Re: Obliterating the Rocket Equation with a Torusail


pgarrone@xxxxxxxxxxx wrote:
> > Controlled fusion is a really tough nut to
> > crack, and I think it's reasonable to assume it won't
> > be developed in the near term. Beyond that, it could
> > plausibly go either way. But the near term? Barring
> > some unexpected science breakthrough (ala Cold Fusion),
> > the rate of progress is too low.

> This is from www.iter.org website. http://www.iter.org/index.htm
> It outlines their aims. They are spending billions of dollars.
> They intend to do all this about 2017 I believe.

Do you need a reminder of how long we've been spending
billions of dollars on fusion research and how long
fusion power has been two decades away?

Even if all of ITER's goals are acheived by 2017, that's
a scientific demonstration rather than an actual power
plant. Any future reactor developed from it would be in
the mid-term, or long-term.

>>The limitation on temperature/pressure is based on the
>>sum of the kinetic energy and potential energy per kg
>>of the hot material. The kinetic energy is determined
>>by the temperature. The potential energy is essentially
>>determined by the pressure and density. This gives you
>>the specific energy of your reactor core. Multiply by
>>two and take the square root to get the exhaust velocity
>>you'd get from simply letting some of the core escape
>>into space.

>Fortescue and Stark "spacecraft systems engineering 2 ed
>1995" in "Principles of rocket propulsion" page 150,
[...]
>If we ignore the pressure term, which only decreases
>the value of Ve, using R = 8314 J/Kg/K and W = 1,
>get
>Ve Tc = Ve**2 / (4*8314)
>0.1C 27 gigaK
>0.05C 6.7 gigaK
>0.01C 270,000,000 K

>Containing a gas at these temperatures is not possible.

You're the one assuming 25th century science/technology;
I didn't want to presume such a feat impossible.

As it is, you don't need to "contain" the gas to acheive
rocket thrust. You only need to direct it, and this can
be done using magnetic fields. You later clarify that
you're thinking in terms of intertial confinement fusion,
in which case you're not actually containing the core
at fusion temperatures at all anyway. For that, you
can direct the high velocity fusion products directly
using a torus shaped magnetic field.

>I'm talking about inertial confinement. Laser beams zap
>the pellets. This is scientifically plausible.

With ICF, it's hard to beat simply directing fusion
products rearward with a magnetic field.

>>The travel distance is something like 71 light years,
>>right? Personally, I think you'd want to make such a
>>long journey at a much higher cruise velocity, but
>>let's assume that's off the table. Starting from the top:

>Yes. 70 lightyears. There is a fuel reserve.
>Trip dynamics are governed by the power of the rocket.

I wouldn't be using a rocket for such a long journey.
I'd go with something like mag-loop propulsion, or a
fusion pellet runway with torusail ram accelerator.
Things depend on your mission goals, of course, but my
gut feeling is that a cruise velocity of maybe .45c
would be a good compromise between mission cost and
trip duration (around 16 decades).

>I get the following table.

>Mission Fuel Mass Trip time
>delV Megatonnes Years
>0.35C 3.622 456.9
>0.375C 4.678 440.4
>0.4C 6.083 431.4
>0.425C 7.988 431.1
>0.45C 10.64 441.9

Wait--you're using the rocket to decelerate, doubling
your delta-v requirements? In that case, your estimate
of rocket performance seems too high. I see later on
that you assume 99% efficiency, which seems rather
optimistic.

>I assume 3 layers in the gas.

>- An inner bubble where the
>temperature is very high and the density is very low.
>The flux of fusion products blows any gas away.

>- A radiating layer that thermalises the neutrons and
>other products. This layer is at the temperature that
>creates a Bremsstruhlung radiation maximum for a given
>pressure. I get 6.170 meters radius, 190000 K plasma
>temperature.

>- An insulating layer that
>protects the delicate optics/semiconductors.
>Neutron wall flux is negligible. Average temperature
>1000 degrees K, but much cooler at the wall.

Personally, I would have called these layers of gas
"shielding" rather than the "core". You're extracting
energy with photo-voltaics, right? Essentially,
what you're doing is creating tiny nuclear explosion
fireballs in an atmosphere, and extracting energy
from the thermal radiation. I don't know the breakdown
for small nuclear explosions, but for larger ones
you'd only get a fraction of the energy in photon
radiation--a lot goes into the shockwave.

Still, maybe you could contain the shockwave and
thermalize all of the energy by containing the entire
"core" within a solid walled container with vacuum on
all sides. In principle, that can still be pretty
efficient, since the "heatsink" side of your heat
engine can in principle be as low as 3K. You're going
to have a low power/weight ratio, though.

Roughly, the walls of the "core" container needs to
be maybe 3000K or less (tungsten walls). This is
surrounded by photo-voltaics at maybe 300K for high
efficiency (I'm going to assume perfect efficiency
PVs; not sure how realistic that is, but we're talking
25th century tech). The photo-voltaics need to be
actively cooled at 300K, maybe with liquid droplet
radiators. Your reactor's performance is going to be
limited by the effectiveness of you heat rejection
system.

Now, in contrast let's say we don't care about the
shockwave energy and just go with radiation from
the hot 190000K "layer". Maybe only 30% of the
reaction's energy is transmitted that way, with the
rest lost to shockwave, heat conduction, and
expansion cooling. However, you can now run your
PV panels nice and HOT. Let's say your PV panels
are made out of tungsten. Okay, maybe not "PV
panels" exactly, but still something heat engine-ish.
These "panels" run at 3000K, which is still plenty
lower than 190000K so thermodynamic efficiency may
be high. Now, you can run your liquid dropled
radiator at 3000K, increasing your heat rejection
rate by 10,0000 compared to the 300K radiator.
Much better!

But what's even better is if you get rid of all the
insulating layers of gas and PV panels and all that
stuff. All you want is a few magnetic coils clustered
radially to direct fusion products rearward, along with
some auxiliary magnetic coils to extract power from
the exhaust pulse.

>>Personally, I like the electric drive concepts where
>>no electrodes physically contact the hot drive plasma.
>>You want your drive to last, and hot plasma arcs aren't
>>friendly to metal electrodes!

>It is a manned mission so maintainence should be possible.

I'm not talking about the sort of unfriendliness that
puts some dings in the metal that you can bang/recast
back into place.

I'm talking about the sort of unfriendliness that vaporizes
metal so it's lost forever into your exhaust. The lost
metal eats into your specific impulse, no matter how
much maintenance you put in.

>With any drive except a maxwellian rocket combustion chamber,
>the fuel has to be ionised. Hence ion drive. That's plain
>english. Plasma is ionised gas.

The term "ion drive" has a standard meaning, distinct
from "plasma drive". In an ion drive, positively
charged ions are accelerated, and the exhaust is
typically neutralized by introducing electrons into
the exhaust stream. In a plasma drive, plasma which
has a bulk neutral charge is accelerated.

>You cant have an electric
>drive unless you ionize the propellant.

Technically, no. For example, one promising electric
drive is electro-thermal. With electro-thermal propulsion,
the propellant is simply heated up and sent through an
exhaust nozzle like any other thermal rocket. For
inner solar system work, this is capable of sufficient
exhaust velocities, and it avoids wasting energy ionizing
the propellant.

Of course, when you talk about relativistic exhaust
velocities, then the ionization energy is comparatively
low. There's no particular reason to avoid ionizing
the propellant.

>Your system has an intrinsic beauty of course.
>Here is my attempt at a derivation of the
>equivalent of the rocket equation.

>The bombs go off generating products normal to the
>velocity line. relative to the bomb. All the kinetic
>energy goes into this.

Yes, that gives you an idea of the performance. Analyzing
the situation when the bomb products have an initial
axial velocity component is a little more complex, but
the results are a bit suprising--the axial velocity
component has virtually no effect one way or the other.
Only the transverse component matters.

Another thing that makes analysis more complex is the fact
that not all bomb products will have the same velocity.
Some will have a larger transverse velocity. Some will
have a lower transverse velocity. This can be compensated
for in an elegant way, by making the torus "thicker"
nearer the edges. This makes the deflection angle
propotional with the radius, causing all of the products
to be deflected by the correct amount.

Finally, there is unfortunately the fact that not every
nucleus will have the exact same charge/mass ratio.
This is determined by the ratio of neutrons to protons.
The deflection angle is proportional to the charge/mass
ratio. Thus, the magnetic field can only be tuned
perfectly for one particular charge/mass ratio. Anything
with a greater or lower ratio will be deflected too much
or too little. Fortunately, efficiency is still good
within a wide range of deflection angles.

> Mp = payload
> U = velocity of bomb particles in y direction.
> m = mass of bomb particles.
> M = mass of unexploeded bombs
> W = energy provided by each particle = 0.5*m*U*U (1)
> Vf = final velocity of bomb products, relative to ship.
> V = Velocity of ship.
> Define theta such that tan(theta) = U/V
> Define K1 s.t. W = K1*M*C*C applicable energy produced.
> Define K2 s.t. m = K2*M bomb that goes into particles

> Now from (1)
> U = sqrt(2*U/m) = sqrt(2*K1/K2)*C (2)
> This means that U is a constant, depending only on K1, K2,
> which are both maximised for best fuel performance.

There may be some compromise between maximum yield
and the amount of shock impulse the starship can
widthstand. A bomb which is big and efficient might
impart and impractically large shock to the starship.

>The magnetic field twists the particles until they are in
>line with the x direction.

>>>From conservation of kinetic energy.
>Vf = sqrt( U*U + V*V) (3)

>>>From conservation of momentum.
>P = Momentum gained from each particle = m * Vf - m * V
>Now cos(theta) = V / Vf
>So P = m * V * (1 - cos(theta))/cos(theta)

> = m * V * ( sqrt((V*V + U*U)/V*V) - 1)
> = (sqrt(V*V + U*U) - V) * m

That's right; using conservation of momentum I don't see
what the point of the middle steps was.

> Now m = K2 * M
> and P = Mp * V
> Differentiate
> Mp * dV = (sqrt(V*V + U*U) - V) * K2 * dM
> Rearranging
> dM/dV = (Mp / K2) * (1.0 / (sqrt(V*V + U*U) - V)

> And integrating from wolfram, V = 0 to V
> M/Mp = (1.0/K2) * (1.0/(2*U*U) * (U*U*log ((V + sqrt(V*V + U*U))/U)
> + V*V + V * sqrt(V*V + U*U))

At this point, I must admit making simplifying
assumptions in my analysis. I just took as a
baseline a particular ineffiency in transfer of
kinetic energy, knowing that the drive gets more
efficient the faster it gets (up to a point, far
above my intended cruise velocity). Thus, my
estimate for final cruise velocity is purposefully
low--but not by too much.

>Using the corresponding values for my mission, i.e.
>K1 = 0.006362
>K2 = 1
>V = half the delV = 0.1875, (acceleration and decelleration),
>I get a ratio of 3.63, verses 5.27 with the rocket equation.
>So you win, OK. Of course the bombtrack fuel will be much more
>difficult and expensive and unproductive, but technology will
>advance i'm sure.

For a delta-v not too much higher than the exhaust velocity,
a rocket is pretty efficient and the operational advantages
may be compelling. You're talking about a delta-v/exhaust
velocity ratio of maybe 1.66, well within what looks good
for a rocket.

However, the benefits of runway propulsion start looking
really good if you shoot for a higher delta-v. Double
the desired velocity, and the rocket's mass ratio shoots
up to 52. The runway's mass ratio only goes up to 15.

>Thanks for answers to the questions.

You're welcome, it's fun and interesting!

I remember in your last big discussion about your story idea,
I had not really developed the bombtrack concept sufficiently
to present a serious alternative with anything more than a
vague idea about performance. What we've come up with since
then is great progress, both qualitatively and quantitatively.

The "torusail propulsion" concept is actually very different
from my original "bombtrack propulsion" concept, and
radically more efficient. My original "bombtrack propulsion"
proposal more or less apexed with "particle puff
propulsion"--technologically challenging and relatively
inefficient.

Isaac Kuo

.

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