Reply to ‘Comment “On the fusion triple product and fusion power gain of tokamak pilot plants and reactors”’

In reply to the Comment by Biel et al (2016 Nucl. Fusion 57 038001) on our recent papers Costley et al (2015 Nucl Fusion 55 033001) and Costley (2016 Nucl. Fusion 56 066003), we point out that the fusion triple product, nTτE, and fusion power gain, Qfus, cannot be expressed solely in terms of independent engineering design variables such as major radius, R, and toroidal field, B; output performance variables such as normalised beta, βN, safety factor, q, and fusion power Pfus, have to be invoked. Further, we show that the density limit has the effect of largely cancelling the size dependence in nTτE and Qfus, which would otherwise be present, when these parameters are expressed in terms of Pfus. Considerations of engineering aspects are also briefly discussed.


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Original content from this work may be used under the terms of the Creative Commons Attribution 3.0 licence. Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI. B and derive our expression, equation (1) in the Comment. The safety factor, q ∝ BR/A 2 I p , where I p is the plasma current, is an aggregate quantity; so several further valid expressions could be derived by appropriate substitutions. There is no mathematical argument for selecting one version over another; they are all valid. In a similar way different possible expressions for Q fus can be derived and the same argument applies for their validity. Any expression can be used but, of course, the performance limits have to be properly respected. They can be applied on the value of the input variables, or as selection criteria on the output calculations.
The Biel et al expression for nTτ E , equation (2) in the Comment, shows a strong dependence on R when expressed in terms of H, R, B, A and q, and demonstrates that nTτ E will increase with R when the values of the other variables are held constant. Because P fus ∝ β N 2 B 4 R 3 /(q 2 A 4 ), a scan in R with the value of the other parameters held constant leads to a simultaneous increase in P fus and Q fus . A good example of such a scan is shown in the paper by Zohm, figure 1 in [4]. The simultaneous increase of nTτ E and Q fus with P fus is inevitable from the conditions of the scan and illustrates the close relationship between these three quantities.
In our investigations, we carry out scans in a different way. Our goal is to find the minimum feasible device size for a set device performance; that is to achieve specified values of Q fus (or P fus ) for a given H factor. Hence we specify Q fus (or P fus ) and H and keep them constant during our scans. Since we want the highest possible performance at any size, we set both the density and the normalised beta to be at fixed (high) fractions of their limits at all points in the scan. This means that our scans represent a set of devices and each is operating at the highest possible performance from a physics perspective, but remains within physics limits. The feasibility of the devices depends, of course, on engineering and technological aspects. These were not included in our initial investigations but we are now extending our system code to include them (below). In contrast to our scans, in the scan carried out by Zohm the density tends to be low at small R and so the devices at this size are relatively under performing.
To illustrate this comparison we repeat the scan made by Zohm with the same fixed parameters that he used and also for the same device using our method; in both cases we use our system code. The results are presented in figure 1. The variation of P fus and Q fus with R is very different for the two types of scans. All operation points are valid at least from a physics perspective in both cases. The relative under performance of the devices at low values of R in the Zohm scan is clearly visible.
It is notable that in our scan nTτ E remains approximately constant with size, and so, from a physics performance perspective, there is no benefit in increasing the device size in terms of this parameter. Biel et al are correct that such a scan implies a reduction in field with R approximately according to B ~ R −3/4 Aq 1/2 / β − N 1 2 but not exactly at this level because the bootstrap current contributes to the current in the safety factor; a calculation of the bootstrap current is included in our code and typically is quite high (~50%). A much more interesting and important question is why is nTτ E approximately The field at R ⩽ 3 for the CHB size scaling is unrealistically high. For comparison we also show results for a low aspect ratio case (A = 1.8, β N = 4.5, H(IPB98y2) = 1.9) (in green). In this case the field is much lower, due to operation at higher β N , but is still a technical challenge. independent of plasma size in our scans given that τ E scales as ~R 2 ? One would have expected nTτ E to go as R 2 too. We submit that the reason is due to the impact of the density limit. Effectively the positive size scaling in the confinement time, which would otherwise lead to an increase of nTτ E and Q fus with size, is significantly diminished due to this limit.
At the simplest level one can understand this by recognising that the density limit scales as ~1/R 2 . Hence, for operation at fixed (high) fractions of the density limit, which is where the highest plasma performance is achieved, nTτ E will not change significantly with size. Effectively the inverse scaling of the density limit negates the positive size scaling in τ E . The situation is actually more complicated because the density limit goes as I p /R 2 and I p has to increase with size in order to keep q > 2. This is handled in the analysis in our paper [3].
It is possible to gain additional insight by repeating our analysis but using a generalised form for the size scaling in the density limit; n lim ∝ I p (A/R) y . One finds: Of course, when the empirical vale of y = 2 is inserted the former result is recovered. This shows the significant impact of the density limit: when nTτ E is expressed in terms of P fus , the residual size dependence becomes very weak. We note also that it effectively eliminates any dependence on A. Similarly, Q fus becomes only weakly dependent on size and has no dependence on A when expressed in terms of P fus . Hence the density limit has the effect of eliminating any significant size impact in terms of the global performance parameters, P fus and Q fus , which are the principal performance parameters of a tokamak fusion reactor. The size scalings in τ E and the density limit are not obviously related so this is almost certainly a coincidence but it does have significant consequences for the design of tokamak pilot plants and reactors, as we discuss in our papers. Turning to the engineering and technological aspects, which Biel et al highlight and which are certainly important. Our specific goal and unusual way of carrying out our performance scans, has led us to develop a novel approach for investigating the engineering aspects. Our approach is to bring into our system code performance param eterisations of the main critical elements, and then for a fixed aspect ratio and set plasma performance param eters (Q fus , P fus , H factor), search for the smallest device that satisfies all engineering requirements by adjusting the radial build for the optimum engineering solution. Since writing our papers [2,3], parameterisations of the engineering aspects of key elements of the central core, which is the most critical area for relatively small size, low A, devices, have been included in the code; particularly, the attenuation of the neutron flux and associated heat deposition in the central core due to an inboard shield (based on MCNP calcul ations of the effectiveness of candidate shield mat erials), estimates of the power requirement of the associated cryoplant, and estimates of the peak crushing stress based on a simplified model. Estimates of divertor loads are also made but not yet fully integrated into the code. For the magnet, we are specifically considering high temperature superconductors (HTS), which can provide relatively high current density and withstand relatively high fields. Key performance parameters of the HTS tape, for example the maximum field on the conductor, are included as boundary conditions. With these additions we can begin to optimise the design of the central core.
The recent developments in the code and some example optimisation results are described in a paper presented at the recent IAEA Fusion Energy Conference [5]. Further code developments are planned and are described in our paper. Thus far encouraging results have been obtained for a device with R 0 ~ 1.5 m and A = 1.8. We believe that these justify invest igation with a more advanced code and this is something that we seek to do in the near future.