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I've seen questions on how small you can make a tokamak. But I haven't yet seen any "physical" upper limit on the tokamak design.

If you take a wind turbine for example, doubling the linear dimensions will increase the sweep area by a factor 4 but the structural mass with a factor 8, which clearly explains why you don't want to make (convensional) wind turbines above certain dimensions.

With a tokamak, I imagine that if you double the linear dimensions, the plasma volume (and hence the power production) will increase eightfold, whereas area that you have to protect against fast neutrons will only quadruple. So once you master the tokamak technology, you would only need to scale it up appropriately to bring down capital costs.

What do I miss? What cannot be scaled up easily in a tokamak?

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The scaling of structural materials with linear dimension has come up several times on this site and I won't quite concede the claim that it scales with a factor of 8. The vanilla skyscraper problem actually gives an exponential. If you assumed the height didn't change with turbine blade length, then you would commit yourself to probably a simple $l$ factor or an exponential. Coincidentally, the vertical air speed profile is also treated as an exponential for boundary layer physics. Naturally, all said relations are wrong, but I don't see the basis for $l^3$ at all. –  AlanSE May 25 '12 at 14:31
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The big problem with controlled fusion is that the equations governing the plasma are highly non-linear. So each time the physicist increase the size of the Tokamak, new effects are discovered. So I guess that the answer is no-one really knows the correct scaling laws !

This contrasts a lot with fission reactors, where the relevant equations are essentially linear (neutron diffusion). It was then possible to 'easily' scale up Enrico Fermi's first nuclear reactor Chicago_Pile-1 which had a power of just 0.5W in 1942 to the design of the B reactor in 1944, which had a power of 250 Megawatt. That is essentially a factor 500 millions between the first and the second nuclear reactor !

EDITED TO ADD

I've just found this wikipedia page about Dimensionless parameters in Tokamaks which is quantitative. It essentially says that constructing a 1:3 model of a power producing tokamak having the same turbulence transport processes is essentially infeasible, because it would need a too high magnetic field. Then, there is a discussion I don't fully understand in order to try to guess the properties of the large machine... In short: the turbulence in the plasma make the use of scaling laws difficult.

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That's a very interesting observation, about non-linearity, which maybe also partly explains why it is taking so long time get fusion power viable as a commercial power source. –  Joel May 25 '12 at 16:18
    
Anyway, so the answer to my question about scale-up is that "it may very well be possible to make huge tokamaks, but we just don't know. in any case, a simple linear scale-up isn't possible due to the large magnetic fields required". That answers my question, thanks! –  Joel May 25 '12 at 16:23
    
@Joel: Actually, it seems (strangely) to be the opposite : it seems that building a small Tokamak is difficult, because the magnetic field would be too big ! Which explain why we have difficulties to make reduced models in order to validate the concept. –  Frédéric Grosshans May 25 '12 at 16:48
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You actually make reference to something which is of crucial importance to the answer to this question:

"With a tokamak, I imagine that if you double the linear dimensions, the plasma volume (and hence the power production) will increase eightfold, whereas area that you have to protect against fast neutrons will only quadruple. So once you master the tokamak technology, you would only need to scale it up appropriately to bring down capital costs."

You suggest that the fusion power of the tokamak scales roughly as $\sim R^{3}$ (where $R$ is the tokamak major radius), but the surface area inside the device onto which the fusion neutrons are incident only scales as $\sim R^{2}$.

This is fairly accurate, although the scaling of the fusion power is closer to $R^4$ or $R^5$, for reasons I will mention later. However, from the context of your remark it sounds like you're implying that this difference in scaling would make it advantageous to scale tokamaks up to an arbitrarily large size. The reality is quite the opposite in fact.

The fact that the internal surface area of the tokamak scales less aggressively with $R$ than the fusion power is quite possibly the most fundamental reason that we cannot build very large tokamaks. This is because the neutron flux on the inner wall of the device scales like the fusion power divided by the surface area, so roughly as $\sim R^2$.

The materials which line the inner wall of a tokamak can only withstand a particular fluence of fusion neutrons before they must be replaced, as the neutrons cause significant structural weakening. Replacing these components is an extremely time consuming and expensive affair, as it must be carried out entirely by remotely controlled robots due to unsafe levels of radioactivity inside the device. The interior wall of the JET tokamak was replaced recently and the project took over a year to complete.

So the the length of time for which you could run a tokamak fusion power plant before a major shutdown is required to replace the wall scales as $\sim R^{-2}$. Clearly this is a serious problem for a very large tokamak, as the inner wall will last an unfeasibly short time, making economically viable electricity generation impossible. After all, the goal of fusion energy research is to solve the looming energy crisis, so we must be able to produce electricity at a cost at least comparable to other renewable sources or there really isn't much point building a reactor in the first place!

Although this thread has been inactive for quite a while, you actually had the answer in the question without realising it so I felt the need to let you know!

A small aside: I mentioned earlier that the fusion power scales more like $R^{4}$. This is because a larger tokamak has a greater distance between the centre of the plasma and the wall, and this allows a higher core plasma pressure to be achieved. This in turn increases the fusion reaction rate, so as you increase $R$ not only do you have a greater plasma volume, but you're also getting more fusion power per unit volume.

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Thanks for this interesting answer. But this neutron flow argument does not hold anymore when considering aneutronic fusion, right? –  Joel Dec 6 '12 at 13:09
    
@Joel: When you say aneutronic fusion what reactions do you have in mind? The only reaction which will produce any meaningful power in a tokamak is the DT fusion reaction which does release a neutron. Probably the more important point is that we need the neutron to carry some of the energy released away from the plasma so that we can capture it as heat, which we then use to drive turbines. –  CBowman Dec 6 '12 at 15:02
    
I was thinking mainly of D/He-3, although I know that it's not really aneutronic. I guess for truly aneutronic reactons, like p/B-11 or He-3/He-3, you are not going to use the tokamak design anyway. But your point that the maximum size of a tokamak is coupled to the acceptable neutron radiation intensity (as well as the choice of material) is very valid, of course. –  Joel Dec 6 '12 at 19:17
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