airbreathing engines and brayton cycle

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idobox
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airbreathing engines and brayton cycle

Postby idobox » Sat Dec 10, 2011 10:24 pm UTC

Hi everyone.

Recently, I have thought a lot about beaming energy via microwave or lasers to a rocket with an air-breathing engine.
And I have realized I have no idea how to do the math.

I suppose thrust will depend on the power, density and heat capacity of air, external and internal temperature, speed of the vehicle and some efficiency factor.

I've found a wikipedia article on the Brayton cycle, which seems to be the best model, but my thermodynamics classes are an old and painful memory, and I don't how to find thrust from there. But I'll show you what I did:

[math]\eta = 1 - \frac{T_1}{T_2}= 1-( \frac{P_1}{P_2})^{(\frac{\gamma-1}{\gamma})}[/math]
According to wikipedia, gamma of air is about 1.4, so [math]\frac{\gamma-1}{\gamma} \approx 0.3[/math]
[math]\frac{T_1}{T_2}= \frac{P_1}{P_2}^0.3 \approx \sqrt[3]{\frac{P_1}{P_2}}[/math]

If I fix the internal temperature, I can get the exhaust pressure.
With the aperture surface and speed of the vehicle (let's say it's a ramjet), I can get the airflow, and the power needed to go to the internal temperature.
I'm not sure of how to get the thrust for the pressure. I suppose the exhaust aperture is smaller that the inlet, so I can't simply multiply by the same surface.

I don't need a very accurate model, just a order of magnitude one, to see if the technology is realistic
(I think it is. The power density limit for beaming microwave in air is above 20 GW/m², so a 1m² dish antenna could receive as much power as burning 500kg/s of kerosen, for a few miliseconds, then it would melt, or burn)

edit: Corrected some stupid latex mistakes, and a few sentences in French. Damn! I'm not used to do maths in English
Last edited by idobox on Sun Dec 11, 2011 1:13 am UTC, edited 1 time in total.
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Re: airbreathing engines and brayton cycle

Postby mfb » Sat Dec 10, 2011 11:33 pm UTC

20GW/m^2 can melt O(1m) of iron per second. With 90% efficiency of your antenna, you still need 10cm per second just to remove waste heat. That does not look like a good idea. However... I don't know if that intensity limit is interesting at all. Your rocket will be far away from the transmitter, even if you have a long line of many transmitters on the ground. With 3cm wavelength, 30m antennas and 30km distance, you can focus the microwaves to ~30km*3cm/30m = 30m (up to factors of ~2). So 1m is a bit optimistic. And I am pretty sure you do not want to send 2 TW to an ~10m x 10m area.

Two formulas have a syntax error, here my estimate for them:

idobox wrote:[math]\eta = 1 - \frac{T_1}{T_2}= 1- \left(\frac{P_1}{P_2}\right)^{\frac{\gamma-1}{\gamma}}[/math]

[math]\frac{T_1}{T_2} = \left(\frac{P_1}{P_2}\right)^{0.3} \approx \sqrt[3]{\frac{P_1}{P_2}}[/math]

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Re: airbreathing engines and brayton cycle

Postby idobox » Sun Dec 11, 2011 1:52 am UTC

Thanks for the syntax correction.

The limit I have given is the breakdown voltage of air. It would be absurd and dangerous to work near that. It was just to show absurd power can be transmitted, and I believe you can fly a rocket if you have absurd power at hand.
For comparison: the thrusters of the shuttle were some of the most powerful rockets ever built, and they burnt 500t of fuel in about 100-120s. The fuel is APCP, and has apparently 31MJ/kg (from wikipedia, I don't know if they counted the oxidizer mass in this). So it burns 500e3 * 31e6 ~ 15e12J in 100s, so has power output of 150GW. The thruster has a diameter of 3.7m, so a surface of 10.7m², and a power density of 14GW/m².
Beaming microwaves at 14GW/m² is impractical, because hot ionized gases would arc, and block the microwaves. But the most powerful thrusters consume power on the same order of magnitude as can be transmitted by microwaves, and they have to carry their own fuel. Also, even if it poses aerodynamical problems, the antenna/collector can be deported from the reactor itself.

With a 1mm wave, and 50m arrays, at 50km distance (maximal altitude of hydrogen balloons, I don't think air breathing engines could be very useful after that), you can focus microwaves at 50e3 * 1e-3 / 50 = 1m. Using a 1m² receiving antenna would be counter productive, and we can imagine having a beam much larger than the ship. In this case, only a part of the power would be used, but it's still better than carrying the fuel with you.

On a side note, what would be the best reaction mass for a beam-powered ship outside the atmosphere? I would say something that is liquid or solid at room temperature, vaporizes at a temperature common materials can support, and has the lowest specific heat capacity and vaporisation energy you can find. Mercury?
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Re: airbreathing engines and brayton cycle

Postby jmorgan3 » Sun Dec 11, 2011 6:20 am UTC

idobox wrote:I don't need a very accurate model, just a order of magnitude one, to see if the technology is realistic

Well, to a first order, you just count combustion as heat addition, so replacing 1GW of burning fuel with 1GW of lasers doesn't change anything too important thermodynamically.

idobox wrote:[math]\eta = 1 - \frac{T_1}{T_2}= 1-( \frac{P_1}{P_2})^{(\frac{\gamma-1}{\gamma})}[/math]
According to wikipedia, gamma of air is about 1.4, so [math]\frac{\gamma-1}{\gamma} \approx 0.3[/math]
[math]\frac{T_1}{T_2}= \frac{P_1}{P_2}^0.3 \approx \sqrt[3]{\frac{P_1}{P_2}}[/math]

If I fix the internal temperature, I can get the exhaust pressure.
With the aperture surface and speed of the vehicle (let's say it's a ramjet), I can get the airflow, and the power needed to go to the internal temperature.
I'm not sure of how to get the thrust for the pressure. I suppose the exhaust aperture is smaller that the inlet, so I can't simply multiply by the same surface.

P1 and T1 are the inlet static conditions, i.e. the ambient temperature and pressure. P1 is also your exhaust pressure, to first approximation. T2 and P2 are the conditions before heat addition-look at the labeled cycle in the wiki page. P2/P1 is called the pressure ratio Pr.

[math]\frac{P2}{P1}=Pr=Pr_i*Pr_c[/math]
[math]Pr_i=(1+(\gamma-1)/2*M^2)^{\frac{\gamma}{\gamma-1}}[/math]
Pr_i is the inlet pressure ratio, and is just a function of flight mach number.
Pr_c is the compressor pressure ratio, and you can just pretend it's constant for a first run. For a ramjet, there is no compressor, so this will be 1.

The efficiency eta is
[math]\eta=\frac{P_T}{\dot{Q}}[/math]
[math]P_T=\frac{1}{2}\dot{m}v_e^2[/math]
Where Q-dot is the heat addition rate, P_T is the thrust power, m-dot is the mass flow rate, and v_e is the exhaust velocity in the engine frame.
Your thrust will be
[math]T=\dot{m}(v_e-v)[/math]
Where v is the flight velocity.

So, for a preliminary analysis, it's probably best to fix inlet diameter, flight speed, ambient temperature and pressure, and compressor pressure ratio (1 for ramjets). Note that approximating mass flow as inlet area times velocity is going to be pretty bad, because the airflow is going to be limited by choking. This gets pretty complicated. Unfortunately, I don't have this book with me, because it has a whole chapter on this stuff.

Ultimately, though, all this analysis will prove is that you can make a ramjet with a particular heat addition rate. The hard part about your proposal is transferring the heat into fast-moving air in a small volume. Limiting yourself to heat transfer over the walls of a duct makes it very difficult to transfer heat in. The primary factor in determining the feasibility of your design will be the weight of the heat exchanger, and you won't know that without at least some preliminary design work on the heat exchanger.

Basically if you're asking "With a perfect heat exchanger and targeting system and everything else, could I make a solar ramjet?", the answer is yes.
*Note: This is all from memory, so there are probably some errors/non-standard notation.

idobox wrote:On a side note, what would be the best reaction mass for a beam-powered ship outside the atmosphere? I would say something that is liquid or solid at room temperature, vaporizes at a temperature common materials can support, and has the lowest specific heat capacity and vaporisation energy you can find. Mercury?

Some useful discussion here and here. Hydrogen seems to be a favorite for efficiency, but keeping it from boiling off seems to be a problem.
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Re: airbreathing engines and brayton cycle

Postby idobox » Sun Dec 11, 2011 2:14 pm UTC

Thank you for your help.

The good thing with microwaves is that you don't need a heat exchanger. You can heat the air by arcing, or even better, dielectric heating (like a in microwave oven).
I love the second idea for its elegance, but it has a few issues. You want the microwaves to heat the gas in the engine, but not air between you and the rocket. I see two ways to do that : emit at a frequency that doesn't heat up air much, and seed the flow with something that resonates at this frequency. The issue is that you would need a lot of seeding material
Second option, emit at a lower frequency, then use frequency doublers (essentially diodes) to reach nitrogen resonant frequency. Here, the issue is that the frequency doublers will be large, and heat a lot.

Focusing and targeting microwaves is not very difficult. A GW microwave emitter is more of a challenge. The power supply would be a headache.
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Re: airbreathing engines and brayton cycle

Postby mfb » Sun Dec 11, 2011 10:02 pm UTC

Another question: Why a ramjet? In that velocity region, there are established airplanes. The tricky part is to come from ~Mach 6 (~2km/s) to ~8km/s.

A silly idea for the microwaves: If you can heat the air (compression?) enough to ionize some parts of it, you have more options to heat it even more.

I prefer ground-based systems :).

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Re: airbreathing engines and brayton cycle

Postby idobox » Mon Dec 12, 2011 2:55 pm UTC

I suggested a ramjet because I thought the compression and airflow would be simpler. And I was wrong.
I we can accelerate a craft to Mach6 at 50km altitude, we solved a big part of the problem. Scramjets for higher velocity and turbojets for lower velocities are nice too, but I don't know if it is worth the weight.

If you heat the air by arcing, significant ionization will occur. I don't know at what temperature significant amounts of gas will be ionized if you use dielectric heating. But at resonant frequency, almost all the power from microwaves will be transferred to heat.
Apparently, there is a resonance of O2 at 60GHz, with 60dB/km absorption at 1 atmosphere. It means 99.9% of the energy is converted to heat in the first meter.
This figure will vary with pressure and temperature, but I don't know how.

In my mind, a 10GW power plant, a 10GW microwave emitter and its 50m antenna array qualifies as a mostly ground based system.
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Re: airbreathing engines and brayton cycle

Postby mfb » Mon Dec 12, 2011 6:46 pm UTC

At least Mach 3.3 at ~25km is done (and it can takeoff from a runway and probably carry several 10 tons). Mach 5 is more speculative. Fuel consumption to reach the height and velocity is small compared to takeoff mass. So if you can add a microwave scramjet to reach ~Mach 15, a smaller rocket could do the rest.

>> In my mind, a 10GW power plant, a 10GW microwave emitter and its 50m antenna array qualifies as a mostly ground based system.
"a[n]" emitter/antenna?
With 1g acceleration (quite a lot for aircrafts), Mach 4 -> Mach 10 requires ~500km flight length. Increase the distance or build several systems.
Ramjet or Scramjet (or even both), it can be used in a certain velocity range only. You still need a rocket and some system to get off the ground. A lot of expensive high-tech flying around there.

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Re: airbreathing engines and brayton cycle

Postby idobox » Mon Dec 12, 2011 8:41 pm UTC

Emitter as in generator, source. The device that turns electricity (or something else, but it would be quite exotic) into microwaves.
The antenna array will have several antennas and pointing systems or phase shifters. The generator need not be in the same place as the antennas, even though it s good to keep the distance short, too avoid transmission losses.

mfb wrote:With 1g acceleration (quite a lot for aircrafts), Mach 4 -> Mach 10 requires ~500km flight length. Increase the distance or build several systems.
Ramjet or Scramjet (or even both), it can be used in a certain velocity range only. You still need a rocket and some system to get off the ground. A lot of expensive high-tech flying around there.

I was thinking of something taking off vertically, with acceleration of a few g, similar to rockets. Is there a significant advantage of a plane-like craft?
My main concern is that a reactor couldn't provide enough thrust, whatever power is available, because air too thin, and no material could withstand the temperatures needed.
The SABRE engine looks like an awesome solution.

idobox wrote:Apparently, there is a resonance of O2 at 60GHz, with 60dB/km absorption at 1 atmosphere. It means 99.9% of the energy is converted to heat in the first meter.

This is an absurd mistake. 60dB/km means 0.06dB/m. Dielectric heating won't work, unless you have pressures of hundreds of atmospheres.
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Re: airbreathing engines and brayton cycle

Postby mfb » Tue Dec 13, 2011 12:22 am UTC

idobox wrote:I was thinking of something taking off vertically, with acceleration of a few g, similar to rockets.

Even then, you need horizontal movement or you leave the atmosphere quite early.
And you need some system to launch it from the ground. A rocket as first stage would ruin a lot of the advantages the microwaves (and airplanes) could potentially give.

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Re: airbreathing engines and brayton cycle

Postby Zamfir » Wed Dec 14, 2011 2:22 pm UTC

Idobox, I am confused on what you are hoping to achieve? An out-of-atmosphere spacecraft? An in-atmosphere plane? A first stage towards orbit?

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Re: airbreathing engines and brayton cycle

Postby idobox » Wed Dec 14, 2011 4:31 pm UTC

The first stage of an orbiter. Being able to reach a few tens of kilometers altitude, with a speed of a few thousands km/s without carrying fuel or fuel tanks would be a great advantage. Of course, you will need something else once you're out of the atmosphere, but beaming energy would still be useful, because you wouldn't need to a power generator for your ion drive or whatever.
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