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#1 2018-05-03 16:21:45

kbd512
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Registered: 2015-01-02
Posts: 1,647

Advanced Aerospike Rocket Engine Design

The linear aerospike engine design produced asymmetric thrust by varying propellant output on one side of the spike or the other.  I was wondering if the same concept could be applied to a massive plug aerospike engine for a spaceship like the BFS, ITS, or perhaps even the boosters, in this instance to reduce asymmetric loads, contend with engine out scenarios, and eliminate the need for engine gimbaling.  This is just future design consideration for oversized launch vehicles.  I see no reason at all to try to redesign BFS or BFR at this point.

Let's say we have a truncated plug aerospike engine variant of the F1 engine that uses 2 sets of turbopumps to push the propellants through the injectors.  Each set pumps feeds propellants to injectors that are evenly space around the edges of the spike, such that the loss of any one set of pumps doesn't produce asymmetric thrust.  The pumps shut down and half the thrust is lost as a result, but the engine continues to produce thrust and it doesn't severely alter the load applied to the vehicle.  To understand what I'm proposing, look at the outermost set of baffle plate separated injectors on the F1 engine.  There are 8 separate injector sections in this ring of the injector plate structure.  Each pair of pumps would feed 4 of the 8 injector sections.

If a pump set fails, the remaining thrust is still pushing straight up on the engine so no gimbaling of the other engines is required to maintain launch vehicle attitude.  The engineering result should be that the engine requires a much abbreviated thrust structure that can more evenly distribute the load to the rest of the vehicle.  If more or less thrust is required in flight to provide normal propulsion or retropropulsion, then it's a matter of spinning up or shutting down pump sets.  Separate engines are not required for sea level atmospheric or vacuum operations.

There may be a million and one reasons why this won't work, it's just a thought experiment to see if there's a better way to build very high thrust engines that don't necessitate more complicated vehicle designs to contend with asymmetric loading caused by engine out scenarios.  We seem to be entering an age of mega rockets, so I thought a bit more engineering could be applied to make the engines as compact and simple to integrate into the vehicle as possible.  I expect a reduction in the mass of interstage and thrust structure components, some of which would be offset by the larger nozzle and extra plumbing.

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#2 2018-05-03 21:36:07

GW Johnson
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From: McGregor, Texas USA
Registered: 2011-12-04
Posts: 2,987
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Re: Advanced Aerospike Rocket Engine Design

I think the idea is fundamentally sound,  but there's a lot of devils in the details.  They have to do with making the aerospike structure,  which is buried deep within all the rocket gas,  survivable at heat transfer recovery temperatures that are very nearly chamber temperature,  all the way to the tip.

I think this can be done,  probably with fuel regenerative cooling.  But the added-weight and complexity-reliability issues may overpower the benefit you would otherwise gain.  I honestly don't know.  But a version of this was the propulsion design on X-30,  so it has to be at least partially beneficial.

As for conventional nozzles,  a sea level nozzle works just fine in vacuum,  it just doesn't get quite as high a thrust and specific impulse as a vacuum design with a larger exit bell.  It even gets more thrust than it did at sea level,  via the positive backpressure vs expanded pressure term on the exit area.  The converse is not true:  a vacuum design usually won't work at all at sea level.  This is way beyond the negative thrust term for backpressure versus expanded pressure on the exit area;  most such failures really are full-blown backpressure-induced flow separation. 

GW


GW Johnson
McGregor,  Texas

"There is nothing as expensive as a dead crew,  especially one dead from a bad management decision"

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#3 2018-05-04 15:00:50

kbd512
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Registered: 2015-01-02
Posts: 1,647

Re: Advanced Aerospike Rocket Engine Design

GW,

Aerospike Construction:

Desktop Metal has a single pass 3D printing solution and MarkForged has very interesting methods of combining composites with metals to structurally reinforce a part using dissimilar materials, to include chopped and continuous filament composite materials.  These solutions are also very fast in operation and produce parts that require little in the way of finishing operations.  I wanted to print the entire nozzle as a single part.  I was thinking that a set of printers could fabricate the major components in a day or two.

Regenerative Cooling:

I fully expect that a channel wall nozzle would be required, it's just that now we have solutions that don't require hundreds of hours of laborious hand brazing of tubing to form the nozzle.  This was a major part of the manufacturing nightmare that the RS-25 presented.  The new RL-10's are supposed to use 3D printed channel wall nozzles to reduce manufacturing time and cost, for example.

Flow Seperation:

My understanding is that there's not enough work done on this aspect of aerospike engine design and that previous thinking on how it would work was proven to be incorrect, both computationally and from actual testing.  However, the NPR involved in the testing wasn't exhaustive so there is still a lot more research to be done.  Supposedly, the flow separation dynamics involved are more similar to those experienced by supersonic nozzles like those used in ramjets and scramjets than conventional bell nozzles.  Please let me know if my understanding about this is incorrect.

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#4 2018-05-04 15:18:03

GW Johnson
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From: McGregor, Texas USA
Registered: 2011-12-04
Posts: 2,987
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Re: Advanced Aerospike Rocket Engine Design

I don't know about any real flow separation theories,  in my time all those were still considered unproven.  All I have is an empirical correlation that seemed to work rather well for tactical rocket motor nozzles,  and for ramjet nozzles. 

Psep/Pc = (1.5 Pe/Pc)^0.83333

in which Pc is the rocket chamber pressure,  essentially the total (stagnation) pressure for the nozzle flow.   Pe is the fully-expanded static pressure at the exit plane of the nozzle.  Psep is the ambient pressure around the nozzle exit which is just high enough to cause flow separation. 

These tactical nozzle designs included both straight conical and full curved bell designs.  We never tried anything else as a production item.  The flapper-lollipop throat insert as a variable-geometry nozzle for ASALM is the exception.  It worked,  but the real-world design restrictions were twofold:  (1) the wake zone behind the lollipop had to close before the exit plane if full nozzle efficiency was to be obtained,  and (2) you could not put a large pressure drop across a structure like that because there were (and are) no materials capable of withstanding such abuse. 

That second restricted the variable-geometry nozzle to ramjet acceleration vs cruise only,  no integral booster pressures.  We did it with silica-phenolic ablative over martensitic stainless structure.  It worked for up to 900 second burns.  Surface roughness was considerable after a burn like that,  but did not seem to affect efficiency in any measurable way,  at the 1% thrust error measurement level. Did not fly in ASALM,  they could not afford the weight.

GW

Last edited by GW Johnson (2018-05-04 15:27:31)


GW Johnson
McGregor,  Texas

"There is nothing as expensive as a dead crew,  especially one dead from a bad management decision"

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#5 2018-05-05 08:45:40

Quaoar
Member
Registered: 2013-12-13
Posts: 366

Re: Advanced Aerospike Rocket Engine Design

GW Johnson wrote:

I think the idea is fundamentally sound,  but there's a lot of devils in the details.  They have to do with making the aerospike structure,  which is buried deep within all the rocket gas,  survivable at heat transfer recovery temperatures that are very nearly chamber temperature,  all the way to the tip.

I think this can be done,  probably with fuel regenerative cooling.  But the added-weight and complexity-reliability issues may overpower the benefit you would otherwise gain.  I honestly don't know.  But a version of this was the propulsion design on X-30,  so it has to be at least partially beneficial.

As for conventional nozzles,  a sea level nozzle works just fine in vacuum,  it just doesn't get quite as high a thrust and specific impulse as a vacuum design with a larger exit bell.  It even gets more thrust than it did at sea level,  via the positive backpressure vs expanded pressure term on the exit area.  The converse is not true:  a vacuum design usually won't work at all at sea level.  This is way beyond the negative thrust term for backpressure versus expanded pressure on the exit area;  most such failures really are full-blown backpressure-induced flow separation. 

GW

Hi, GW,

What about a dual-bell nozzle?

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#6 2018-05-05 10:03:20

GW Johnson
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From: McGregor, Texas USA
Registered: 2011-12-04
Posts: 2,987
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Re: Advanced Aerospike Rocket Engine Design

Not familiar with "dual-bell".  Do you mean some sort of retractable bell-extension?

That could work,  but imposes added weights for the machinery,  and some very severe high-temperature sealing problems,  which in turn imposes more weights and risks of reduced reliability. 

Not sure that kind of variable geometry is worth it.  Has never been worth it with missiles.  Has never been used on production launch vehicles,  either.

Look at Spacex's new Raptor engines.  The sea level version is listed on their website (at the current 250 bar chamber pressure) as 1700 KN thrust at Isp = 330 s,  sea level,  with a 1.3 m dia exit plane.  Operating that same engine in vacuum,  they claim Isp 356 s,  and I calculate vacuum thrust 1834 KN thrust. 

Putting the vacuum bell on the same chamber,  Spacex says vacuum thrust is 1900 KN at Isp = 375 s.  You get almost as much thrust out of the sea level engine,  just at a loss of Isp.  And it's really not that bad. 

These engines have a super-high thrust/engine weight ratio,  somewhere near 100:1.  You put any more variable geometry than simple gimballing on that,  and you lose that super-high thrust to weight ratio.  Which raises your stage inert weight,  and cuts your delta-vee potential.  Hardly seems worthwhile.

What they are already doing with two stage to orbit seems to me the most practical solution.  Sea level engines in the first stage for somewhat less than half the delta-vee to orbit at the lower average Isp,  and vacuum engines in the second stage for a bit over half the delta-vee to orbit at the higher Isp.  That way both stages can stay around 5% inert,  and maximize their delivered delta-vee potential.  The key to this is high staging altitude,  in super-thin air around 150,000+ feet.

Why bother changing what already works really well,  except to make it reusable as fly-back stages?

GW

Last edited by GW Johnson (2018-05-05 10:05:30)


GW Johnson
McGregor,  Texas

"There is nothing as expensive as a dead crew,  especially one dead from a bad management decision"

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#7 2018-05-05 14:53:08

Quaoar
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Registered: 2013-12-13
Posts: 366

Re: Advanced Aerospike Rocket Engine Design

GW Johnson wrote:

Not familiar with "dual-bell".  Do you mean some sort of retractable bell-extension?


GW

No, i intended a fixed nozzle, without moving parts, constituted in a base nozzle and a nozzle extension, linked by an abrupt wall angle change at the contour inflection, like this picture.

Sketch-of-a-dual-bell-nozzle.jpg

At low altitude the flow detaches at the exit of the first bell and the nozzle work at low area ratio, but in the vacuum the flow expands in the second bell, giving an high area ratio. Probably it is best suited for a SSTO: nine of such big bells cannot be fitted in a Falcon 9 first stage.

Last edited by Quaoar (2018-05-05 16:06:49)

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#8 2018-05-06 09:29:38

GW Johnson
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From: McGregor, Texas USA
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Re: Advanced Aerospike Rocket Engine Design

I've never seen a thing like that proposed before.  I honestly don't know how it might really perform.  It in effect gives the flow separation a definite point to happen,  rather than the typical oscillating-around unsteadily.  Whether that would really work,  I dunno.

GW


GW Johnson
McGregor,  Texas

"There is nothing as expensive as a dead crew,  especially one dead from a bad management decision"

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#9 2018-05-06 20:24:11

kbd512
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Registered: 2015-01-02
Posts: 1,647

Re: Advanced Aerospike Rocket Engine Design

GW,

Is there any merit to creating these truncated plug aerospike to function in the manner I described?

Would it simplify the overall design of the launch vehicle?

If a family of such engines were created that simply add pump sets and appropriate plumbing, is this be a feasible way to assure that the flight loads applied to the vehicle, thus where and how the tankage must be structurally reinforced, remains consistent as the size of the vehicle increases or decreases?

I wanted to create something that scales up and down, between Falcon 9-sized and ITS-sized booster cores.  It would be great if we had a sort of universal all-in-one solution that permits 2.5m to 25m diameter cores by adding appropriate numbers of turbos and plumbing.  If the turbos and nozzles were created with 3D printing technology, I'm thinking that the result would be major engine components that get fabricated in a day or so, finished the next day, and assembled on the third day.  Hopefully a continuous supply of these engines can be created to provide a steady supply of source stock for companies like SpaceX.  I was thinking the same thing about the robotically wound composite propellant tanks.  The humans are present to simply monitor machinery, feed in materials, remove finished components, and assemble the components.

Rather than designing individual rockets that are inflexible in their ability to increase payload without total redesign unless originally designed to handle greater loads, as SpaceX had to do with Falcon Heavy's center core, we could have a scalable propulsion and tankage hardware set to produce payload tonnage class appropriate boosters that do not require specialized production and assembly hardware.  There need not be a stock of a specific type of vehicle and spare parts at that point.  Rocket components are created on-demand, consumed in launch activities, refurbished when possible or completely replaced if not possible, and total turn around times are measured in hours or days.  There's no waiting for specialized parts or maintaining expensive part stocks because all of those parts are 3D printed.

Think of this as the Japanese kanban system for rocket vehicle fabrication and refurbishment.  This 2.5m core is back today from its satellite delivery, pump 1 needs a new impeller, the piping from pump 3 is corroded or damaged, and the injector plate from section 4 has some damaged nozzles, so print these parts, reinstall them, and then we have a 10m core that just delivered a new inflatable ISS module with some other damage requiring various repairs.  Both of these boosters need to be back in service in a week for the next set of payloads.  Make sense, or not really?

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#10 2018-05-07 10:40:38

Quaoar
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Registered: 2013-12-13
Posts: 366

Re: Advanced Aerospike Rocket Engine Design

GW Johnson wrote:

I've never seen a thing like that proposed before.  I honestly don't know how it might really perform.  It in effect gives the flow separation a definite point to happen,  rather than the typical oscillating-around unsteadily.  Whether that would really work,  I dunno.

GW

There are a lot of numerical simulation but very few real tests. So - please correct me if I'm wrong - at the moment the best solution for a SSTO may be a high-pressure chamber rocket engine with a quite high area ratio conventional bell nozzle, like the old SSME, which is lighter and simpler than an aerospike.
For example, can a rocket with a chamber pressure of 300 bar take off from sea level with a 100:1 nozzle?

Last edited by Quaoar (2018-05-07 10:44:17)

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#11 2018-05-08 12:34:27

GW Johnson
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From: McGregor, Texas USA
Registered: 2011-12-04
Posts: 2,987
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Re: Advanced Aerospike Rocket Engine Design

Hi Quaoar:

I think the answer to your question is a resounding no.  Here's why,  and it's more than you asked for.  For one thing,  I reverse-engineered the Raptor engine. 

What’s reported on Spacex’s website for the Raptor engine is chamber pressure 250 bar throttleable 20%-100% in both versions. 

For the sea level Raptor,  the sea level thrust 1700 KN,  and sea level Isp = 330 sec.  They say the exit diameter is 1.3 m,  and that this engine’s vacuum Isp = 356 sec without giving a vacuum thrust or operating pressure. 

For the vacuum Raptor,  they claim 1900 KN at Isp 375 sec,  with an exit diameter = 12.4 m.  They give no pressure,  but the same 250 bar is implied.  The posted exit diameter must be incorrect,  as a single 12.4 m diameter bell will not fit behind a 9 m diameter spacecraft body.  This error casts some doubt over the sea level diameter figure as well. 

There are 860 metric tons of liquid oxygen in that tank,  and 240 metric tons of liquid methane in that tank.  On the assumption that tank-contained masses are driven by the operating ratio r of the engines,  that is r=3.58 oxidizer/fuel by mass. 

From my 1970-vintage Pratt & Whitney Vest-Pocket Aeronautical Handbook,  the performance potential of liquid oxygen-liquid methane is listed for 1000 psia chamber pressure (and 100% nozzle efficiency) as c* = 6120 ft/sec and sea level Isp (for perfect expansion) = 310 sec.  This also assumes rather fuel rich at r = 3.15. 

The characteristic velocity c* correlates empirically as c* = k P^m,  where m is a number in the vicinity of 0.01.  Using that value,  the r = 3.15 c* at 250 bar = 3625 psia is right at 6200 ft/sec.  If we increase the heat release by raising oxygen to r = 3.58 that value might possibly fall in the 6250-6300 ft/sec range.  There has to be a loss somewhere,  so I use 6200 ft/sec to be as realistic as I can.

Now,  using my ancient charts for thrust coefficient CF that have a 98.3% nozzle efficiency already built in,  and picking the one for specific heat ratio = 1.20,  at Pc/Psea level = 3625 psia/14.70 psia = 246.6,  the optimum expansion ratio is 24.0,  at CF = 1.713.

Now 1700 KN = 382,000 lb of thrust.  For F = CF Pc At,  with those figures for thrust,  CF,  and chamber pressure,  I get a throat area At = 61.52 sq.in,  for a throat dia = 8.85 inch = 0.2248 m.  Using that throat area in the massflow equation with that chamber pressure and a 6200 ft/sec c*,  I get a massflow of 1157 lbm/sec,  which assumes all the turbopump drive flow gets re-fed back into the cycle at the indicated chamber pressure. 

Now for F = 1700 KN = 382,000 lb,  and sea level Isp = 330 sec,  the indicated massflow through the nozzle is 1158 lbm/sec.  That close correspondence pretty much confirms my ballistic estimates. 

The CF goes with an ideal expansion ratio of 24.0.  Using that on the throat area I found,  I get an ideally-expanded exit plane area of 1476 sq.in,  corresponding to an exit plane inside diameter of 43.36 inch = 1.101 m.  This is the part that does not match up perfectly with what Spacex reports on its site.  If I were to use their 1.3 m figure,  the bell is overexpanded,  not perfectly expanded.  CF would be nearer an even 1.70,  which means my ballistics are still in the ballpark.  I’d guess the 1700 KN is also rounded.

The Pe/Pc expanded pressure ratio is 14.7 psia/3625 psia = .004055.  Using my correlation for nozzle separation Psep/Pc = (1.5 Pe/Pc)^0.8333,  I get a Psep/Pc = .014.  If the Psep is Pbaro = 14.7 psia,  then the min chamber pressure at which separation is imminent is 1032 psia = 71.2 bar.  That’s for the sea level engine. 

If we assume the obvious error in reported exit diameter for the vacuum engine is a typo,  the exit diameter that might possibly fit in the illustrations they show might be 2.3 to 2.4 m dia max.  Using 2.3 vs throat dia 0.2248 m,  the indicated area ratio is 104 to 105.  That’s pretty close to your 100:1.

I don’t have an area ratio – Mach chart at specific heat ratio 1.2,  but for air’s 1.4,  a nominal 100: 1 area ratio corresponds to an exit plane Mach of 6.9.  At such a Mach number,  and a specific heat ratio of 1.2,  the Pe/Pc is near 0.000027.  Running that through my separation correlation gives Psep/Pc = 0.00022.

For Psep = 14.7 psia,  the corresponding min chamber pressure at separation is 66,515 psia = 4587 bar.  It doesn’t really matter that my expansion Mach estimate isn’t right,  this just isn’t anywhere near the ballpark.  Spacex says it would like to “grow” the Raptor chamber pressure to 300 bar.  There is just no way to operate the vacuum Raptor at sea level,  without massive flow separation essentially killing over half the thrust,  and risking hardware damage from unforeseen heat transfer conditions.

Looked at another way,  the ambient atmospheric pressure that would risk separation at the nominal operating Pc = 250 bar is 0.098 psia.  That’s about 110,000 feet = 33.5 km altitude.

GW

Last edited by GW Johnson (2018-05-08 12:37:56)


GW Johnson
McGregor,  Texas

"There is nothing as expensive as a dead crew,  especially one dead from a bad management decision"

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#12 2018-05-09 11:28:57

Quaoar
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Registered: 2013-12-13
Posts: 366

Re: Advanced Aerospike Rocket Engine Design

GW Johnson wrote:

Hi Quaoar:

I think the answer to your question is a resounding no.  Here's why,  and it's more than you asked for.  For one thing,  I reverse-engineered the Raptor engine.  [...]
GW

Thanks GW, it's ever a pleasure to hear your lessons.

So I argue that the Space Shuttle Main Engines, which have an area ratio near 70:1 and almost 200 bar of chamber pressure, are not very efficient at sea level, and it wouldn't be possible to build a real SSTO (without the solid rocket busters) with such kind of engines.

Is it correct?

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#13 2018-05-09 13:08:00

GW Johnson
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From: McGregor, Texas USA
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Re: Advanced Aerospike Rocket Engine Design

SSTO really isn't so much about nozzle expansion,  it's about mass ratio.  That takes low inert weight (and low payload) to get high mass ratio. 

The performance of the sea level Raptor in vacuum is not all that far from the vacuum Raptor.  If instead of a conventional bell nozzle you use what is called an aerospike,  you can bring sea level and vacuum Isp a little closer together while eliminating the separation risk. 

The downsides of aerospike designs lie in two disparate areas.  (1) they inherently add as much weight or more than a conventional bell,  being harder to cool to the point of survivability.  They are immersed in the supersonic blast,  rather than containing it.  The other is real-world nozzle efficiency.  Not containing the stream allows it to spread very far laterally during the expansion,  when backpressures are low.  That's jet streamlines at off-angle to the thrust axis,  and that's low nozzle kinetic energy efficiency. 

Those two hard-to-solve problems I think are why aerospike hasn't been used very much.

As for SSTO,  the 1956-vintage Atlas was almost an SSTO.  It had one set of tanks,  and 3 engines.  On the way up,  after pitching over outside the sensible air,  they staged off two engines to lower inert mass.  That rocket was capable of sending a 2-ton Mercury capsule into very low Earth orbit.  Or an early 5-ton warhead halfway across the world.  More than 6 decades ago. 

While fitting out on its side before launch,  it had to be pressurized so that the propellant tanks did not crush under their own weight.  That structural vulnerability is how it achieved low enough inert weight to reach orbit as almost an SSTO with first-generation LOX-kerosene sea-level engines of about 300 s Isp.

Now,  using your numbers for the pressure of the space shuttle main engines,  I show a sea-level optimal expansion CF (with 98.3% nozzle efficiency) of 1.690 at an expansion ratio of 20 not 70.  That's for a pressure ratio of 200:1,  at specific heat ratio 1.20. 

Now,  at 200:1 pressure ratio and 70:1 area ratio,  I'm showing CF only 1.545 if not separated,  and operation just beyond the separation warning on my old chart.  I'd be surprised if they really operated them that way,  but maybe they did.  The ratio of CF's is the ratio of Isp's,  for the same pressure and flow rate.  You really want to be operating on the flatter part of the curve where the off-design losses are much lower.

On the other hand,  my old nozzle separation criterion evaluated at 1/200 expanded pressure for design,  shows a chamber pressure nearer 60 bar for nozzle separation with a 1 bar backpressure.  That would say the shuttle engine could be operated at sea level,  just with low thrust and efficiency.

Operating on the verge of separation really reduces thrust and Isp rather steeply below well-expanded values.  Most of us favor designing for perfect expansion at launch,  and letting thrust and Isp rise as you climb (a relatively small effect,  but real),  just not quite as high as "perfect" expansion would have been. 

Usually launch against the heaviest weight is where you need the most thrust you can get.   If you are still short,  that's where the solids are so useful.  Those usually have even higher frontal thrust density (thrust per unit motor or stage cross section) than the liquids,  just lower Isp,  so they stage off earlier than the liquid first stages.

GW

Last edited by GW Johnson (2018-05-09 13:10:05)


GW Johnson
McGregor,  Texas

"There is nothing as expensive as a dead crew,  especially one dead from a bad management decision"

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#14 2018-05-11 16:59:44

Quaoar
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Registered: 2013-12-13
Posts: 366

Re: Advanced Aerospike Rocket Engine Design

GW Johnson wrote:

Operating on the verge of separation really reduces thrust and Isp rather steeply below well-expanded values.  Most of us favor designing for perfect expansion at launch,  and letting thrust and Isp rise as you climb (a relatively small effect,  but real),  just not quite as high as "perfect" expansion would have been. 

GW

So, just out of curiosity, which is a good area ratio for the rocket nozzle of a lander that has to take off from Titan where the surface atmospheric pressure is almost 2 Bar?

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#15 2018-05-12 08:23:01

GW Johnson
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From: McGregor, Texas USA
Registered: 2011-12-04
Posts: 2,987
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Re: Advanced Aerospike Rocket Engine Design

Hi Quaoar:

It’s not just about area ratio,  it’s area ratio and pressure ratio acting together.  My ancient chart has thrust coefficient (which you would like to maximize,  or operate near the max) on the vertical axis,  and area ratio on the horizontal axis. 

At any one pressure ratio (chamber/ambient),  there is a concave downward curve,  sort of resembling a downward-opening parabola. Its vertex is the best thrust coefficient available,  located at the optimum area ratio,  all for that particular pressure ratio.

There is a whole family of these curves,  one for each pressure ratio.  The vertices trend upwards to the right a little.  My particular chart has the kinetic energy efficiency of the nozzle built-in,  for the 98.3% of a 15 degree half-angle conical nozzle,  very common in missile work.  Most curved bell nozzles fall very close to that same efficiency,  once the designs are optimized. 

There is a chart like that for every different value of the rocket gas’s specific heat ratio.  The chart I use most often is sp. heat ratio = 1.2,  as “typical” of most rockets.  Reading that chart,  the optimum vertices of these thrust coefficient CF curves locate at values of area ratio Ae/At (Ae = exit plane area,  At = throat plane area),  for each of several pressure ratios:

For Pc/Pa………………..CFopt………………..Ae/At
5…………………………….1.071…………………1.47
10…………………………..1.247…………………2.25
20…………………………..1.383…………………3.60
40…………………………..1.497…………………6.00
70…………………………..1.573………………..9.00
100…………………………1.616………………..11.9
200…………………………1.690………………..20.0
400…………………………1.757………………..34.8

The vertices of these curves are fairly broad,  so you can be off of optimum some (as long as you don't risk backpressure-induced separation),  and still have the higher thrust coefficient that in part confers higher Isp.  Higher Pc acts directly through the pressure ratio to confer higher CF and thus higher Isp.  Pc also acts indirectly to confer higher Isp by way of higher characteristic velocity c* at higher pressures.  That c* is not a constant as most people assume,  but is in fact a weak power function of Pc with an exponent on the order of 0.01.  If all the massflow of the engine passes through the nozzle to produce thrust,  then there is a shortcut way to figure Isp:

Isp = CF c*/gc, 

where gc is the “gravity constant” to make the units work out consistently.

The optimum Pc to make this work is pressure ratio multiplied by backpressure.  It’s also a minimum,  in the sense that if you are a bit low,  your exit plane expanded pressure is under the ambient backpressure,  and so your exit area pressure correction term is negative.  That reduces thrust and Isp. 

If you are a bit high,  the exit area pressure correction term is positive,  and your thrust and Isp are higher,  but not as high as they would have been if you were perfectly expanded at that higher pressure.  If your Pe/Pa were 100,  and your Pa 2 bar as you indicated,  then you need a Pc = 200 bar or greater. Your expansion ratio Ae/At = 11.9,  but it’s different for every pressure ratio,  as is your min Pc.

F = mV + (Pe-Pa)Ae = CF Pc At  (note zero P term when Pe = Pa “perfect expansion”)

where m = mass flow rate,  V is the exit velocity,  Pe is the expanded exit plane pressure,  and Pa is the ambient backpressure.

Depending upon how your propellant is pumped and the details of how the tapped stream is treated,  will affect your Isp.  If the pump drive stream is not re-introduced into the propulsion stream before it goes through the nozzle,  then your propellant use rate is larger than what went through the nozzle.  This not only reduces Isp = F/wtotal,  it makes Isp = CF c* / gc inaccurate.   

In solids,  everything goes through the nozzle,  there are no pumps,  so we got to use Isp = CF c*/gc as quite accurate a lot,  with solids.  I recommend against that procedure generally,  for liquids.  Do the real ballistics instead.  These are based upon:

Obtain CF and Ae/At at proposed Pc/Pa from the chart
Characteristic velocity correlation from tests c* = kP^m
Thrust F = CF Pc At
Flow rate through the nozzle wnoz = Pc At gc/c*
Total flow rate from the engine cycle design wtot > wnoz
Isp = F/wtot  (not F/wnoz !!!)

Hope that helps.  Rocket chamber/nozzle ballistics are fairly easy.  For solids,  the propellant burning surface ballistics can get quite hard,  by around an order of magnitude.  Ramjet cycle analysis calculations are another order of magnitude more complicated still. 

GW

Last edited by GW Johnson (2018-05-12 08:32:15)


GW Johnson
McGregor,  Texas

"There is nothing as expensive as a dead crew,  especially one dead from a bad management decision"

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#16 2018-05-15 08:13:23

Quaoar
Member
Registered: 2013-12-13
Posts: 366

Re: Advanced Aerospike Rocket Engine Design

GW Johnson wrote:

Hope that helps.  Rocket chamber/nozzle ballistics are fairly easy.  For solids,  the propellant burning surface ballistics can get quite hard,  by around an order of magnitude.  Ramjet cycle analysis calculations are another order of magnitude more complicated still. 

GW

Thanks GW. You are a very good teacher.

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#17 2018-05-15 10:18:51

GW Johnson
Member
From: McGregor, Texas USA
Registered: 2011-12-04
Posts: 2,987
Website

Re: Advanced Aerospike Rocket Engine Design

It shows, I guess,  because most of what I did after getting laid off out of defense work was to teach.  Everything from public school to university graduate school.

GW


GW Johnson
McGregor,  Texas

"There is nothing as expensive as a dead crew,  especially one dead from a bad management decision"

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#18 2018-05-26 08:02:02

Quaoar
Member
Registered: 2013-12-13
Posts: 366

Re: Advanced Aerospike Rocket Engine Design

GW Johnson wrote:

It shows, I guess,  because most of what I did after getting laid off out of defense work was to teach.  Everything from public school to university graduate school.

GW

It's not easy to teach clearly. Only people who knows the matter very well can do it. It's a shame that they laid off you.

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