Nuclear fusion scaling with reactor size

I am an engineer, not a physicist. But Prokop's last equation suggests that particle confinement time scales with the cube of radius, a, but only in linear proportion to magnetic field strength, B. Also, as the reactor scales up, the radius of curvature of the tokamak walls increases, so the local magnetic field strength needed to steer particles away from the walls decreases. It should be possible to allow magnetic field strength to decrease with increasing scale, even as confinement time increases. This may negate the need for ever increasing current density in the field coils.

If a spherical vessel is large enough and can operate close to ambient pressure, it should be possible to build a fusion reactor that achieves breakeven without any magnetic field. Contamination of the plasma can be prevented by injecting cold deuterium gas through pores in the vessel of the chamber. The cold gas would form a sacrificial boundary layer that would interact with the plasma preventing the ions from impacting the walls. The plasma would develop a temperature profile, with the highest temperatures and fusion taking place at the centre. The only question is how large the vessel would need to be to achieve breakeven. If the answer is 100m in diameter, it is achievable. If it is kilometre scales, then it probably isn't.

Power density may be a key limiting factor. We need an economic power source; it needs to do more than simply work from a technical viewpoint. In my opinion, that is going to be the ultimate sticking point for nuclear fusion. It will be competing with other energy sources, especially nuclear fission.

You are referring to scaling laws for the energy confinement time ($tau_{E}$), which is a key performance parameter for a fusion reactor. For example, a stellarator currently has begin{equation} tau_{E} propto , a^{2.33} B^{0.85}, end{equation} where $a$ is the minor radius and $B$ is the toroidal magnetic field. This particular scaling is of the Bohm type, which is found during low confinement operation. During high confinement operation, an improved scaling of the gyro-Bohm type is present.

To answer your question, I will derive the origin of the above scaling using general principles (see sec. 7.6.4, here). Exponential degradation of confinement is generally assumed, which gives the following confinement time for particles in a cylindrical device with minor radius $a$ and length $L$, begin{equation} tau_E approx frac{N}{dN/dt}= frac{n pi a^2 L}{Gamma_{perp} 2 pi a L} = frac{n a}{2 Gamma_{perp}},, end{equation} where $N$ is the number of ion-electron pairs, $n$ is the number density and $Gamma_{perp}$ is the cross-field particle flux with diffusion coefficient D, begin{equation} Gamma_{perp}=- D ,nabla n,. %= v_{perp} n,. end{equation} The normalized density gradient scales with the machine size as $frac{nabla n}{n} propto frac{1}{a}$, giving begin{equation} tau_E propto frac{a^2}{D},. end{equation} Physically, the particle diffusion in strongly magnetized plasmas is carried by turbulence that is driven by gradients such as the ion temperature gradient or density gradient. This so-called drift wave turbulence can be analytically shown (see Eq. 21.39, here) to have a diffusion coefficient begin{equation} Dapprox frac{1}{k_{perp}a}frac{k_B T_e}{e B}proptofrac{1}{k_{perp}a}frac{T_e}{B} ,, end{equation} where $k_{perp}$ is the wavenumber of turbulent fluctuations perpendicular to the magnetic field.

In the worst-case scenario, the fluctuations occur on the scale of the minor radius due to global effects, $k_{perp}approxfrac{1}{a}$. This gives the Bohm diffusion, begin{equation} tau_E propto frac{a^2 B}{T_e},. end{equation}

In the best-case scenario, the fluctuations occur on the ion gyro-radius scale, $k_{perp}approxfrac{1}{rho_i}$, due to micro-turbulence that is much smaller than the machine size, where the ion gyro-radius is given by begin{equation} rho_i=frac{sqrt{k_B T_i m_i}}{e B},. end{equation} In this case, we get the gyro-Bohm scaling, which is more favorable by factor $frac{a}{rho_i}gtrsim 1000$, begin{equation} tau_E propto frac{a^2 B}{T_e} left(frac{a}{rho_i}right),. end{equation} Due to this very favorable scaling with size, ITER is projected to become the first machine to get 10 times more fusion power out than heating power in (with $^2H$+$^3H$), and you probably don't need to make the device several kilometers large for $^1H$+$^{11}B$ fusion.

In general, you are right. Make the plasma big enough, and for a given diffusion rate, the particles will be confined long enough to fuse before they exit of the plasma volume.

This is an extended comment on magnets.

Typically one of the more expensive components of a tokamak are the field coils. The bigger the tokamak, the bigger the coils. Also, for high performance tokamaks, you need a large field, say 5 T on axis. ITER's toroidal field is ~5 T on axis and ~12 T at the coils, with a minor radius of 2 m. ITER's magnets are barely doable with our technology.

Based on a simple current loop model, a 10x increase in magnet size results in a 10x increase in required current for the same on axis field. 10 times the current in the coil means an estimated 10 times the magnetic field near the coil as well. Assuming ITER's magnets are already at the limits of our technology, one that is 10x bigger with 10x the currents and even higher fields at the coil itself is not feasible technology. To put some perspective on that, I think the highest generated magnetic fields by humans are in the 100 T range. These setups are pulsed with tiny test volumes. Large scale steady state coils suitable for tokamaks are limited to 10-20 T.

Also, toroidal field strength is inversely proportional to major radius. Your maximum field is on the inboard side. Low aspect ratio tokamaks (small hole in the center) utilize field coils more efficiently than large aspect ratio ones (large hole in the center). Aspect ratio is major radius over minor radius. This leads to even larger magnets for a given major radius, and corresponding higher cost. This also has implications on plasma performance.

This probably sounds like just an engineering problem, but it is really about being limited by available magnet technology and the optimization that needs to be done for the right magnetic field structure. Some in the fusion community already view ITER as following the brute force method of "If you make it big enough, you'll get fusion."