Thinking about the physics of thermonuclear fusion, I have always had an intuitive sense that making fusion feasible is matter of reactor size.
In other words I feel like:
If the fusion reactor is big enough you can achieve self-sustaining nuclear fusion of $^2$H+$^3$T but perhaps also of $^1$H+$^{11}$B (even if it means that such a device should be several kilometres large).
Some arguments on why it should be so:
- Energy is generated by volume while losses should be proportional to surface (this is probably not true for TOKAMAKs where plasma is not optically thick for bremsstrahlung X-ray, but it is true for inertial confinement)
- Big stars can burn almost any fusion fuel because the released energy cannot escape from its core very quickly. Can a similar effect be used in a practical device? (like a TOKAMAK with $1~\mathrm{km}$ toroidal vessel)
- In magnetic confinement many problems are connected with magnetic field and temperature gradients leading to Rayleigh–Taylor-like instability. If the reactor is larger these gradients are smaller.
- History says that TOKAMAKs are made bigger over time in order to achieve breakeven. I understand the practical point that a big plasma vessel is expensive so people try to make it as small as possible. But if the cost of one device wasn't an issue, would it be possible (based on just the same physics and scaling law) to build a large TOKAMAK that can burn $^1\mathrm{H}+^{11}$B fuel?
I was searching the literature to get some general idea about scaling laws for nuclear fusion. I found several different empirical expressions for TOKAMAKs, how it scales with radius of torus, temperature and magnetic field, however it was quite specialized and device specific (there was no single general expression).
I would rather like to get just a very rough idea about the scaling as general as possible, and derived from basic physical principles.
Best Answer
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} D\approx \frac{1}{k_{\perp}a}\frac{k_B T_e}{e B}\propto\frac{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, $k_{\perp}\approx\frac{1}{a}$. 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, $k_{\perp}\approx\frac{1}{\rho_i}$, 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.