ) / area (i.e., it's a flux)
reflectance + absorptance = 1 (dimensionless constants)
Emission = emissivity * boltzmannConstant * Temperature⁴ (flux)

PowerAbsorbed = (Irradiance*absorptance - Emission) * area (power)

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Re: Starship On-orbit refueling - Options and Discussion
« Reply #2997 on: 01/19/2025 12:31 pm »
Yep, that all looks right to me.

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Re: Starship On-orbit refueling - Options and Discussion
« Reply #2998 on: 01/19/2025 05:08 pm »

This is further pessimistic because it assumes you don't point the nose at the Sun, which is something you probably want to do. You can stuff a bunch of MLI under the nose cone (where won't freeze up on the pad, unlike trying to surround the tanks).

I think this is the minimal implementation of etudiant's "shade."  You still need the foam, however, in order to manage thermal heating and reflected sunlight from the Earth.

Is there any reason the nose wouldn't be pointed at the sun? I'd always assumed that would be the orientation. The cross sectional area would be more like 70 sq m.

If that were true, and the nose had MLI under a vacuum, wouldn't the heat transfer to the tanks be mainly through the skin and structure connecting the nose to the rest of the 2nd stage? In that case the conduction cross sectional area would be pretty small I'd think. Perhaps some kind of "chomper" opening on the lee side of the nose cone would radiate some of that in addition to decreasing the cross sectional area. Maybe the chomper door actually could be a boom extending forward to put a reflective shield in front of the nose?

In that case, I'd think the Earth's radiation would be the main effect. That's something like a quarter of the Sun's and would vary depending on where the ship was in the orbit with the maximum heat flux being at local noon with the tail pointed at Earth. Obviously at night it would be less.
« Last Edit: 01/19/2025 07:44 pm by RDoc »

Online InterestedEngineer

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Re: Starship On-orbit refueling - Options and Discussion
« Reply #2999 on: 01/19/2025 09:34 pm »
https://docs.google.com/spreadsheets/d/1esN27mUh7s2gFvXzmvM6x1aU_AnVjfvRYwnGs5U2plw/edit?usp=sharing

Just checked out the spreadsheet.

Absorption shouldn't change with exterior temperature.

Net should be equal to absorption minus emission, not absorption plus emission.

The "rows of temperature" is an interesting approach. I would have used the goal seek feature, selecting the temperature and seeking the error term to zero.

Thanks! 

I renamed the column "transmission through insulation", I really meant "absorption by the spacecraft", but that's too confusing.  I think that should resolve items (1) and (2).  The numbers didn't change.

I wanted to see what the slope of convergence was so that's why I went with rows of temperature.


Tags: HLS 
 

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