How big would a fusion reactor need to be to produce a constant triple digit terawattage?

Question

And how much fuel would you need to carry? I'm trying to determine what kind of engine you'd need to accelerate a good-size spacecraft (thousands to millions of tons) at a constant 1g in a race across the solar system, and what kind of weight-to-fuel ratio you'd end up with (as well as what ratio would be ideal). Obviously fusion rockets are currently the most plausible and efficient method of propulsion we have in "harder" science fiction (aside from nuclear propulsion and lasers), so I need to know what kind of reactor you'd need to be packing to push a racing vessel equipped with crew, shielding and weapons down the cosmic cul-de-sac, so to speak.

I'd really appreciate it if anyone could tell me how to solve for this problem, because I do need to apply it to other constant rates of thrust (most of the racers are altered transhumans who can handle abnormal g-loads and one aptly-named "Machinehead" has cut himself down to a cybernetically-augmented head in a jar of suspension fluid in order to push the smallest possible craft at a constant 43gs).

Answer 1

Warning: We haven't reached hard science level for such question, so I'd treat it as science based, tops.

For inspiration of design I'd suggest looking at "Stellarator":

Quite possibly a realistic fusion reactor would include even more twisted design to contain plasma in magnetic trap. With possibility to direct part of plasma through nozzle.

Realistic engine would have to be huge. Big part of energy is being produced in form of neutrons that escape any magnetic containment. In a normal power plant they would be used to heat water and produce electricity... you would presumably have to make it big enough, that such neutrons would hit some Li-6 and H-1, thus producing you further fuel (deutrium and tritium).

Needless to say, regardless how efficient such process would be, you'd need huge radiators to dispose waste heat.

As nicely put by project rho:

http://www.projectrho.com/public_html/rocket/basicdesign.php#radiators

Answer 2

variables and constants

$a$ - is the craft acceleration, in m/s²
$g_0$ - is the acceleration at the Earth's surface, in m/s²
$I_{sp}$ - is the specific impulse of engines, in seconds
$\Delta v$ - delta-v, value you have to define
$m_0$ - dry mass of the craft, kg

Equations

$\Delta v = g_0 I_{sp} \ln\left(\frac{m_{fuel}+m_0}{m_0} \right)$ - Tsiolkovsky rocket equation

Q/A

how much fuel would you need to carry?

$m_{fuel} = m_0\left(\exp \left(\frac{\Delta v}{g_0 I_{sp}}\right) - 1\right)$

what kind of weight-to-fuel ratio you'd end up

$\frac{m_{fuel}}{m_0} = \exp \left(\frac{\Delta v}{g_0 I_{sp}}\right) - 1$

well as what ratio would be ideal

cite from wiki specific impulse :

Theoretically, for a given delta-v, in space, among all fixed values for the exhaust speed the value $v_e = 0.6275 \Delta v$ is the most energy efficient for a specified (fixed) final mass, see energy in spacecraft propulsion.

Ideal what what ideal what?

Obviously fusion rockets are currently the most plausible and efficient method of propulsion we have in "harder" science fiction...

No, it is not obvious, it depends on the situation(goals) and definition of efficiency(which parameter is optimized).

An example energy wise mass-drivers are very efficient, as they use planets and other very massive bodies as reactive mass.

Because of conservation of momentum and connection between momentum and kinetic energy, kinetic energy in the system will be distributed not evenly:

$$\Delta v_{planet}M_{planet} = \Delta v_{craft}m_{craft} = p_0$$

$${E_p} = \frac{p_0^2}{2\cdot M_{planet}} ,\, {E_c} = \frac{p_0^2}{2\cdot m_{craft}} ,\, \frac{E_p}{E_c} = \frac {m_{craft}}{M_{planet}}$$

Obvious here is they have to have a massive body, but if the goal is to reach a certain speed, which may be perfectly fine for travel cross the solar system.

Can be the system be applied to the race it depends on rules and the goals of the race - if it is about to maneuvering in dense star war asteroid fields then with proper tech it is the only viable option because the owner will win the race.

If it is about of reaching $\alpha$-Centauri (or Oort cloud for the inner system race) it might be also a valid option.

Reactor energy

$\eta$ - engine efficiency, defined as $\frac {\text{kinetic energy of exchaust}}{\text{Total energy produced by reaction used in the engine}}$

$P_r$ - reactor power, total

$$P_r = \frac{1}{2} \left(g_0 I_{sp}\right)^2\cdot\dot{m}_{fuel}, \, \dot{m}_{fuel} = \frac{a}{g_0 I_{sp}}\cdot \left(m_0 + m_{fuel}\right)$$

$$ \begin{align} P_r &= \frac{1}{2\eta} a\cdot g_0 I_{sp} \left(m_0 + m_{fuel}\right) \\ &=\frac{1}{2\eta} a\cdot g_0 I_{sp} m_0 \exp \left(\frac{\Delta v}{g_0 I_{sp}}\right) \end{align} $$

For an thermonuclear rocket engine only charged products are useful, not charged and any gamma radiation during the reaction are the waste.

Thus, D+T, D+D are so so fuels, D+³He - also have its own problems(needs sophisticated setup, to prevent D+D react first, as D+D is faster reaction than D+³He)

$\eta$ for D+D fuel is less than about 0.66, $I_{sp}$ is about 826'000 seconds

So for one 100'000 tons (dry mass) ship, with acceleration of $g_0$, with delta-v 100km/s, D+D fuel

$P_r=$ 6e+15 W or 6000 TW

$m_{fuel} \approx 0.0124\cdot m_0$, or 1.24% of dry mass of the ship, or 1240 tons

The race will go on about 2h 50min, max distance covered in the time will be about 509'683 km with resulting speed 100 km/s.

Reactor size

ITER projected to be a 500MW reactor, plasma volume 840 cubic meters (https://www.iter.org/factsfigures)

So about 0.6 MW/m³, it is with D+T fuel. Lawson criterion for D+D is about 2 orders of magnitude higher(http://hyperphysics.phy-astr.gsu.edu/hbase/NucEne/lawson.html), it means when established stable state it has to produce 2 orders of magnitude more per volume to work, so we may expect higher output from a D+D reactor (it must if it works).

so for a 100'000 tons ship, with a 6000TW reactor, we might expect something like a cube $2150 \times 2150 \times 2150$ meters (or an equivalent volume of another shape), for a D+D reactor $460 \times 460 \times 460$ meters (or an equivalent volume of another shape).

Pro tip

Use lesser acceleration with thermonuclear engine or use massdriver catapult to put your cyborgs on the edge of their acceleration capabilities.