In fusion power plants, larger devices are said to achieve fusion more easily, next step tokamaks like DEMO are even larger than ITER. In a quantitative way, what exactly isn't scale invariant in a fusion power plant and why?


To get a self-sustaining (burning) plasma, the triple product of plasma density $n$, ion temperature $T_i$, and energy confinement time $\tau_E$ needs to surpass a certain value: $$ n T_i \tau_E > F $$ This triple product, which is often referred to as Lawson criterion, tells us that we have three possibilities to achieve a burning plasma, since there are three factors. Physics, however, taught us that density and temperature are both limited for various reasons (stability, optimum temperature for the fusion process). Optimum values for the Deuterium-Tritium fusion, which is the easiest to achieve, are roughly $$ n \approx 10^{20}\,\mathrm{m}^{-3}, \quad T_i=15\,\mathrm{keV}. $$ Increasing those values further creates problems, so we need to get $\tau_E$ large. This is basically the time it takes for the energy to "leave" the plasma when the heating processes are switched-off.

Energy is lost through the surface, so in order to keep the energy inside, we need a large volume, as the surface increases with the radius squares whereas the volume increases to the radius cubed. This is similar to a mammal: there is a minimum size below which no mammal can exists due to the bad surface to volume ratio.

Since the energy confinement time is such an important quantity in magnetic confinement fusion, scaling laws had been developed using experimental data from dozens of tokamaks across the world. Doing a regression analysis, the energy confinement scaling can be simplified to $$ \tau_E \propto f_H V B^{0.8} P^{-0.6}, $$ with $V$ the plasma volume, $B$ the magnetic field, $P$ the applied heating power, and $f_H$ a factor characterizing the confinement quality (this is where the so-called H-mode comes into play).

A larger plasma volume obviously helps. And this is why tokamaks need to be big. Note that a related approach is to use higher magnetic fields, which seems to be possible due to novel, high-temperature superconducting magnets which can be operated at higher field strengths.

Also note that this question has been partly answered here or here or here.


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