Corresponding author: Igor A. Chusov ( igrch@mail.ru ) Academic editor: Boris Balakin
© 2020 Igor A. Chusov, Vladimir G. Pronyayev, Grigory Ye. Novikov, Nikolay A. Obysov.
This is an open access article distributed under the terms of the Creative Commons Attribution License (CC BY 4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.
Citation:
Chusov IA, Pronyayev VG, Novikov GYe, Obysov NA (2020) Correlations for calculating the transport and thermodynamic properties of leadbismuth eutectic. Nuclear Energy and Technology 6(2): 125130. https://doi.org/10.3897/nucet.6.55232

The paper presents recommended correlations for calculating the thermodynamic and transport properties of PbBi eutectic (44.5% Pb + 55.5% Bi), namely: density, dynamic viscosity, specific heat, thermal conductivity, surface tension, specific electrical resistance, and local speed of sound as a function of temperature. These correlations are based on calculated data presented in 39 experimental studies performed in our country and abroad and published during the period from 1923 to 2015. The authors had information on 1103 experimental points; however, a direct assessment was performed on 1076 points. The main difficulty in processing the data was that the experiments considered in the work were performed at different times using a variety of measurement methods, nonunified methods of statistical processing, varying degrees of eutectic purity, etc. The basis of the data estimation technique was the modified least square method, which made it possible to take into account the errors of the experimental data involved. The paper gives the error values of the proposed correlations and the temperature ranges of their applicability. The paper was prepared based on the results of the work of the Thermodynamic Data Center (TDC INPE NRNU MEPhI) of Rosatom State Corporation.
PbBi eutectic, heat capacity, thermal conductivity, local speed of sound, regression, density, viscosity
The eutectic leadbismuth alloy has relatively recently begun to be used as a coolant mainly in transportable nuclear power plants (TNPP) (
A generalized systematic analysis of the experimental results related to the transport and thermodynamic properties of PbBi eutectic was performed by P.L. Kirillov and his colleagues (
There are foreign publications summarizing the results obtained by various authors (
The emergence of new data resulted in the need to clarify and correct existing correlations. A calculation analysis was performed for the following parameters: density, dynamic viscosity, specific heat, thermal conductivity, surface tension, specific electrical resistance, and speed of sound in the alloy.
Since 2020, the correlations proposed in this work have been recommended by the Rosatom State Atomic Energy Corporation for calculating the thermodynamic properties of PbBi eutectic.
The thermodynamic and transport properties of the leadbismuth eutectic alloy were evaluated on the basis of experimental data presented in 39 works. The authors of this work had information about 1103 experimental points. A direct assessment was performed on 1076 points. Table
Parameter  Number of works*  Period, yrs  Temperature range, K  Total number of points  Number of evaluated points 

Density  12  1952–2015  400–1225  594  594 
Dynamic viscosity  8  1954–2015  400–1300  155  137 
Specific heat  3  1954–1973  400–950  19  11 
Thermal conductivity  9  1923–2008  400–1000  122  121 
Surface tension  10  1951–2008  400–1250  77  77 
Specific electrical resistance  3  1954–2008  400–1050  24  24 
Speed of sound  4  1975–2009  400–1400  112  112 
To determine the comparative accuracy of the calculated properties according to the recommended correlations, calculations were performed using the formulas from the reference book published by the Nuclear Energy Agency (NEA) (
Relative errors of regression equations (1)–(7) for the leadbismuth eutectic parameters in comparison with the data from (
Parameter  This paper  Kirillov ( 
NEA ( 

Density  0.25  0.33  0.29 
Dynamic viscosity  3.22  24.0  4.95 
Specific heat  1.83  2.07  37.86 
Thermal conductivity  5.3  –  3.87 
Surface tension  1.21  1.17  1.4 
Specific electrical resistance  2.43  3.99  2.5 
Local speed of sound  0.4  –  0.8 
1. The regression equation for the density of PbBi eutectic was obtained based on an analysis of 12 experimental works (
The recommended correlation for the density of PbBi eutectic is written as
ρ = 1.1·10^{4} – 1.223·T, (1)
where T is the temperature, K; ρ is the density, kg/m^{3}. The applicability range is 400–1225 K. A graphic presentation of the experimental material with a regression curve is shown in Fig.
2. The equation for calculating the dynamic viscosity of PbBi eutectic in the liquid phase was obtained on the basis of the experimental data from (
µ = 8.65×10^{–4} + 1.77·10^{–2}×exp(–T/200.5), (2)
where µ is the dynamic viscosity, Pa×s. Relation (2) is applicable in the temperature range of 400–1300 K.
The calculation results for relation (2) are graphically presented in Fig.
3. The correlation for the specific heat c_{p} of PbBi eutectic in the liquid phase was obtained on the basis of the processed data from (
c_{p} = 147.0 J/(kg×K). (3)
The applicability range is 400–950 K. The experimental data are shown in Fig.
It should be especially noted that obtaining experimental data on the specific heat of PbBi eutectic is associated with great technical difficulties and requires the use of rather expensive equipment. This is due to the complexity of measuring heat fluxes at high temperatures, when heat losses significantly increase as a result of convective and radiation transfers.
Currently, there is no unambiguous theory of thermal conductivity. In the most general case, thermal conductivity depends on temperature, chemical composition, etc.
4. The recommended correlation for calculating the thermal conductivity of PbBi eutectic in the liquid phase, obtained on the basis of the experimental data from (
λ = 3.615 + 0.0172 T – 0.405·10^{–5}T ^{2}, (4)
where λ is the thermal conductivity, W/(m×K). The correlation is applicable in the range of 400–1000 K. As a whole, 122 experimental points were obtained in the above studies. The calculated estimation was performed on 121 experimental points.
The graph for equation (4) is shown in Fig.
It should be noted that the experimental data presented in this figure diverge on average by 25%. This spread is apparently caused by a high content of impurities in bismuth. Unfortunately, in the majority of the experimental studies, there is no information on the percentage of impurities in the initial experimental material. The authors of this work could only assume their presence. Based on this, the fundamental arguments in selecting data for the assessment were the facts of the deviation of the trend of the experimental data presented in a particular work from the general trend and the explicit “loss” of individual points from the general data array.
5. The regression equation for the surface tension coefficient of leadbismuth eutectic was obtained on the basis of the experimental data from (
σ = (441.1 – 0.0711·T)·10^{–3} , (5)
where σ is the surface tension coefficient, N/m. The temperature range is 400–1370 K.
The graph for equation (5) and experimental data are shown in Fig.
6. The regression equation for the electrical resistance of leadbismuth eutectic in the liquid phase was obtained based on an analysis of the experimental data from (
r = (88.71 + 0.052·T)×10^{–8}, (6)
where r is the specific electrical resistance, Ohm×m. The temperature range is 400–1050 K.
The graph for equation (6) and experimental data are shown in Fig.
7. The local speed of sound in PbBi eutectic was assessed based on an analysis of 112 experimental points from (
u = 1,855·10^{3} – 0,257 T , (7)
where u is the sound of speed, m/s. The temperature range is 400–1400 K.
The calculation results for relation (7) are presented in Fig.
The values of relative errors for relations (1)–(7) and the errors given in (
The paper presents semiempirical dependences for calculating several transport and thermodynamic properties of PbBi eutectic in the liquid phase. These dependencies were obtained on the basis of an analysis of the experimental data given in 39 works published over the period from 1923 to 2015. The authors indicate the temperature ranges of applicability of the recommended correlations and calculation errors for the seven main parameters characterizing PbBi eutectic, namely: density, dynamic viscosity, specific heat, thermal conductivity, surface tension, specific electric resistance and local sound velocity.