thermodynamic properties of lead-bismuth eutectic for use in reactor safety analysis

13th International Conference on Nuclear Engineering Beijing, China, May 16-20, 2005

ICONE13-50813

1 ICONE13-50813 Copyright © 2005 by CNS

THERMODYNAMIC PROPERTIES OF LEAD-BISMUTH EUTECTIC

FOR USE IN REACTOR SAFETY ANALYSIS

Koji Morita* Kyushu University, Institute of Environmental Systems

6-10-1 Hakozaki, Higashi-ku, Fukuoka 812-8581, Japan Phone/Fax: +81-92-642-3788, E-mail: [email protected]

Werner Maschek

Forschungszentrum Karlsruhe Institut für Kern- und Energietechnik

Michael Flad D.T.I. GmbH

Yoshiharu Tobita, Hidemasa Yamano

Japan Nuclear Cycle Development Institute, O-arai Engineering Center

Keywords: Lead-bismuth eutectic (LBE); Thermodynamic properties; Equation of state (EOS); Liquid-metal-cooled reactor (LMR); Reactor safety analysis ABSTRACT

A consistent set of thermodynamic properties of lead-bismuth eutectic (LBE) alloy was developed for use in safety analysis of LBE liquid-metal-cooled reactor systems. The thermodynamic states of LBE, of which composition is 44.5 weight % lead and 55.5 weight % bismuth, were evaluated up to the critical point based on a van-der-Waals type equation, which is extended to a reacting system. We assumed that the LBE vapor is composed of monatomic lead and bismuth, and diatomic bismuth components, and that the liquid LBE is a non-ideal mixture of lead and bismuth. Reasonable agreement was obtained for the present set of thermodynamic properties of LBE, which includes a vapor pressure curve, liquid enthalpy and enthalpy of vaporization, with available experimental data in literature.

1. INTRODUCTION Use of liquid lead-bismuth eutectic (LBE) alloy as

coolant in nuclear systems has been investigated for more than half a century. Recently, a reactor system featuring a fast-spectrum lead or LBE liquid metal-cooled reactor (LMR) was selected as one of the concepts for new nuclear systems by the Generation IV International Forum. Accelerator driven systems, which use LBE as their coolants, are also of special interest for incinerating/transmuting plutonium and minor actinides, as they inherently offer a large flexibility in design and in coping with safety problems. The main interest in these reactor systems comes from the fact that LBE has many unique nuclear, thermophysical and chemical features. A low melting point, very high boiling temperature, and chemical inertness are characteristics of LBE that can contribute to the safety inherent with LMRs.

Numerical simulation of postulated severe-accident sequences in nuclear reactors requires thermodynamic and transport properties of reactor-core materials such as fuel, structure and coolant over wide temperature and pressure ranges. In particular, an accident analysis code such as SIMMER-III (Maschek et al., 2000; Tobita et al., 2002) requires thermodynamic properties up to the critical point to complete an equation-of-state (EOS) model. In a series of our previous studies (Morita and Fischer, 1998; Morita et al., 1998), thermophysical properties and EOS for mixed oxide fuel, stainless steel and sodium have been successfully developed as a standard data basis for the safety analysis of LMRs. However, for LBE, there are rather scarce experimental data, especially in a high temperature region, and less theoretical based recommendation, which are necessary for the reactor safety analysis.

In this paper, a consistent set of thermodynamic properties of LBE alloy will be developed for use in the safety analysis of liquid LBE cooled reactor systems. The thermodynamic states of LBE, of which composition is 44.5 weight % lead and 55.5 weight % bismuth, will be evaluated up to the critical point based on a van-der-Waals type equation extended to a reacting system.

2. LIQUID PROPERTIES 2.1 Liquid Density

Recently, Alchagirov et al. (2003) measured the density of molten LBE in a temperature range of 410–726 K, and proposed a linear function of temperature. We recommend their linear function and extrapolate it up to 2 3T

c. The resultant set of


equations is

!!

+ !m= 1"1.07978 #10

"4(T " T

m) ,

Tm! T ! 2 3T

c, and (1)

!!

+ !c = 1+ 6.35972 "10#2(Tc # T )

1/2

!8.08885 "10!8(Tc ! T )

5/2 ,

2 3Tc< T ! T

c, (2)

where Tm

and !m

are the melting temperature and the density at T

m, respectively, and T

c and !

c are

the critical temperature and the critical density, respectively, which are discussed later. According to Alchagirov et al. (2003), the melting point of LBE varies depending on the experimental data from 397.7–398.1 K. Here, we adopt the well-known value of 125 ˚C recommended by Lyon (1952). Therefore,

Tm

= 398.15 K and !m

= 10529 kg m–3.

The coefficients appearing in Eq. (2) are determined so as to satisfy the continuous conditions at the connecting point of 2 3T

c. Figure 1 shows the

recommended linear equation (1) for the density of liquid LBE in comparison with values given by Lyon (1952) and Kutateladze et al. (1959).

8,000

8,500

9,000

9,500

10,000

10,500

11,000

0 500 1,000 1,500 2,000

Lyon (1952)

Kutateladze et al. (1959)

Alchagirov et al. (2003)

Extrapolation

Density (

kg m

-3)

Temperature (K)

Fig. 1. Recommended equation for the density of liquid LBE.

2.2 Liquid Compressibility

The adiabatic compressibility !S,!

can be estimated from the speed of sound

v! in the liquid

state using the relation:

!S,!

=1

"!

+v!

2. (3)

Measurements of the speed of sound in liquid LBE have not been reported. Konyuchenko (1972) demonstrated that the following formula is applicable to the calculation of the speed of sound in liquid Pb-Sn alloys:

v!

2= x

A

WA

xAW

A+ x

BWB

!"#

$%&vA

2

+xB

WB

xAW

A+ x

BWB

!"#

$%&vB

2 , (4)

where indexes A and B mean the two components of a binary alloy. In the present study, we applied Eq. (4) to estimate the speed of sound in liquid LBE. The experimental correlations of the speed of sound in liquid lead (Tsuchiya and Takahashi, 1990) and bismuth (Kanai and Tsuchiya, 1993) above their melting points are given by

vA= 1820.0 ! 0.263" (T ! 600.6) ,

T ! 1273.2 K for lead, and (5)

vB = 1656.0 ! 0.141" (T ! 544.2) ,

T ! 733.2 K for bismuth. (6)

To represent the correct behavior of !S,!

at the critical point, the following expression, which has been used by Fink and Leibowitz (1995) for sodium, is introduced:

!S,!

= !S,m

1+"B

#$%

&'(1

1)", (7)

where !S,m

is the adiabatic compressibility at Tm

, B is the dimensionless constant, and the dimensionless temperature difference ! is defined by

! =T " T

m

Tc" T

m

. (8)

For liquid LBE, the values of !S,m

and B are determined using the values of

!S,!

and its temperature derivative at T

m by extrapolating Eqs.

(5) and (6) to Tm

. The resultant values are

!S,m

= 3.042×10–11 Pa–1 and B = 2.027.

In Fig. 2, the adiabatic compressibility of liquid LBE calculated with the proposed equation, (7), is compared with the values given by Eq. (3).


10-11

10-10

10-9

10-8

0 1,000 2,000 3,000 4,000 5,000 6,000

Present evaluation

Proposed equation

Adia

batic c

om

pre

ssib

ility

(P

a-1

)

Temperature (K)

Fig. 2. Proposed equation for the adiabatic compressibility of liquid LBE.

3. VAPOR PRESSURE CURVE 3.1 Saturation Pressure

The vapor pressure of LBE has been reported by Prasad et al. (1977a) from 1123–1323 K, by Tupper et al. (1991) from 508–823 K, by Orlov (1997) from 508–1943 K, by Michelato et al. (2003) from 723–923 K, and by Ohno et al. (2005) from 723–1023 K. However, due to the large scatter and the lack of internal consistency in the data, we estimate the saturation vapor pressure of LBE based on a theoretical equation using vapor pressure data of its constituents. It is known that bismuth exists mainly as monatomic Bi and diatomic Bi2 components in the vapor state. On the other hand, diatomic Pb2 component appears to a minor extent at least near and below the boiling point of lead (Carlson, et al., 1960; Kim and Cosgarea, 1966; Hultgren, et al., 1973), and hence it might be neglected in the lead vapor. Here, we assume that the LBE vapor is composed of monatomic Pb and Bi, and diatomic Bi2 components, and that the liquid LBE is a non-ideal mixture of lead and bismuth. Therefore, the saturation vapor pressure of LBE might be calculated using the following equation:

p+= !

PbxPbpPb

++ !

BixBipBi

++ (!

BixBi)2pBi2

+ , (9)

where pPb

+ is the saturation vapor pressure of pure liquid lead, p

Bi

+ and pBi2

+ , respectively, are the partial pressures of Bi and Bi2 over the pure liquid bismuth, xPb

(= 0.447) and xBi

(= 0.553), respectively, are the atomic concentrations of lead and bismuth in the liquid LBE, and !

Pb and !

Bi, respectively, are the

activity coefficients of lead and bismuth in their alloy. The activity coefficients of lead and bismuth have been obtained by Moser (1973) from 943–1033 K and by Prasad et al. (1977b) from 1150–1320 K as a function of temperature and composition. In the present study, we use the following correlations by

Prasad et al. (1977b), which cover the higher temperature range:

ln!Pb= " 0.2693+ 391.5 T( ) 1" xPb( )

2 , and (10)

ln!Bi= " 0.2693+ 391.5 T( ) 1" xBi( )

2 . (11)

Although the above correlations have been obtained in the rather narrow temperature range, we extrapolate them out of the range. For the calculation of Eq. (9), vapor pressures of Pb, Bi and Bi2 are taken from the "best" values tabulated by Hultgren et al. (1973) in a temperature range of 700–2000K. The results are fit to a four-parameter vapor-pressure equation as follows:

ln p+= 35.773+ 2.8006 !10

"4T

!24053

T!1.6402 lnT . (12)

The above equation yields the normal boiling point of 1944 K, which agrees well with the usually referred boiling temperature, 1943 K at 98.066 kPa (Orlov, 1977).

10-8

10-6

10-4

10-2

100

102

104

106

108

0.0 0.5 1.0 1.5 2.0

Proposed equation

Present evaulation

Ohno et al. (2005)

Michelato et al. (2003)

Orlov (1997)

Tupper et al. (1991)

Vapor

pre

ssure

(P

a)

Reciprocal temperature x103 (K

-1)

Fig. 3. Proposed vapor-pressure curve of

saturated LBE.

Assuming an ideal liquid mixture of lead and bismuth, an alternative theoretical vapor pressure could be given by Raoult’s law:

p+= x

PbpPb

++ x

BipBi

++ (x

Bi)2pBi2

+ . (13)

Equation (13) indicates 29–13 % higher values than Eq. (9) between 700–2000 K. The negative deviation of the LBE vapor pressure from Raoult's law was also suggested by Michelato et al. (2003). Figure 3 shows the natural logarithm of the proposed vapor pressure curve, Eq. (12), over the saturated LBE as a function of inverse temperature in comparison with experimental data (Tupper et al., 1991; Orlov, 1997;


Michelato et al., 2003; Ohno et al., 2005). Better agreement exists between Eq. (12) and the values measured by Ohno et al. (2005) from 723–1023 K. On the other hand, around 500–550 K, vapor pressure data (Tupper et al., 1991; Orlov, 1997) indicate quite high values even in comparison with Raoult's law. However, we propose Eq. (12) as the saturation vapor pressure of LBE for consistent representation of vapor thermodynamic states as discussed later.

3.2 Critical Parameters There appears to be neither measured nor

calculated values available for critical parameters for LBE. However, it appears that a method proposed by Martynyuk (1998) and Martynyuk and Tamanga (1999) is one of the reliable approaches for estimating critical constants of metals. It is a method based on a generalized van-der-Waals equation using experimental data on liquid phase density and enthalpy of vaporization. The experimental data required for the estimation are taken from the liquid density equation (1) and the vapor pressure curve, Eq. (12). The resultant critical parameters for LBE are

Tc

= 4890 K, !c

= 2170 kg m–3 and pc

= 87.8 MPa.

These yield the critical compressibility

Zc =pc!c

RgTc= 0.21 . (14)

This value is close to an estimated value for lead (Martynyuk, 1998) and seems to be reasonable as a liquid metal. The obtained critical temperature can be compared with other traditional approaches, which show reasonable agreement with what is know about the values of the critical temperature of metals. A method proposed by Watson (1931) using the liquid density at the normal boiling temperature gives the critical temperature of 4750 K. Kopp's equation (Kopp, 1967) based on the general relation between the critical temperature and the heat of vaporization yields 4860 K as the critical temperature. These results suggest that the proposed critical constants for LBE be reasonable.

4. VAPOR EQUATIONS OF STATE 4.1 Vapor Pressure

In SIMMER-III, a modified Redlich-Kwong (MRK) equation proposed by Fischer (1992) is used to represent vapor thermodynamic states. The MRK equation is practically simple and similar to the well-known van-der-Waals equation, but it can be made reasonably accurate especially at high temperatures. The MRK equation has been also extended to a reacting system, where dimmer and monomer components consist of vapor (Morita and

Fischer, 1998). The extension was done by developing a partition function for a vapor mixture, where the vapor components obey the MRK equation. With this extended formalism, properties of sodium were described satisfactorily (Morita et al., 1998). Its function form is

p =RgT

(1+ y2 )(!g " a1 )"

a(T )

!g (!g + a3 ), (15)

with

a(T ) = a2

T

Tc

!"#

$%&

a4

, T ! Tc

, and (16)

a(T ) = a2+da

dTTc

(T -Tc) , T > T

c, (17)

where y2

is the dimer fraction, and a1, a

2, a

3 and

a4

are the EOS parameters. From the definition, y2

is related to the equilibrium constant k

2 given by

k2=

p2

(p1)2=

y2

(1–y2)2p

, (18)

where the total pressure p is the sum of the monomer pressure p

1 and the dimer pressure p

2:

p = p1+ p

2. (19)

We apply Eq. (15) to LBE vapor assuming again that the LBE vapor consists of monatomic and diatomic components. Therefore, we define

p1= p

Pb+ p

Bi, and (20)

p2= p

Bi2

. (21)

Although application of Eq. (15) to LBE vapor is not physically correct in the original sense, it could be a good approximation since monatomic lead and bismuth are close in the molecular weight. The gram molecular mass of LBE is calculated as W

LBE =

208.2 by taking as a mole-weighted mean of molecular weights of monatomic lead (W

Pb = 207.2)

and bismuth ( WBi

= 209.0). This yields the gas constant of LBE:

Rg = 39.935 J kg–1 K–1.

Figure 4 gives a comparison of the mole fraction data of dimer in pure bismuth vapor obtained from vapor pressure measurements (Aldred and Pratt, 1963; Fischer, 1966; Kim and Cosgarea, 1966;


Voronin et al., 1967; Kohl et al., 1967; Hultgren et al., 1973; Prasad et al., 1977a). We can see that a certain quantity of diatomic bismuth in the vapor is in equilibrium with liquid bismuth. However, the portion of dimer in the vapor varies from about 35–80 mole % in a temperature range of 700–2000K, and the temperature dependency does not necessarily indicate a similar tendency. Here, however, the equilibrium constant of LBE, Eq. (18), is evaluated based on the vapor pressure data of Pb, Bi and Bi2 by Hultgren et al. (1973) as used to determine the vapor pressure curve. The results are fitted to the following equation:

k2 = exp !24.611+23511

T

"#$

%&'

. (22)

0.30

0.40

0.50

0.60

0.70

0.80

0 500 1,000 1,500 2,000 2,500

Aldred and Pratt (1963)

Fischer (1966)

Kim and Cosgarea (1966)

Voronin et al. (1967)

Kohl et al. (1967)

Hultgren et al. (1973)

Prasad et al. (1977a)

Mole

fra

ction o

f dim

er

in B

i vapor

Temperature (K)

Fig. 4. Comparison of mole fraction data of dimer in pure bismuth vapor.

10-8

10-6

10-4

10-2

100

102

104

106

108

0 500 1,000 1,500 2,000 2,500

Present evaluation

Proposed equation

Equili

brium

consta

nt (P

a-1

)

Temperature (K)

Fig. 5. Equilibrium constant of diatomic bismuth in LBE vapor.

Figure 5 shows the equilibrium constant of

diatomic bismuth in the LBE vapor calculated by the proposed equation (22), and the values evaluated from the vapor pressure data in the temperature range of 700–2000 K.

The EOS parameters, a1, a

2 and a

3, in Eq. (15)

are determined from the critical constants and the fact that the critical isotherm on a pressure-volume p–υ diagram has an inflection point at the critical point. The parameter a

4 is fit to the slope of the vapor

pressure curve at the critical temperature. The resultant parameters are

a1 = 6.26824×10–5, a

2= 1.59328×102,

a3

= 8.11866×10–4 and a4

= 3.78359×10–1.

104

105

106

107

108

109

10-5

10-4

10-3

10-2

10-1

100

101

102

500 K2000 K4890 K (Tc)9000 Ksaturation curvespinodal volumecritical point

Pre

ssu

re (

Pa

)

Specific volume (m3 kg

-1)

Fig. 6. LBE vapor isotherm shapes on p–υ diagram calculated by the MRK equation.

0.15

0.20

0.25

0.30

0.35

0.40

0 500 1,000 1,500 2,000 2,500

Present evaluation

Evaluation by the MRK equation

Mole

fra

ction o

f B

i dim

er

in L

BE

vapor

Temperature (K)

Fig. 7. Mole fraction of diatomic bismuth in LBE vapor.

Figure 6 shows the vapor isotherm shapes of LBE

on a p–υ diagram calculated by the MRK equation. The spinodal line and the saturation curve are also indicated in the figure. In Fig. 7, the mole fraction of diatomic bismuth in the LBE vapor along the saturation curve calculated by the MRK equation is compared with the values evaluated from Eq. (18) using the vapor pressure data in the temperature range of 700–2000 K. It can bee seen that the LBE vapor contains about 20–35 mole % of diatomic bismuth in the temperature range of 700–2000K.


4.2 Vapor Internal Energy The equation for the specific internal energy eg of

vapor is derived from the MRK equation for pressure to satisfy the following thermodynamic relation:

!eg!"g

#

$%

&

'(T

= T! pg!T

#$%

&'("g

) pg . (23)

This gives

eg = eg++

a(T )–Tda

dT

a3

ln

1+a3

!g+

1+a3

!g

"

#

$$$$

%

&

''''

+y2RT

1+ y2

1+1

k2

dk2

dT

!"#

$%&

, (24)

where eg+ and !g

+ are reference values. Using the specific volume and specific internal energy of infinitely dilute vapor as the reference values, the specific internal energy of vapor is given as a function of T and !g :

eg = cv,g (T ! Tm ) + eg,mD

+y2RT

1+ y2

1+1

k2

dk2

dT

!"#

$%&

!a2 (1! a4 )"

a3

ln 1+a3

#g

$

%&

'

() , (25)

with

! = T Tc( )

a4 , T ! T

c, and (26)

! = 1 , T > Tc

, (27)

where eg,mD is the specific internal energy of infinitely

dilute vapor at Tm

and cv,g is the heat capacity at constant volume.

The specific internal energy eg,m+ of saturated

vapor at Tm

is determined from the Clapeyron equation:

hlg = hg+! h

!

+= ("g

+!"

!

+ )Tdp

+

dT, (28)

where hlg is the enthalpy of vaporization,

hg+= eg

++ p

+!g

+ , and (29)

h!

+= e

!

++ p

+!!

+ . (30)

Using the fact !g,m+ >>

!!,m

+ , eg,m+ is given by

eg,m+ ! e

!,m

++"g,m

+Tm

dp+

dTm

# p+(Tm )

$

%&

'

() , (31)

where the specific volume !g+ of saturated vapor is

calculated by equating Eqs. (12) and (15), and solving numerically with respect to !g . The value of

eg,mD calculated from Eq. (25) is

eg,mD = 9.74215×105 J/kg.

The value of cv,g is set to the heat capacity at constant volume of monatomic gas:

cv,g = 1.5R = 59.9 J/kg/K.

5. LIQUID AND VAPOR ENTHALPIES The specific enthalpy of saturated liquid is

calculated from the Clapeyron equation (28):

h!

+= hg

+! ("g

+!"

!

+)Tdp

+

dT. (32)

For comparison of the specific enthalpies of saturated liquid and vapor, as described below, their values are also calculated from available data in literature.

110

120

130

140

150

160

200 400 600 800 1,000 1,200

40 wt.% Pb - 60 wt.% Bi : Flinn et al. (1974)

60 wt.% Pb - 40 wt.% Bi : Flinn et al. (1974)

LBE : Hultgren et al. (1973)

Proposed equation

Heat capacity a

t consta

nt pre

ssure

(J k

g-1

K-1

)

Temperature (K)

Fig. 8. Proposed equation for the specific heat of liquid LBE at constant pressure.

The specific enthalpy of saturated liquid is

evaluated from heat capacity values tabulated by Hultgren et al. (1973) for the liquid LBE with x

Pb =

0.453 from 400–1100 K, which might be correlated by a linear function of temperature:

cp,! = 159.0 ! 2.302 "10

!2T . (33)


Figure 8 gives a comparison of the proposed equation (33) with recommendation by Hultgren et al. (1973) and measurements by Flinn et al. (1974). There is no clear dependence on the composition from 40–60 weight % Pb. Equation (33) results in the following expression of the specific enthalpy of the liquid LBE:

h!

+! h

l

+(T

m) = !6.14812 "10

4

+159.0 ! T "1.151!10"2T2 (34)

The quasi-chemical method (Golden and Tokar, 1967) is used to calculate the specific enthalpy of vaporization. In this method, assuming W

Pb ≈ W

Bi (=

WLBE

), the heat of vaporization is calculated by

hlg =NPb!hPb + NBi!hBi + NBi2

!hBi2

WLBE (NPb +NBi +2NBi2)

, (35)

where NPb

, NBi

, and NBi2

are, respectively, the mole fractions of Pb, Bi and Bi2 components, and !h

Pb, !h

Bi and !h

Pb are, respectively, the

contribution to the enthalpy of vaporization for Pb, Bi and Bi2 components. Each of mole fractions is obtained as a ratio of the partial pressure of the component to the total pressure assuming the ideal gas law. The values of !h

Pb, !h

Bi and !h

Pb are

calculated from values of heat of formation of Pb, Bi and Bi2 tabulated by Hultgren et al. (1973).

0.00

0.20

0.40

0.60

0.80

1.00

1.20

0 1,000 2,000 3,000 4,000 5,000 6,000

Liquid phase

Vapor phase

Critical point

Quasi-chemical data

Hultgren et al. (1973)

Specific

enth

alp

y (

J k

g-1

) x1

0-6

Temperature (K)

Fig. 9. Specific enthalpy of liquid and vapor LBE along the saturation curve.

Figure 9 shows the specific enthalpy of the vapor

and liquid from Eqs. (29) and (32), respectively, along the saturation curve. The values are relative to the liquid at T

m. Comparison of the present liquid

enthalpy with values calculated by Eq. (34) shows good agreement in a temperature range of 400–1100 K. Good agreement is also obtained for the vapor enthalpy with values from the quasi-chemical method, which are calculated by adding Eq. (35) to Eq. (32).

In Fig. 10, the enthalpy of vaporization of LBE calculated by the extended MRK equation for a reacting system is compared with the values from the MRK equation neglecting the bismuth dimerization. From this comparison, the bismuth dimer component in the LBE vapor is found to have a large effect on the value of the enthalpy of vaporization of LBE.

0

200

400

600

800

1,000

0 1,000 2,000 3,000 4,000 5,000 6,000

MRK EOS without Bi dimerization

MRK EOS with Bi dimerization

Quasi-chemical data

Enth

alp

y o

f vaporization (

J k

g-1

) x1

0-3

Temperature (K)

Fig. 10. Effect of bismuth dimerization on enthalpy of vaporization of LBE.

6. CONCLUSION

A consistent set of thermodynamic properties of LBE has been developed for use in the reactor safety analysis. Due to a lack of experimental data published in literature, the basic properties such as the liquid density, vapor pressure and liquid adiabatic compressibility, were estimated up to the critical point using semi-empirical models based on the extrapolation of low temperature data of LBE or its constituents. The thermodynamic states of LBE were evaluated up to the critical point based on the MRK equation extended to a reacting system with bismuth dimerization. Reasonable agreement was obtained for the present set of thermodynamic properties with available experimental data. We expect that that the present set of thermodynamic properties of LBE could be utilized as a standard data basis for use in safety analysis of LBE LMR systems.

ACKNOWLEDGMENTS A part of this work was performed as a joint

research between Kyushu University, Forschungszentrum Karlsruhe and Japan Nuclear Cycle Development Institute.

NOMENCLATURE a1 ~ a4 parameters of the MRK equation B dimensionless constant cv specific heat at constant volume (J kg–1 K–1) cp specific heat at constant pressure (J kg–1 K–1) e specific internal energy (J kg–1) h specific enthalpy (J kg–1) hlg enthalpy of vaporization (J kg–1) k2 equilibrium constant (Pa–1)


N mole fraction p pressure (Pa) Rg gas constant (J kg–1 K–1) T temperature (K) W molecular weight (g mol–1) x atomic concentration y2 dimer fraction Zc critical compressibility Greek letters βS adiabatic compressibility (Pa–1) γ activity coefficient ν speed of sound (m s–1) θ dimensionless temperature difference ρ density (kg m–3) υ specific volume (m3 kg–1) Subscripts 1 monomer component 2 dimer component A, B components of a binary alloy Bi monatomic bismuth component Bi2 diatomic bismuth component c critical point g vapor liquid m melting point Pb monatomic lead component Superscripts + saturation quantity or reference value D infinitely dilute vapor REFERENCES Alchagirov, B.B., et al., 2003, High Temp., Vol. 41, No. 2, pp. 210–215. Aldred, A.T., Pratt, N.J., 1963, J. Chem. Phys., Vol. 38, No. 5, pp. 1085–1087. Carlson, C.M., et al., 1960, Proc. National Academy of Science of America, Vol. 46, No. 5, pp. 649–659. Fink, J.K., Leibowitz, L., 1995, ANL/RE-95/2. Fischer, A.K., 1966, J. Chem. Phys., Vol. 45, No. 1, pp. 375–377. Fischer, E.A., 1992, KfK 4889. Flinn, J.M., et al., 1974, J. Chem. Phys., Vol. 60, No. 11, pp. 4390–4395. Golden, G.H., Tokar, J.V., 1967, ANL-7323. Hultgren, R., et al., 1973, “Selected Values of the Thermodynamic Properties of the Elements,” American Society for Metals, Ohio. Kanai, S., Tsuchiya, Y., 1993, J. Phys. Soc. Jpn.Vol. 62, No. 7, pp. 2388–2394. Kim, J.H., Cosgarea, A., 1966, J. Chem. Phys., Vol. 44, No. 2, pp. 806–809. Konyuchenko, G.V., 1972, High Temp., Vol. 10, No.

2, pp. 272–274. Kopp, I.Z., 1967, Russ. J. Phys. Chem., Vol. 41 No.6, pp. 782–783. Kutateladze, S.S., et al., 1959, “Liquid-Metal Heat Transfer Media,” translated from Russian, Supplement No.2, of Soviet J. Atomic Energy, Academic Press, Moscow, 1958, Consultants Bureau, Inc., New York. Lyon, R.N., 1952, “Liquid-Metals Handbook,” Government Printing Office, Washington. Martynyuk, M.M., 1998, Russ. J. Phys. Chem., Vol. 71, No. 1, pp. 13–16. Martynyuk, M.M., Tamanga, P.A., 1999, High Temp.-High Press., Vol. 31, pp. 561–566. Maschek, W., et al., 2000, “SIMMER-III code development for analyzing transients and accidents in accelerator driven systems (ADS),” Proc. of AccApp00, Washington. Michelato, P., et al., 2003, “The TRASCO-ADS project windowless interface: theoretical and experimental evaluation,” Proc. Int. Workshop on P&T and ADS Development 2003, Mol, Belgium. Morita, K., Fischer, E.A., 1998, Nucl. Eng. Des., Vol. 183, No. 3, pp. 177–191. Morita, K., et al., 1998, Nucl. Eng. Des., Vol. 183, No. 3, pp. 193–211. Moser, Z., 1973, Z. Metallkd., Vol. 64, pp. 40–46. Kohl, F.J., et al., 1967, J. Chem. Phys., Vol. 47, No. 8, pp. 2667–2676. Ohno, S, et al., 2005, JNC TN9400 2004-72. Orlov, Y.L., 1997, “Technology of PbBi and Pb coolants,” Paper presented to the Seminar on the Concept of Lead-Cooled Fast Reactor, Cadarashe, France. Prasad, R., et al., 1977a, J. Chem. Thermodyn., Vol. 9, pp. 593–601. Prasad, R., et al., 1977b, J. Chem. Thermodyn., Vol. 9, pp. 765–772. Tobita, Y., et al., 2002, “The development of SIMMER-III, an advanced computer program for LMFR safety analysis,” Proc. A Joint IAEA/NEA Technical Meeting on the Use of Computational Fluid Dynamics Codes for Safety Analysis of Reactor Systems (including Containment), Pisa, Italy. Tsuchiya, Y., Takahashi, T., 1990, J. Phys. Soc. Jpn., Vol. 59, No. 7, pp. 2382–2386. Tupper, R.B., et al., 1991, Proc. Int. Conf. on Fast Reactors and Related Fuel Cycles, Vol. 4, pp. 5.6-1–5.6-9, Kyoto Japan. Voronin, G.F., et al., 1967, Russ. J. Phys. Chem., Vol. 41, No. 12, pp. 1637–1640. Watson, K.M., 1931, Ind. Eng. Chem., Vol. 23, No. 4, pp. 360–364.

thermodynamic properties of lead-bismuth eutectic for use in reactor safety analysis

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