Fable, E.

Koechl, F.

Lang, P.

Maviglia, F.

Pautasso, G.

Siccinio, M.

English

MPG.PuRe

Modelling of shattered pellet ablation: a discussionG. Pautasso1, E. Fable1, F. Koechl2, P. Lang1, M. Siccinio3 and F. Maviglia3(1) Max-Planck-Institut für Plasmaphysik, 85748 Garching, Germany(2) CCFE, Cuhlam Science Centre, Abingdon, OX143DB,United Kingdom(3) EUROfusion Programme Management Unit, 85748 Garching, Germany1. IntroductionThe electron density (ne) required to unconditionally suppress the runaway electrons(REs), generated during the disruption of a reactor-relevant tokamak plasma, is 2-3orders of magnitude larger than the ne of the target plasma. An estimate for a worstcase ITER plasma disruption at full current and magnetic field can be found in [1]. Thedensity needed to reduce the RE seed and the final RE current to an acceptable level (e.g.< 1 MA) is smaller, i.e. O(10 - 100)×ne, and it was calculated by R. Martin-Solis forITER [2] and for DEMO plasmas [3]. Shattered pellet injectors [4] are being consideredfor disruption mitigation and particularly for RE suppression in ITER; it is also beingdebated, whether they can fulfill this purpose or not.In modelling the RE generation, the density increase can be imposed with a Heavisidefunction. Nevertheless, in the reality, the matter must be injected into the plasma inthe form of pellets from the edge and must cross the plasma to reach its center. Thespatial distribution of the material deposited in the plasma is determined by mostly-known physics mechanisms and it depends on pellet and plasma parameters. Since thedeposited matter cools the plasma, induces the fast growth of tearing modes and causesthe thermal quench, it must be delivered in the plasma within the so called pre-thermalquench time interval, which lasts a few milliseconds. The pellet velocity depends on thetype of injector, it is limited and it ranges from a few 100s m/s to a few km/s.This contribution presents a discussion of what is required, in terms of number of frozendeuterium pellet, of pellet radius and pellet velocity, to increase the plasma density of afactor of (let’s say) 20 within a few milliseconds over the whole plasma cross section ofa DEMO-like plasma [5]. The Neutral Gas Shielding (NGS) ablation model [6] is usedin the following section for parametric studies of the ablation of deuterium pellets. Asimple model for the density increase following multiple pellet injection is discussed insection 3. More detailed simulations of the ablation of one single deuterium pellet havebeen carried out with the HPI2 code [7] and are briefly discussed in section 4.2. Parametric study of matter deposition with the NGS modelAlthough it is oversimplified, the NGS model predicts the parametric dependence ofthe ablation rate of pellets in existing devices over a wide range of plasma and pelletparameters. Different additional ablation mechanisms, not included in the model, havebeen found to cancel out [8]. According to the model, the ablation rate of a sphericalpellet is described bydNpdt= C∗ n1/3e r4/3p T5/3e , C∗ = 4.12× 1016 [9] (1)Np =43πηpr3p,dNpdt= 4πηpr2pdrpdt(2)146th EPS Conference on Plasma Physics P4.1045or equivalently bydrpdt= C n1/3e r−2/3p T5/3e , C =C∗4πηp, rp(r) =(rp(a)5/3 − 53Cvp∫ ran1/3e T5/3e dr)3/5(3)∆ne(r) ≡ ∆Np(r)∆Vol=dNp(r)ZZdrZZdrdt14π2rR0kvp(4)Np in the number of deuterium atoms in the pellet of radius rp; ηp is the pellet numberdensity (6 × 1028 atoms/m3 for frozen deuterium); vp is the pellet velocity; Te is thebackground temperature (expressed in eV throughout the paper) at the radial positionr; a and R0 are the minor and major radius respectively; k is the plasma elongation; ∆neis the increase of the density on the flux surface.Fig. 1 shows the density increase achieved by injecting a large D2 pellets (rp = 2.1 cm andNp = 2.3×1024 atoms) into a burning DEMO plasma (ne(0) = 1020 m−3, Te(0) = 40 keV).The pellet velocity was scanned from 500 m/s up to 10 km/s, velocity at which the pelletreaches ρ = r/a ∼ 0.2. Therefore, according to the NGS model, one single deuteriumpellet, able to increase the electron density by a factor of circa 20, must be launched ata velocity higher than 10 km/s in order to reach the plasma center. This upper velocityis extremely large if compared to the velocity of the present cryogenic pellets (300-1200m/s) or e.g. to the muzzle velocity of selected pistols and rifle guns (<1400 m/s [10]),and it is probably not realizable. Increasing only rp at a realistic velocity, e.g. 1200 m/s,does not foster the deposition of the ablated material beyond ρ ∼ 0.5 (calculation notshown). These results suggest to investigate the injection of multiple pellets.During the current quench of a disruption, the electric field is maximum in the plasmacore, where the toroidal current is the largest. Therefore, the density increase (and thematerial deposition) required to suppress the REs, must take place in the plasma center.The pellet penetrates up to the plasma center when the following condition is satisfiedrp(0) = 0 → rp(a) = Te(0) 53Cvp∫ 0an1/3e(TeTe(0))5/3dr3/5∝ Te(0) (5)(from eq. 3). Let’s assume that the Te of the target plasma can vary in the range 1 - 40keV and that the pellets do not interact with each other while ablating. It follows that adisruption mitigation system, built to inject simultaneously a number Npellets of pellets,must be designed to vary this number within a factor of 6.4×104 according to the targetTe. In factNpellets = Nneeded/Np ∝ Te(0)−3, [(maxTe(0))−3, (minTe(0))−3] ∝ [1, 64000] (6)Eq.s 5 and 6 are strong conditions aimed to optimize the use of the injected gas (i.e.Nneeded) and minimize the gas inventory in the torus and pumping/exhaust system. Theassumption, that many pellets can ablate simultaneously, is not realistic and the ablationof a packet of pellets is considered in the following.3. Multiple pellet injectionA realistic injection scheme will not be simultaneous or sequential but a combination ofthe two. Let’s assume that a number Np of pellets are launched as a rectangular packetof dimension D×D×L, and that the ablation of the single pellet is not influenced by thepresence of the other pellets. Then, the rate of density rise on the flux surface becomes246th EPS Conference on Plasma Physics P4.10450 0.5 1 (adim)77.588.599.510ne (m-3)10 190 0.5 1 (adim)01234Te (eV)10 40 0.5 1 (adim)00.0050.010.0150.020.025r p (m)0 0.5 1 (adim)01234 ne (m-3)10 210 0.5 1 (adim)0123456 tfly(s)10 -3vp = 0.5 - 10 (km/s)rp(a) = 2.1 cmNp = 2.3e24 atomsFigure 1:Radial profiles(ρ = r/a) ofthe densityincrease (∆ne),of the pelletradius (rp) andthe penetrationtime (∆tfly) forpellets of dif-ferent velocities(vp).0 0.5 1 (adim)00.511.522.53ne (m-3)10 210 0.5 101002003004005006007008009001000Te (eV)0 0.5 1 (adim)00.511.522.53r p (m)10 -4Figure 2: Time evo-lution of the den-sity (ne), of the tem-perature (Te) and ofthe pellet radius (rp)following the injec-tion of Np = 8.4 ×105 (rp = 0.29 mm)pellets.0 0.5 1Norm. minor radius050100150200250300350ne [1019 m-3]0 0.5 1Norm. minor radius050100150200250300ne [1019 m-3]Figure 3: Three cases of final density (ne) calculated by the HPI2 code after the abla-tion and assimilation of one massive deuterium pellet in a DEMO-like plasma (Np =1.0, 1.3, 1.5× 1024 deuterium atoms) without (left) and with (right) plasmoid drift.346th EPS Conference on Plasma Physics P4.1045∂ne∂t=dNpdtnpD2∆r4π2rR0k∆r=dNpdtNp4π2rR0kL, np =NpD2L(7)The temperature on the flux surface can be recalculated assuming instantaneous matterredistribution and thermal equilibration of the electrons: ne(r, t)Te(r, t) ' ne(r, 0)Te(r, 0).Calculations of ∆ne following the injection of multiple pellets were carried out for differentTe(0). Fig. 2 illustrate the case with Te(0) = 1 keV: a packet of Np = 8.4 × 105 pelletswith vp = 1200 m/s and rp = 0.29 mm could increase the core ne of a factor of circa 20.The same total density increase and pellet penetration up to ρ = 0.05 (final ne(r) morepeaked toward the center in this case; not shown) is obtained with Np = 837 (rp = 2.9mm) pellets into a plasma with Te(0) = 10 keV. These calculations confirm the scalingrp(a) ∝ Te(0).4. Modelling massive pellet ablation with HPI2The HPI2 pellet code was used to simulate the ablation of single massive deuteriumpellets in DEMO-like plasmas. Fig. 3 shows the time traces of the evolution of thedensity profile for three different pellets with Np = 1.0, 1.3, 1.5 × 1024, vp = 1200 km/s,into the DEMO-like plasma of fig. 1 and 2 (Te(0) = 40 keV) without (left) and with (right)drift. If the calculation of the plasmoid drift is suppressed, the largest pellet generates a∆ne(0) ∼ 27 × 1020 close to the plasma center. Nevertheless, the strong outward drift,due to the high temperature of the target plasma, prevents the penetration of the ablatedmaterial. The NGS model predicts a penetration of the largest pellet up to ρ = 0.65.Therefore, this benchmark shows that the NGS model is not adequate enough for thecalculation of the density increase in a DEMO-like plasma.5. Discussion and outlookThe ablation of cryogenic deuterium pellets and the deposition of matter in a plasmais determined by pellet (material, radius, velocity and number) and plasma (dimension,electron density and temperature) parameters. Most of these parameters are either fixedor constrained. For a given device, the plasma dimension, the density, the plasma current- and therefore the density increase required for RE suppression - are fixed. The maximumpellet velocity is limited by the injector technology and the minimum by the pre-thermalquench duration; reasonable velocities for reactor relevant plasmas are in the range 1-1.5 km/s since they translate into a pellet transit time from edge to plasma center - of2-3 ms. Essentially, only the pellet radius, and consequently the number of pellets, canand must be varied to obtain the required density increase. The NGS model indicates astrong dependence of the ablation rate on the plasma temperature. Since the temperatureof the target plasma - to be shut-down by the disruption mitigation system - can varyconsiderably (e.g. by a factor of 40 in DEMO, and rp(a) must vary proportionally, seeeq. 6), the injection system must be extremely (unrealistically?) flexible.Future work will address the validity of the NGS model (by benchmarking it with theHPI2 code and against experiments), the validity of the non-interacting multiple pelletinjection model and the presence of impurities in the plasma or/and in the pellets.References[1] T C Hender et al. Progress in the ITER Physics Basis Chapter 3: MHD stability, operational limits and disruptions 2007 Nucl. Fusion 47S128-S202[2] J.R. Martin-Solis et al., Nuclear Fusion (2017) 57 066025[3] R. Martin Solis et al., DEMO report (2018)[4] L. Baylor et al., 27th IAEA Fusion Energy Conference FIP/P1-1 (Gandhinagar, 2018)[5] G. Federici et al., Fusion Eng. Des., 136 72941 (2018)[6] P.B. Parks et al., Phys. Fluids 21 10 (1978) 1735[7] F. Koechel et al., Modelling of Pellet Particle Ablation and Deposition: The Hydrogen Pellet Injection code HPI2, Report EFDA-JET-PR(12)57[8] B. Pegourie, Plasma Physics and Control Fusion, 49 (2007) R87-R160[9] K. Gal et al., Nuclear Fusion 48 (2008) 085005[10] https : //en.wikipedia.org/wiki/Table of handgun and rifle cartridges446th EPS Conference on Plasma Physics P4.1045

Modelling of shattered pellet ablation: a discussion

http://hdl.handle.net/21.11116/0000-0008-777D-7

Modelling of shattered pellet ablation: a discussion

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