Fusion Engineering and Design 39–40 (1998) 1015–1020

Effect of gravity and intermolecular potential on 3He–4Heseparative performances of ‘cryogenic-wall’ thermal

diffusion column

Hiroshi Yamakawa *, Akira Segi, Noboru Kobayashi, Takayuki Matsunaga,Youichi Enokida, Ichiro Yamamoto

Department of Nuclear Engineering, Graduate school of Engineering, Nagoya Uni6ersity, Furo-cho, Chikusa-ku,Nagoya 464-8603, Japan

Abstract

The effect of gravity and intermolecular potential on 3He–4He separative performances of the thermal diffusioncolumn were analysed by using the two-dimensional rigorous numerical solution of convection-diffusion equationswith cold wall temperature Tc, pressure P and temperature difference DT between the hot and cold walls beingparametrically changed, while the feed flow rate F and the cut u being a constant 1 cm3 min−1 (converted at 288.15K, 0.1 MPa) and 0.1, respectively. On the moon, the optimum pressure Popt, where the total separation factor ab

takes a maximum, is about 6 times greater than that of the earth. The way in which the thermal diffusion factoraT varies with temperature depends on the intermolecular potential model. Therefore, separative performances of thethermal diffusion column vary according to the intermolecular potential with Tc and DT being changed. In designanalyses, it is important to evaluate the temperature dependence of aT of helium. © 1998 Elsevier Science S.A. Allrights reserved.

1. Introduction

Recently, helium-3 has had attention as a fuelfor D–3He nuclear fusion. Helium-3 is one of thetwo stable isotopes of helium (3He and 4He) andhas a natural abundance of merely 1.3×10−6.Adequate resources of 3He may be obtained fromthe moon as mixture of 3He–4He. Accordingly,the separation of the helium on the moon will beespecially important for enrichment of 3He in thenear future.

The thermal diffusion column has been usedwith particular success for the qualitative separa-tion of isotopes in a gas phase because the processhas the advantage of a large separation factor in arelatively simple apparatus [1] and with a smallinventory. A system composed of the thermaldiffusion column can be applied to the recovery oftritium from a used gas mixture of hydrogenisotopes and for purification of tritium from pro-duction in a fusion fuel cycle. The remarkableenhancement of the separation factor was verifiedby an axisymmetric two dimensional separativeanalysis [2] and experiments [3–5] by using athermal diffusion column with a ‘cryogenic-wall’* Corresponding author.

0920-3796/98/$19.00 © 1998 Elsevier Science S.A. All rights reserved.

PII S0920-3796(98)00243-9

H. Yamakawa et al. / Fusion Engineering and Design 39–40 (1998) 1015–10201016

[6] cooled by liquid nitrogen. The effect of acold-wall temperature of less than 77 K on theHT–H2 separative performances of the columnwas studied [7]. From the results, the authorsfound that the enhancement of the separationfactor is not so large, because the value of thethermal diffusion factor aT decreases in lowertemperature region and is probably negative un-der �70 K. In the previous paper [8], analyticalresults were shown for 3He–4He separative per-formance of the column in which a cold-walltemperature Tc of less than 77 K was used. Thethermal diffusion factor of a helium mixture isestimated to never be negative just in case to usethe Dymond–Alder numerical potential, so anenhancement of the total separation factor, due tofurther reduction of Tc, becomes larger. However,other potentials have the possibility of a signinversion for aT in the lower temperature region.According to the sign inversion, the total separa-tion factor would be smaller.

The purpose of the present paper is (1) toexamine the effect of gravity, terrestrial or lunarand (2) to study the effect of intermolecular po-tential on separative performances of the thermaldiffusion column.

Details of the notation are summarised inNomenclature.

2. Fundamentals

2.1. Diffusion coefficient and thermal diffusionfactor

Helium is a monatomic gas and the diffusioncoefficient DAB and the thermal diffusion factoraT are obtained from the kinetic theory of gases;DAB (cm2 s−1) and aT (− ) are expressed in theChapman–Cowling approximation scheme as fol-lows [9]:

DAB=0.002680

PsAB2

'T3 MA+MB

2MAMB

·fd

V(1,1)* (1)

aT=r

c2MAMB

·DA

T

DAB

(2)

=(6CAB* −5) · (SAxA−SBxB)(QAxA

2 +QBxB2 +QABxAxB)

#MA−MB

MA+MB

a0,

a0=15(2A*+5) (6C*−5)

2A*(16A*−12B*+55), (3)

where DAT is the thermal diffusion coefficient.

Collision integrals and reduced collision integralratios are defined as follows:

V(l,s)*

=2

(s+1)!(T*)s+2 ·&�

0

e− (g*)2/T (g*)2s+3Q (l)* (g*)

×dg*, (4)

Q (l)*(g*)

=2

[1−{(1+ (−1)l)/2(1+ l)}]·&�

0

(1−cosl x)

× b*db*, (5)

x(g*, b*)

=p−2b*&�

rm*

dr*/(r*)2

1− (b*/r*)2−f*(r*)/(g*)2.

(6)

A*=V(2,2)*

V(1,1)*, B*=5V(1,2)*−4V(1,3)*

V(1,1)* ,

C*=V(1,2)*

V(1,1)*, (7)

2.2. Intermolecular pair potential model

Several theoretical and experimental effortshave been carried out to determine the pair poten-tial for helium. In principle, helium is probablythe simplest molecule that the authors can applyto direct quantum mechanical calculations. Inpractice, however, the potential function of he-lium is not well established.

Intermolecular potentials available in the litera-ture [10] are used for present calculations. Table 1displays intermolecular potentials used in thepresent calculations. The potential parametersand details of the above potentials are sum-marised by Maitland et al. [10]. The values of thecollision integrals that are needed for evaluatinga0 and DAB are calculated by using the FOR-TRAN program appearing in Maitland et al. [10]

H. Yamakawa et al. / Fusion Engineering and Design 39–40 (1998) 1015–1020 1017

and expressed as a polynomial function. Theoreti-cal values of aT and DAB, obtained by five differ-ent intermolecular potential models, are examinedalong with experimental values of aT by Hurly etal. [11]:

aT(T)

= −�

0.06368+0.10856

T−

0.15366T

−0.12312

×10−4 · T�

. (8)

Because the value of DAB is not given by Hurly etal., DAB based on the Dymond–Alder potential isused for the calculations with aT by Hurly et al.

3. Results and discussion

3.1. Effect of changes in gra6ity

Computations are carried out for a 3He–4Hegas mixture in a thermal diffusion column as afunction of gravitational acceleration. The dimen-sion of the analysed column in the present calcu-lation is an inner hot radius of 0.15 mm, an outercold radius of 15 mm and an effective height of1500 mm. Heads and tails separation factors a, b

are defined as follows:

Fig. 1. Effect of gravity.

a xP

1−xP

/xF

1−xF

, b xF

1−xF

/xW

1−xW

, (9)

where the quantity x is the mole fraction of 3He,subscript P, F and W denote the value of en-riched, feed and depleted flows, respectively. Pres-sure dependences of the total separation factor ab

in the case where Tc=77.35 K, F=1 cm3 min−1

and Cut u=0.1 (the ratio of the enriched to thefeed rate) obtained from the rigorous numericalcalculation are shown in Fig. 1 with DT beingchanged, where n(r)−6 potential is used. Valuesof the maximum separation factor (ab)max are thesame for both acceleration of gravity (981, 164cm−1 s2). On the moon, the value of the optimumpressure Popt, which gives a maximum ab is ap-proximately 6 times as large as that on theearth. The lunar gravity is about one sixth that ofterrestrial gravity. The value of Popt for the totalreflux operation was predicted by the simplifiedanalytical model [13]

Popt=T( �384 6mD %Rrc

3MgDTn1/2

·� (ln d)2 (1+ ln d)2

15(ln d)2+56 ln d+53n8�1

gn1/2

.

(10)

The present analytical results were explained byEq. (10) because F is small enough.

Table 1Potential energy function

Potential energy functionPotential name

f(r)=4e [(s/r)12−(s/r)6]Lennard–Jones(12–6)

exp–6 f(r)

=e

1−6/g·�6

gexp

!g�

1−r

rm

�"−�rm

r

�6nn(r)−6

f(r)=!� 6

n−6

�r (−n)−

� n

n−6

�r (−n)"

Hartree–Fock f(r)=A exp(−ar)-dispersion

+ (C6*r (−n)+C8*r (−n)+C10* r (−n))

×F(r)Numerical potential for Ar [12] used inDymond–Alderthe previous paper [8]

H. Yamakawa et al. / Fusion Engineering and Design 39–40 (1998) 1015–10201018

Fig. 2. (a) Pressure dependence of total separation factor (Tc=10 K); (b) pressure dependence of total separation factor (Tc=77.35K); and (c) pressure dependence of total separation factor (Tc=288.15 K).

3.2. Effect of intermolecular potentials

Fig. 2a,b,c show the pressure dependence of ab

of each intermolecular potential with Tc (10,77.35, 288.15 K) and DT (400, 800, 1200 K) beingparametrically changed in the case where F=1.0cm3 min−1 and u=0.1. The optimum pressuredoes not depend on the intermolecular potentialas predicted by the simplified analytical model. InFig. 2a (Tc=10 K), the pressure dependences ofab with DT=1200 K are almost the same exceptthat of the Lennard–Jones (12–6) potential, butwith DT=400 K, the pressure dependence of ab

resulting in Hurly et al. is different from others.

In Fig. 2b (Tc=77.35 K), the value of (ab)max

resulting in Lennard–Jones (12–6) potential withDT=1200 K is about seven times as large as thatfrom the Dymond–Alder potential. In Fig. 2c(Tc=288.15 K), the pressure dependences of ab

and the value of (ab)max are almost the sameexcept that of the Lennard–Jones (12–6) and theDymond–Alder potential. When DT is large, thedifferences of total separation factors obtainedfrom the intermolecular potentials became large.

In the design of a system composed of a ther-mal diffusion column for helium isotope separa-tion, it is indispensable to evaluate thetemperature dependence of aT.

H. Yamakawa et al. / Fusion Engineering and Design 39–40 (1998) 1015–1020 1019

4. Conclusions

Conclusions are summarised as follows: (1) Onthe moon, Popt, where ab is a maximum, is about6 times as large as that on the earth; and (2)The temperature of aT is affected by the inter-molecular potential. When the combination of DTand Tc is changed, it is revealed that the separa-tive performances of the thermal diffusion columndepend on the intermolecular potential. In designanalyses, it is important to evaluate the tempera-ture dependence of aT.

Appendix A. Nomenclature

A*, B*, C* collision integral ratioDAB diffusion coefficient (cm2 s−1)

diffusion coefficient in 0.1 MPaD %F feed flow rate (cm3 min−1)

molecular weight (g g-mol−1)Mpressure (atm)Preduced transport cross sectionQ (l)*

in collisionR gas constant

functions of various type of col-SA, QA, QAB

lision integralstemperature (K)T

T( reference mean temperaturecold wall temperatureTc

total molecular concentrationccorrection factor from higher tofd

the first approximationg acceleration of gravityg* reduced relative kinetic energyk Boltzmann constant, 1.380662×

10−23 (J K−1)rc radii of cold wallrm* reduce classical distance of clos-

est approach in collisionx mole fraction

Greek lettersheads separation factora

thermal diffusion factoraT

total separation factorab

tails separation factorb

parameter in exp-6 potentialg

d ratio of hot and cold wallcutu

m viscositytotal densityr

sAB intermolecular parameter (A)intermolecular potential (J)f

reduced intermolecular potentialf*energy (=f e−1)

x deflection angletemperature difference of hotDTand cold wallsdimensionless collision integralV(1,1)*

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