PSFC/RR-18-4 Comparing Different Scalings of Parallel Heat Flux with Toroidal Magnetic Field [q with BT] M.L. Reinke February, 2018 Plasma Science and Fusion Center Massachusetts Institute of Technology Cambridge MA 02139 USA This work supported by DOE contract: DE-AC05-00OR22725. Reproduction, translation, publication, use and disposal, in whole or in part, by or for the United States government is permitted.
Comparing Different Scalings of q with B T M.L. Reinke Oak Ridge National Laboratory, Oak Ridge, TN 37831, USA E-mail: reinkeml@ornl.gov 2/13/2018 Abstract. High field tokamak concepts are being explored as a means to accelerate the development net energy production from magnetic confinement fusion. One aspect of these designs currently being investigated is the impact of size and field on the heat exhaust and as well as solutions that use impurity radiation or interaction with high pressure neutral clouds. In [1], the neutron wall loading was used to argue that q B T R while in [2], the H-mode threshold was shown to lead to q BT 2.52 R 0.16. Both are numerically correct, but it is shown that scaling q using a fixed neutron wall flux, q nw, does not capture expected exhaust scenarios. At low field, this leads to cases where the power into the scrape-off-layer, P SOL, is well in excess of that needed to sustain the pedestal, thus over burdening the divertor. At higher fields, eventually more power is required to get into H-mode than is obtained from P SOL based on neutron wall loading. It is argued that q and q nw should be seen as independent, although related parameters, with the former being set by the pedestal needs and exhaust solution and the later being set by the reactor economics and material capabilities. To further utilize this threshold approach to q, recent estimates of the L/I-transition scaling from [3] are used to derive q,i BT 2.25 R 0.00 while f z,i BT 0.57 R 1.14, following the procedure of [2]. This results in the same conclusion as reached for H-mode, that a 0-D optimization for detachment in I-mode is aligned with optimizing the core in terms of maximizing the field before increasing the device size. Submitted to: PSFC Library 1. Section 1 The present focus for the feasibility of the high-field path to fusion lies primarily in the design and demonstration of magnets that use high temperature superconductors increase the maximum on-axis field. Core scenarios with relatively modest nondimentional parameters [4] are expected. The field scaling of the exhaust problem has received arguably less attention, with 0-D scalings or estimates used to try and scope if this known problem gets better or worse at higher field. Two approaches which have been used in the literature are quantitatively compared. In [1], the neutron wall flux is used along with the Eich scaling [5] for the upstream heat flux width, resulting in q B T R. In [2] the Eich scaling is also used, but with the power into the scrape
Comparing Different Scalings of q with B T 2 off-layer, P SOL set by the some margin above the Martin L-H threshold [6], leading to q BT 2.52 R 0.16. In Section 2, the implications of each are compared, arguing that q should be scaled using a threshold power. In Section 3, the scaling approach used for H-modes repeated using an estimate for the L/I threshold to show that the f z to induce detachment scales favorably with magnetic field. 2. Section 2 The methodology in [2] is summarized, using P SOL that set by the L/H threshold and then compared with P SOL set by the neutron wall loading. The upstream parallel heat flux is q = P SOLB (1) 2πRλ q B p where no assumption is made regarding how much is spread across how many divertors. This means quantitative values of the heat flux are systematically higher than would be encountered in practice, but this does not impact the comparison that is the focus. Using the Eich scaling for λ q [5] and ignoring small exponent terms this reduces to q = 0.1179 P SOLB T (2) Rε 0.42 where q is in GW/m 2. Using the L/H threshold to replace P SOL, detailed in [2], results in q,h = 0.112f LH ( fgw q ) 0.72 ε 0.52 ( 1 + κ 2) 1.19 B 2.52 T R 0.16 (3) The neutron wall loading constraint means that P SOL = 0.25P N (1 + 5/Q fus ), where P N is the power carried by neutrons which can be approximated as P N = q nw S wall where q nw is the neutron wall flux in MW/m 2 and for simplicity we ll assume S wall = S plasma = 2 2π 2 R 2 ε (1 + κ 2 ) 0.5. Carrying through the algebra results in q,wl = 0.822 (1 + 5/Q fus ) ( 1 + κ 2) 0.5 ε 0.58 q nw B T R (4) where q,wl B T R is recovered, but now there is a exact relation, allowing quantitative comparison to q,h. To examine the benefits of size and field, it is common to look at some fusion relevant target like the triple product and claim since nt τ BT a R b, that R can be replaced by B a/b. For [1], a = 3.0 and b = 1.3 and (4) reduces to q,wl BT 1.3. while q,h BT 2.15. These two results indicate qualitatively opposing trends, worth further investigating further. Taking the ratio of the two terms results in q,wl q,h = P SOL,wl = 7.34 (1 + 5/Q fus) q nw (1 + κ 2 ) 0.69 (q /f gw ) 0.72 0.06 R0.84 ε f LH BT 1.52 Different first generation reactor concepts can be used to complete the terms in (5). For the ARC concept, these values are given in Table 1 in [7]. There is no f LH assumed, but this taken to be f LH = 1.2 in all cases for consistency. This results in P SOL,wl = 2.77. For the EU-DEMO1 and DEMO2 concepts, these values are in Table 1 of [8] and result (5)
Comparing Different Scalings of q with B T 3 DEMO1 DEMO2 ARC TTA-1 TTA-2 B T [T] 5.7 5.6 9.2 12 12 Q fus 41 24 13.6 3.6 2.0 q wl [MW/m 2 ] 1.05 1.91 2.5 1.7 1.1 ε 0.32 0.38 0.34 0.30 0.30 R [m] 9.1 7.5 3.3 1.65 1.65 κ 1.5 1.8 1.84 1.8 1.8 f LH 1.2 1.2 1.2 1.2 1.2 f GW 1.2 1.2 0.67 0.51 0.26 I p [MA] 19.6 21.6 7.8 7.5 7.5 q 2.24 3.34 4.9 2.57 2.57 Ratio (H-Mode) 1.89 3.37 2.77 0.95 1.47 Ratio (I-Mode) 0.97 1.69 1.82 0.55 1.02 Table 1: Inputs used in calculating ratios of P SOL necessary for neutron wall loading and to sustain confinement regime for H-mode (5) and I-mode (6) in in P SOL,wl = 1.89 and 3.37, respectively. Two operating points for a Twelve Tesla ASDEX (TTA) are given in [4] and results in P SOL,wl = 0.95 and 1.47 for the high Q fus and high P loss /P thresh cases, respectively. The inputs used are summarized in Table 1 For EU-DEMO1, EU-DEMO2 and ARC, this ratio is well over unity, meaning that maintaining the H-mode pedestal takes much less power than will be generated to hit the Q fus and q nw targets. Moving to more higher q nw makes this even worse. Thus, mantle radiation using a high-z impurity like xenon can be used to reduce the power crossing the separatrix without impacting the dilution. For this reactor design space, the q,wl scaling should not be used to estimate the scaling of the heat exhaust. For the TTA designs, the ratio is closer to unity, and might even be < 1.0 depending upon the specific requirement to operate at f LH > 1. When P SOL,wl < 1.0 occurs, this means that the power flux crossing the last closed flux surface is likely to be insufficient to maintain the H-mode, despite meeting the Q fus and q nw requirements. This is an indication that the field is too high for the configuration since it demands a higher neutron wall loading. There is also a risk that changes or uncertainty in the radiated power due to intrinsic impurities or estimations of synchrotron losses put the core device mission at risk. In Section 3 the derivation of the q required to sustain the I-mode regime is derived and following the same assumptions the ratio for the wall loading can be derived to be q,wl q,i = P SOL,wl (1 + 5/Q fus )q wl R = 1.86 P SOL,I (1 + κ 2 )f LI (f GW /q ) BT 1.25 For the same data used in Table 1 and somewhat arbitrarily, f LI = 1.2, this results in the ratio below unity for DEMO1, and both TTA designs. The results are summarized graphically in Figure 1 with the clear trend that more loss power is required for I-mode, (6)
Comparing Different Scalings of q with B T 4 4 H-Mode DEMO2 3 I-Mode ARC P SOL,wl /P SOL,X 2 DEMO1 TTA-2 1 TTA-1 0 0 2 4 6 8 10 12 B T [T] Figure 1: Graphical summary of results from (5) and (6) using data from Table 1 thus making designs closer to requiring nearly all the power characterized by the wall loading to sustain the confinement regime. These examples indicate how assuming the parallel heat flux scales along with the neutron wall loading is ill-advised since it does not capture what is sufficient to sustain the confinement regime. Looking at the ratio of P SOL,wl with the P SOL for the relevant confinement regime and ensuring that this is greater than unity, ensures that the neutron wall flux will be an upper limit to the exhaust power, giving the reactor design contingency against unexpectedly high core radiation. While not comprehensive, there is a general trend, expected from the 1/BT 1.52 dependence in (5), that higher field regimes will generally be pushed to smaller values of P SOL,wl /. 3. Section 3 The methodology used in [2] to derive q and the impurity fraction, f z for H-modes is not exclusive to that confinement regime. If a scaling for the power needed to sustain the pedestal is known or can be estimated to be within an order unity constant of a known scaling this can be included to look at the results for I-mode. Qualitatively, I-modes run with topologies in the so-called unfavorable B drift direction exist with P SOL above some L/I threshold and below the I/H threshold. The latter is generally estimated to be twice the conventional L/H threshold, which if one used the Martin scaling [6], then (3) and (10) in [2] are multiplied by 2.0 and 2.2, respectively for an upper limit on the I-mode q and f z. New, multi-machine results from [3] suggest the
Comparing Different Scalings of q with B T 5 L/I threshold may scale as P SOL = 0.2 n e S (B T /2.0) 0.25 (7) where P SOL is in MW, n e is in 10 20 m 3. Following [2] as done for the L/H scaling, the density is replaced with the Greenwald fraction and the current replaced by the external kink limit. This results in P SOL = 3.76f LI f GW q ( 1 + κ 2 ) 1.5 B 1.25 T εr (8) which when inserted into (1) results in a parallel heat flux, in GW/m 2, of q,i = 0.440f LI f GW q ε 0.58 B 2.25 T (9) and an impurity fraction to induce detachment of f z = 0.069 BT 0.57 fli 1.14 R 1.14 ε 0.66 fsep 2 fgw 0.86 (1 + κ 2 ) 0.29 ˆl0.86 m L (Z, n e τ) making f Z,I BT 0.57 R 1.14 while f Z,H BT 0.88 R 1.33 as shown in [2]. The I-mode f z scaling also exhibits no dependance on q and weaker dependance on f GW than H-mode. With a defined scaling of nt τ with field and current, using the derivation from [1], this results in f Z,I BT 2.1 and f Z,H BT 2.2 and f Z,I R 0.89 and f Z,H R 0.95. Thus the basic interpretation of this analysis holds for I-mode as well as H-mode, that optimization of detachment scaling is aligned with the core in maximizing field prior to increasing the device size to reach the desired triple product. Since first published in [2], this result for H-modes has been confirmed with more detailed core simulations with a similar boundary model [9]. (10) 4. Acknowledgments Thank you members of the MIT-PSFC team for useful, on-going discussions regarding the optimization of detachment in high-field reactor concepts. This work supported by DOE contract: DE-AC05-00OR22725 5. References [1] D. Whyte, et al. J. Fusion Energ., 35:41, 2016. [2] M.L. Reinke. Nucl. Fusion, 57:034004, 2017. [3] A. Hubbard, et al. Submitted to Nucl. Fusion, 2017. [4] R. Mumgaard, et al. Scoping study for compact high-field superconducting net energy tokamaks. In APS-DPP Meeting Abstracts, 2016. [5] T. Eich, et al. Nucl. Fusion, 53:093031, 2013. [6] Y.R. Martin, et al. J. Phys: Conf. Ser., 123:012033, 2008. [7] B.N. Sorbom, et al. Fusion Eng. Des, 100:378, 2015. [8] R. Wenniger et al. Nucl. Fusion, 57:016011, 2017. [9] M. Siccinio, et al. Nucl. Fusion, 58:016032, 2018.