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ARTICLE IN PRESSG ModelUSION-7265; No. of Pages 6

Fusion Engineering and Design xxx (2014) xxx–xxx

Contents lists available at ScienceDirect

Fusion Engineering and Design

jo ur nal home p age: www.elsev ier .com/ locate / fusengdes

imulation of neutral gas flow in a tokamak divertor using the Directimulation Monte Carlo method

ristian Gleason-González ∗, Stylianos Varoutis, Volker Hauer, Christian Daynstitute for Technical Physics, Karlsruhe Institute of Technology, 76344 Karlsruhe, Germany

i g h l i g h t s

Subdivertor gas flows calculations in tokamaks by coupling the B2-EIRENE and DSMC method.The results include pressure, temperature, bulk velocity and particle fluxes in the subdivertor.Gas recirculation effect towards the plasma chamber through the vertical targets is found.Comparison between DSMC and the ITERVAC code reveals a very good agreement.

r t i c l e i n f o

rticle history:eceived 16 September 2013eceived in revised form 3 February 2014ccepted 4 February 2014vailable online xxx

a b s t r a c t

This paper presents a new innovative scientific and engineering approach for describing sub-divertorgas flows of fusion devices by coupling the B2-EIRENE (SOLPS) code and the Direct Simulation MonteCarlo (DSMC) method. The present study exemplifies this with a computational investigation of neutralgas flow in the ITER’s sub-divertor region. The numerical results include the flow fields and contours of

eywords:ivertor gas flowsSMCacuum flows

the overall quantities of practical interest such as the pressure, the temperature and the bulk velocityassuming helium as model gas. Moreover, the study unravels the gas recirculation effect located behindthe vertical targets, viz. neutral particles flowing towards the plasma chamber. Comparison betweencalculations performed by the DSMC method and the ITERVAC code reveals a very good agreement alongthe main sub-divertor ducts.

arlsr

nudsen number © 2014 K

. Introduction

In future fusion reactors efficient particle exhaust will beeeded so as to sustain long-pulsed operation. Thus, fusion producthelium) removal, impurity and density control are critical issuesor any reactor. In ITER and also in DEMO, a pumped divertor will besed to maintain the essential vacuum conditions and to provideood particle control. Moreover, one of the key issues affecting theerformance and achievable burn time of a fusion reactor is theontrol of the gas throughput that contains the unburned fuel, viz.he helium ash produced in the D–T fusion reactions and impuri-ies generated via plasma-wall interaction. Thus, it is important totudy the mechanism of particle exhaust and its direct effects onhe plasma performance.

Please cite this article in press as: C. Gleason-González, et al., SimulaSimulation Monte Carlo method, Fusion Eng. Des. (2014), http://dx.do

Also from a technical point of view, especially with regard tohe exhaust vacuum pumping systems, the understanding of theub-divertor gas flow in different tokamak operational scenarios

∗ Corresponding author. Tel.: +49 72160828105.E-mail address: cristian.gleason@kit.edu (C. Gleason-González).

ttp://dx.doi.org/10.1016/j.fusengdes.2014.02.005920-3796/© 2014 Karlsruhe Institute of Technology. Published by Elsevier B.V. All rights

uhe Institute of Technology. Published by Elsevier B.V. All rights reserved.

is of vital importance. However, the description of gas flow in thedivertor and the vacuum systems for fusion devices is a challengingtask since the flows cover a wide range of the Knudsen (Kn) number.Starting from continuum and slip regimes in the plasma and neutralgas sources, e.g. the divertor private region and along the divertorsurface, covering transitional flow in the sub-divertor ducts andending up in the free molecular flow regime inside the cryopumps[1]. A reliable assessment of the macroscopic properties of such acomplex system therefore requires a tool to describe the flow inthe whole range of gas rarefaction.

In this paper, the neutral helium gas flow behavior throughthe ITER divertor is investigated by coupling two well-establishedand reliable numerical approaches, namely the B2-EIRENE (SOLPS)code package [2,3,5] and the direct simulation Monte Carlo, orDSMC, method [4]. The former comprises a two-dimensional fluiddescription for the ions and the electrons of the plasma edge and athree-dimensional kinetic Monte Carlo neutral model whereas the

tion of neutral gas flow in a tokamak divertor using the Directi.org/10.1016/j.fusengdes.2014.02.005

latter is a particle-based stochastic numerical approach, which cir-cumvents the Boltzmann equation by simulating group of modelparticles that statistically mimic the behavior of real molecules. Inthe past, neutral flows in the divertor region have been modeled

reserved.

IN PRESSG ModelF

2 ngineering and Design xxx (2014) xxx–xxx

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sing the B2-EIRENE code [5]. Nevertheless, within this approachhere is no clear prescription on how to select the grid for bothhysical and velocity spaces, crucial for any kinetic approach. Onhe other hand, in the DSMC method neither the grid in the velocitypace nor the finite difference scheme are necessary. Complemen-ary to DSMC, B2-EIRENE can simulate ion-molecule collisions.owever, a crucial shortcoming of the BGK approximation of

he Boltzmann equation, which is used in the B2-EIRENE, is thempossibility to fit simultaneously the correct values of viscosityoefficient and thermal conductivity by means of a single param-ter. This means that the Prandtl (Pr) number is equal to unity,hich of course differs from the correct value equal to 2/3. For-

unately, the DSMC method is capable of reproducing the thermalonductivity of flows up to several thousands kelvin [6], showing itsapability of calculating transport coefficients for both isothermalnd non-isothermal flows.

The present work provides numerical results of 2D DSMC cal-ulations of the flow field in the ITER sub-divertor geometry. Theesults include detailed information of overall quantities of prac-ical interest assuming neutral helium as model gas. Based on theresent numerical approach, an insight of the helium flow recir-ulation is performed for ITER scenarios, namely a case where theivertor’s total gas pressure is 2.6 Pa and a second high-pressurease with 9.9 Pa. It should be stressed that in the foreseen ITERcenarios, helium particles will represent only a fraction of theotal exhausted gas, being D or T the main gas component. There-ore, it is expected that collisions between helium gas and theulk (deuterium/tritium) will take place, so the helium particleransport will be dominated by the interaction with the D and Ttoms and molecules. Although a 2D modeling is presented, theres no principle limitation for the DSMC method to perform 3D- and

ultispecies-calculations. In addition to the above mentioned, aomparison is performed between the DSMC results and the cor-esponding ITERVAC calculations. ITERVAC [7] is a semi-empiricaleterministic numerical code, developed at Karlsruhe Institute ofechnology, where the divertor geometry is modeled as a complexetwork of channels with various lengths and cross-sections [8].reviously [8], a model of the full network of the ITER torus vacuumystem has been built with ITERVAC, this opens the possibility toenchmark the DSMC modeling.

. Theoretical and computational approach

.1. Divertor flow configuration

The scope of this work is to examine the 2D behavior of theelium gas flow inside the geometry of the ITER divertor, based onhe 2008 design [9,10]. Fig. 1 illustrates the performed 2D-cut (red)f the 3-dimensional CATIA model of ITER’s divertor cassette.

The red area indicated in Fig. 1 was chosen as the flow domainor the DSMC calculations. Although the selected geometry is sim-lified, it still preserves the high degree of complexity of the innerivertor ducts and slots.

As it is pointed out in Section 1, the present study is done by cou-ling the B2-EIRENE (SOLPS) code and the DSMC algorithm. For theake of clarity, the corresponding parameters used by B2-EIRENEre first introduced. The SOLPS edge modeling corresponds to anTER scenario for different average divertor gas pressures, wherehe plasma consists of D (representing both D and T), He, C ionsnd atoms. The ratio of fusion power to auxiliary input power waset to QDT = 10 and the power crossing the magnetic separatrix and

Please cite this article in press as: C. Gleason-González, et al., SimulSimulation Monte Carlo method, Fusion Eng. Des. (2014), http://dx.do

ntering the scrape-off layer (SOL) was PSOL = 100 MW (the primaryhysics goal of ITER is to achieve a sustained burning plasma oper-tion with QDT ≥ 10). The used B2-EIRENE output includes particleux densities, temperature and pressure of D, He, and D2. These

Fig. 1. Diagram of the ITER divertor geometry [9] indicating the selected flowdomain for the simulations (red). (For interpretation of the references to color inthis figure legend, the reader is referred to the web version of this article.)

output quantities are defined along the divertor vertical targets,reflectors, slots and in the area below the dome. The correspondingSOLPS information for helium is then coupled as boundary con-ditions (BC) along the interfaces between the above mentioneddivertor surface locations and the corresponding flow domain usedin the DSMC modeling.

2.2. DSMC modeling

In the present work the DSMC algorithm based on the No-TimeCounter (NTC) scheme [4] is implemented for simulating the gasflow. Briefly, in a DSMC simulation a large number of model par-ticles are evolved in time steps of duration �t in which their freemotion and collisions between them are uncoupled. Typically, thetime step is selected such that its value is a fraction of the meancollision time of the particle. The state of the system is given by thepositions and velocity vectors of the model particles. Each modelparticle in the simulation represents an effective number of realatoms (or molecules) in the physical system. The details of thealgorithm are well documented and extensively covered [4,11],therefore its detailed description is here omitted and only specificmatters of the implementation are provided for completeness andclarity.

In order to achieve less statistical scattering and to increase theaccuracy of the numerical results, in all calculations the averagenumber of 30 particles per cell was chosen. The variable hard-sphere (VHS) model [4,5] was employed for the simulation of theintermolecular potential in all calculations. The time increment wastaken as �t = 100 �s, while a purely diffuse gas–surface interactionmodel was implemented on the sub-divertor walls.

Regarding the selection of the time step �t, other values smallerthan the selected were disregarded, as they did not essentiallychange the results. By selecting higher values of time increment, i.e.�t > 100 �s, it is not satisfied a fundamental criterion of the DSMCmethod, where the time step �t should be smaller than the meancollision time �VHS and only then, the decoupling between freemotion and collisions will take place. The decoupling is achieved bysetting the time step �t as a fraction of the mean collision time �VHS

defined as �VHS = �VHS /〈v〉, where �VHS is the mean free path [12]

and 〈v〉 =√

8kT/�m is the average molecular velocity of the parti-

ation of neutral gas flow in a tokamak divertor using the Directi.org/10.1016/j.fusengdes.2014.02.005

cle. In the present work the selected time step fulfills the condition�t/�

VHS< 1 for all the cells in the computational domain.

The computational mesh shown in Fig. 2 is unstructured witha total number of triangular cells of 2 × 104. Several grids with

ARTICLE ING ModelFUSION-7265; No. of Pages 6

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moving towards the EPD, thus being exhausted to the pumping

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ig. 2. A 2D ITER sub-divertor unstructured grid and a close-up to it are shown.

ifferent number of elements were examined; the one that hasrovided numerical accuracy of 2 significant digits was chosen.ig. 2 also shows the interfaces, where the BCs, coupled to theOLPS output, were implemented: below the dome, at slots andoth divertor arm gaps. In addition, the pressure at the entranceo the vacuum port was taken as 0 Pa (expansion into vacuum).his BC corresponds to an ideal pump that has an infinite pumpingpeed. Although this BC represents the ideal situation that oneay achieve during ITER’s operation, in reality this pressure

s not equal to zero and its value depends on the conductancend pumping speed at the downpipes (cryopump), which in this

Please cite this article in press as: C. Gleason-González, et al., SimulaSimulation Monte Carlo method, Fusion Eng. Des. (2014), http://dx.do

wo-dimensional model is not taken into account.Finally, a few definitions are necessary for later use. The left-

ost duct that connects the inner divertor arm gap with the

ig. 3. (a) Helium gas recirculation towards the plasma chamber is found through the DIVowards the plasma chamber through DIVT and DOVT for the case where the total diverhown in (d), is plotted in blue (Low-p) and red (High-p) symbols.(For interpretation of thf this article.)

PRESSring and Design xxx (2014) xxx–xxx 3

reference mark A, see Fig. 2, will be referred as the duct behindthe inner vertical target (DIVT). Similarly, the right-most duct thatconnects the outer divertor arm gap with the reference mark B willbe referred as the duct behind the outer vertical target (DOVT).Additionally, the location referred as the entrance of the pumpingduct (EPD) is indicated with a dashed line.

3. Results and discussion

The DSMC modeling comprises two study cases, namely thestudy of neutral helium gas flow resulting from a total divertor gaspressure of 2.6 Pa (Low-p case) and a second case where the totaldivertor gas pressure is 9.9 Pa (High-p case), i.e. the helium gas rep-resents only a fraction of the total gas in the divertor. Hence, neutralhelium gas flow will be only referred as gas flow in what follows.Moreover, isothermal conditions between the helium gas and thesub-divertor walls are considered and set to 420 K.

3.1. DSMC simulations

The gas flow across the sub-divertor ducts results from the pres-sure difference between the gaps, slots and EPD. However, far fromintuitively are the velocity flow fields shown in Fig. 3 for the twostudy cases. It is seen that the DIVT acts as path towards the plasmachamber for the He particles for the Low- and High-p cases, seeFig. 3(a) and (b) respectively. As revealed in Fig. 3(a) for the Low-p case, the flow pattern in DOVT shows that the He particles are

tion of neutral gas flow in a tokamak divertor using the Directi.org/10.1016/j.fusengdes.2014.02.005

system. On the other hand, as seen in Fig. 3(b) only for the High-pcase helium particles flow through the DOVT towards the plasmachamber leaving the sub-divertor duct.

T for the case where the total divertor pressure is 2.6 Pa. (b) Helium particles flowtor pressure is 9.9 Pa. (c) Shows the arc length where the bulk velocity profile, ase references to color in this figure legend, the reader is referred to the web version

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4 C. Gleason-González et al. / Fusion Engineering and Design xxx (2014) xxx–xxx

Fi

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with a high gas pressure level and constant pumping speed, 10times more pumping speed will be required in order to establish thesame recycle flow as the Low-p case. DSMC calculations regardingdeuterium as standalone gas and the binary mixture of D and He

Table 1Particle flux densities.

Case Helium flux density [m−1 s−1]

ig. 4. (a) Isotherms (K) of helium gas for the case where divertor total gas pressures 2.6 Pa.

Additionally, in Fig. 3(c) and (d) the bulk velocity profile alongn arc length is presented. The arc length of 2.44 m defined inig. 3(c) covers a region below the dome until the pumping duct.he numbering designates a specific location, where the arc lengthtarts (I), ends (VII) and the particular points where the velocityrofile suffers a change on its slope (II–VI). As one can expect,ue to the pressure difference between the points I and VII, theas flow is accelerated towards the EPD reaching a velocity at theuct entrance of 973 and 978 m/s, for the High- and Low-pressurease respectively. Although the profile remains mostly unchangedor both study cases, it is interesting to note that in the regionearby the point V a small difference between the two local maximavLow

max = 0.96vHighmax ) is found.

From the temperature plot in Fig. 4 it is seen that the tempera-ure field is not uniform across the sub-divertor geometry for theow-p case. Moreover, the expected temperature drop is observedt the EPD since the gas is expanding into vacuum. Similar quali-ative explanation for the temperature field in the High-p case isound.

The characteristic isobars are shown in Fig. 5. It is recalledhat the presented values involve only local and partial pressurealues of helium, since the simulation only deals with single gasalculations. The pressure contours for the Low-p case are shownn Fig. 5(a). Pressure values of the order of 10−2 Pa are observednder the dome (2.05 × 10−2 Pa) and below the throat of the innerlot (4.00 × 10−2 Pa), whereas near the EPD the pressure valuerops two orders of magnitude when compared to the pressureound below the dome (2.05 × 10−2 Pa → 8.7 × 10−4 Pa). In the sameig. 5(a), it is noted that through the DIVT a pressure drop of onerder of magnitude is found between the inner arm gap located athe top of the DIVT and the bottom part of the DIVT, where a smallottleneck is formed. In other words, through the DIVT helium par-icles flow from the bottleneck (pInner arm

bottleneck∼10−2 Pa) towards the

nner arm gap (pInner armgap = 2.72 × 10−3 Pa) and thus recirculation

f helium gas into the plasma chamber takes place. This is consis-ent with the velocity field pattern shown in Fig. 3(a). Continuingith the Low-p case in Fig. 5(a), it is observed that along the outer

rm the pressure difference between both ends of the DOVT is suchhat a flow is developed from the arm gap toward the bottom partf the outer arm, i.e. the pressure value located at the arm gap5.32 × 10−3 Pa) is greater than the pressure value at the bottom ofhe outer arm (2.72 × 10−3 Pa). This means that particles will flowowards the EPD contributing to the particle exhaust and again, thiss consistent with the velocity field shown in Fig. 3(a).

In Fig. 5(b) are shown the isobars for the High-p case. In a similarashion as explained above, the gas recirculation into the plasma

Please cite this article in press as: C. Gleason-González, et al., SimulSimulation Monte Carlo method, Fusion Eng. Des. (2014), http://dx.do

hamber through the DIVT relies on the local pressure differenceound on the inner gap of the divertor arm (2.08 × 10−3 Pa) andhe spatial region nearby the inner slot (1.86 × 10−2 Pa), this is

Fig. 5. Isobars (Pa) of helium gas flow for (a) divertor total gas pressure of 2.6 Pa and(b) divertor total gas pressure of 9.9 Pa.

confirmed by the velocity field seen in Fig. 3(b). Moreover, inthe same Fig. 5(b) it is found that the pressure drop in the DOVTimposes the condition of particle flow towards the plasma chamber(pOuter arm

gap = 3.65 × 10−4 Pa < pOuter armbottom = 1.70 × 10−3 Pa),

which contrasts the situation of the Low-p case where the particlesmove from the top of the DOVT to the EPD. The above findingswere double-checked in terms of the net particle flux densitycalculated at the bottom part of the DIVT, DOVT and at the EPD.The calculated values are found in Table 1 for both study cases.

The adopted sign convention in this work is defined as follows,the particle flux contributing to the exhaust towards the pumpingduct is considered as positive, otherwise negative (particles flowingtowards the plasma chamber).

To provide a better idea of the impact of these values, it is com-pared the ratio

RLow-p =∣∣∣∣

�to Pump

�to Plasma

∣∣∣∣ ∼= 18.7 (1)

with

RHigh-p =∣∣∣∣

�to Pump

�to Plasma

∣∣∣∣ ∼= 1.84. (2)

It is seen that RLow-p/RHigh-p∼= 10, which means that for a divertor

ation of neutral gas flow in a tokamak divertor using the Directi.org/10.1016/j.fusengdes.2014.02.005

DIVT DOVT EPD

Low-p −1.22 × 1019 6.72 × 1018 2.28 × 1020

High-p −8.55 × 1019 −1.54 × 1018 1.60 × 1020

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ig. 6. Distribution of probes along the sub-divertor geometry where the partial andocal pressure of helium gas is calculated.

re underway in order to quantify the recycling levels. On the otherand, this result exemplifies the great value the DSMC simulationpproach can have for a physics-integrated design of the vacuumumping system [13]. To the best knowledge of the authors, such

quantitative relation between recycle flows and pumping speeds established for the very first time.

.2. Comparison with ITERVAC

This section presents the comparison between the correspond-ng DSMC modeling and ITERVAC results. The local pressure valuest specific locations of the sub-divertor geometry, which are

Please cite this article in press as: C. Gleason-González, et al., SimulaSimulation Monte Carlo method, Fusion Eng. Des. (2014), http://dx.do

eferred to as probe locations, are compared. Since ITERVAC mod-ls the divertor via a complex network of interconnected channels,t is possible to extract the pressure at in- and outlet for everyhannel of the network. Thus, for consistency of the results, the

ig. 7. Comparison between DSMC simulations and ITERVAC calculations. The localnd partial pressure of helium gas is calculated at specific locations in the sub-ivertor region for the cases of divertor total gas pressure: (a) 2.6 Pa and (b) 9.9 Pa.

PRESSring and Design xxx (2014) xxx–xxx 5

location of the ITERVAC channel outlet coincides with the probelocation, ensuring that the spatial location is the same for bothnumerical approaches. 11 probe locations were distributed alongthe sub-divertor geometry (see Fig. 6).

A pair of probes is located at each divertor arm; one nearby thebottom of each slot, a pair below the dome and 3 horizontal co-linear probes toward the pumping duct are placed.

Simulations have been performed with ITERVAC considering thesame ITER scenarios used in the DSMC modeling, viz. the same BCsare applied. Additionally, ITERVAC typically performs calculationsconsidering the 54 ITER divertor cassettes, however for the presentcomparison, ITERVAC runs have been performed considering onlyone divertor cassette. As seen in Fig. 7(a) and (b), very good agree-ment is found between the approaches at the probes located belowthe dome and at all the probes located at the right-most part of thedivertor geometry (DOVT and EPD).

Regardless the case study, the ITERVAC pressure values at theDIVT (probes 4 and 5) and the one just below the entrance of theinner slot (probe 3) differs by a numerical factor equal or greaterthan 2, when comparing to the results of the DSMC modeling at thesame locations. This probably suggests that toroidal effects, whichare not taken into account in the present DSMC modeling, but arein the ITERVAC model, do play a role for the gas flow pressure. Thestatement can be confirmed with a 3D-DSMC simulation, which isout of scope for the present study.

4. Conclusions and outlook

In this paper the numerical investigation of neutral heliumgas flow through the ITER 2008 divertor is studied. The presentapproach concerns the coupling of the DSMC method and the B2-EIRENE code. Two ITER relevant divertor-scenarios are studied,namely the Low- and High-p cases where the total divertor gaspressure level is 2.6 Pa and 9.9 Pa, respectively. The calculation ofthe flow fields, i.e. temperature, pressure and bulk velocity of thehelium gas flow are presented. The pressure driven flow insidethe divertor geometry reveals gas flow recirculation towards theplasma chamber. It was found that for the High-p case the gapsbehind both divertor targets act as paths for recycle flows towardsthe plasma chamber, whereas in the Low-p case, the gap behind theouter target acts as a gas sink and transports the neutral particlestowards the pumping duct. This phenomenon could be quantifiedusing pressure contours and particle fluxes.

Also the present DSMC modeling is compared with the ITERVACcode showing very good agreement in most of the locations wherethe pressure is calculated. However, only in few locations in thedivertor cassette, the calculated pressures differ by a factor of 2. Itis speculated that toroidal flow effects, which are not consideredin the DSMC calculations, may take place. Further studies on 3Dsimulations are required to prove this statement.

In continuation of this work, the DSMC modeling could be fur-ther developed for describing gas mixture flows (binary mixturesof helium and DT) under relevant fusion reactor scenarios. Addi-tionally, the present approach will be extended to the currentITER divertor design and to divertor configurations of other fusionmachines.

Acknowledgments

The authors are particularly grateful to Vladislav Kotov, FZJ,

tion of neutral gas flow in a tokamak divertor using the Directi.org/10.1016/j.fusengdes.2014.02.005

Germany for his fruitful help and contribution to this work byproviding the SOLPS data. This work was carried out withinthe framework of the European Fusion Development Agreement(EFDA) under the ITER Physics Support activities. The views and

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[11] C. Shen, Rarefied Gas Dynamics, Springer, Berlin, Heidelberg, 2005.

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pinions expressed herein do not necessarily reflect those of theuropean Commission.

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