24P143-P
Evaluating Thermal Conductivity of Dense Tungsten Plasma
toward Thermal Transport in Divertor
ダイバータの熱輸送特性評価に向けた高密度タングステンプラズマの	 熱伝導率解析	 
Toru Sasaki, Takuya Takahashi, Yusuke Amano, Yasutoshi Miki, Takashi Kikuchi, Nob. Harada,
and Kazuhiko Horioka1)
佐々木徹，高橋拓也，天野祐介，三木康稔，菊池崇志，原田信弘，堀岡一彦1)
Department of Electrical Engineering, Nagaoka University of Technology
Kamitomioka 1603-1, Nagaoka, Niigata 940-2188, Japan
長岡技術科学大学 〒940-2188	  新潟県長岡市上富岡町1603-11)

Department of Energy Sciences, Tokyo Institute of Technology
Nagatsuta 4259, Midori-ku, Yokohama, 226-8502, Japan1)
東京工業大学 〒226-8502	  神奈川県横浜市緑区長津田町4259
Understanding properties of warm dense tungsten, we have proposed a semi-empirical evaluation methodof transport coefficients as a thermal conductivity. The electrical conductivity of warm dense tungsten has
been evaluated by the exploding wire discharge in water and the isochoric heating of wire in a vacuum.
The results indicate that the thermal conductivity derived by the electrical conductivity is from 0.1 to 100
WK-1m-1 at temperature of 5000 K.

1. Introduction
Type of Tokamak in magnetic confinement
fusion such as ITER and DEMO requires adivator. The divator is irradiated by the highly
energetic particles from confined plasmas.
Heating load on divator is estimated to be 10
MW/m 2 at steady state and 1000 MW/m 2 at
plasma disruption condition. To prolong thelifetime, the divator is going to be made bytungsten. Ablated tungsten affects to the fusion
plasmas because of high-Z elements. Therefore,
we should understand the effect of fusion plasma
induced by the ablated tungsten. The ablated
tungsten achieves warm dense state, which affects
the coupled ions and the degenerated electrons
including phase-transition such as liquid-vapor,triplet points. The thermal transport and the heat
capacity in ablated tungsten should be evaluated.
We proposed the evaluation of thermaltransport coefficients and the heat capacity inablated tungsten by using pulsed-power discharge
and semi-empirical estimation. The pulsed-power
discharge can evaluate the electrical conductivity
in warm dense state, directly. Thermal transportcoefficients in ablated tungsten may berecommended the ordinary relations.
2. Experimental setupThe experimental layout is previously publishedstudy using exploding wire discharge in water [3]and isochoric heating confined by sappier vessel [4].

To drive a wire explosion, we arranged low
inductance capacitors cylindrically (8×0.4 µF). The
capacitor bank was charged up to 10 kV to ensure
vaporization of the wire.
The mass density and temperature assuming local
thermodynamic equilibrium is evaluated by theratio of radius and the black body emission. The
input energy history and the electrical conductivity
is evaluated by the voltage-current waveforms andthe expansion radius.
3. Results and Discussions
Figure 1 shows the electrical conductivity as afunction of density at constant temperature intungsten. Theoretical electrical conductivities basedon the models of Spitzer [5], Lee-More [6], andIchimaru [7] at 5000 K are also shown in Fig. 1.
Figure 1 also indicates COMPTRA04 [8,9] madeby Kuhlbrodt and Redmer at 10000 K. These resultscover a wide density range for a material, from aliquid metal having highly degenerated electrons toplasma that approaches to ideal plasma conditions.
COMPTRA04
indicates,
fairlywell,thedependence of high density regime of log10 (ρ/ρs) >−1.5.
From these results, we reconstruct the thermal
conductivity as following equation [10],! π 2 $ ! k $2
κ = # &# B & Tσ ," 3 %" e %(1)where, κ is the thermal conductivity, kB is the3

4

3-3.5

-3-2.5-2-1.5

Density log10(ρ/ρs)

-1

-0.5

0(b) Aluminum(1)

7

Thermal Conductivity log10g[W/K/m]

Conductivity log10σ [S/m]

8(2)
(3)

65(4)
43-3.5-3-2.5-2-1.5

Density log10(ρ/ρs)-1-0.50(c) Tungsten
Fig. 1 Electrical conductivityof tungsten as a function ofdensity at 5000K±10%. The solid line as (1) indicatesthe Spitzer conductivity model [5], the dash-dotted line
FIG.
2. !Color
online"conductivities
of copper,aluminum,as (2)
revealstheElectrical
Ichimaru
conductivity
model[6], andthetungsten as a function of density at 5000 K " 10%. The solid line as !1"
dashedline
as(3)denotestheLee-Moreconductivity
indicates the Spitzer conductivity model !Ref. 33", the dot-dashed line as !2"modelthe[7],and conductivity
the dottedline(4)the Kuhlbrodt
reveals
Ichimaru
model!Ref.as34",
dashed line asand!3"denotesthe Lee–Moreconductivity
model !Ref. 35",
dotted line as !4"
Redmermodel fromCOMPTRA04
atthetemperatureofshows the Kuhlbrodt and Redmer model from COMPTRA04 !Ref. 13" at10000K.
temperature of 10 000 K, and the dot-dot-dashed line as !5" indicates the2.5
Lee–More–Desjarlais
model !Ref. 14" at temperature of 6000 K in copperand 10 000 K in aluminum. The squares, which are shown in !a", denote
2
DeSilva’s electricalconductivity results !Ref. 24" at 6000 K.
Thermal Conductivity log10g[W/K/m]

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0
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Density log10(l/ls)
21
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Fig. 3 Thermal conductivity of tungsten as a function ofConf. Ser. 112, 042026 !2008".
22
at Morikami,5000K±10%.
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K. Ueda, and R. M. More, Phys. Rev. Lett. 91,075004 !2003".
23
A. W. DeSilva and H.-J. Kunze, Phys. Rev. E 49, 4448 !1994".
-1 -1
24 conductivity is estimated to be 0.1 WK m .
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25
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26 4. Summary
S. H. Glenzer, O. L. Landen, P. Neumayer, R. W. Lee, K. Widmann, S. W.
Pollaine,R. J. Wallace,Phys. Rev. Lett. 98, evaluation
065002 !2007". method
Weandproposedsemi-empirical
27
V. N. Korobenko, A. D. Rakhel, A. I. Savvatimski, and V. E. Fortov, Phys.
forthermalconductivityofdense
tungsten forRev. B 71, 014208 !2005".
28
V.
N. Korobenkofusion
and A. D. Rakhel,Phys.by
Rev. using
B 75, 064208!2007".
disrupted
plasmapuled-power
29
V. E. Fortov, R. I. Ilkaev, V. A. Arinin, V. V. Burtzev, V. A. Golubev,discharges. This method can predict the thermal
I. L. Iosilevskiy, V. V. Khrustalev, A. L. Mikhailov, M. A. Mochalov,
4conductivity from solid density to log10 (ρ/ρs) > −1.5.orThe
predicted
thermal conductivity reduces atDownloaded 21 Sep 2011 to 131.112.203.217. Redistribution subject to AIP license
copyright;
see http://pop.aip.org/about/rights_and_permissions
1log10 (ρ/ρs) > −1.5 with 5000 K of temperature. The
results indicate that the thermal conductivity
0.5
derived by the electrical conductivity is from 0.1 to0100 WK-1m-1 at temperature of 5000 K. We will
demonstrate the numerical simulation of thermal-0.5
conductions and crosschecked evaluated thermal-1
conductivity.
-3.5-3-2.5
-2
-1.5
-1
-0.5
0
1.5

Density log10(l/ls)

Fig. 2 Thermal conductivity of tungsten as a function ofdensity at 10000K ± 10%. The solid line indicates
COMPTRA04.

Boltzmann constant, e is the elementally charge, T
is the temperature, and σ is the electricalconductivity.
Figure 2 shows the thermal conductivity as afunction of density at constant temperature of10000 K. Theoretical thermal conductivity based on
the model of COMPTRA04. The results indicatedthat the thermal conductivity from empiricalevaluation is almost comparative to the theoreticalmodel of COMPTRA04 in high density regime oflog10(ρ/ρs) > -1.5. Thus, the method of empiricalthermal conductivity estimation by using electricalconductivity is valid for the high density regime.
Figure 3 shows the thermal conductivity as afunction of density at constant temperature of 5000
K. The predicted thermal conductivity reduces atlog10(ρ/ρs)>-1.5.
Theminimum
thermalAcknowledgments
This work was partly supported by Grant-in-Aid forYoung Scientists (B) from Japan Society for thePromotion of Science (23740406) and the NIFS
Collaboration Research program (NIFS1KEMF019).

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