An extrapolated equation of state for xenon for use in fuel swelling calculations

JOURNAL

OF NUCLEAR

MATERIALS

AN EXTRAPOLATED

31 (1969) 99-106.

EQUATrON

0 NORTH-HOLLAND

PUBLISHING

CO., AMSTERDAM

OF STATE FOR XENON FOR USE

IN FUEL SWELLING

CALCULATIONS

J. W. HARRISON

Received 3 1 October 1968

An extrapolation is made of the existing argon equation of state to higherpressuresand temperatures. Because there is good ex~r~en~l evidence that the theorem of corresponding states is obeyed for argon and xenon over the range of existing experimental measurements the extrapolated argon equation may be used for xenon with suitable scaling. Examples are given of the use of the new xenon equation of state in the oalculation of gas contents of bubbles and uranium carbide swelling and compared with similar calculations using the modified van der Waals’ law showing how the latter underestimatesthe gas content of bubbles and overestimates gas swelling.

Qchelle convenable. On donne des exemples de l’utilisation de la nouvelle equation d&at pour le xenon au calcul du gonfiement du oarbme d’uranium et on compare. les r&ultats aux oal&3 similairef3 utilisant la loi moditiee de Van der Waal, ce qui montre que la derniere loi surestime le gonflement dd au gaz.

On a extrapole I’equation d&at concernant I’argon aux pressions et aux temperatures plus Levees. En raison du fait qu’il y a une bonne evidence experimentale, que l’argon et le xenon obeissentau theoreme d’etat correspondant a l’intervalle de mesures experimentales a&ue&3s, 1’6quation extrapolee pour t’argon peut dtre utilisee pour fe xenon en utilisant une

Die vorhandeneZustandsgleichungvon Argon wird zu hoheren Driicken und Temperaturen extrapoliert. Diese Zuatandsgleiohung kann mit entsprechenden Korrekturen auoh fiir Xenon verwendet werden, da gem&s Experimenten der Satz der korre@pondierenden Zust&nde sowohl f?. Argon als such fiir Xenon im untersuchten Bereioh gilt. Es wird anhand von Beispielen gezeigt, wie die neue Zustandsgleichung fiir Xenon bei der Bereohmmg von UC-Sohwellung angewendet werden kmn. Ein Vergleioh mit &n&hen Bereobnungen mittels mod&ierter Van der WaalsGleichung zeigt, daeediem ftir die G~sohwe~~~n zu grosse Werte liefert.

1.

2.

Introduction

While current theories of gas swelling in fissile materials present an overall picture of the swelling process the detailed comparison of experiment and theory is beset by the difficulty that the precise behaviour of the gas (mainly xenon) is unknown at the pressures existing in the bubbles. In this paper it is pointed out that since recent work on inert gas beha~o~ has shown that the theorem of corresponding states is closely obeyed for argon and xenon, data from argon may be used to extend the tabulated equation of state of xenon and then reasonably extrapolated to cover the whole region of pressure and temperat~e of technical importanoe for fuel swelling calculations.

The argon and xenon data

In 1960 Levelt 1) published tables of the reduced equations of state of argon and xenon, summaris~g the work of Michels and his associates s), and showed that if certain scaling factors for the reduced temperature and density were used the argon and xenon data agreed to about one percent over the entire region of overlap of the experimental meas~ements. While it was pointed out by Levelt that a small discrepancy existed between the scaling factors calculated from the critical parameters of argon and xenon and those necessary to give agreement, this descrepancy was largely removed in two later papers a*4) and the theorem of corresponding states was found to 99

100

J.

W.

HARRISON

hold very well for the two gases. It is therefore assumed in the present paper that all of the

made for Tr = 1.9 to Tr = 2.8 and were linear at

argon

about such a plot is that if it may be assumed

data

may

be transformed

into

data. This allows one immediately a tabulated range

of

investigated tempt

equation

pressure

here

to construct

of state for xenon for a

and

temperature

experimentally.

is made

xenon

to

not

Further,

make

yet

an at-

a reasonable

extrapolation of the argon data to higher temperatures and pressures and thus construct a tabulated equation of state for xenon extending up to temperatures of about 1700 “C and pressures of lOlo d/cm2. Finally examples are given comparing swelling calculations using the equation of state derived here and the van der Waals’ equation. 3.

Extrapolation

of the argon

data

From table 7 of Levelt’s paper pressures were computed and plotted as a function of reduced temperature from T, = 1.9 to 2.8 for reduced densities from 0.2 to 2.0. For argon this covers a pressure range of 60 to 2100 atm (1 atm= 1.013 x lo6 dyn/cm2). The plots were very linear and similar to those described by Beattie and Stockmeyer 5) which cover a range T, from 1.0 to 2.0 and pr from 1.0 to 5.0 and which describe the behaviour of the seven lower polar hydrocarbons and the non-polar gases other than hydrogen and helium with an overall accuracy of a few percent 6). This type of plot is also suggested

by the equations

of state of

Beattie and Bridgeman or Benedict, Webb and Rubin recommended by Beattie 6) for gas compressibility at temperatures above the critical temperature and densities up to about twice the critical density. In the extrapolation used in the present paper the results between T, = 2.2 to 2.8 were used to give greater weight to the high temperature experimental data. extended the tabulated This extrapolation equation of state from Tr=2.8 to 7.0 and &=0.2 to 2.0. 4.

The density

extrapolation

Levelt’s table was again used and plots made of pressure p versus c=pvlRT. The plots were

high pressures to extend density

(c> 1.3). The interesting

to higher

is implied

densities

point

a limiting

at infinite pressure.

gas

For if

p=a(T)+b(T)c, and since p =cdRT

where d is the gas density,

then d approaches b(T)/RT at infinitely high pressures. This, of course, is a reflection of the fact that by compressing a gas at very high pressures a density is attained which approaches the density of the liquid form of the element. Unfortunately there is not any experimental measurement which gives this limiting density at high temperatures but there are two important experimental observations available which allow one to modify the extrapolation slightly and give some confidence in accepting its consequences. The first piece of experimental evidence is due to Bridgeman 7) and consists of p-v measurements on argon up to pressures of 15 000 atm at a temperature of 55°C (T,= 2.174). We can judge how acceptable the linear extrapolation of Levelt’s table is by comparing the extrapolated plots of p versus c with Bridgeman’s data. Fig. 1 shows Bridgeman’s data and the extrapolation for T,= 2.2. The error in the pressure at 15 000 atm is about 13% and it errs on the low side. We have therefore corrected the simple linear extrapolation in the following way. Although the Bridgeman data show slight curvature over the range 2000 to 15 000 atm a straight line may be drawn which agrees with his data to about 2%. From this straight line is found a limiting density of 0.0476 mole/cm3. This may be regarded as a pseudo-limiting density valid for the purpose of extrapolating data up to 15 000 atm for argon. The Levelt table was therefore extrapolated from cl, = 2.0 using Bridgeman’s pseudo-limiting density. This limit, corresponding to a molar volume of 21 cm3/mole may be compared with some other argon data. from high pressure Stewart 8* B, concluded measurements on solid argon at 77 “K that the molar volume was about 17.5 cm3/mole and

AN

EXTRAPOLATED

EQUATION

5. IS-

13-

converted

iz-

8” 7L

5432I-

I

I

I

I

I

I

I

I

I

I

I

2

3

4

5

6

7

8

9

IO

I II II

12

13

c=p-J/l?T

Fig.

1. 0

equa-

to a corresponding

table for xenon

reduced temperatures are multiplied by 1.005 and reduced densities by 1.021). To use the

6-

Comparison

extrapolation

tabulated

by the scaling factors used by Levelt (in fact table 1 may be used as it stands provided

IIIO 9

An equation of state for xenon

tion of state of argon containing the experimental and extrapolated data. This is easily

14 -

PXld

101

STATE

Table 1 shows the complete

16~

Kg/cm2

OF

of

Extrapolated

of Bridgeman’s Levelt’s

table.

argon data with

l

data from Levelt’s

Bridgeman. table.

insensitive to pressure beyond 2000 atm. Clusius 10) also reports the increase in volume on the melting of argon as being 24.6 to 28.1 ems/mole. If we assume the same fractional increase of volume on melting occurs at high pressures then we find, using Stewart’s data, a molar volume of 20 cma/mole for liquid argon at high pressures comparing well with 21 cm”/ mole with the pseudo-limiting density. It is well known that at extremes of pressure gases do begin to behave like liquids 11~7) so one may have some confidence in accepting the above correspondence of limiting densities as a real one. The density extrapolation was made then for Tr=1.9 to Tr=2.8 up to a reduced density of 3.0 (the pseudo-limiting density is equivalent to dr= 3.55). After these temperature and density extrapolations have been made there still remains to fill in the extrapolated table for T,=2.8 to 7.0 and c&=2.0 to 3.0. This was done by extrapolating the already extrapolated temperature data to higher densities using the same pseudo-limiting density as before. The lack of temperature dependence of the limiting density is implicit in the density extrapolation but this is believed to be reasonable at the high pressures involved.

equation of state it is more convenient to have c=pv/RT expressed as a function of gas pressure p. Table 2 summarises the xenon pressures corresponding to table 1. [The constants used in calculating the xenon pressures were as follows. The critical density and temperature were taken from Levelt’s paper viz d,= 186.3 amagat units, Tc = 289.74 “K. For xenon 1 amagat unit of density is 4.4927 x IO-5 mole/cm3 is).] (Note that the ordinates and abscissa of table 2 must have the same scaling factors of table 1.) It would be most convenient to have a single equation expressing G as a function of T and p but initial attempts to do this have failed. The scheme attempted was to first express each line of table 1 as an 11th order polynomial in p. Thus C(p, Tz) =

:

aj(Tt)pf-l,

for i = 1,27,

i=l

i.e. the 27 tabulated values of T in table 1 and then for the coefficients uj(Ta) either express them as a polynomial

in Tt or interpolate

them using a cubic polynomial. The expansion in p gave an excellent fit to the individual lines in table 1. Unfortunately when the pressure expansion was combined with either method of describing the temperature variation of the al coefficients and used for interpolating the table very large errors were obtained at large pressures. For the purpose of the illustrations to be described below computation was restricted to tabulated temperatures and interpolation at intermediate pressures was done by fitting a cubic polynomial in pressure between tabulated G values such that the cubic had a continuous first derivative at the tabulated pressures and temperatures. Since the values

\‘r

Wl87

1.0341‘ 1.0748

0.9M

0.9928

1.0021

l.OloO

24

2.6

2.8

0.9937

I.1355

1.m

1.1534

1.15%

1.0686

l.mO

1.0733

Lo725

6.4

6.6

6.8

7.0

l.vO8

1.1453

6.2

*.t422

t.0655

I.0671

6.0

1.13w

1.1317

Ls0-l

5.4

t.0621

1.1277

Lo581

52

I.0639

1.1233

1.05%

5.0

5.8

1.*te6

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I.4779

1.3439 t.3512

4.2532

1.2497 1.3699

1.3653

l.?Jt60 t.36&

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I.2379 4.3497

1.5099

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I.4983

1.49t9

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1.3376

1.22w

1.2334

t-w9

I.2235 1.3309

t.4434

1.433o

t.4218

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1.3159

h3w5

u984

1.2180 1.3236

L212o

1.2056

t.3987

t.W97

L 1912 1.2885

4s8

1.3964

1.1829 1.2777

l.lWb

WS7

4.6

l.lo78

1.&n

4-4

t.3658 I.3818

1.2528

1.1739 1.2658

i.oais

LlOV

1.0112

l&47

4.0

b2

5-6

t.3284

t.16SP

1.2225 1.3481

u875

1.0374

t.3065

I.2385

1.1409

1.0792

1.0331

3.6

X8

t.2w

1.2817

t.2537

IA217

t.t8%

I.1426

t.0907,

1.0268

0.9894

1.0

t.t53t

l.a69!3 1.1273

1.0284

3.4

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I.0594

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I.1358

t.1064

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o-9777

0.9474

0.8

t.ogb

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Lot69

1.0230

3.0

3.2

TABLE

1

I.4473

1.2965

I.3766

I.8572 t.8657 I.8737 1.8813

I.6575 1.6650 1.6720 1.6787

1.8481

1.8384

U409 1.6495

1.8169 1.8280

1.6220

1.6311

2.3338

2.coa6 2.W93

2.1225

2.3915

2-3817

2.3714

2.3603

2.1143

2.1055

2.0962

2.0663

2.a157

2.3485

2.0522 2.0644

2.3222

2.3076

LeosO

2.0392

2.0252

1.6116

1.7922

2.2917

2.0101

1.6003

2.2746

1.9937

2.2560

2.2357

2.7463

2.7379

2.7289

2.7f93

2.7091

2.6983

26067

2.6742

2.6609

2.6465

2.6309

2.5958 26t4.1

2.5759

2.550

L57&J3 1.7633 1.5881 I.7783

1.9564

2.5300

2.2134

2.50%

2.1889

2.4739

2.1318 2.1619

1.9758

1.7292

1.9351

1.9117

I.8858

2.w

2.4w8

2.3617

4.0352

43620

3.6037

3.6507

3.1527

3. wc

3.1326

3.7008

3.6894

3.6773

3. $217 3.6644

3.1100

4.4797

h4658

4.4532

4.4356

k4W

7.6594

5.6737

5.6%2

7.73%

7.117

7.6863

5.6178 5.6376

7.6307

7.6001

7.5674

7.5324

5.5968

5.5% 5.5744

5.5247

7.49@

7.4543

5.4971

5.4674 4.3168 4.34@

7.4106

7.%32

5.w 5.4353

7.3116

7.2554

5. %28

5.3215

7.t261

5.2267

7.1939

7.0513

5.2764

6.9681

5.1loB

6.8752

6.m

6.6521

6.5167

6.3739

5.1718

5.w

I. %59

48790

k7797

4.6750

5.9627 6.19%

45w

5.6649

5.4ey

3.0

L 3733

4935%

LO260

2.8

k29j5

4&w

42342

4-1660 4.2016

6. -67

Lw4

3.9%

3.5663

3.5453

3.5227

3.49w

3.4711

3.4447

3.4093

3.3735

3.3337

3.9814

3.2892

3.6204 k-3923 3.0975 ‘3.6360 kW2

Low

xc699

3.0546

3.0321

3*ooo9 3.0202

2.9799

2.9570

2.9319

2.9@3

2.8738

2.8399

2.8020

3.9209

3.8522

3.7738

3.6911

3.5867

3.2392

3.1825

2.7111 2.7594

3.1177

3.w

2.9631

3.4530

3.2ti

2.8526

3.1771

2.7102

2.6

2.6yB

2.4

Tc= 150~36~ K and d, = 300.4

2.6559

2.5977

2.31%

2.2629

2.4301

2.1q69 2.5243

2.3088

2.1989

2.2360

1.9478

2.2

2.0112

2.0

2.0982

2.0604

2.0176

1.9686

I.9155

1.65oO

I.7683

I.6648

1.6031

1.8

I.7470

1-H

1.5442

1.6882 1.7097

1.w 1.5276

1.6382 1.6645

U&647 1.4879

1.8570

1.8248

1.6088

I.4388

I.7477 t.7887

I.5381

1.W

1.6492

I.5863

1.5091

I.4124

I.3551

1.6

1.57%

1.W96

1.4952

I.3898

1.3388

1.3197

1.*!57

I.2327

l-W3

1.1814

l*?c85

1.4

1.0641

1.2

of state c= pv/RT of argon. Reduction with critical parameters amagat units.

1.1120

1.0523

l.oocr;l 1.0258

0.9788

0.9%

2.2

0.9520

0.9679

0.9310

0.6

0.95%

c-9361

9.4

1.9

O-2

2.0

%

The equation

23.329

22.6722

21.w

20.7368

20.0031

3.4

27.2871, 27.5732 27.8206

27.7011

12.3517 a3m

2.0742 28.1364 12.1141 28.22s

27.9328 1.*9 12.0339 28.03a2

1.8874 i.9389

ii.8325

‘1.7733 27.4352

N.70%

26.93% 27.1271

Il.5666 Il.6410

26.w

26.592

Il.3007 Il.3974 ii&El56

26.oegB 26.3w

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25.5416 25.8116

MY766

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IO.6357 24.7843 10.8aoo 25.1671

IO.4496 24e3m

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IO.0126

9.7293

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8.8988

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0.10&$03

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O.W+6M3

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0.6714803

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0.279303

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o.i2g2saq 0,1386EO&

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0.1632W

0.~43oEO4

0.122UO4

0.112OBO4

1.e

0.62OOEO4

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0.5603IW

o.W5EOq

0.52cGw

0.5007B04

0.4808Lo4

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o.&l1W,

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0.242uw

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0.8301X03

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0.5433EO3

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0.2089EOl, 0.2786304

0.1467~11+ 0.2019EO4

0.1417~04

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0.13f7~01, 0.181rEoq

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0.1218~04

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0.111&04

0.~06BLolc O.&~~ECN,

TABLE

0.2657304 0.3076EOq

0.24cmof+

0.9816BOh o.1owo:

0.7486EO4 0.7887m4

0.58hfm 0.6159EO!

~0.77w?Ol

I0.74~mn

0.7098m

0.835om4

0.7852w

0.7604.W

0.7356w

0.7108rO4

0.1109805

0.1069~05

o.io29805

0.989Qm4

0.94gOm4

0.9915mw

0.9602X04

0.9289W

O.lWO5

0.198&30:

0.2271105

0.1612.EO5 0.2199EO5

0,123OEO5

0.2128x05

0.3560~05

0.1665BO5

0.4155m5

0.4-5

0.331m5

0.3213X05

0.3m3EO5

0.300jnO5

0.2899x05

0.1029m6

0.*53w

0.94mm5

0.8983SO5

0.8549IW

o.a11l$ao5

0.7m

0.7245m

0.68llrg

0.6376X05

0.~159s

O.lllfm6

0.5559ao5

0.538W5

0.5208sO5

0.503=05

0.1376tO6

0.433s

0.128ym6

o.t?46m6

0.4857m50.1203x%

o.I913Bo: '0.2794Eo5 0.468QO5

0.1190~05

0.127OBO5

0.2mm5

0.3979ro5

0.3804w5

0.55m 0.594lm5

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0.1769IW

O.?507lK35 0.2056X05

0.145YO5

0.1402ao5

0.1350x05

o.t?gm5

0.1697EO:

0.227om5 0.2375BO5

0.1626BO5 0.1192x05

0.1554w

0.89766xX1 0.1149BO5

0.6860~0& '0.8663rn

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0.345W5 0.3628~

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0.32T7w5

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0.2575105

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0.19ghrO5

o.go8gm4

O.lL&EO5

O.l411RO!

0.1339EO5

0.1267'~0! 0.1e51ro5

0.8688~11, 0.1139~0:

0.6472801 0.8288BO4

0.1087x0:

0.9291PO4

0.5533m

0.708304

0.52201301 o.6681,~cu,0.8765~

0.1746EO5

0.1662305

O.li2J&O~ 0.11g6E0:

0.1537Eo5

0.1052B0J

0.628jEOJ+ 0.8239Eo4

0.4638E05

0.42GSO5

O-2224305

0.1873EO5

0.3778BO5

0.3334EO5

0.2O49EO5 0.507uo5

O.llWiEO5

0.7653X04

0.17OlEO5

0.1526EO5

0.1327EO5

0.101%05

0.6950~04

3.4

0.2650~05

I

0.1163~05 0.2880~05

0.1070~05

3.2

0.1347X05

I

0.122JE05

0.9t07~04

0.6234~04

0.837Om

0.8037ECh

0.5502m4

0.4593Eol0.5882l304 0.77i3lw

0.6365~04 '0.8037Bu

0.611i'm

0.586m

0.5621304

3.0

0.638&04

I

0.4752~04, 0.6941~4

0.4372~04

0.6662~11, 0.9087xol

0.6136E04

0.56~1~

0.5096W

0.4571E04

o.LoWOq

0.3@4JW

2.8

0.143x05

0.5080~04

2.6

0.3205X04

I

106 dyn/cma

0.&28oEoL 0.5&31x04 0.7188Eo4 0.980&Q

0.3%7s@

0.3654.Eol o.@o~o4

t

0.53773904 0.6785EOd

0.5125Eo4

0.4877xu

0.4429m

0.438fm

0.4r33m

0.3886~4

O.~J~IBCU, 0.4279~04

0.3035~04

0.388hBol 0.4906301

0.3638W

0.339ow

0.3wm

0.2894.W

0.26mOL

0.2403Im

0.2156~ol ,0.2722~-~& 0.3486x04

0.1902m

0.2444W4

2.4

0.2075604

I

O.I~O~W&

2.2

1.013.

units.

1 atm=

amagat

0.1643Bo1

2.C

c&=186.3

equation of state. “K,

o.1512Eol

calculated from the tabulated

O.i255EOl, 0.1666EO4

0.1185~04

0.1116~0&

O.m,.6mX

0.9767~0:

0.9072~03

0.8376~0~

0.768lEO3

0.6986BO!

0.1018~04 0.1394~04

0.9683~03

0.63781503 0.9185~03

0.604iro3

0.479OI!O3 0.7723tO3

0.2156303 0.4588BO3

0.2247no3

5.2

0.7190~03

0.6692~03

0.61gBo3

0.5694~03

0.5369~03 0.768gEO3

0.5033EO3

0.4696EO3

0.4386EO3 0.705OEo3

0.4msO3

0.378~~03

5.0

5.4

0.4024~03

o.i&OEO3

0.5600~0:

0.49oOEO:

0.4194Bo3

0.3839EO:

0.12343

0.47001tO3 0.6295EO3

0.3688E03 0.5196BO3

0.3352BO3

0.3578EO3 0.5705~03

0.2972EO3

0.2769m3

0.2567X03

0.2164~03

o.&z01~03

0.3700~03

2.6

0.3017~03

0.2679~03

0.9732~02 0.1962~03

0.175gm3

oA820~02

2.2

2.4

0.2941303

0.233SE.03 0.3195Bo3

0.7906802 0.1555Eo3

O.i453EO3 0.2167BO3

0.7447102

1.9

(Notation

pressures in atmospheres

2.0

Xenon

104

J.

W.

HARRISON TABLE 3

Number

of atoms per equilibrium bubble computed

from Van

der Wads

law and extrapolated

equation

of

stats (y= 1000 erg/c&). Temp.

Radius

(“C)

(cm)

No. of atoms

No. of atoms

Radius

ext.

(cm)

(vdw)

No.

of atoms

No. of atoms

(vdw)

axt. 4.3 x 103

309

1.7 x

2.3 x 10%

3.4 x 102

4.15 x 10-7

3.0 x 103

600

1.6 x 10-7

1.9 x 102

2.7 x 102

4.2 x 10-7

2.9 x 103

3.9 x 103

891

1.1 x

6.6 x 101

9.3 x 10’

4.8 x 10-7

3.8 x 103

4.7 x 103 4.0 x 103

IO-7

10-7

2.2 x 102

2.9 x 102

4.7 x 10-7

3.3 x 103

1474

1.85 x

10-7

2.5 x 102

3.3 x 102

4.8 x 10-T

3.4 x 103

3.9 x 103

1765

1.5 x

IO-7

1.5 x 10’

1.9 x 102

4.1 x 10-e

2.0 X I03

2.3 x 103

1182

1.7 x 10-V

309 600

9.5 x 10-7

2.9 x 104

4.1 x 104

1.6x 10-S

1.2 x 105

1.6x 105

8.9x 10-7

2.1 x 104

2.7 x 104

1.5 x 10-B

8.1 x 104

1.0 x 105

891 1182

1.0x 10-G

2.7 x 104

3.2 x 104

1.8 x 10-e

1.0 x 105

1.2 x 105

1.0x 10-e

2.4 x lo4

2.7 x 104

1.85 x 10-G

9.8 x 104

1.1 x 105

1474

1.1 x 10-S

2.5 x 104

2.8 x 104

1.5 x 10-B

5.3 x 104

5.8 x 104

1765

9.1 x 10-T

1.5 x 104

1.6 x 104

1.8 x 10-a

7.6 x 104

8.0 x 10”

IO

‘r

9

8

7

6 34 x 10”

I

i

2

4

II,,

6

I

8

I

I

I

I

I

I

I

I

I

EUBBLWCC

I

I

I

I

IO 12 14 16 I8 20 22 24 26 28 30 32 34 36 38 40t10X) FISSONSKC

Fig.

2.

Comparison of modified Van der Waals equation and extrapolated

of G are t&bulated at intervals of reduced temperature 4 0.2 corresponding to M 58 “C this is not thought to be a great inconvenience. Indeed a table could easily be constructed using an even smaller temperature interval.

6.

equation of state for UC at 927 “K.

Use of the equation in computing gas contents

of bubbles

It is apparent that xenon is

from the c values in table 1, more compressible at high

AN

pressures

than

~(v - b) = RT

EXTRAPOLATED

the van der Waals’

(with b = 51.0 cma/mole),

EQUATION

OF

105

STATE

equation, suggests

and one consequence of this is that gas contents of small bubbles will be underestimated if this latter equation of state is used. Table 3 illustrates

gas

extrapolated

contents

equation

computed of state

and

using

the

van

der

Waals’ equation up to bubble radii of about 180 A over a wide temperature range. For 40 A to 50 A radius bubbles the gas content computed from the extrapolated equation is 44% higher at low temperatures than the value obtained from van der Waals law. At high temperatures M 1200 “C this difference drops to 20%. 7.

Use of the equation calculations

of state in swelling

Representing the tabulated values of c by c = f (T, p) we may use it in swelling calculations as follows. Knowing the fission gas generation rate G (atom/cma) and assuming g the gas bubble density to be known then the following equation must hold P =

nfP’, PW’,

where n is the number of gas atoms per bubble. Since n=Gt/g we can compute the bubble radius as a function of time t by solving for r p(T)v =

nfV’,PM'.

A simpler approach is to solve for t as a function of r since this eliminates the search for roots of the interpolation polynomial and was the procedure adopted in this paper. 8.

Numerical

/ cc

results

Since the departures from perfect gas behaviour become worse the higher the gas pressure then it is obvious that the higher the gas bubble density in a particular fuel then the more important it is to use an accurate equation of state for the gas within the bubbles. To illustrate the difference between the use of the extrapolated equation of state and the modified van der Waals’ equation p(v -b) = RT regimes of temperature and bubble density have been chosen for their relevance to uranium

I

I

I

I

I

I

I

I

I

I

I

I

I

2

4

6

8

IO

12

14

16

18

20

22

24

26

FISSIONS

Fig.

3.

Comparison

of

equation and extrapolated

xlOzo

/ CC

modified equation

Van

der

Waals

of state for UC

at 1332 “K.

carbide swelling. Bubble densities were calculated using a simple homogeneous nucleation model 1s). Figs. 2 and 3 show computed gas swelling for temperatures of 927 “K and 1332 “K with associated bubble densities of 3.6 x 1017/ems and 3.3 x lOle/cma. The overestimate of swelling at low temperature

using the van der Waals to about 17% at

law is about 40% decreasing the higher temperature.

Since the lower temperature corresponds to a technically important temperature of operation of uranium carbide fuel elements it is important to be able to make accurate predictions of gas swelling in optimising fuel element design. 9.

Summary

An extrapolation of the existing argon equation of state is made to higher temperatures and pressures. Because the theorem of corresnonding states is obeved for argon and xenon

106

J.

the tabulated

equation

W.

HARRISON

suitable scaling may be used directly for xenon. Examples are given of the use of the new xenon equation of state in the calculation carbide

swelling

calculations

A. Michels, J. M. H. Levelt and G. J. Walkers,

of state of argon with

and

compared

using a modified

of uranium with

similar

Physica

3) 4,

overestimates

1958)

A. Michels and C. Prins, Physica 28 (1962) Phys.

and

W.

7 (1940)

Thermodynamics

H.

Stockmeyer,

101

Rept.

195

and Physics of Matter 1, High

Speed

Aerodynamics

F. R.

Rossini,

7) P. W.

References

Thesis (Amsterdam,

G. Boato and G. Casanova, Physica 27 (1961) 571

Progr.

9

769

Levelt,

5) J. A. Beattie

van der Waals

law, showing how the latter swelling due to gas bubbles.

24 (1958)

J. M. H.

and

Oxford

Bridgeman,

Jet

Propulsion

(ed.

U.P.)

Collected

papers

(Harvard

U.P.)

1) J. M. H. Levelt, Physica 26 (1960) 361

9

J. W.

A. Michels, H. Wijker and H. Wijker,

8)

C. A. Swenson, Solid State Physics 11 (Academic

2)

(1949)

A.

Physica A.

R. J. Lunbeck

15 (1949)

Michels,

T.

J. Dawson,

T.

Physica

24 (1958)

and

P.

Louwerse,

G. J. Wolkers

22 (1956)

659

10

)

Phys.

Chem.

Sol. 1 (1956)

146

Z. Clusius, Z. Phys. Chem. B31 (1936) 459; Also G. Borelius,

99

Wassenaar,

A. Michels, J. M. H. Levelt Physica

and G. J. Wolkers,

689

Wassenaar

20 (1954)

Michels,

Stewart,

Press)

627

A. Michels, Physica

Physica 15

and

de Graaft,

Physics

15 (Academic

11) J. Fenkel, Kinetic theory of liquids (Dover Publ.) 12) A. Michels, T. Wassenaar and P. Louwerse, Physica

17

and W.

Solid State

Press)

20 (1954)

99

13) G. W. Greenwood, A. J. E. Foreman and D. E. Rimmer,

J. Nucl.

Mat.

1 (1959)

305